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Impulse Response Analysis of Structural Nonlinear Time Series Models

August 2023

DOI:10.48550/arXiv.2305.19089

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University of St. Gallen

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Linear time series models are the workhorse of structural macroeconometric analysis. However, economic theory as well as data suggest that nonlinear and asymmetric effects might be key to understand the potential effects of sudden economic changes. Taking a dynamical system view, this paper proposes a new semi-nonparametric approach to construct impulse responses of nonlinear time series. Estimation of autoregressive models with sieve methods is discussed under natural physical dependence assumptions, and uniform consistency results for structural impulse responses are derived. Simulations and two empirical exercises show that the proposed method performs well and yields new insights in the dynamic effects of macroeconomic shocks.

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Impulse Response Analysis of Structural

Nonlinear Time Series Models

Giovanni Ballarin∗

University of Mannheim

October 13, 2023

First version: May 30, 2023

Link to most recent version

Abstract: Linear time series models are the workhorse of structural macroeconometric

analysis. Yet, economic theory as well as data suggest that nonlinear and asymmetric eﬀects

might be key to understanding the potential eﬀects of sudden economic changes. This paper

proposes a new semi-nonparametric sieve approach to estimate impulse response functions of

nonlinear time series within a general class of structural models. Using physical dependence

conditions, I prove that a two-step procedure can ﬂexibly accommodate nonlinear speciﬁ-

cations, avoiding the choice of ﬁxed parametric forms. Sieve impulse responses are proven

to be consistent by deriving uniform estimation guarantees, while an iterative algorithm

makes it straightforward to compute them in practice. Simulations show that the proposed

semi-nonparametric approach provides insurance against misspeciﬁcation at minor eﬃciency

costs. In a US monetary policy application, I ﬁnd that the sieve GDP response associated

with a rate hike is, at its peak eﬀects, 16% larger than that of a linear model. Finally, when

studying interest rate uncertainty shocks, sieve responses imply up to 54% and 71% stronger

contractionary eﬀects on production and inﬂation, respectively.

∗E-mail: giovanni.ballarin@gess.uni-mannheim.de. I thank Otilia Boldea, Timo Dimitriadis, Juan Car-

los Escanciano, Lyudmila Grigoryeva, Klodiana Istreﬁ, Marina Khismatullina, So Jin Lee, Yuiching Li, Sarah

Mouabbi, Andrey Ramirez, Christoph Rothe, Carsten Trenkler and Mengshan Xu, as well as the participants

of the Econonometrics Seminar at the University of Mannheim, the 2023 ENTER Jamboree, the 10th HKMEt-

rics Workshop, the GSS Weekly Seminar at Tilburg University and the Internal Econometrics Seminar at Vrije

Universiteit Amsterdam for their comments, suggestions and feedback.

1 Introduction

This paper presents a semi-nonparametric method to study the structural dynamic eﬀects of

unpredictable shocks in a class of nonlinear time series models.

Linear models are the foundation of economic structural time series modeling. The nature

of linear models makes them especially tractable and apt at describing fundamental interactions

and processes. For example, large classes of macroeconomic models in modern New Keynesian

theory can be reduced to linear VARMA form via linearization techniques. This often justiﬁes the

application of the linear time series toolbox from a theoretical point of view. Concurrently, the

work of Sims (1980) on VARs reinvigorated the strain of macroeconometric literature that seeks

to study dynamic economic relationships. Brockwell and Davis (1991), Hamilton (1994b) and

Lütkepohl (2005) provide detailed overviews of linear time series modeling and its developments.

When the objects of interest are solely dynamic eﬀects, the local projection (LP) approach of

Jordà (2005) has also gained popularity as an alternative thanks to its ﬂexibility and ease of

implementation. LPs do not directly impose a linear model on the conditional distribution

of the time series, but rather consist of linear lag regressions. Throughout this paper, the

key dynamic eﬀect under discussion will be the impulse response function (IRF), which is the

common inference object of both linear VARMA and LP analyses.

Nonlinear methods seek to ﬂexibly study the dependence structure between variables of

interest by accommodating a potentially complex model structure. In recent years, research in

nonlinear and asymmetric eﬀects has grown, partly due to the increasing availability of data,

making it feasible to estimate more elaborate models (Fuleky,2020). From a macroeconomic

perspective, one can imagine at least three broad categories of nonlinearities that may be impor-

tant to study. Sign-dependence of impulse responses is a potential key factor in the evaluation

of monetary policy, as the speciﬁc eﬀects of an interest rate change might be mitigated if the

central bank implements a rate drop rather than a rate hike, while some others might be en-

hanced (Debortoli et al.,2020). If impulse responses are size-sensitive, large shocks and small

shocks can have vastly diﬀerent economic impacts, meaning that the policymaker must account

for nonlinear scaling in the intensity of an intervention (Tenreyro and Thwaites,2016). Finally,

if the researcher’s objective lies in studying exogenous changes impacting a variable that is non-

linear by deﬁnition, such as volatility indexes, any valid structural model should account for this

feature.

The main contribution of this work is the development of an approach that allows estimating

structural IRFs which can account for general nonlinear eﬀects. This goal entails solving two

related issues: ﬁrst, structural identiﬁcation of shocks, so that it is possible to give a valid

economic interpretation to impulse responses; second, estimation of nonlinear functions in the

setting of dependent data. In a linear setup, identiﬁcation and estimation can be considered as

distinct problems, but when working with nonlinear models these questions become intertwined.

Without speciﬁc assumptions, nonlinear model classes are much too vast in terms of complexity:

there are too many channels for any variable to aﬀect any other. Disentangling such channels

thus becomes impossible, and one cannot structurally interpret IRFs and dynamic eﬀects such

as multipliers. This problem can be solved by being more precise about the classes of models

one is willing to entertain. I consider the structural nonlinear framework originally proposed by

Gonçalves et al. (2021), which involves selecting one variable to identify the structural shocks of

interest, Xt, and treating it separately from all other series, a vector Yt, included in the model.

By imposing a few additional assumptions on the dependence structure of innovations, one is able

to include general nonlinear eﬀects of Xtand its lags onto Yt. By further allowing the lags of Ytto

inﬂuence Xt, this setup permits nonlinear dynamics to propagate to all variables over time. The

signiﬁcant upside of this paradigm is that structural identiﬁcation is built-in, instead of being

treated as a separate step. The latter path is most often taken in the literature by implementing

the generalized impulse response function (GIRF) proposed by Koop et al. (1996). Kilian and

Lütkepohl (2017) have, however, highlighted that common linear identiﬁcation strategies such as

long-term and sign restrictions are generally impossible to impose in general nonlinear models,

since closed-form expressions are not available but in a handful of special cases.

A weakness of the framework in Gonçalves et al. (2021) is that it requires choosing a speciﬁc

functional form for the nonlinear components of the model, such as the negative-censoring

map or a cubic map. These are used to tease out the sign and size eﬀects of shocks.1Yet,

correct prior knowledge of such terms is often unreasonable, especially in multivariate, multi-lag

models. The natural way to avoid selecting a parametric nonlinear speciﬁcation is to resort

to semi-nonparametric techniques. Nonparametric time series methods have a long history in

econometrics (Härdle et al.,1997), but until recently not much progress has been made in

applying them to studying dynamic eﬀects. Impulse response functions are objects that depend

on the global properties of the model and, to be more precise, deﬁning an IRF requires iterating

shock perturbations over time. In a nonlinear model, the perturbation depends on the variables’

state, so that one must consider the shock’s eﬀects across possible states. That is, diﬀerent

features of the nonlinear model such as level, slope, curvature must be evaluated over a range

of values. Therefore, in this setting, an econometrician must provide error guarantees that are

uniform over the variables’ domain. In this work, I combine the uniform inference framework

of Chen and Christensen (2015) with the structural nonlinear time series scheme discussed

above. The general idea is to resort to semi-nonparametric series estimation and work in a

physical dependence setup (Wu,2005). On the one hand, I argue that physical dependence is a

natural way of imposing assumptions that lead to estimable models, being more transparent than

standard mixing conditions. On the other hand, the series approach makes it easy to estimate

models with linear and nonlinear components of the type considered in this paper. It also

provides well-developed theoretical results to study uncertainty. Under appropriate regularity

assumptions, I show that a two-step semi-nonparametric series estimation procedure is able

to consistently recover the structural model in a uniform sense. This result encompasses the

generated regressors’ problem, which arises in the second step due to the structural identiﬁcation

strategy. Lastly, I prove that the nonlinear impulse response function estimates obtained are

themselves asymptotically consistent and, thanks to an iterative algorithm, straightforward to

1The negative-censoring map applied to variable ais aÞÑ maxpa, 0q.

compute in practice.

To validate the proposed methodology, I provide simulation evidence. The ﬁrst set of results

shows that, with realistic sample sizes, the eﬃciency costs of the semi-nonparametric procedure

are small compared to correctly-speciﬁed parametric estimates. A second set of simulations

demonstrates that whenever the nonlinear parametric model is mildly misspeciﬁed the large-

sample bias is large, while for semi-nonparametric estimates it is negligible. Finally, I study how

the IRFs computed with the new method compare with the ones from two previous empirical

exercises. In a small, quarterly model of the US macroeconomy, I ﬁnd that the parametric non-

linear and nonlinear appear to underestimate by intensity the GDP responses by 13% and 16%,

respectively, after a large exogenous monetary policy shock. Moreover, sieve responses achieve

maximum impact a year before their linear counterparts. Then, I evaluate the eﬀects of interest

rate uncertainty on US output, prices, and unemployment following Istreﬁ and Mouabbi (2018).

In this exercise, the impact on industrial production of a one-deviation increase in uncertainty

is approximately 54% stronger according to semi-nonparametric IRFs than the comparable lin-

ear speciﬁcation. These ﬁndings suggest that structural impulse responses predicated on linear

speciﬁcations might be appreciably underestimating shock eﬀects.

Related Literature. Nonlinear models for dependent data have been extensively developed

with the aim of analyzing diverse types of series, see e.g. the monographs of Tong (1990), Fan

and Yao (2003), Gao (2007), Tsay and Chen (2018). Teräsvirta et al. (2010) provide a thorough

discussion of nonlinear economic time series modeling, but, by only presenting the generalized

IRF (GIRF) approach proposed by Koop et al. (1996), Potter (2000) and Gourieroux and Jasiak

(2005), they do not explicitly address structural analysis.

Parametric nonlinear speciﬁcations are common prescriptions, for example, in time-varying

models (Auerbach and Gorodnichenko,2012,Caggiano et al.,2015) and state-depend models

(Ramey and Zubairy,2018). They have been and are commonly used in time-homogeneous

models. Kilian and Vega (2011) provide a structural analysis of the eﬀects of GDP on oil

price shocks and, in contrast to previous literature, ﬁnd that asymmetries play a negligible role:

they do this by including a negative-censoring transformation of the structural variable and

testing for signiﬁcance. Caggiano et al. (2017), Pellegrino (2021) and Caggiano et al. (2021) use

interacted VAR models to estimate eﬀects of uncertainty and monetary policy shocks. From a

ﬁnance perspective, Forni et al. (2023a,b) study the economic eﬀects of ﬁnancial shocks. Their

generalized VMA speciﬁcation, which is based on that of Debortoli et al. (2020), sets that

innovations be transformed with the quadratic map.2Gambetti et al. (2022) study news shocks

asymmetries by imposing that news changes enter their autoregressive model with a pre-speciﬁed

threshold function.

Extension of nonparametric methods to nonlinear time series have already been discussed in

the recent literature. For example, Kanazawa (2020) proposed to use radial basis function neural

networks to estimate a nonlinear time series model of the US macroeconomy. This work focuses

2I will discuss how their nonlinear model setup compares to the one I consider below.

on estimating the GIRF of Koop et al. (1996), with its structural limitations: productivity is

assumed to be a fully exogenous variable. Gourieroux and Lee (2023) provide a framework for

nonparametric kernel estimation and inference of IRFs via local projections. Yet, they primarily

work in the one-dimensional case and only mention economic identiﬁcation in multivariate setups

from the perspective of linear VARs. The work possibly closest to the present paper seems to be

that of Lanne and Nyberg (2023), who develop a nearest-neighbor approach to impulse responses

estimation that builds on the local projection idea and the GIRF concept. These papers, save

for Gourieroux and Lee (2023), do not fully develop an asymptotic theory for their estimators,

which makes it hard to judge the econometric assumptions under which they are applicable.

Outline. The remainder of this paper is organized as follows. Section 2provides the general

framework for the structural model. Section 3describes the two-step semi-nonparametric esti-

mation strategy, provides a thorough treatment of physical dependence assumptions and reports

the key uniform consistency guarantees. Section 4is devoted to the discussion of nonlinear im-

pulse response function computation, validity and consistency. In Section 5, I report simulation

results that show the performance of the proposed method, while in Section 6I discuss empirical

applications. Finally, Section 7concludes. All proofs and additional technical results, as well as

secondary plots, can be found in Appendices Band C, respectively.

Notation. A (vector) random variable will be denoted in capital or Greek letters, e.g. Yt

or ϵt, while its realization will be in lowercase Latin letters, that is yt. For a process tYtutPZ,

we write Yt:s“ pYt, Yt`1, . . . , Ys´1, Ysq, as well as Y˚:t“ p...,Yt´2, Yt´1, Ytqfor the left-inﬁnite

history and Yt:˚“ pYt, Yt`1, Yt`2, . . .qfor its right-inﬁnite history. The same notation is also

used for random variable realizations. For a matrix APRdˆdwhere dě1,∥A∥is the spectral

norm, ∥A∥8is the supremum norm and ∥A∥rfor 0ără 8 is the r-operator norm. For a

random vector or matrix, I will use ∥¨∥Lrto denote the associated Lrnorm.

2 Model Framework

In this section, I introduce the nonlinear time series model that will be considered throughout

the paper. This model setup will be a generalization of the one developed in Gonçalves et al.

(2021) by letting the form of nonlinear components to remain unspeciﬁed until estimation. The

idea behind the partial structural identiﬁcation scheme is simple: if Ztis the full vector of time

series of interest, one must choose one series, call it Xt, as the structural variable, and add

speciﬁc assumption on its dynamic eﬀects on the remaining series, vector Yt. The central goal

will be the estimation of the impulse responses of Ytdue to a shock in Xt.

2.1 Model

Let Zt:“ pXt, Y 1

tq1where XtPXĎRand YtPYĎRdY, and let d“1`dYbe the dimension

of Zt. The structural nonlinear data generating process has form

B0Zt“b`BpLqZt´1`FpLqXt`ϵt,(1)

where b“ pb1, b1

2q1PRdand ϵt“ pϵ1, ϵ1

2q1PEĎRdare partitioned accordingly. I assume that

model (1) imposes a linear dependence of observables on Ytand its lags, while series Xtcan

enter nonlinearly. That is, BpLq “ B1`B2L`. . . `BpLp´1and FpLq “ F0`F1L`. . . `FpLp

are linear and functional lag polynomials, respectively.3

Matrices pF0, . . . , Fpqare functional in the sense that their entries consist of real univariate

functions. The product between FpLqand Xtis to be interpreted as functional evaluation, i.e.

FpLqXt“»

—

–

f0,1pXtq

f0,dpXtq

ﬁ

ﬃ

ﬂ`»

—

–

f1,1pXt´1q

f1,dpXt´1q

ﬁ

ﬃ

ﬂ`. . . `»

—

–

fp,1pXt´pq

fp,dpXt´pq

ﬁ

ﬃ

ﬂ,

where tfj,lu P Λfor j“0, . . . , p,l“1, . . . , d, and Λis a suﬃciently regular function class.4The

modeling choice to remain within the autoregressive time series class with additive lag struc-

ture has two core advantages. First, it yields a straightforward generalization to classical linear

models (Lütkepohl,2005,Kilian and Lütkepohl,2017). Second, it keeps semi-nonparametric es-

timation of nonlinear components feasible. Additivity in variables and lags means that the curse

of dimensionality involved with multivariate nonparametric estimation is eﬀectively mitigated

(Fan and Yao,2003).

Let the lag polynomials be given by

BpLq “ «B11pLqB12 pLq

B21pLqB22 pLqﬀ, F pLq “ «0

F21pLqﬀ.

This structural formulation means that the model equation for Xtis restricted to be linear in

all regressors. It also implies that Xtdoes not depend contemporaneously on itself. Note that

as long as B12pLq “ 0,Xtstill depends upon nonlinear functions of its own lags, which enter

via lags of Yt. Next, I impose that B0PRdYˆdYhas the form

B0“«1 0

´B0,12 B0,22ﬀ,

where B0,22 is non-singular and normalized to have unit diagonal. The structural model is thus

3This is a minor abuse of notation compared to e.g. Lütkepohl (2005). The choice to use a matrix notation is

due to the ease and clarity of writing a (multivariate) additive nonlinear model such as (1) in a manner consistent

with standard linear VAR models. In cases where a real matrix APRdˆdis multiplied with a conformable

functional matrix F, I simply assume the natural product of a scalar times a function, e.g. Aij Fkℓ , where Fkℓ is

a function, returning a new real function.

4To ﬁx ideas, one may think of ΛqpMq, the Hölder function class of smoothness qą0and domain MĎR.

We shall make more precise assumptions regarding Λin Section 3when discussing model estimation.

given by

Xt“b1`B12pLqYt´1`B11 pLqXt´1`ϵ1t,

B0,22 Yt“b2`B22pLqYt´1`B21 pLqXt´1`B0,12Xt`F21 pLqXt`ϵ2t.

Moreover, it follows that B´1

0exists and has form

B´1

0“«1 0

B21

0B22

0ﬀ.

The constraints on B0yield a structural identiﬁcation assumption and require that Xtbe

pre-determined with respect to Yt(Gonçalves et al.,2021). By introducing

µ:“B´1

0b, ApLq:“B´1

0BpLqand GpLq:“B´1

0FpLq,

one thus obtains

Xt“µ1`A12pLqYt´1`A11 pLqXt´1`ϵ1t,

Yt“µ2`A22pLqYt´1`A21 pLqXt´1`G21pLqXt`B21

0ϵ1t`B22

0ϵ2t,(2)

or, equivalently,

Zt“µ`ApLqYt´1`GpLqXt`ut,(3)

where ut“ ru1t, u2ts1,u1t”ϵ1tand u2t:“B21

0ϵ1t`B22

0ϵ2t. Given the structure of B´1

0, one can

see that A12pLq ” B12 pLq,A11pLq ” B11 pLqand G11pLq “ 0. Importantly, one must also notice

that A12pLqand G21 pLq “ B22

0F21pLqmight now be not properly identiﬁed without further

assumptions. Since A21 pLqis not necessarily zero, linear eﬀects of lags of Xton Ytcan enter

by means of both lag polynomials. To resolve this issue, I therefore assume that the functional

polynomial G21pLqcontains, at lags greater than zero, only nonlinear components.5

Example 2.1. (A Simple Bivariate Model). To give a concrete example of (2), assume that

one wants to model the eﬀects of monetary policy shocks on U.S. GDP growth following Romer

and Romer (2004). Then, let

Xt“ϵ1t,

Yt“µ2`A2Yt´1`GpXtq ` B21

0ϵ1t`ϵ2t,

where Xtare the policy shocks, which are assumed to be i.i.d., while Ytis a macroeconomic

variable whose responses the researcher is interested in, e.g. GDP growth or PCE inﬂation.

This setup is very minimal, and I assume here, for the sake of simplicity, that endogeneity of

ϵ2tdoes not pose a problem. Then, the term GpXtq ` B21

0ϵ1t”Gpϵ1tq ` B21

0ϵ1t“:Hpϵ1tqfully

captures any contemporaneous eﬀect of monetary policy shocks on Yt. When Gpϵ1tq “ 0,Hpϵ1tq

5When using a semi-nonparametric estimation strategy with B-splines, this will be feasible to implement

numerically. When using wavelets, this also is a natural approach. In practice, however, some care must be taken

to avoid constructing collinear regression matrices.

and the model are linear. If Gpϵ1tq “ β0maxp0, ϵ1tqfor some β0“ 0, function His piece-wise

linear: contractionary and expansionary shocks have, in general, diﬀerent eﬀects on Yt, but

shocks with the same sign have proportional impact. As a ﬁnal example, if Gpϵ1tq “ β0ϵ3

1tthen

Hpϵ1tqis a third-degree polynomial, so that both sign and size of monetary policy shocks are

fundamental determinants of Yt’s impulse response function. In principle, to correctly quantify

the repercussions of a speciﬁc monetary intervention a researcher must model all of these eﬀects,

unless they have a strong prior belief that either or both can be safely ignored. More complex

nonlinear and asymmetric relations are also possible. A more robust strategy - as proposed in

the present work - is to avoid choosing G(or H) as part of the model’s speciﬁcation, but rather

to empirically estimate it jointly with all other coeﬃcients.

Remark 2.1. (Constrained Models). The general approach of leaving FpLqunconstrained is

appealing when no precise economic intuition or information is available. However, there might

be cases where the functional form of the nonlinear component is either partially known, or can

be restricted. A simple restriction is that of a uniform functional over lags,

FpLq “ F`F L `F L2`. . . `F Lp.

This is a constraint eﬀectively imposed by e.g. Gonçalves et al. (2021), Kilian and Vega (2011)

and other references. They do this by fully specifying F, but nonparametric constraints may be

desired, e.g. monotonicity. Constrained estimation of FpLqis addressed in Remark 3.2 below.

The system of equations in (2) provides the so-called pseudo-reduced form model. By design,

one does not need to identify the model fully, meaning that fewer assumptions on Ztand ϵtare

needed to estimate the structural eﬀects of ϵ1ton Yt. This comes at the cost of not being able

to simultaneously study structural eﬀects with respect to ϵ2t. An associated problem is that,

in general, G21pLqXtis correlated with innovation u2tthrough B21

0ϵ1t. The main challenge to

structural shock identiﬁcation of ϵ1tthus lies in the fact that if B21

0“ 0and G21p0q “ 0, there

is endogeneity in the equations for Ytsince Xtdepends linearly on ϵ1t.Gonçalves et al. (2021)

address the issue by proposing a two-step estimation procedure wherein one explicitly controls

for ϵ1tby using regression residuals pϵt. In Section 3below, I show that this approach also allows

for consistent semi-nonparametric estimation of structural impulse responses.

Remark 2.2. (Identiﬁcation Schemes). Forni et al. (2023a,b) provide an alternative nonlinear

structural identiﬁcation framework to that of Gonçalves et al. (2021). Their approach was

originally introduced in Debortoli et al. (2020) and is based on the VMA form of the time series.

Using the current notation, suppose that the structural representation of Ztis given by

Zt“b`QpLqFpϵ1tq ` BpLqϵt

where ϵtare independent structural shocks with zero mean and identity covariance, while ϵ1t

identiﬁes, e.g., ﬁnancial innovations and shocks. QpLqand BpLqare both linear lag polynomials

and Fis a nonlinear function to be speciﬁed by the researcher. Imposing some additional

assumptions, the reduced-form assumed by Forni et al. (2023a) is

Zt“µ`ApLqZt`Q0Fpϵ1tq ` B0ϵt,(4)

where Fpxq “ x2in their baseline speciﬁcation. Forni et al. (2023b) use an analogous model,

while Debortoli et al. (2020) also consider more general setups where Q0is replaced by a general

lag polynomial DpLq. These kinds of structural assumptions are similar but not identical to

the ones imposed in Gonçalves et al. (2021) and this paper. For (4) to overlap with (2), one

must assume that Xtis exogenous and independently distributed, so that its level does not

aﬀect the mapping of ϵ1tthrough F. That is, (4) requires that only the shocks have nonlinear

eﬀects, not the structural variable itself. The upside of this approach is that one can directly

and explicitly model asymmetry in the innovation process. The drawbacks are that, without a

clear identiﬁcation of a structural variable, one must fully identify B0. Moreover, function F

remains to be speciﬁed a priori. Note, however, that if innovation sequence ϵ1tis observable, a

generalization of the semi-nonparametric estimation results of this paper to the framework of

Debortoli et al. (2020) would be straightforward.

I now state some preliminary assumptions for the model.

Assumption 1. tϵ1tutPZand tϵ2tutPZare mutually independent time series such that

«ϵ1t

ϵ2tﬀi.i.d.

„˜0,«σ2

0 Σ2ﬀ¸

where Σ2is a diagonal positive deﬁnite matrix.

Assumption 2. tZtutPZis strictly stationary, ergodic and such that suptEr|Zt|să8.

Assumption 3. The roots of equation detpId´ApLqLq “ 0are outside the complex unit circle.

Assumption 1follows Gonçalves et al. (2021). Assumption 2is a high-level assumption

on the properties of process tZtutPZand is common in the analysis of structural time series.

Assumption 3ensures that it is possible to invert lag polynomial pI´ApLqLqin order to deﬁne

impulse responses, as done below. However, Assumption 2and 3will not be suﬃcient to make

sure that (2) is estimable from data, and in Section 3additional constraints on ApLqand GpLq

will be required in order to apply semi-nonparametric estimation. Moreover, Assumption 2is

not easily interpretable: functional lag polynomial GpLqmakes it impossible to reduce semi-

structural equations (2) to an explicit inﬁnite moving average form.

I will resolve both the former (suﬃciency) and latter (interpretability) issue by using the

nonlinear dynamic model framework outlined by Pötscher and Prucha (1997). It will allow

introducing regularity assumptions on the dependence of Ztwhich enable the derivation of

consistency of impulse response estimates.

2.2 Structural Nonlinear Impulse Responses

Starting from pseudo-reduced equations (2), by letting ΨpLq“pId´ApLqLq´1one can further

derive that

Zt“η`ΘpLqϵt`ΓpLqXt,(5)

where

µ:“Ψp1q«µ1

µ2ﬀ,ΘpLq:“ΨpLqB´1

0,and ΓpLq:“ΨpLq«0

G21pLqﬀ.

To formally deﬁne impulse responses, it is useful to partition the polynomial ΘpLqaccording to

ΘpLq:“”Θ¨1pLqΘ¨2pLqı,

where Θ¨1pLqrepresents the ﬁrst column of matrices in ΘpLq, and Θ¨2pLqthe remaining dY

columns.

Given impulse δPRat time t, deﬁne the shocked innovation process as ϵ1spδq “ ϵsfor

s“ tand ϵ1tpδq “ ϵt`δ, as well as the shocked structural variable as Xspδq “ Xtfor sătand

Xspδq “ XspZt´1, ϵt`δ, ϵt`1...,ϵsqfor sět. Further, let

Zt`h:“η`Θ¨1pLqϵ1t`h`Θ¨2pLqϵ2t`h`ΓpLqXt,

Zt`hpδq:“η`Θ¨1pLqϵ1t`hpδq ` Θ¨2pLqϵ2t`h`ΓpLqXtpδq,

be the time-tbaseline and shocked series, respectively. The unconditional impulse response is

given by

IRFhpδq “ ErZt`hpδq ´ Zt`hs.(6)

The diﬀerence between shock and baseline is clearly

Zt`hpδq ´ Zt`h“Θh,¨1δ`ΓpLqXtpδq ´ ΓpLqXt

“Θh,¨1δ`pΓ0Xt`hpδq ´ Γ0Xt`hq ` . . . ` pΓhXtpδq ´ ΓhXtq,

therefore the unconditional IRF reduces to

IRFhpδq “ Θh,¨1δ`ErΓ0Xt`hpδq ´ Γ0Xt`hs ` . . . `ErΓhXtpδq ´ ΓhXts.(7)

Notice that, in (7), while one can linearly separate expectations in the impulse response

formula, terms ErΓjXt`jpδq ´ ΓjXt`jsfor 0ďjďhcannot be meaningfully simpliﬁed. Coef-

ﬁcients Γjare functional, therefore it is not possible to collect them across Xt`jpδqand Xt`j.

Moreover, these expectations involve nonlinear functions of lags of Xtand cannot be computed

explicitly. To address this issue, Section 4provides an iterative procedure that makes computa-

tion of nonlinear impulse responses in (7) straightforward.

Remark 2.3. (Local Projection Approaches). As mentioned in the introduction, in recent years

there has been growing interest in nonlinear IRF estimation procedures, and, accordingly, ways to

generalize the LP framework. Jordà (2005) already suggested that nonlinear impulse responses

can, in principle, be directly estimated with local projections via the so-called ﬂexible local

projection approach. The ﬂexible LP method relies on the Volterra expansion of time series to

account for nonlinearities. There are multiple issues with this method. First, Jordà (2005) does

not directly state how the validity of Volterra series implies the autoregressive form used in the

LP regression. Second, the ﬂexible LP proposal is fundamentally equivalent to adding polynomial

factors to the linear regression speciﬁcation. Thus, it is eﬀectively a semi-nonparametric method,

yet Jordà (2005) does not provide a theoretical analysis from this viewpoint. Moreover, no

criterion or empirical rule-of-thumb for selecting the truncation order of the Volterra expansion

are suggested, which becomes a key issue in practice. Due to these concerns, application of

ﬂexible LPs seems hard to justify from an econometric perspective.6Lanne and Nyberg (2023)

propose to nonparametrically recover the conditional mean function with a nearest-neighbor

(k-NN) regression estimator. Their method is very ﬂexible, but requires appropriately choosing

the neighborhood size kand a distance measure for histories of realizations, and the authors do

not theoretically address these issues. Very recently, Gourieroux and Lee (2023) have considered

nonlinear IRF estimation with kernel-based methods by means of a novel conditional quantile

representation of the process. They prove kernel LP estimators based on such representation

are consistent, and that the direct estimator is asymptotically normal. The theory is developed

only for the univariate case, with an autoregressive structure of lag order one, limiting the

applicability of their procedure.

3 Estimation

Pseudo-reduced form model (2) can be compactly rewritten as

Xt“Π1

1W1t`ϵ1t,

Yt“Π1

2W2t`u2t,(8)

where

Π1:“`η1, A1,11,¨¨¨ , Ap,11, A1

1,12,¨¨¨ , A1

p,12˘1PR1`pd ,

Π2:““η2G1,21 ¨¨¨ Gp,21 A1,22 ¨¨¨ Ap,22 B21

0‰1,

Zt´1:t´p:“`Xt´1, . . . , Xt´p, Y 1

t´1, . . . , Y 1

t´p˘1PRpd,

W1t:“`1, Z1

t´1:t´p˘1PR1`pd,

W2t:“`1, Xt, Z1

t´1:t´p, ϵ1t˘1PR3`pd.

Additionally, let W1“ pW11, . . . , W1nq1and W2“ pW21, . . . , W2nq1be the design matrices for Xt

and Yt, respectively.

6Moreover, the complexity of estimating Volterra kernels grows exponentially with the kernel order, and thus

more sophisticated approaches have been proposed to make estimation feasible, see e.g. Sirotko-Sibirskaya et al.

(2020) and Movahedifar and Dickhaus (2023).

Two-step Estimation Procedure. Since W2tis an infeasible vector of regressors, to estimate

Π2one can use x

W2t“ p1, Xt, Z1

t´1:t´p,pϵ1tq1, which now contains generated regressors in the form

of residual pϵ1t. This approach is an adaptation of the two-step procedure put forth by Gonçalves

et al. (2021), where I allow for semi-nonparametric estimation:

1. Regress Xtonto W1tto get estimate p

Π1and compute residuals pϵ1t“Xt´p

Π1

1W1t.

2. Fit Ytusing x

W2tto get estimate p

Π2. Since G1,21 , . . . , Gp,21 contain functional parameters,

a semi-nonparametric estimation method is required.

3. Compute coeﬃcients in p

ΘpLqand p

ΓpLqfrom p

Π1and p

Π2.

4. Consider the two paths with time tshocks ϵt`δversus ϵt: to construct the unconditional

IRF, average over histories as well as future shocks by using the algorithm detailed in

Proposition 4.1 or Proposition 4.2.

Gonçalves et al. (2021) only allow for pre-determined nonlinear transforms of Xt. The core

contribution of this paper is allowing G1,21, . . . , Gp,21 to be estimated in a nonparametric way.

I focus on series estimation in order to build on the extensive theory available in the setting of

dependent data (Chen,2013,Chen and Christensen,2015). This further adds to the framework

of Gonçalves et al. (2021), as their regularity assumptions are stated only as preconditions for

a uniform LLN to hold and are not easy to interpret.

Remark 3.1. (Alternative Estimation Approaches). One does not need to limit estimation of

the nonlinear functional parameters G1,21, . . . , Gp,21 to series-type estimators. The literature on

nonparametric regression is mature, and thus kernel (Tsybakov,2009), nearest-neighbor (Li and

Racine,2009), partitioning (Cattaneo et al.,2020) and deep neural network (Farrell et al.,2021)

estimators are all potentially valid alternatives. For example, Huang et al. (2014) use kernel

regression to perform density estimation and regression under physical dependence. However,

thanks to both availability of uniform inference results (see also Belloni et al. 2015) and ease

of implementation, series methods stand out as a choice for semi-nonparametric time series

estimation and nonlinear impulse response computation.

In the reminder of this section, I ﬁrst introduce the semi-nonparametric series estimation

strategy in detail. Then, I outline the core assumptions of the sieve setup. Special focus is

put on the dependence structure of the data: rather than directly assuming β-mixing as in

Chen and Christensen (2015), I shall consider physical dependence assumptions (Wu,2005).

to provide transparent conditions on the model itself that, if satisﬁed, ensure consistency. I

prove that the proposed two-step semi-nonparametric procedure is uniformly consistent under

physical dependence assumptions. These assumptions can be imposed directly on the model,

and, as such, may be empirically checked, if necessary. The uniform asymptotic guarantees

are ﬁrst stated for the infeasible estimator involving true innovations ϵ1tand later extended to

encompass feasible estimator p

Π2.

3.1 Semi-nonparametric Series Estimation

Starting from (8), one can introduce the ith-row coeﬃcient matrices

G21

i“ rG1,21 ¨¨¨ Gp,21si,

A22

i“ rA1,22 ¨¨¨ Ap,22si,

and B21

0iaccordingly. Consider now the regression problem for each individual component of Yt,

Yit “G21

iXt:t´p`A22

iYt´1:t´p`B21

0iϵ1t`u2it,

where Xt:t´p:“ pXt, . . . , Xt´pq1and i“1, . . . , dY. For simplicity of notation, I suppress inter-

cept η2i, but this is without loss of generality. Since G21

iconsists of 1`pfunctional coeﬃcients

and A22

ican be segmented into prow vectors of length dY, it is possible to rewrite the above as

Yit “

j“0

g21

ij pXt´jq `

j“1

A22

ij Yt´j`B21

0iϵ1t`u2it.(9)

I will use π2,i :“ rG21

iA22

iB21

0is1to identify the vector of coeﬃcients in the equation for the ith

component of Yt. From (9), Π1

2can be decomposed in dYrows of coeﬃcients, i.e.

—

–

Y1t

YdYt

ﬁ

ﬃ

ﬂ“»

—

–

π2,1

π2,dY

ﬁ

ﬃ

ﬂW2t`u2t

and one can treat each equation separately.

A semi-nonparametric series estimator for (9) is built on the idea that, if functions g21

belong to an appropriate functional space, one can construct a growing collection of sets of basis

functions – called a sieve – which, linearly combined, progressively approximate g21

ij . That is,

one can reduce the inﬁnite dimensional problem of estimating the functional coeﬃcients in π2,i

to a linear regression problem. Although (9) features a sum of possibly nonlinear functions in

tXt´jup

j“0, as well as linear functions of tYt´jup

j“1and ϵ1t, constructing a sieve is straightfor-

ward.7

Assume that g21

ij PΛ, where Λis a suﬃciently regular function class to be speciﬁed in the

following, and let BΛbe a sieve for Λ. Let b1κ, . . . , bκκ be the collection of κě1sieve basis

functions from BΛand deﬁne

bκpxq:“ pb1κpxq, . . . , bκκpxqq1,

Bκ:“ pbκpX1:1´pq, . . . , bκpXn:n´pqq1.

The sieve space for π2,i is B1`p

ΛˆR1`pdY, where here Ridentiﬁes the space of linear functions.

Since the nonparametric components of Π2are linearly separable in the lag dimension, I take

7See Chen (2007) for a comprehensive exposition of sieve estimation. Chen and Shen (1998) and Chen (2013)

also provide additional examples of partially linear semi-nonparametric models under dependence.

B1`p

Λto be a direct product of sieve spaces.8Importantly, the same sieve can be used for all

components of Yt, as I assume the speciﬁcation of the model does not change across i.

Let bπ,1K, . . . , bπ,K K be the sieve basis in B1`p

ΛˆR1`pdYwhich, for κě1and K“

pκ ` p1`pdYq, is given by

bπ,1KpW2tq “ b1κpXtq,

bπ,ppκqKpW2tq “ bκκpXt´pq,

bπ,ppκ`1qKpW2tq “ Yt´1,1,

bπ,pK´1qKpW2tq “ Yt´p,dY,

bπ,K K pW2tq “ ϵ1t,

where κﬁxes the size of the nonparametric component of the sieve. Note that K, the overall size

of the sieve, grows linearly in κ, which itself controls the eﬀective dimension of the nonparametric

component of the sieve, bπ,1κ, . . . , bπ,κκ. In all theoretical results, I will focus on the growth rate

of Krather than κ, as asymptotically they diﬀer at most by a constant multiplicative factor.

The regression equation for π2,i is

Yi“π1

2,iW2`u2i,

where Yi“ pYi1, . . . , Yinq1and u2i“ pu2i1, . . . , u2in q1. The estimation target is the conditional

expectation π2,ipwq “ ErYit |W2t“wsunder the assumption Eru2it |W2ts “ 0. By introducing

πpwq:“ pbπ,1Kpwq, . . . , bπ,K K pwqq1,

Bπ:“`bK

πpW21q, . . . , bK

πpW2nq˘1,

the infeasible least squares series estimator p

π˚

2,ipwqis given by

π˚

2,ipwq “ bK

πpwq1pB1

πBπq´1B1

KYi.

Similarly, consider the feasible series regression matrices

πpwq:“ pbπ,1Kpwq, . . . , bπ,K K pwqq1,

Bπ:“´bK

πpx

W21q, . . . , bK

πpx

W2nq¯1.

8It is not necessary to consider the more general case of tensor products of 1D sieve functions, as it would

be the case for a general p1`dYq-dimensional function G21

ipXt, Xt´1,...,Xt´pq. As previously discussed, the

additive structure avoids the curse of dimensionality which in nonlinear time series modeling if often a primary

concern when working with moderate sample sizes (Fan and Yao,2003).

Thus, the feasible least squares series estimator is

π2,ipwq “ bK

πpwq1pp

πp

Bπq´1p

KYi.

Given that the semi-nonparametric estimation problem is the same across i, to further

streamline notation, where it does not lead to confusion I will let π2be a generic coeﬃcient

vector belonging to tπ2,iup

i“1, as well as deﬁne pπ2,Yand u2accordingly.

Remark 3.2. (Constrained Sieve Estimation). The idea of constrained estimation was only

brieﬂy touched upon in Remark 2.1. In fully parametric nonlinear models, constraints are often

imposed out of necessity or simplicity. If, say, G1,21 is constituted only of the negative-censoring

map, it is unclear why G2,21 would be constituted instead of quadratic or cubic functions, for

example. That is, speciﬁc parametric assumptions can be either unreasonable or hard to justify

in practice.9Yet, constrained semi-nonparametric estimation might be desirable at times.

If the shape of the regression function is to be constrained to ensure e.g. non-negativity,

monotonicity or convexity, Chen (2007) gives examples of shape-preserving sieves, like cardinal

B-spline wavelets. Constraints on a generic sieve can also be imposed at estimation time. For

example, for simplicity suppose dY“1and p“2, and that one wants to impose G1,21 “G2,21.

The constrained sieve estimator then solves

min

t“p`1`Yt´β1bK

πpW2tq˘2subject to “Iκ,´Iκ,0κˆp1`pdYq‰β“0.

Analysis of restricted or constrained estimators, however, is still a challenging problem in non-

parametric theory, c.f. Horowitz and Lee (2017), Freyberger and Reeves (2018), Chetverikov

et al. (2018). Misspeciﬁcation in particular is complex to address. Accordingly, I will not be

imposing any speciﬁc restrictions on the nonlinear functions in Π2outside the ones necessary to

derive uniform asymptotic theory.

Spline Sieve. The B-spline sieve BSplpκ, r0,1sdY, rqof degree rě1over r0,1sdYcan be con-

structed using the Cox-de Boor recursion formula. Alternatively, an equivalent way of construct-

ing the spline sieve is as follows. For simplicity, let dY“1and let 0ăm1ă. . . ămκ´r´1ă1

be a set of knots. Then

bκ

splinepxq:“`1, x, x2, . . . , xr,maxpx´m1,0qr,...,maxpx´mκ´r´1,0qr˘1.

The resulting spline sieve is piece-wise polynomial of degree r. Moreover, notice that in practice

the spline sieve already contains a linear and constant term, so care must be taken to avoid

collinearity (for example, by not including an additional intercept and linear term in Xtin the

series regression).

9For more precise examples and a more in-depth discussion, see Section 2.1 of Chen (2013).

3.2 Distributional and Sieve Assumptions

To develop the asymptotic uniform consistency theory, I rely on the general theoretical frame-

work established by Chen and Christensen (2015). Basic distributional and sieve assumptions

can be carried over from their setup mostly unchanged.

Assumption 4. (i) tϵtutPZare such that ϵt

i.i.d.

„ p0,Σq, (ii) tϵ1tutPZand tϵ2tutPZare mutually

independent, (iii) ϵtPEfor all tPZwhere EĂRdYis compact, convex and has nonempty

interior.

Assumption 5. (i) tZtutPZis a strictly stationary and ergodic time series, (ii) XtPXfor all

tPZwhere XĂRis compact, convex and has nonempty interior, (iii) YtPYfor all tPZwhere

YĂRdYis compact, convex and has nonempty interior.

Assumptions 4(i)-(ii) are a repetition of Assumption 1. As W2tdepends only on Xt:t´p,

Yt´1:t´p, and ϵ1t, Assumption 1also implies that entries of u2tare independent of W2t, so that

Eru2it |W2ts “ 0.10 Assumption 5(i) also follows from Assumption 2. However, thanks to the

results derived in Section 3.3, below I will impose more primitive conditions on the model for Zt

that allow to recover 5(i). Assumption 4(iii) and Assumptions 5(ii)-(iii) imply that Xt,Yt, as well

as ϵtare bounded random variables. In (semi-)nonparametric estimation, imposing that Xtbe

bounded almost surely is a standard assumption. Since lags of Ytand innovations ϵtcontribute

linearly to all components of Zt, it follows that they too must be bounded. Unbounded regressors

are more complex to handle when working in the nonparametric setting. Generalization from

bounded to unbounded domains under dependence has already been discussed by e.g. Fan and

Yao (2003). Chen and Christensen (2015) also allow for an expanding support by using weighted

sieves. I leave this extension for future work.

It is, however, important to highlight that bounded support assumptions are relatively

uncommon in time series econometrics. This is clear when considering the extensive litera-

ture available on linear models such as, e.g., state-space, VARIMA and dynamic factor mod-

els (Hamilton,1994a,Lütkepohl,2005,Kilian and Lütkepohl,2017,Stock and Watson,2016).

Avoiding Assumptions 4(iii) and 5(iii) can possibly be achieved with a change in the model’s

equations – so that, for example, lags of Ytonly eﬀect Xteither via bounded functions or not

at all – so I do not discuss this approach here. In practice, Assumptions 4(ii) and 5(ii)-(iii)

are not excessively restrictive, as most credibly stationary economic series often have reasonable

implicit (e.g. inﬂation) or explicit bounds (e.g. employment rate).11

Let Ft“σp...,ϵ1t´1, u2t´1, Yt´1, ϵ1t, u2t, Ytqbe the natural ﬁltration deﬁned up to time t.

Thanks to Assumptions 4and 5the following moment requirements hold trivially.

10Moreover, for any given i, the sequence tu2itutPZis i.i.d. over time index t.

11This is not true, of course, when modeling extreme events like natural disasters, wars or ﬁnancial crises. To

study these types of series, however, researchers often apply specialized models. Thinking in this direction, a

future development could be to extend the framework presented here to allow for innovations with unbounded

support.

Assumption 6. (i) Eru2

2it|Ft´1sis uniformly bounded for all tPZalmost surely, (ii) Er|u2it|2`δs ă

8for some δą0, (iii) Er|Yit|2`δsis uniformly bounded for all tPZalmost surely, and (iv)

ErY2

it |Ft´1s ă 8 for any δą0.

Now let W2ĂRdbe the domain of W2t. By assumption, W2is compact and convex and is

given by the direct product

W2“X1`pˆYpˆE1,

where E1is the domain of structural innovations ϵ1ti.e. E”E1ˆE2.

Assumption 7. Deﬁne ζK,n :“supwPW2∥bK

πpwq∥and

λK,n :“ rλminpErbK

πpW2tqbK

πpW2tq1sqs´1{2.

It holds:

(i) There exist ω1, ω2ě0s.t. supwPW2∥∇bK

πpwq∥Ànω1Kω2.

(ii) There exist ω1ě0,ω2ą0s.t. ζK,n Ànω1Kω2.

(iii) λminpErbKpW2tqbKpW2tq1sq ą 0for all Kand n.

Assumption 7provides mild regularity conditions on the families of sieves that can be

used for the series estimator. More generally, letting W2be compact and rectangular makes

Assumption 7hold for commonly used basis functions (Chen and Christensen,2015).12 In

particular, Assumption 7(i) holds with ω1“0since the domain is ﬁxed over the sample size.

In the proofs, it is useful to consider the orthonormalized sieve basis. Let

πpwq:“E“bK

πpW2tqbK

πpW2tq1‰´1{2bK

πpwq,

Bπ:“´r

πpW21q,...,r

πpW2nq¯1

be the orthonormalized vector of basis functions and the orthonormalized regression matrix,

respectively.

Assumption 8. It holds that ∥pr

πr

Bπ{nq ´ IK∥“oPp1q.

Assumption 8is the key assumption imposed by Chen and Christensen (2015) to derive

uniform converges rates under dependence. They prove that if tW2tutPZis strictly stationary

and β-mixing – with either geometric or algebraic decay, depending on the sieve family of interest

– then Assumption 8holds. Let pΩ,Q,Pqbe the underlying probability space and deﬁne

βpA,Bq:“1

2sup ÿ

pi,jqPIˆJ

|PpAiXBiq ´ PpAiqPpBiq|

where A,Bare two σ-algebras, tAiuiPIĂA,tBjujPJĂBand the supremum is taken over all

ﬁnite partitions of Ω. The h-th β-mixing coeﬃcient of process tW2tutPZis deﬁned as

βphq “ sup

βpσp...,W2t´1, W2tq, , σpW2t`h, W2t`h`1, . . .qq,

12See Chen (2007), Belloni et al. (2015) for additional discussion and examples of sieve families.

and W2tis said to be geometric or exponential β-mixing if βphq ď γ1expp´γ2hqfor some γ1ą0

and γ2ą0. The main issue with mixing assumptions is that they are, in general, hard to

compute and evaluate. Therefore, especially in nonlinear systems, assuming that βphqdecays

exponentially over himposes very high-level assumptions on the model. There are, however,

many setups in which it is known that β-mixing holds under primitive assumptions (see Chen

(2013) for examples).

In the next subsection, I will argue that using a diﬀerent concept of dependence - one rooted

in a physical understanding of the underlying stochastic process - leads to imposing transparent

assumptions on the model’s structure.

3.3 Physical Dependence Conditions

Consider now a non-structural model of the form

Zt“GpZt´1, ϵtq.(10)

This is a generalization of semi-reduced model (3) where linear and nonlinear components are

absorbed into one functional term and B0is the identity matrix.13 Indeed, note that models

of the form Zt“GpZt´1, . . . , Zt´p, ϵtqcan be rewritten as (10) using a companion formulation.

If ϵtis stochastic, (10) deﬁnes a causal nonlinear stochastic process. More generally, it deﬁnes

a nonlinear diﬀerence equation and an associated dynamical system driven by ϵt. Throughout

this subsection, I shall assume that ZtPZĎRdZas well as ϵtPEĎRdZ.

Relying on the framework of Pötscher and Prucha (1997), I now introduce explicit conditions

that allow to control dependence in nonlinear models by using the toolbox of physical dependence

measures developed by Wu (2005,2011). The aim is to use a dynamical system perspective to

address the question of imposing meaningful assumptions on nonlinear dynamic models. This

makes it possible to give more primitive conditions under which one can actually estimate (8)

in a semi-nonparametric way.

Stability. An important concept for dynamical system theory is that of stability. Stability

turns out to play a key role in constructing valid asymptotic theory, as it is well understood

in linear models. It is also fundamental in developing the approximation theory of nonlinear

stochastic systems.

Example 3.1. (Linear System). As a motivating example, ﬁrst consider the linear system

Zt“BZt´1`ϵt

where we may assume that tϵtutPZ,ϵtPRdZ, is a sequence of i.i.d. innovations.14 It is well-

known that this system is stable if and only if the largest eigenvalue of Bis strictly less than

13In this speciﬁc subsection, shock identiﬁcation does not play a role and, as such, one can safely ignore B0.

14One could alternatively think of the case of a deterministic input, setting ϵt„Ptpatqwhere Ptpatqis a Dirac

density on the deterministic sequence tatutPZ.

one in absolute value (Lütkepohl,2005). For a higher order linear system, Zt“BpLqZt´1`ϵt

where BpLq “ B1`B2L`. . . `BpLp´1, stability holds if and only if |λmax pBq| ă 1where

B:“

—

–

B1B2¨¨¨ Bp

IdZ0¨¨¨ 0

0IdZ¨¨¨ 0

.¨¨¨ .

0¨¨¨ IdZ0

ﬁ

ﬃ

ﬂ

is the companion matrix.

Extending the notion of stability from linear to nonlinear systems requires some care. Pötscher

and Prucha (1997) derived generic conditions allowing to formally extend stability to nonlinear

models by ﬁrst analyzing contractive systems.

Deﬁnition 3.1 (Contractive System).Let ZtPZĎRdZ,ϵtPEĎRdZ, where tZtutPZis

generated according to

Zt“GpZt´1, ϵtq.

The system is contractive if for all pz , z1q P ZˆZand pe, e1q P EˆE

∥Gpz, ϵq ´ Gpz1, ϵ1q∥ďCZ∥z´z1∥`Cϵ∥e´e1∥

holds with Lipschitz constants 0ďCZă1and 0ďCϵă 8.

Suﬃcient conditions to establish contractivity are

sup #



stackdZ

i“1„BG

BZpzi, eiqȷi



ˇˇˇˇziPZ, eiPE+ă1(11)

and 



BG

Bϵ



ă 8,(12)

where the stacking operator stackdZ

i“1r¨siprogressively stacks the rows, indexed by i, of its

argument (which can be changing with i) into a matrix. Values pzi, eiq P ZˆEchange with

index ias the above condition is derived using the mean value theorem, therefore it is necessary

to consider a diﬀerent set of values for each component of Zt.

It is easy to see, as Pötscher and Prucha (1997) point out, that contractivity is often a

too strong condition to be imposed. Indeed, even in the simple case of a scalar AR(2) model

Zt“b1Zt´1`b2Zt´2`ϵt, regardless of the values of b1, b2PRcontractivity is violated. This is

due to the fact that in a linear AR(2) model studying contractivity reduces to checking ∥B∥ă1

instead of |λmaxpBq| ă 1, and the former is a stronger condition than the latter.15 One can

weaken contractivity – which must hold for Gas a map from Zt´1to Zt– to the idea of eventual

15See Pötscher and Prucha (1997), pp.68-69.

contractivity. That is, intuitively, one can impose conditions on the dependence of Zt`hon Zt

for hą1suﬃciently large. To do this formally, I ﬁrst introduce the deﬁnition of system map

iterates.

Deﬁnition 3.2 (System Map Iterates).Let ZtPZĎRdZ,ϵtPEĎRdZwhere tZtutPZis

generated from a sequence tϵtutPZaccording to

Zt“GpZt´1, ϵtq.

The h-order system map iterate is deﬁned to be

GphqpZt, ϵt`1, ϵt`2, . . . , ϵt`hq:“GpGp¨¨¨GpZt, ϵt`1q¨ ¨¨ , ϵt`h´1q, ϵt`hq

“Gp¨, ϵt`hq ˝ Gp¨, ϵt`h´1q ˝ ¨ ¨¨ ˝ GpZt, ϵt`1q,

where ˝signiﬁes function composition and Gp0qpZtq “ Zt.

To shorten notation, in place of GphqpZt, ϵt`1, ϵt`2, . . . , ϵt`hqI shall use GphqpZt, ϵt`1:t`hq.

Additionally, for 1ďjďh, the partial derivative

BGph˚q

Bϵj

for some ﬁxed h˚is to be intended with respect to ϵt`j, the j-th entry of the input sequence.

This derivative does not dependent on the time index since by assumption Gis time-invariant

and so is Gphq.

Taking again the linear autoregressive model as an example,

Zt`h“GphqpZt, ϵt`1:t`hq “ Bh

1Zt`

h´1

i“0

1ϵt`h´i

since Gpz, ϵq “ B1z`ϵ. If B1determines a stable system, then ∥Bh

1∥Ñ0as hÑ 8 since Gh

converges to zero, and therefore ∥Bh

1∥ďCZă1for hsuﬃciently large. It is thus possible to

use system map iterates to deﬁne stability for higher-order nonlinear systems.

Deﬁnition 3.3 (Stable System).Let ZtPZĎRdZ,ϵtPEĎRdZ, where tZtutPZis generated

according to the system

Zt“GpZt´1, ϵtq.

The system is stable if there exists h˚ě1such that for all pz, z 1q P ZˆZand pe1, e2,...eh˚,

1, e1

2, . . . , e1

h˚q P Ś2h˚

i“1E

∥Gph˚qpz, e1:h˚q ´ Gph˚qpz1, e1

1:h˚q∥ďCZ∥z´z1∥`Cϵ∥e1:h˚´e1

1:h˚∥

holds with Lipschitz constants 0ďCZă1and 0ďCϵă 8.

It is important to remember that this deﬁnition encompasses systems with an arbitrary ﬁnite

autoregressive structure, i.e., Zt“GpZt´p`1, . . . , Zt´1, ϵtqfor pě1, thanks to the companion

formulation of the process. An explicit stability condition, similar to that discussed above for

contractivity, can be derived by means of the mean value theorem. Indeed, for a system to be

stable it is suﬃcient that, at iterate h˚,

sup #



stackdZ

i“1«BGph˚q

BZpzi, ei

1:h˚qﬀi



ˇˇˇˇziPZ, ei

1:h˚P

h˚

i“1

E+ă1(13)

and

sup #



BGph˚q

Bϵjpz, e1:h˚q



ˇˇˇˇzPZ, e1:h˚P

h˚

i“1

E+ă 8, j “1, . . . , h˚.(14)

Pötscher and Prucha (1997) have used conditions (11)-(12) and (13)-(14) as basis for uniform

laws of large numbers and central limit theorems for Lr-approximable and near epoch dependent

processes.

Physical Dependence. Wu (2005) ﬁrst proposed alternatives to mixing concepts by propos-

ing dependence measures rooted in a dynamical system view of a stochastic process. Much work

has been done to use such measures to derive approximation results and estimator properties,

see for example Wu et al. (2010), Wu (2011), Chen et al. (2016), and references within.

Deﬁnition 3.4. If for all tPZ,Zthas ﬁnite rth moment, where rě1, the functional physical

dependence measure ∆ris deﬁned as

∆rphq:“sup



Zt`h´GphqpZ1

t, ϵt`1:t`hq



Lr

where ∥¨∥Lr“ pEr∥¨∥r

rsq1{r,Z1

tis due to F1

t“ p...,ϵ1

t´1, ϵ1

tqand tϵ1

tutPZis an independent copy

of tϵtutPZ.

Chen et al. (2016), among others, show how one may replace the geometric β-mixing as-

sumption with a physical dependence assumption.16 They show that the key suﬃcient condition

is for ∆rphqto decay suﬃciently fast as hgrows.

Deﬁnition 3.5 (Geometric Moment Contracting Process).tZtutPZis geometric moment con-

tracting (GMC) in Lrnorm if there exists a1ą0,a2ą0and τP p0,1ssuch that

∆rphq ď a1expp´a2hτq.

GMC conditions can be considered more general than β-mixing, as they encompass well-

known counterexamples, e.g., the known counterexample provided by Zt“ pZt´1`ϵtq{2for ϵt

i.i.d. Bernoulli r.v.s (Chen et al.,2016). In the following proposition I prove that if contractivity

or stability conditions as deﬁned by Pötscher and Prucha (1997) hold for Gand tϵtutPZis an

i.i.d. sequence, then process tZtutPZis GMC under weak moment assumptions.

16I adapt the deﬁnitions of Chen et al. (2016) to work with a system of the form Zt“GpZt´1, ϵtq.

Proposition 3.1. Assume that tϵtutPZ,ϵtPEĎRdZare i.i.d. and tZtutPZis generated according

Zt“GpZt´1, ϵtq,

where ZtPZĎRdZand Gis a measurable function.

(a) If contractivity conditions (11)-(12)hold, suptPZ∥ϵt∥Lră 8 for rě2and ∥Gpz , ϵq∥ă 8

for some pz, ϵq P ZˆE, then tZtutPZis GMC with

∆rpkq ď aexpp´γhq

where γ“ ´logpCZqand a“2∥Zt∥Lră 8.

(b) If stability conditions (13)-(14)hold, suptPZ∥ϵt∥Lră 8 for rě2and ∥BG{BZ∥ďMZă

8, then tZtutPZis GMC with

∆rpkq ď ¯aexpp´γh˚hq

where γh˚“ ´logpCZq{h˚and ¯a“2∥Zt∥LrmaxtMh´1

Z,1u{CZă 8.

Proposition 3.1 is important in that it links the GMC property to transparent conditions on

the structure of the nonlinear model. It also immediately allows handling multivariate systems,

while previous work has focused on scalar systems (c.f. Wu (2011) and Chen et al. (2016)).

Finally, it is now possible to show that if tW2tutPZsatisﬁes physical dependence assumptions,

then Assumption 8is fulﬁlled, c.f. Lemma 2.2 in Chen and Christensen (2015) for β-mixing

assumptions.

Lemma 3.1. If Assumption 7(iii) holds and tW2tutPZis strictly stationary and GMC then

one may choose an integer sequence q“qpnq ď n{2with pn{qqr`1qKρ∆rpqq “ op1qfor ρ“

5{2´ pr{2`2{rq ` ω2and rą2such that

∥pr

πr

Bπ{nq ´ IK∥“OP˜ζK,nλK,n cqlog K

n¸“oPp1q

provided ζK,nλK,n apqlog Kq{n“op1q.

It can be seen that Lemma 3.1 holds by setting aKplogpnqq2{n“op1qand choosing qpnq “

γ´1logpKρnr`1q, where γis the GMC factor introduced in Proposition 3.1. Therefore, the

rate is the same as the one derived by Chen and Christensen (2015) for exponentially β-mixing

regressors. As shown in Proposition 3.1, system contractivity and stability conditions both

imply geometric moment contractivity, meaning that in place of Assumption 8one may require

the following.

Assumption 9. For rą2it holds either:

(i) tZtutPZis GMC in Lrnorm,

(ii) tZtutPZis generated according to Zt“ΦpZt´1, . . . , Zt´p;ϵtqwhere suptPZ∥ϵt∥Lră 8 and

Φis either contractive according to Deﬁnition 3.1 or stable according to Deﬁnition 3.3.

It is straightforward to prove that if GMC conditions are imposed on tZtutPZ, this im-

plies that tW2tutPZis also GMC.17 Therefore, Lemma 3.1 applies and Assumption 8as well as

Assumption 5(i) are veriﬁed.

3.4 Uniform Convergence and Consistency

Since the key asymptotic condition of Chen and Christensen (2015) is upheld under GMC

assumptions, their uniform convergence bound on the approximation error of the series estimator

can be applied. In order to do so, one must also impose some regularity conditions on π2.

Without loss of generality, let X“ r0,1sand let ∥π2∥8:“supwPY|π2pwq|be the sup-norm

of the conditional mean function π2pwq.

Assumption 10. The unconditional density of Xtis uniformly bounded away from zero and

inﬁnity over X.

Assumption 11. For all 1ďiďdYand 0ďjďp, the restriction of g21

ij to r0,1sbelongs to

the Hölder class Λspr0,1sq of smoothness sě1.

Assumptions 10 and 11 are standard in the nonparametric regression literature. One only

needs to restrict the complexity of functions g21

ij since, for any i, the remainder of π2,i consists

of linear functions. More precisely, what is really needed is that the nonparametric components

of the sieve given by bπ,1K, . . . , bπ,K K are able to approximate g21

ij well enough.

Assumption 12. Sieve Bκbelongs to BSplpκ, r0,1sdY, rq, the B-spline sieve of degree rover

r0,1sdY, or Wavpκ, r0,1sdY, rq, the wavelet sieve of regularity rover r0,1sdY, with rąmaxts, 1u.

In the remainder of the paper, I will consider the cubic spline sieve (r“3), but theoretical

results are stated in the more general setting. Moreover, dwill be the eﬀective dimension of the

joint estimation domain for G21

Theorem 3.1 (Chen and Christensen (2015)).Let Assumptions 4,5,6,7,9,10,11 and 12

hold. If

K— pn{logpnqqd{p2s`dq,

then

∥pπ˚

2´π2∥8“OP´pn{logpnq´s{p2s`dq¯

provided that δě2{s(in Assumption 6) and dă2s.

In Theorem 3.1 the sup-norm consistency rate generally depends on the dimension dand

thus, in principle, the curse of dimensionality slows down convergence compared to parametric

estimation. Fortunately, under the current strctural model assumptions, the nonlinear functional

components in π2are linearly separable in the lag dimension, and thus one may take d“1as

eﬀective dimension. This also means that condition dă2sis trivially satisﬁed.

17A formal argument can be found in Appendix B.

Two-step Consistency. The following theorem ensures that the two-step estimation proce-

dure produces consistent estimates. Since for impulse response functions one needs to study

the iteration of the entire structural model, this results is stated in terms of the full coeﬃcient

matrices.

Theorem 3.2. Let tZtutPZbe determined by structural model (1). Under Assumptions 1,4,

5,6,7,9,10,11 and 12, let p

Π1and p

Π2be the least squares and semi-nonparametric series

estimators for Π1and Π2, respectively, based on the two-step procedure. Then,

∥p

Π1´Π1∥8“OPpn´1{2q

and

∥p

Π2´Π2∥8ďOPˆζK,nλK,n

?n˙`∥p

Π˚

2´Π2∥8,

where p

Π˚

2is the infeasible series estimator involving ϵ1t.

Sup-norm bounds for ∥p

Π˚

2´Π2∥8follow immediately from Lemma 2.3 and Lemma 2.4

in Chen and Christensen (2015). In particular, choosing the optimal nonparametric rate K—

pn{logpnqqd{p2s`dqfor the infeasible estimator would yield

∥p

Π˚

2´Π2∥8“OP´pn{logpnqq´s{p2s`dq¯

as per Theorem 3.1. The condition for consistency in Theorem 3.2 reduces to

K3{2

?n“op1q,

since for B-spline and wavelet sieves λK,n À1and ζK,n À?K. It simple to show that if for the

feasible estimator p

Π2the same rate pn{logpnqqd{p2s`dqis chosen for K, the consistency condition

in the above display is fulﬁlled assuming sě1and d“1.18

Remark 3.3. (Hyperparameter Selection). An important practical question when applying any

series or kernel-type methods is the selection of hyperparameters. For the former, this entails

the choice of the sieve’s size K. Although theory provides only asymptotic rates, a number

of methods can be used to select K, such as cross-validation, generalized cross-validation and

Mallow’s criterion (Li and Racine,2009). In the case of piece-wise splines, once size is selected,

knots can be chosen to be the Kuniform quantiles of the data. This ensures knots are not

located in regions of the domain with very few observations. In simulations and applications,

for simplicity, I select sieve sizes manually and locate knots approximately following empirical

quantiles. In unreported numerical experiments, I check that results are robust to moderate

changes in the number and approximate locations of spline knots.

18The rate for Kmay be optimized by balancing the uniform (infeasible) rate with the error due to residuals.

Since this paper is not concerned with ﬁnding the optimal rate, I do not perform this exercise here.

4 Impulse Response Analysis

Once the model’s linear, functional and structural coeﬃcient are consistently estimated, compu-

tation of nonlinear impulse responses must be addressed. As discussed in Section 2, nonlinear

IRFs are generally hard to lay hands on, since the functional MA(8) form of the process is

highly non-trivial. In this section, I will provide an explicit, iterative algorithm to compute

responses that is numerically straightforward and does not require the construction of moving

average functional coeﬃcients. Moreover, since to derive uniform bounds it is assumed that

the data has compact support, I will introduce a novel yet familiar IRF deﬁnition, called the

relaxed impulse response function, which is compatible with boundedness. Lastly, I prove that

semi-nonparametric IRF estimates are consistent with respect to their population counterparts.

4.1 Computation

Recall from equation (7) in Section 2.1 that impulse responses involve two moving average

lag polynomials, ΘpLqfor the linear model component and ΓpLqfor the nonlinear component,

respectively. As a ﬁrst step, one can derive a semi-explicit recursive algorithm for computing

IRFhpδqin a manner that does not involve simulations of the innovations process.

Proposition 4.1 (Gonçalves et al. (2021), Proposition 3.1).Under Assumptions 1,2and 3,

for any h“0,1, . . . , H, let

Vjpδq:“ErΓjXt`jpδqs ´ ErΓjXt`js.

To compute

IRFhpδq “ Θh,¨1δ`

j“0

Vjpδq,

the following steps can be used:

(i) For j“0, set Xtpδq “ Xt`δand V0pδq “ ErΓ0Xtpδqs ´ ErΓ0Xts.

(ii) For j“1, . . . , h, let

Xt`jpδq “ Xt`j`Θj,11δ`

k“1pΓk,11Xt`j´kpδq ´ Γj,11 Xt`j´kq

“γjpXt`j:t;δq,

where γjare implicitly deﬁned and depend on ΘpLqand ΓpLq.

(iii) For j“1, . . . , h, compute

Vjpδq “ ErΓjγjpXt`j:t;δqs ´ ErΓjXt`js.

The proof of Proposition 4.1 is identical to that in Gonçalves et al. (2021), with the only

variation being that in the current setup it is not possible to collect the nonlinear function across

Xt`j´kpδqand Xt`j´k. Computation of Xt`jpδqin step (ii) involves recursive evaluations of

nonlinear functions, which is why the algorithm is semi-explicit. For each horizon h, one needs

to evaluate h`1iterations of Xtpδq. Importantly, however, this approach dispenses from the need

to simulate innovations tϵt`juh´1

j“1as the joint distribution of tXt`h´1, Xt`j´1, . . . , Xtualready

contains all relevant information. Gonçalves et al. (2021) naturally argue that the algorithm

outlined in Proposition 4.1 is signiﬁcantly more eﬃcient than schemes involving Monte Carlo

simulations like e.g. the one used by Kilian and Vigfusson (2011).

However, tΓjuh

j“1are combinations of real and functional matrices and closed-form deriva-

tion is numerically impractical. Note that, by the deﬁnition of IRFs, the following explicit

iterative algorithm is also valid.

Proposition 4.2. In the same setup of Proposition 4.1, to compute IRFhpδqthe following steps

can be used:

(i 1) For j“0, let Xtpδq “ Xt`δand

IRF0pδq “ «δ

B21

0δﬀ`E«0

G21,0Xtpδqﬀ´E«0

G21,0Xtﬀ.

(ii 1) For j“1, . . . , h, let

Xt`jpδq “ µ1`A12pLqYt`j´1pδq ` A11 pLqXt`j´1pδq ` ϵ1t`j,

Yt`jpδq “ µ2`A22pLqYt`j´1pδq ` H21 pLqXt`jpδq ` B21

0ϵ1t`j`u2t`j,

where H21pLq:“A21pLqL`G21 pLqand u2t`j:“B22

0ϵ2t`j. Setting Zt`jpδq“pXtpδq, Ytpδqq1

it holds

IRFhpδq “ ErZt`jpδqs ´ ErZt`js.

Proposition 4.2 follows directly from the deﬁnition of the unconditional impulse response

(6) combined with explicit iteration of the semi-reduced form (2) and sidesteps the MA(8)

formulation in (7). Step (i1) is trivial in nature. Step (ii1) may not seem useful when compared

to (ii), since, in practice, innovations ϵ1tand u2tare not available. However, let

µ, p

A11pLq,p

A12pLq,p

A21pLq,p

H11pLq,p

B21

be estimates of the model’s coeﬃcients derived, for example, from series estimator p

Π1and p

Π2.

In sample, one can compute residuals pϵ1tand pu2t, and by deﬁnition it holds

Xt“pµ1`p

A12pLqYt´1`p

A11pLqXt´1`pϵ1t,

Yt“pµ2`p

A22pLqYt´1`p

H21pLqXt`p

B21

ϵ1t`pu2t.

This means that one can readily construct the shocked sequence recursively as

Xt`jpδq “ pµ1`p

A12pLqp

Yt`j´1pδq ` p

A11pLqp

Xt`j´1pδq ` pϵ1t`j,

Yt`jpδq “ pµ2`p

A22pLqp

Yt`j´1pδq ` p

H21pLqp

Xt`jpδq ` p

B21

0pϵ1t`j`pu2t`j,

for j“1, . . . , h where p

Xtpδq “ Xt`δ,p

Xt´s“Xt´sfor all sě1and similarly for p

Ytpδq. To

evaluate a structural IRF, over a sample of size none can compute

IRFhpδq “ 1

n´j

t“1”p

Yt`jpδq ´ Ytı,

which is still considerably less demanding than Monte Carlo simulations. Additionally, the

advantage in implementing steps (i1)-(ii1) over the procedure in Proposition 4.1 is that, when

H21pLqis a semi-nonparametric estimate, iterating model equations is numerically much more

straightforward than handling functional MA matrices tp

Γjuh

j“1.

4.2 Nonlinear Responses with Relaxed Shocks

Following Proposition 4.1, the sample impulse response would be

IRFhpδq:“p

Θh,¨1δ`

j“0

Vjpδq,(15)

where

Vjpδq:“1

n´j

t“1”p

Γjp

γjpXt`j:t;δq ´ p

ΓjXt`jı

and p

Θ,p

Γand p

γjare plug-in estimates of the respective quantities based on p

Π1and p

Π2. However,

under Assumptions 4and 5, the construction of impulse response (15) is improper. This can be

immediately seen by noticing that, at impact,

Xtpδq “ γjpXt;δq “ Xt`δ,

meaning that PpXtpδq R Xq ą 0since there is a translation of size δin the support of Xt.

The problem is rooted in the fact that the standard deﬁnition of IRF involves a translation of

the distribution of time tstructural innovations, which is incompatible with the assumptions

imposed in Section 3to derive semi-nonparametric consistency.

There are multiple ways to address this issue. One option, which would require substantial

technical work, is to extend Theorem 3.2 to encompass regressors with unbounded or expanding

domains. A potential direction could be coupling the weighted sieves of Chen and Christensen

(2015) with appropriately deﬁned shocks. Instead, I propose to take a more direct approach by

changing the type of structural shock one studies in a way consistent with bounded domains for

all variables.

Deﬁnition 4.1. A mean-shift structural shock ϵ1tpδqis a transformation of ϵ1tsuch that

Ppϵ1tpδq P E1q “ 1and Erϵ1tpδqs “ δ.

A mean-shift shock is such that the distribution of time tinnovations is shifted to have

mean δ, while retaining support Ealmost surely. This deﬁnition is natural in that it makes

evaluating the eﬀect of the MA(8) component of the unconditional IRF straightforward. With

−2−1 1 2

Figure 1: Example of symmetric shock relaxation. Unperturbed (left, blue) versus shocked

(right, orange) densities of innovations ϵ1t. The shock relaxation function (right, gray) and δ

together determine the form of the relaxed shock used to compute the IRF.

a mean-shift shock, at impact it holds

Xtpδq “ Xt`ϵ1tpδq ´ ϵ1t,

yet ϵ1tpδq ´ ϵ1tis not known unless the transformation for the mean-shock is itself known.

Unfortunately, the assumption that the mean of ϵ1tpδqis exactly equal to δrequires that the

distribution of ϵ1tbe known to properly choose a mean-shift transform. If instead one is willing

to assume only that Erϵ1tpδqs « δ, it is possible to sidestep this requirement by introducing a

shock relaxation function.

Deﬁnition 4.2 (Shock Relaxation Function).A shock relaxation function is a map ρ:E1Ñ

r0,1ssuch that ρpzq “ 0for all zPRzE1,ρpzq ě 0for all zPE1and there exists z0PE1for

which ρpz0q “ 1.

In general, choosing a shock relaxation function without taking into account the shape

of domain E1does not necessarily imply that the relaxed shocks will not push the structural

variable out-of-bounds. Therefore, I also introduce the notion of compatibility.

Deﬁnition 4.3 (Compatible Relaxation).Consider a shock δPRand let E1“ ra, bs.

(i) If δą0,ρis said to be right-compatible with δif

ρpzq ď b´z

|δ|for all zPE.

(ii) If δă0,ρis said to be left-compatible with δif

ρpzq ď a`z

|δ|for all zPE.

(iii) Given shock size |δ| ą 0,ρis said to be compatible if it is both right- and left-compatible.

By setting

ϵ1tpδq “ ϵ1t`δρpϵ1tq

where ρis compatible with δ, it follows that Xtpδq “ Xt`δρpϵ1tqand |Erϵ1tpδqs|“|δErρpϵ1tqs|ď

|δ|since Erρpϵ1tqs P r0,1qby deﬁnition of ρ. If ρis a bump function, a relaxed shock is a structural

shock that has been mitigated proportionally to the density of innovations at the edges of E1

and the squareness of ρ. For better intuition, Figure 1provides a graphical rendition of shock

relaxation of a symmetric error distribution with a bump function.

Remark 4.1. The deﬁnition of compatible relaxation function is static, as it considers only the

impact eﬀect of a shock. Nonetheless, the assumption that XtPXfor all tmust also hold for

Xtpδq, the shocked structural variable. In theory, given δ, one can always either expand Xor

strengthen ρso that compatibility is enforced at all horizons 1ďhďH. For simulations, where

one has access to the data generating process, the choice of domains and relaxation functions

can be done transparently. In practice, some care is required. When working with empirical

data, unless one is willing to assume Xtis wholly exogenous – as in Section 6.1 with monetary

policy shocks – or strictly autoregressive, some scenarios are more amenable to analysis with

the framework presented here than other. In Section 6.2, following Istreﬁ and Mouabbi (2018),

I will let Xtbe a non-negative uncertainty measure, so that negative shocks are harder to study

without producing sequences that contain negative uncertainty values. Thus, I will focus on

positive, contractionary shocks.

For a given Xt, transformation Xt`δρpϵ1tqis not directly applicable since ϵ1tis not observed.

In practice, therefore, I will consider

Xtpδq:“Xt`δρppϵ1tq.

For simplicity of notation, let r

δt:“δρpϵ1tq. Similarly to Step (ii) of Proposition 4.1, given a

path Xt`j:tone ﬁnds

Xt`jpr

δtq “ Xt`j`Θj,11r

δt`

k“1pΓk,11Xt`j´kpr

δtq ´ Γk,11Xt`j´kq

“γjpXt`j:t;r

δtq,

The relaxed-shock impulse response is thus given by

IRFhpδq:“ErZt`jpr

δtq ´ Zt`js “ Θh,¨1δErρpϵ1tqs `

k“1

E”ΓkXt`j´kpr

δtq ´ ΓkXt`j´kı.

In what follows, I show that by replacing r

δtwith p

δt“δρppϵ1tqit is possible to consistently

estimate unconditional expectations involving Xt`jpr

δtqas well as Xt`j, and thus Ą

IRFhpδq, by

averaging over sample realizations.

4.3 Relaxed Impulse Response Consistency

For a given δPRand compatible shock relaxation function ρ, vector Vjpδqis the nonlinear

component of impulse responses. One can focus on a speciﬁc variable’s response by introducing,

for 1ďℓďd,

Vj,ℓpδq:“1

n´j

t“1”Γj,ℓγjpXt`j:t;r

δtq ´ Γj,ℓXt`jı,

where Vj,ℓpδqis the horizon jnonlinear eﬀect on the ℓth variable and Γj,ℓ is the ℓth component

of functional vector Γj. For the sake of notation I also deﬁne

vj,ℓpXt`j:t;r

δtq:“Γj,ℓγjpXt`j:t;r

δtq ´ Γj,ℓXt`j.

Let pvj,ℓpXt`j:t;p

δtqbe its sample equivalent, so that

vj,ℓpXt`j:t;p

δtq “ p

Γj,ℓp

γjpXt`j:t;p

δtq ´ p

Γj,ℓXt`j,

Vj,ℓpδq “ 1

n´j

t“1p

vj,ℓ`Xt`j:t;p

δt˘

and

IRFh,ℓpδq “ Θh,¨1δ n´1

t“1

ρppϵ1tq `

j“0p

Vj,ℓpδq

for 1ďℓďd.

Theorem 4.1. Let y

IRFh,ℓpδqbe a semi-nonparametric estimate for the horizon hrelaxed shock

IRF of variable ℓ. Under the same assumptions as in Theorem 3.2

IRFh,ℓpδqP

ÑĄ

IRFh,ℓpδq

for any ﬁxed integers 0ďhă 8 and 1ďℓďd.

5 Simulations

This section is devoted to analyzing the empirical performance of the two-step semi-nonparametric

estimation strategy discussed above. I will consider the two simulation setups employed by

Gonçalves et al. (2021), with focus on bias and MSE of the estimated relaxed shocked impulse

response functions. Additionally, I provide simulations under a modiﬁed design which high-

light how in larger samples the non-parametric sieve estimator consistently recovers impulse

responses, while a least-squares estimator constructed with a pre-speciﬁed nonlinear transform

does not. In all simulations, I use a B-spline sieve of order 1.

5.1 Benchmark Bivariate Design

The ﬁrst simulation setup involves a bivariate DGP where the structural shock does not directly

aﬀect other observables. This is a simple environment to check that indeed the two-step estimator

recover the nonlinear component of the model and impulse responses are consistently estimated,

and that the MSE does not worsen excessively.

I consider three bivariate data generation processes. DGP 1 sets Xtto be a fully exogenous

innovation process,

Xt“ϵ1t,

Yt“0.5Yt´1`0.5Xt`0.3Xt´1´0.4 maxp0, Xtq ` 0.3 maxp0, Xt´1q ` ϵ2t.(16)

DGP 2 adds an autoregressive component to Xt, but maintains exogeneity,

Xt“0.5Xt´1`ϵ1t,

Yt“0.5Yt´1`0.5Xt`0.3Xt´1´0.4 maxp0, Xtq ` 0.3 maxp0, Xt´1q ` ϵ2t.(17)

Finally, DGP 3 add an endogenous eﬀect of Yt´1on the structural variable by setting

Xt“0.5Xt´1`0.2Yt´1`ϵ1t,

Yt“0.5Yt´1`0.5Xt`0.3Xt´1´0.4 maxp0, Xtq ` 0.3 maxp0, Xt´1q ` ϵ2t.(18)

Following Assumption 1, innovations are mutually independent. To accommodate Assumptions

4and 5, both ϵ1tand ϵ2tare drawn from a truncated standard Gaussian distribution over

r´3,3s.19 All DGPs are centered to have zero intercept in population.

I evaluate bias and MSE plots using 1000 Monte Carlo simulation. For a chosen horizon

H, the impact of a relaxed shock on ϵ1tis evaluated on Yt`hfor h“1, . . . , H. To compute the

population IRF, I employ a direct simulation strategy that replicates the shock’s propagation

through the model and I use 10 000 replications. To evaluate the estimated IRF, the two-step

procedure is implemented: a sample of length nis drawn, the linear least squares and the semi-

nonparametric series estimators of the model are used to estimate the model and the relaxed

IRF is computed following Proposition 4.2. For the sake of brevity, I discuss the case of δ“1

and I set the shock relaxation function to be

ρpzq “ exp ˜1`„



´1ȷ´1¸

over interval r´3,3sand zero everywhere else.20 Choices of δ“ ´1and δ“ ˘0.5yield similar

results in simulations, so I do not discuss them here.

Figure 2contains the results for sample size n“240. This choice is motivated by considering

the average sample sizes found in most macroeconometric settings: it is equivalent to 20 years

of monthly data or 60 yearly of quarterly data (Gonçalves et al.,2021). The benchmark method

is an OLS regression that relies on a priori knowledge of the underlying DGP speciﬁcation.

Given the moderate sample size, to construct the cubic spline sieve estimator of the nonlinear

component of the model I use a single knot, located at 0. The simulations in Figure 2show that

while the MSE is slighlty higher for the sieve model, the bias is comparable across methods. Note

19Let eit „Np0,1qfor i“1,2, then the truncated Gaussian innovations used in simulation are set to be

ϵit “minpmaxp´3, eitq,3q. The resulting r.v.s have a non-continuous density with two mass points at -3 and 3.

However, in practice, since these masses are negligible, for the moderate sample sizes used this choice does not

create issues.

20It can be easily checked that this choice of ρis compatible with shocks of size 0ď |δ| ď 1.

0 5 10 15 20 25

15 10-3 n = 240

Sieve OLS

0 5 10 15 20 25

-0.02

-0.015

-0.01

-0.005

0.005

0.01 n = 240

Sieve OLS

0 5 10 15 20 25

15 10-3 n = 240

Sieve OLS

0 5 10 15 20 25

-0.02

-0.015

-0.01

-0.005

0.005

0.01 n = 240

Sieve OLS

0 5 10 15 20 25

0.05

0.1

0.15 n = 240

Sieve OLS

0 5 10 15 20 25

-0.4

-0.3

-0.2

-0.1

0n = 240

Sieve OLS

Figure 2: Simulations results for DGPs 1-3.

that for DGP 3, due to the dependence of the structural variable on non-structural series lags,

the MSE and bias increase signiﬁcantly, and there is no meaningful diﬀerence in performance

between the two estimation approaches.

5.2 Structural Partial Identiﬁcation Design

To showcase the validity of the proposed sieve estimator under the type of partial structural

identiﬁcation discussed in the paper, I again rely on the simulation design proposed by Gonçalves

et al. (2021). All speciﬁcations are block-recursive, and require estimating the contemporaneous

eﬀects of a structural shock on non-structural variables, unlike in the previous section.

The form of the DGPs is

B0Zt“B1Zt´1`C0fpXtq ` C1fpXt´1q ` ϵt,

where in all variations of the model

B0“»

—

–

1 0 0

´0.45 1 ´0.3

´0.05 0.1 1 ﬁ

ﬃ

ﬂ, C0“»

—

–

´0.2

0.08ﬁ

ﬃ

ﬂ,and C1“»

—

–

´0.1

0.2ﬁ

ﬃ

ﬂ.

I focus on the case fpxq “ maxp0, xq, since this type of nonlinearity is simpler to study. DGP 4

treats Xtas an exogenous shock by setting

B1“»

—

–

000

0.15 0.17 ´0.18

´0.08 0.03 0.6ﬁ

ﬃ

ﬂ;

DGP 5 add serial correlation to Xt,

B1“»

—

–´0.13 0 0

0.15 0.17 ´0.18

´0.08 0.03 0.6ﬁ

ﬃ

ﬂ;

and DGP 6 includes dependence on Yt´1,

B1“»

—

–´0.13 0.05 ´0.01

0.15 0.17 ´0.18

´0.08 0.03 0.6ﬁ

ﬃ

ﬂ.

For these data generating processes, I employ the same setup of simulations with DGPs 1-3,

including the number of replications as well as the type of relaxed shock. as well as the sieve

grid. Here too I evaluate MSE and bias of both the sieve and the correct speciﬁcation OLS

estimators with as sample size of n“240 observations. The results in Figure 3show again that

there is little diﬀerence in terms of performance between the semi-nonparametric sieve approach

and a correctly-speciﬁed OLS regression.

0 2 4 6 8 10 12

15 10-3 n = 240

0 2 4 6 8 10 12

-4

-2

410-3 n = 240

0 2 4 6 8 10 12

15 10-3

Sieve OLS

0 2 4 6 8 10 12

-4

-2

410-3

Sieve OLS

0 2 4 6 8 10 12

15 10-3 n = 240

0 2 4 6 8 10 12

-4

-2

410-3 n = 240

0 2 4 6 8 10 12

15 10-3

Sieve OLS

0 2 4 6 8 10 12

-4

-2

410-3

Sieve OLS

0 2 4 6 8 10 12

15 10-3 n = 240

0 2 4 6 8 10 12

-4

-2

410-3 n = 240

0 2 4 6 8 10 12

15 10-3

Sieve OLS

0 2 4 6 8 10 12

-4

-2

410-3

Sieve OLS

Figure 3: Simulations results for DGPs 4-6.

−3−2−1 1 2 3

−1

Figure 4: Plot of nonlinear function φpxqused in DGP 7.

5.3 Model Misspeciﬁcation

The previous sections report results that support the use of the sieve IRF estimator in a sample of

moderate size, since it performs comparably to a regression performed with a priori knowledge

of the underlying DGP. I now show that the semi-nonparametric approach is also robust to

model misspeciﬁcation compared to simpler speciﬁcations involving ﬁxed choices for nonlinear

transformations.

To this end, I modify DGP 2 to use a smooth nonlinear transformation to deﬁne the eﬀect

of structural variable Xton Yt. That is, there is no compounding of linear and nonlinear eﬀects.

The autoregressive coeﬃcient in the equation for Xtis also increased to make the shock more

persistent. The new data generating process, DGP 21, is, thus, given by

Xt“0.8Xt´1`ϵ1t,

Yt“0.5Yt´1`0.9φpXtq ` 0.5φpXt´1q ` ϵ2t.(19)

where φpxq:“ px´1qp0.5`tanhpx´1q{2q, which is plotted in Figure 4.

To emphasize the diﬀerence in estimated IRFs, in this setup I focus on δ“ ˘2, which

requires adapting the choice of innovations and shock relaxation function. In simulations of

DGP 21,ϵ1tand ϵ2tare both drawn from a truncated standard Gaussian distribution over

r´5,5s. The shock relaxation function of this setup is given by

ρpzq “ exp ˜1`„



3.9

´1ȷ´1¸

over interval r´5,5sand zero everywhere else. This form of ρis adapted to choices of δsuch

that 0ă |δ| ď 2. The sieve grid now consists of 4 equidistant knots within p´5,5q. I use the

same numbers of replications as in the previous simulations. Finally, the regression design is

identical to that used for DGP 2 under correct speciﬁcation.

The results obtained with sample size n“2400 are collected in Figure 5. I choose this larger

sample size to clearly showcase the inconsistency of impulse responses under misspeciﬁcation:

as it can be observed, the simple OLS estimator involving the negative-censoring transform

0 5 10 15 20 25

0.01

0.02

0.03

0.04

0.05

0.06 n = 2400

Sieve OLS

0 5 10 15 20 25

-0.25

-0.2

-0.15

-0.1

-0.05

0.05 n = 2400

Sieve OLS

(a) δ“ `2

0 5 10 15 20 25

0.005

0.01

0.015 n = 2400

Sieve OLS

0 5 10 15 20 25

-0.02

0.02

0.04

0.06

0.08

0.1 n = 2400

Sieve OLS

(b) δ“ ´2

Figure 5: Simulations results for DGP 7.

produces IRF estimates with consistently worse MSE and bias than those of the sieve estimator

at almost all horizons. Similar results are also obtained for more moderate shocks δ“ ˘1,

but the diﬀerences are less pronounced. These simulations suggest that the semi-nonparametric

sieve estimator can produce substantially better IRF estimates in large samples than methods

involving nonlinear transformations selected a priori.

In this setup, it is also important to highlight the fact that the poor performance of OLS

IRF estimates does not come from φpxqbeing “complex”, and, thus, hard to approximate by

combinations of simple functions. In fact, if in DGP 21function φis replaced by rφpxq:“

φpx`1q, the diﬀerences between sieve and OLS impulse response estimates become minimal in

simulations, with the bias of the latter decreasing by approximately an order of magnitude (see

Figure 8in Appendix C). This is simply due the fact that rφpxqis well approximated by maxp0, xq

directly. However, one then requires either prior knowledge or sheer luck when constructing the

nonlinear transforms of Xtfor an OLS regression. The proposed series estimator, instead,

just requires an appropriate choice of sieve. Many data-driven procedures to select sieves in

applications have been proposed, see for example the discussion in Kang (2021).

6 Empirical Applications

In this section, I showcase the practical utility of the proposed semi-nonparametric sieve estima-

tor by considering two applied exercises. First, I revisit the empirical analysis of Gonçalves et al.

(2021), which is itself based on the work of Tenreyro and Thwaites (2016). This provides both

linear and nonlinear benchmarks for the monetary policy responses within a compact econo-

metric model. I ﬁnd that, although the diﬀerences between approaches are mild, nonparametric

IRFs in fact provide counter-evidence to the conclusions reported by Gonçalves et al. (2021). In

the second application, I compare the linear and nonlinear impulse responses that are produced

by uncertainty shocks in the setup studied by Istreﬁ and Mouabbi (2018). Here, sieve-estimated

IRFs show diﬀerences in shape, timing and intensity, chieﬂy when the sign of the shock changes.

6.1 Monetary Policy Shocks

The objective of the empirical analysis in Gonçalves et al. (2021) is to analyze the eﬀects of a

monetary policy shock on a model of the US macroeconomy. Structural identiﬁcation is achieved

via a narrative approach, following the seminal work of Romer and Romer (2004).

The four-variable model is set up identically to the one of Gonçalves et al. (2021), Section

7. Let Zt“ pXt,FFRt,GDPt,PCEtq1, where Xtis the series of narrative U.S. monetary policy

shocks, FFRtis the federal funds rate, GDPtis log real GDP and PCEtis PCE inﬂation.21 As

21In Gonçalves et al. (2021) p. 122, it is mentioned that CPI inﬂation is included in the model, but both

in the replication package made available by one the authors (https://sites.google.com/site/lkilian2019/

research/code) from which I source the data, and Tenreyro and Thwaites (2016), PCE inﬂation is used instead.

Moreover, the authors say that both the FFR and PCE enter the model in ﬁrst diﬀerences, yet in their code these

variables are kept in levels. I keep their original formulation to allow for a proper comparison between estimation

methods.

a pre-processing step, GDP is transformed to log GDP and then linearly detrended. The data

is available quarterly and spans from 1969:Q1 to 2007:Q4. As in Tenreyro and Thwaites (2016),

I use a model with one lag, p“1. Narrative shock Xtis considered to be an i.i.d. sequence,

i.e. Xt“ϵ1t, therefore I assume no dependence on lagged variables when implementing pseudo-

reduced form (2). Like in Gonçalves et al. (2021), I consider positive and negative shocks of size

|δ| “ 1. As such, I choose

ρpzq “ It|z| ď 4uexp ˜1`„



´1ȷ´1¸

to be the shock relaxation functions. Figure 10 in Appendix Cprovides a check for the validity of

ρgiven the sample distribution of Xt. Knots for sieve estimation are located at t´1,0,1u. The

model is block-recursive, and the structural formulation of Section 2.1 allows identifying the U.S.

monetary policy shocks without the need to impose additional assumptions on the remaining

shocks. Gonçalves et al. (2021), following Tenreyro and Thwaites (2016), use two nonlinear

transformations, Fpxq “ maxp0, xqand Fpxq “ x3, to try to gauge how negative versus positive

and large versus small shocks, respectively, aﬀect the U.S. macroeconomy. For clarity, below

I refer to this approach as “parametric nonlinear method”. Since the authors ﬁnd that the

inclusion of a cubic term does not meaningfully change impulse responses, I focus on comparing

the IRFs estimated via sieve regression with the ones obtained by setting Fpxq “ maxp0, xq, as

well as by not including nonlinear terms (i.e. linear IRFs).

Figure 6shows the estimated impulse response to both a positive and negative unforeseen

monetary policy shock. The impact on the federal funds rate is consistent across all three

procedures, but there are important diﬀerences in GDP and inﬂation responses. In case of an

exogenous monetary tightening change, the parametric nonlinear response for GDP, unlike in

the case of linear and parametric nonlinear IRFs, is nearly zero at impact and has a monotonic

decrease until around 10 quarters ahead. The change is shape is meaningful, as the procedure

of Gonçalves et al. (2021) still yields a small short-term upward jump in GDP when a monetary

tightening shock hits. Moreover, after the positive shock, the sieve GDP responses reaches its

lowest value 4 and 2 quarters before the linear and parametric nonlinear responses, while its size

is 13% and 16% larger, respectively.22 Finally, the sieve PCE response is positive for a shorter

interval, but looks to be more persistent once it turns negative also 10 months after impact.

When the shock is expansionary, sieve IRFs show a pronounced asymmetry, even more than

that of parametric nonlinear responses. One can observe that semi-nonparametric federal funds

rate IRF is marginally mitigated compared to the alternative estimates. An important puzzle

is due to the clearly negative impact on GDP. Indeed, both types of nonlinear responses show

a drop in output in the ﬁrst 5 quarters. Also note that the PCE inﬂation has a positive spike

the ﬁrst couple of quarters after impact. Such a quick change seems unrealistic, as one does not

expect inﬂation to suddenly reverse sign, but, as Gonçalves et al. (2021) also remark, the overall

22The strength of this eﬀect changes across diﬀerent shocks sizes, as Figure 12 in Appendix Cproves. As

shocks sizes get smaller, nonlinear IRFs, both parametric and sieve, show decreasing negative eﬀects.

0 5 10 15 20

Quarters

-0.5

0.5

1.5

2.5 Fed Funds Rate

Linear

Nonlin-par

Sieve

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

(a) δ“ `1

0 5 10 15 20

Quarters

-2

-1.5

-1

-0.5

0Fed Funds Rate

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

(b) δ“ ´1

Figure 6: Eﬀect of an unexpected U.S. monetary policy shock on federal funds rate, GDP and

inﬂation. Linear (gray, dashed), parametric nonlinear with Fpxq “ maxp0, xq(red, point-dashed)

and sieve (blue, solid) structural impulse responses. For δ“ `1, the lowest point of the GDP

response is marked with a dot.

impact on inﬂation of both shocks is small when compared to the change in federal funds rate.

This comparison between methods, and speciﬁcally the nature of nonparametric impulse

responses, provides evidence that a small econometric model, such as the one studied by Tenreyro

and Thwaites (2016), may be inadequate to fully capture the dynamic eﬀects of monetary policy

shocks. In both setups, however, impulse response interpretation is only suggestive, as conﬁdence

bands are missing and only pointwise IRFs are available. Whether the puzzles highlighted above

would persist after accounting for estimation uncertainty is an important research question that

I leave for future analysis.

6.2 Uncertainty Shocks

Uncertainty in interest rates appears to be a signiﬁcant factor in recent economic history. Start-

ing with the fundamental changes brought forth by the unprecedented measures of unconven-

tional monetary policy after the 2007-2008 ﬁnancial crisis, to the powerful economic stimuli

during the COVID-19 pandemic, and ﬁnally the subsequent interest rate tightening and in-

ﬂation phenomenon of 2022, central banks and institutional agents are often very concerned

about uncertainty. Since traditional central bank policymaking is heavily guided by the prin-

ciple that the central bank can and should inﬂuence expectations, controlling the (perceived)

level of ambiguity in current and future commitments is key.

Istreﬁ and Mouabbi (2018) provide an analysis of the impact of unforeseen changes in the

level of subjective interest rate uncertainty on the macroeconomy. They derive a collection of

new indices based on short- and long-term profession forecasts. Their empirical study goes in

depth into studying the diﬀerent components that play a role in transmitting uncertainty shocks,

but here I will focus on re-evaluating their structural impulse response estimates under the light

of potentially-missing nonlinear eﬀects. For the sake of simplicity, my evaluation will focus only

on the 3-months-ahead uncertainty measure for short-term interest rate maturities (3M3M) and

the US economy.23

Like in Istreﬁ and Mouabbi (2018), let Zt“ pXt,IPt,CPIt,PPIt,RTt,URtq1be a vector

where Xtis the chosen uncertainty measure, IPtis the (log) industrial production index, CPIt

is the CPI inﬂation rate, PPItis the producer price inﬂation rate, RTtis (log) retail sales and

URtis the unemployment rate. The nonlinear model speciﬁcation is given by

Zt“µ`A1Zt´1`A2Zt´1`F1pXt´1q ` F2pXt´2q ` DWt`ut,

where Wtincludes a linear time trend and oil price OILt.24 The data has monthly frequency and

spans the period between May 1993 and July 2015.25 Note here that, following the identiﬁcation

strategy of Gonçalves et al. (2021), nonlinear functions F1and F2are to be understood as not

eﬀecting Xt, which is the structural variable. The linear VAR speciﬁcation of Istreﬁ and Mouabbi

(2018) is recovered by simply assuming F1“F2“0prior to estimation. Since they use recursive

identiﬁcation and order the uncertainty measure ﬁrst, this model too is block-recursive.

I consider a positive shock with intensity δ“σϵ,1, where σϵ,1is the standard deviation of

structural innovations. In this empirical exercise, the relaxation function is given by

ρpzq “ I"|z| ď 1

4*exp ˆ1`”|4x|8´1ı´1˙

and I set t0.1,0.3uto be the cubic spline knots. As 3M3M is a non-negative measure of un-

certainty, some care must be taken to make sure that the shocked paths for Xtdo not reach

negative values. Figure 14 in Appendix Cshows that the relaxation function is compatible, and

also that the shocked nonlinear paths of Xtwith impulse δand δ1all do not cross below zero.

Figure 7presents both the linear and nonlinear structural impulse responses obtained.

Importantly, even though Istreﬁ and Mouabbi (2018) estimate a Bayesian VAR model and here

23Istreﬁ and Mouabbi (2018) also provide comparisons with results obtained with the other uncertainty mea-

sures, which they comment are all very similar to the ones obtained with 3M3M. Their paper additionally evaluates

a number of other highly developed countries.

24Inclusion of linear exogenous variables in the semi-nonparametric theoretical framework detail in Section 3

is straightforward as long as one can assume that they are stationary and weakly dependent. The choice of using

p“2is identical to that of the original authors, based on BIC.

25I reuse the original data employed by the authors, who kindly shared it upon request, but rescale retail sales

(RTt) so that the level on January 2000 equals 100.

0 20 40 60

Months

-0.05

0.05 3M3M Uncertainty

Linear

Sieve

0 20 40 60

Months

-0.5

0.5

0 20 40 60

Months

-0.1

-0.05

0.05

0.1

CPI Inflation

0 20 40 60

Months

-0.5

0.5 PP Inflation

0 20 40 60

Months

-1

-0.5

0.5

1Retail

0 20 40 60

Months

-0.3

-0.2

-0.1

0.1

0.2

Unemployment

Figure 7: Eﬀect of an unexpected, one-standard-deviation uncertainty shock to US macroeco-

nomic variables. Linear (gray, dashed) and sieve (blue, solid) structural impulse responses. The

extreme points of the responses are marked with a dot.

I consider a frequentist vector autoregressive benchmark, the shape of the IRFs is retained, c.f.

the median response in the top row of their Figure 4. When uncertainty increases, industrial

production drops, and the size and extent of this decrease is intensiﬁed in the nonlinear responses.

In fact, the sieve IP response reaches a value that is 54% lower than that of the respective linear

IRF.26 A similar behavior holds true for retail sales (38% lower) and unemployment (23% higher),

proving that this shock is more profoundly contractionary than suggested by the linear VAR

model. Further, CPI and PP inﬂation both show short-term ﬂuctuations which strengthen the

short- and medium-term impact of the shock. CPI and PP nonlinear inﬂation responses are

76% and 41% stronger than their linear counterpart, respectively. These diﬀerences suggest

that linear IRFs might be both under-estimating the short-term intensity and misrepresenting

long-term persistence of inﬂation reactions. From another perspective, Nowzohour and Stracca

(2020) presented evidence that consumer consumption growth, credit growth and unemployment

do not co-move with the policy uncertainty index (EPU) of Baker et al. (2016), but are negatively

correlated with ﬁnancial volatility. Given the strength of nonlinear IRFs, this discrepancy may

also suggest that the 3M3M uncertainty measure partially captures the ﬁnancial channel, too.

The introduction of nonlinear terms in the structural VAR of Istreﬁ and Mouabbi (2018)

thus provides evidence that fundamental impulse response features might otherwise be missed.

Indeed, Figure 13 in Appendix C- which plots regression functions of endogenous variables

26Figure 15 in Appendix Cconﬁrms that this diﬀerence is consistent over a range of shock sizes, too.

with respect to Xt- proves that high and low uncertainty levels may have signiﬁcantly diﬀerent

eﬀects on endogenous economic variables. In particular, at the second lag, tail eﬀects appear to

be milder, while at low levels changes in uncertainty have more pronounced impact.

7 Conclusion

This paper studies the application of semi-nonparametric series estimation to the problem of

structural impulse response analysis for time series. After ﬁrst discussing the partial identiﬁ-

cation model setup, I have used the conditions of system contractivity and stability to derive

physical measures of the dependence for nonlinear systems. In turn, these allow to derive

primitive conditions under which series estimation can be employed and structural IRFs are

consistently estimated. The simulation results prove that this approach is valid in moderate

samples and has the added beneﬁt of being robust to misspeciﬁcation of the nonlinear model

components. Finally, two empirical applications showcase the utility in departing from both

linear and parametric nonlinear speciﬁcations when estimating structural responses.

There are many possible avenues for extending the results I have presented here. A key

aspect that I have not touched upon is inference in the form of conﬁdence intervals: the theory

of Chen and Christensen (2015) does not encompass uniform inference, and, as such, additional

results have to be developed. Indeed, (uniform) conﬁdence bands are necessary to properly

quantify the uncertainty of IRF estimates. Belloni et al. (2015) give a uniform asymptotic

inference theory, but their derivations are limited to non-dependent data. Li and Liao (2020)

and Cattaneo et al. (2022) provide theoretical coupling results that could be exploited in order

to handle time series data. Chen and Christensen (2018) give a theory of uniform inference for

panel IV setups, which could possibly be generalized to handle nonlinear IRFs. In the spirit

of Kang (2021), it would be also important to derive inference results that are uniform in the

selection of series terms, as, in practice, a data-driven procedure for selecting Kshould be used.

Studying other sieve spaces, such as neural networks or shape-preserving sieves (Chen,2007),

would also be highly desirable. The latter can be especially useful in contexts where economic

knowledge suggests that the nonlinear components of the model are e.g. strictly monotonic

increasing or convex. Finally, sharpening of convergence rates used in the main proofs is of

independent interest.

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Appendix

A Preliminaries

Matrix Norms. Let

∥A∥r:“max ␣∥Ax∥rˇˇ∥x∥rď1(

be the r-operator norm of matrix APCd1ˆd2. The following Theorem establishes the equivalence

between diﬀerent operator norms as well as the compatibility constants.

Theorem A.1 (Feng (2003)).Let 1ďp, q ď 8. Then for all APCd1ˆd2,

∥A∥pďλp,qpd1qλq,ppd2q∥A∥q,

where

λa,bpdq:“$

1if aěb,

d1{a´1{bif aăb.

This norm inequality is sharp.

In particular, if pąqthen it holds

pd2q1{q´1{p∥A∥pď∥A∥qď pd1q1{q´1{p∥A∥p.

B Proofs

B.1 GMC Conditions and Proposition 3.1

Lemma B.1. Assume that tϵtutPZ,ϵtPEĎRdZare i.i.d., and tZtutPZis generated according

Zt“GpZt´1, ϵtq,

where ZtPZĎRdZand Gis a measurable function. If either

(a) Contractivity conditions (11)-(12)hold, suptPZ∥ϵt∥Lră 8 and ∥Gpz, ϵq∥ă 8 for some

pz, ϵq P ZˆE;

(b) Stability conditions (13)-(14)hold, suptPZ∥ϵt∥Lră 8 and ∥BG{BZ∥ďMZă 8;

then

sup

∥Zt∥Lră 8 w.p.1.

Proof.

(a) In a ﬁrst step, we show that, given event ωPΩ, realization Ztpωqis unique with probability

one. To do this, introduce initial condition z˝for ℓą1such that z˝PZand ∥z˝∥ă 8.

Deﬁne

Zp´ℓq

tpωq “ Gpℓqpy˝, ϵt´ℓ`1:tpωqq.

Further, let Z1p´ℓq

tbe the realization with initial condition z1

˝“ z˝and innovation realiza-

tions ϵt´ℓ`1:tpωq. Note that



Zp´ℓq

tpωq ´ Z1p´ℓq

tpωq



ďCℓ

Z

z˝´z1

˝

,

which goes to zero as ℓÑ 8. Therefore, if we set Ztpωq:“limℓÑ8 Zp´ℓq

tpωq,Ztpωqis

unique with respect to the choice of z˝w.p.1. A similar recursion shows that



Zp´ℓq

tpωq



ďCℓ

Z∥z˝∥`

ℓ´1

k“0

ZCϵ∥ϵt´kpωq∥.

By norm equivalence, this implies



Zp´ℓq



LrďCℓ

Z∥z˝∥r`

ℓ´1

k“0

ZCϵ∥ϵt´k∥Lr

ďCℓ

Z∥z˝∥r` p1´CZq´1Cϵsup

tPZ

∥ϵt∥Lră 8,

and taking the limit ℓÑ 8 proves the claim.

(b) Consider again distinct initial conditions z1

˝“ z˝and innovation realizations ϵt´ℓ`1:tpωq,

yielding Z1p´ℓq

tpωqand Zp´ℓq

tpωq, respectively. We may use the contraction bound derived

in the proof of Proposition 3.1 (b) below, that is,



Zp´ℓq

tpωq ´ Z1p´ℓq

tpωq



rďCℓ

ZC2∥z˝´z1

˝∥r,

where C2ą0is a constant. With trivial adjustments, the uniqueness and limit arguments

used for (a) above apply here too.

Proof of Proposition 3.1.

(a) By assumption it holds that for all pz, z1q P ZˆZand pe, e1q P EˆE

∥Gpz, ϵq ´ Gpz1, ϵ1q∥ďCZ∥z´z1∥`Cϵ∥e´e1∥

holds, where 0ďCZă1and 0ďCϵă 8. The equivalence of norms directly generalizes

this inequality to any r-norm for rą2. We study ∥Zt`h´Z1

t`h∥rwhere Z1

t`his constructed

with a time-tperturbation of the history of Zt`h. Therefore, for any given tand hď1it

holds that



Zt`h´GphqpZ1

t, ϵt`1:t`hq



rďCZ∥Gph´1qpZt, ϵt`1:t`h´1q ´ Gph´1qpZ1

t, ϵt`1:t`h´1q∥r

ďCh

Z∥Zt´Z1

t∥r,

since sequence ϵt`1:t`his common between Zt`hand Z1

t`h. Clearly then



Zt`h´GphqpZ1

t, ϵt`1:t`hq



rď2∥Zt∥rexpp´γhq

for γ“ ´logpCZq. Letting a“2∥Zt∥rand shifting time index tbackward by h, since

supt∥Zt∥Lră 8 w.p.1 from Lemma B.1 the result for Lrfollows with τ“1.

(b) Proceed similar to (a), but notice that now we must handle cases of steps 1ďhăh˚.

Consider iterate h˚`1, for which



Zt`h`1´Gph`1qpZ1

t, ϵt`1:t`h`1q



rďCZ∥GphqpGpZt, ϵt`1q, ϵt`2:t`hq ´ GphqpGpZ1

t, ϵt`1q, ϵt`2:t`hq∥r

ďCh

Z∥GpZt, ϵt`1q ´ GpZ1

t, ϵt`1q∥r

ďCh

ZMZ∥Zt´Z1

t∥r

by the mean value theorem. Here we may assume that MZě1otherwise we would fall

under case (a), so that MZďM2

Zď. . . ďMh˚´1

Z. More generally,



Zt`h`1´Gph`1qpZ1

t, ϵt`1:t`h`1q



rďCjphq

ZmaxtMh˚´1

Z,1u∥Zt´Z1

t∥r

for jphq:“th{h˚u. Result (b) then follows by noting that jphq ě h{h˚´1and then

proceeding as in (a) to derive GMC coeﬃcients.

Companion and Lagged Vectors. The assumption of GMC for a process translates natu-

rally to vectors that are composed of stacked lags of realizations. This, for example, is important

in the discussion of Section 3when imposing Assumption 9, since one needs that series regressors

tW2tutPZbe GMC.

Recall that W2t“ pXt, Xt´1, . . . , Xt´p, Yt´1, . . . , Yt´p, ϵ1tq. Here we shall reorder this vector

slightly to be

W2t“ pXt, Xt´1, Yt´1, . . . , Xt´p, Yt´p, ϵ1tq.

For hą0and 1ďlďh, let Z1

t`j:“ΦplqpZ1

t, . . . , Z1

t´p;ϵt`1:t`jqbe the a perturbed version of

Zt, where Z1

t, . . . , Z1

t´pare taken from an independent copy of tZtutPZ. Deﬁne

2t“ pX1

t, X1

t´1, Y 1

t´1, . . . , X1

t´p, Y 1

t´p, ϵ1tq.

Using Minkowski’s inequality

∥W2t`h´W1

2t`h∥Lrď∥Xt`h´X1

t`h∥Lr`

j“1

∥Zt`h´j´Z1

t`h´j∥Lr

j“0

∥Zt`h´j´Z1

t`h´j∥Lr,

thus, since pą0is ﬁxed ﬁnite,

sup

∥W2t`h´W1

2t`h∥Lrď

j“0

∆rph´jqďpp`1qa1Zexpp´a2Zhq.

Above, a1Zand a2Zare the GMC coeﬃcients of tZtutPZ.

B.2 Lemma 3.1 and Matrix Inequalities under Dependence

In order to prove Lemma 3.1, the idea is to modify the approach of Chen and Christensen (2015),

which relies on Berbee’s Lemma and an interlaced coupling, to handle variables with physical

dependence. Chen et al. (2016) provide an example on how to achieve this when working with

self-normalized sums. In what follows I modify their ideas to work with random dependent

matrices.

First of all, I recall below a Bernstein-type inequality for independent random matrices of

Tropp (2012).

Theorem B.1. Let tΞiun

i“1be a ﬁnite sequence of independent random matrices with dimensions

d1ˆd2. Assume ErΞis “ 0for each iand max1ďiďn∥Ξi∥ďRnand deﬁne

ς2

n:“max #



i“1

E“Ξi,nΞ1

j,n‰



i“1

E“Ξ1

i,nΞj,n ‰



+.

Then for all zě0,

P˜



i“1

Ξi



ěz¸ď pd1`d2qexp ˆ´z2{2

nqς2

n`qRnz{3˙.

The main exponential matrix inequality due to Chen and Christensen (2015), Theorem 4.2

is as follows.

Theorem B.2. Let tXiuiPZwhere XiPXbe a β-mixing sequence and let Ξi,n “ΞnpXiqfor

each iwhere Ξn:XÞÑ Rd1ˆd2be a sequence of measurable d1ˆd2matrix-valued functions.

Assume that ErΞi,ns “ 0and ∥Ξi,n∥ďRnfor each iand deﬁne

n:“max ␣E“∥Ξi,nΞ1

j,n∥‰,E“∥Ξ1

i,nΞj,n ∥‰(.

Let 1ďqďn{2be an integer and let I‚“qtn{qu, . . . , n when qtn{quănand I‚“ H otherwise.

Then, for all zě0,

P˜



i“1

Ξi,n



ě6z¸ďn

qβpqq ` P˜



ÿ

iPI‚

Ξi,n



ěz¸`2pd1`d2qexp ˆ´z2{2

nqS2

n`qRnz{3˙,

where ∥řiPI‚Ξi,n∥:“0whenever I‚“ H.

To fully extend Theorem B.2 to physical dependence, I will proceed in steps. First, I

derive a similar matrix inequality by directly assuming that random matrices Ξi,n have physical

dependence coeﬃcient ∆Ξ

rphq. In the derivations I will use that

pd2q1{2´1{r∥A∥rď∥A∥2ď pd1q1{2´1{r∥A∥r.

for rě2.

Theorem B.3. Let tϵjujPZbe a sequence of i.i.d. variables and let tΞi,n un

i“1,

Ξi,n “GΞ

np...,ϵi´1, ϵiq

for each i, where Ξn:XÞÑ Rd1ˆd2, be a sequence of measurable d1ˆd2matrix-valued functions.

Assume that ErΞi,ns “ 0and ∥Ξi,n∥ďRnfor each iand deﬁne

n:“max ␣E“∥Ξi,nΞ1

j,n∥‰,E“∥Ξ1

i,nΞj,n ∥‰(.

Additionally assume that ∥Ξi,n∥Lră 8 for rą2and deﬁne the matrix physical dependence

measure ∆Ξ

rphqas

∆Ξ

rphq:“max

1ďiďn



Ξi,n ´Ξh˚

i,n



Lr,

where Ξh˚

i,n :“GΞ

np...,ϵ˚

i´h´1, ϵ˚

i´h, ϵi´h`1, . . . , ϵi´1, ϵiqfor independent copy tϵ˚

jujPZ. Let 1ďqď

n{2be an integer and let I‚“qtn{qu, . . . , n when qtn{quănand I‚“ H otherwise. Then, for

all zě0,

P˜



i“1

Ξi,n



ě6z¸ďnr`1

qrpd2qr{2´1zr∆Ξ

rpqq`P˜



ÿ

iPI‚

Ξi,n



ěz¸`2pd1`d2qexp ˆ´z2{2

nqS2

n`qRnz{3˙,

where ∥řiPI‚Ξi,n∥:“0whenever I‚“ H.

Proof. To control dependence, we can adapt the interlacing block approach outlined by Chen

et al. (2016). To interlace the sum, split it into

i“1

Ξi,n “ÿ

jPKe

Jk`ÿ

jPJo

Wk`ÿ

iPI‚

Ξi,n,

where Wj:“řqj

i“qpj´1q`1Ξi,n for j“1,...,tn{quare the blocks, I‚:“ tqtn{qu`1, . . . , nuif

qtn{quănand Jeand Joare the subsets of even and odd numbers of t1,...,tn{quu, respectively.

For simplicity deﬁne J“JeYJoas the set of block indices and let

j:“E“Wj|ϵℓ, qpj´2q ` 1ďℓďqj ‰.

Note that by construction tW:

jujPJeare independent and also tW:

jujPJoare independent. Using

the triangle inequality we ﬁnd

P˜



i“1

Ξi,n



ě6z¸ďP˜



ÿ

jPJpWj´W:



`



ÿ

jPJ



`



ÿ

iPI‚

Ξi,n



ě6z¸

ďP˜



ÿ

jPJpWj´W:



ěz¸`P˜



ÿ

jPJe



ěz¸

`P˜



ÿ

jPJo



ěz¸`P˜



ÿ

iPI‚

Ξi,n



ěz¸

“I`II `III `IV.

We keep term IV as is. As in the proof of Chen and Christensen (2015), terms II and III

consist of sums of independent matrices, where each W:

jsatisﬁes ∥W:

j∥ďqRnand

max !E”∥W:

jW:1

j∥ı,E”∥W:1

jW:

j∥ı)ďqS2

Then, using the exponential matrix inequality of Tropp (2012),

P˜



ÿ

jPJe



ěz¸ď pd1`d2qexp ˆ´z2{2

nqS2

n`qRnz{3˙.

The same holds for the sum over Jo. Finally, we use the physical dependence measure ∆Ξ

rto

bound I. Start with the union bound to ﬁnd

P˜



ÿ

jPJpWj´W:



ěz¸ďP˜ÿ

jPJ



Wj´W:



ěz¸

ďn

qP´



Wj´W:



ěq

nz¯,

where we have used that tn{quďn{q. Since Wjand W:

jdiﬀer only over a σ-algebra that is q

steps in the past, by assumption



Wj´W:



Lrďq∆Ξ

rpqq,

which implies, by means of the rth moment inequality,

P´



Wj´W:



ěq

nz¯ďP´pd2q1{r´1{2



Wj´W:



rěq

nz¯ďnr

qr´1pd2qr{2´1zr∆Ξ

rpqq.

where pd2q1{r´1{2is the operator norm equivalence constant such that ∥¨∥ě pd2q1{r´1{2∥¨∥r

(Feng,2003). Therefore,

P˜



ÿ

jPJpWj´W:



ěz¸ďnr`1

qrpd2qr{2´1zr∆Ξ

rpqq

as claimed.

Notice that the ﬁrst term in the bound is weaker than that derived by Chen and Christensen

(2015). The β-mixing assumption and Berbee’s Lemma give strong control over the probability

Pp∥řjPJpWj´W:

jq∥ězq. In contrast, assuming physical dependence means we have to explic-

itly handle a moment condition. One might think of sharpening Theorem B.3 by sidestepping

the rth moment inequality (c.f. avoiding Chebyshev’s inequality in concentration results), but

I do not explore this approach here.

The second step is to map the physical dependence of a generic vector time series tXiuiPZ

to matrix functions.

Proposition B.1. Let tXiuiPZwhere Xi“Gp...,ϵi´1, ϵiq P Xfor tϵjujPZi.i.d. be a sequence

with ﬁnite rth moment, where rą0, and functional physical dependence coeﬃcients

∆rphq “ sup



Xi`h´GphqpX˚

i, ϵi`1:i`hq



Lr

for hě1. Let Ξi,n “ΞnpXiqfor each iwhere Ξn:XÞÑ Rd1ˆd2be a sequence of measurable

d1ˆd2matrix-valued functions such that Ξn“ pv1, . . . , vd2qfor vℓPRd1. If ∥Ξi,n ∥Lră 8 and

CΞ,ℓ :“sup

xPX

∥∇vℓpxq∥ďCΞă 8,

then matrices Ξi,n have physical dependence coeﬃcients

∆Ξ

rphq “ sup



Ξi,n ´Ξh˚

i,n



Lrďad1ˆd2

d1˙1{r

CΞ∆rphq,

where Ξh˚

i,n “ΞnpGphqpX1

i, ϵi`1:i`hqq.

Proof. To derive the bound, we use ΞnpXiqand ΞnpXh˚

iqin place of Ξi,n and Ξh˚

i,n, respectively,

where Xh˚

i“GphqpX˚

i, ϵi`1:i`hq. First we move from studying the operator r-norm (recall,

rą2) to the Frobenius norm,



ΞnpXiq ´ ΞnpXh˚

iq



rď pd2q1{2´1{r



ΞnpXiq ´ ΞnpXh˚

iq



F.

where as intermediate step we use the 2-norm. Let Ξn“ pv1, . . . , vd2qfor vℓPRd1and ℓP

1, . . . , d2, so that

∥Ξn∥F“g

ed2

ℓ“1

∥vℓ∥2

where vℓ“ pvℓ1, . . . , vℓd1q1. Since vℓ:XÞÑ Rd1are vector functions, the mean value theorem

gives that



ΞnpXiq ´ ΞnpXh˚

iq



Fďg

ed2

ℓ“1

Ξ,ℓ ∥Xi´Xh˚

i∥2ďad2CΞ∥Xi´Xh˚

i∥.

Combining results and moving from the vector r-norm to the 2-norm yields



ΞnpXiq ´ ΞnpXh˚

iq



rď pd2q1´1{rpd1q1{2´1{rCΞ∥Xi´Xh˚

i∥r.

The claim involving the Lrnorm follows immediately.

The following Corollary, which speciﬁcally handles matrix functions deﬁned as outer prod-

ucts of vector functions, is immediate and covers the setups of series estimation.

Corollary B.1. Under the conditions of Proposition B.1, if

ΞnpXiq “ ξnpXiqξnpXiq1`Qn

where ξn:XÞÑ Rdis a vector function and QnPRdˆdis nonrandom matrix, then

∆Ξ

rphq ď d3{2´2{rCξ∆rphq,

where Cξ:“supxPX∥∇ξnpxq∥ă 8.

Proof. Matrix Qncancels out since it is nonrandom and appears in both ΞnpXiqand ΞnpXh˚

iq.

Since ΞnpXiqis square, the ratio of row to column dimensions simpliﬁes.

The following Corollaries to Theorem B.3 can now be derived in a straightforward manner.

Corollary B.2. Under the conditions of Theorem B.3 and Proposition B.1, for all zě0

P˜



i“1

Ξi,n



ě6z¸ďnr`1

qrzrpd2q2´pr{2`1{rqpd1q1{2´1{rCΞ∆rpqq ` P˜



ÿ

iPI‚

Ξi,n



ěz¸

`2pd1`d2qexp ˆ´z2{2

nqS2

n`qRnz{3˙.

where ∆rp¨q if the functional physical dependence coeﬃcient of Xi.

Corollary B.3. Under the conditions of Theorem B.3 and Proposition B.1, if q“qpnqis chosen

such that nr`1

qrpd2q2´pr{2`1{rqpd1q1{2´1{rCΞ∆rpqq “ op1q

and Rnaqlogpd1`d2q “ opSn?nqthen



i“1

Ξi,n



“OP´Snanq logpd1`d2q¯.

This result is almost identical to Corollary 4.2 in Chen and Christensen (2015), with the only

adaptation of using Theorem B.3 as a starting point. Condition Rnaqlogpd1`d2q “ opSn?nq

is simple to verify by assuming, e.g., q“opn{logpnqq since logpd1`d2q À logpKqand K“opnq.

Note that when d1“d2”K, which is the case of interest in the series regression setup, the

ﬁrst condition in Corollary B.3 reduces to

K5{2´pr{2`2{rqCΞ∆rpqq “ op1q,

which also agrees with the rate of Corollary B.1. Assumption 7(i) and a compact domain further

allow to explicitly bound factor CΞby

CΞÀKω2,

so that the required rate becomes

Kρ∆rpqq “ op1q,where ρ:“3

2´r

2`ω2.

Proof of Lemma 3.1.The proof follows from Corollary B.3 by the same steps of the proof of

Lemma 2.2 in Chen and Christensen (2015). Simply take Ξi,n “n´1pr

pbqK

πpXiqr

pbqK

πpXiq1´IKq

and note that Rnďn´1p1`ζ2

K,nλ2

K,nqand Snďn´2p1`ζ2

K,nλ2

K,nq.

For Lemma 3.1 to hold under GMC assumptions a valid choice for qpnqis

qpnq “ γ´1logpKρnr`1q

where γas in Proposition 3.1. This is due to

ˆn

q˙r`1

qKρ∆rpqq À nr`1

qrKρexpp´γqq

Ànr`1Kρ

logpKρnr`1qrpKρnr`1q´1

“1

logpKρnr`1qr“op1q.

Note then that, if λK,n À1and ζK,n À?K, since

ζK,nλK,n cqlog K

nÀcKlogpKρnr`1qlogpKq

nÀcKlogpnρ`r`2qlogpnq

nÀcKlogpnq2

to satisfy Assumption 8we may assume aKlogpnq2{n“op1qas in Remark 2.3 of Chen and

Christensen (2015) for the case of exponential β-mixing regressors.

B.3 Theorem 3.2

Before delving into the proof of Theorem 3.2, note that we can decompose p

Π2´Π2as

Π2´Π2“ pp

Π2´p

Π˚

2q`pp

Π˚

2´r

Π2q`pr

Π2´Π2q,

where r

Π2is the projection of Π2onto the linear space spanned by the sieve. The last two terms

can be handled directly with the theory developed by Chen and Christensen (2015). Speciﬁcally,

their Lemma 2.3 controls the second term (variance term), while Lemma 2.4 handles the third

term (bias term). This means here we can focus on the ﬁrst term, which is due to using generated

regressors pϵ1tin the second step.

Since p

Π2can be decomposed in dYrows of semi-nonparametric coeﬃcients, i.e.,

Yt“»

—

–

π2,1

π2,dY

ﬁ

ﬃ

ﬂW2t`ru2t,

we further reduce to the scalar case. Let π2be any row of Π2and, with a slight abuse of notation,

Ythe vector of observations of the component of Ytof the same row, so that one may write

pπ2pxq ´ pπ˚

2pxq “ r

πpxq`p

πp

Bπ˘´`p

Bπ´r

Bπ˘1Y`r

πpxq„`p

πp

Bπ˘´´`r

πr

Bπ˘´ȷr

πY

“I`II

where r

πpxq “ Γ´1{2

B,2bK

πpxqis the orthonormalized sieve according to ΓB,2:“ErbK

πpW2tqbK

πpW2tq1s,

Bπis the infeasible orthonormalized design matrix (involving ϵ1t) and p

Bπis feasible orthonor-

malized design matrix (involving p

ϵ1t). In particular, note that

Bπ“Bπ`Rn,where Rn:“»

—

–

0 0 p

ϵ11 ´ϵ11

.¨¨¨ .

0 0 pϵ1n´ϵ1n

ﬁ

ﬃ

ﬂPRnˆK,

which implies p

Bπ´r

Bπ“RnΓ´1{2

B,2“:r

Rn.

The next Lemma provides a bound for the diﬀerence pp

πp

Bπ{nq´pr

πr

Bπ{nqthat will be

useful in the proof of Theorem 3.1 below.

Lemma B.2. Under the setup of Theorem 3.1, it holds



pp

πp

Bπ{nq´pr

πr

Bπ{nq

“OPpaK{nq.

Proof. Using the expansion p

πp

Bπ“r

πr

Bπ` p r

πr

Rn`r

Bπq ` r

Rn, one immediately ﬁnds

that



pp

πp

Bπ{nq´pr

πr

Bπ{nq

ď2

r

πr

Rn{n

`

r

Rn{n

.

The second right-hand side factor satisﬁes 

r

Rn{n

ďλ2

K,n

R1

nRn{n

. Moreover,



R1

nRn{n

“



t“1ppϵ1t´ϵ1tq2



“



t“1pΠ1´p

Π1q1W1tW1

1tpΠ1´p

Π1q



ď

Π1´p

Π1



2

W1

1W1{n



“OPpn´1q,

since ∥W1

1W1{n∥“OPp1q. Under Assumption 12,λ2

K,n{n“oPpaK{nqsince B-splines and

wavelets satisfy λK,n À1. Consequently, 

r

Rn{n

“oPpaK{nq.

Factor ∥r

πRn{n∥is also straightforward, but depends on sieve dimension K,



r

πRn{n

ď



t“1r

πpW2tqpp

ϵ1t´ϵ1tq



“



t“1r

πpW2tqW1

1tpΠ1´p

Π1q



ď

Π1´p

Π1



r

πW1{n



“OPpaK{nq,

since ∥r

πW1{n∥“OPp?Kqas the column dimension of W1is ﬁxed. The claim then follows by

noting OPpaK{nqis the dominating order of convergence.

Proof of Theorem 3.2.Since p

Π1the least squares estimator of a linear equation, the rate of

convergence is the parametric rate n´1{2. The ﬁrst result is therefore immediate.

For the second step, we consider



p

Π2´Π2

8ď

p

Π2´p

Π˚

2

8`

p

Π˚

2´Π2

8,

and bound explicitly the ﬁrst right-hand side term. For a given component of the regression

function,

|pπ2pxq ´ pπ˚

2pxq| ď |I|`|II|.

We now control each term on the right side.

(1) It holds

|I| ď ∥r

πpxq∥

`p

πp

Bπ{n˘´



`p

Bπ´r

Bπ˘1Y{n



ďsup

xPW2

∥r

πpxq∥

`p

πp

Bπ{n˘´



`p

Bπ´r

Bπ˘1Y{n



ďζK,nλK,n 

`p

πp

Bπ{n˘´



`p

Bπ´r

Bπ˘1Y{n

.

Let Andenote the event on which 

p

πp

Bπ{n´IK

ď1{2, so that 

`p

πp

Bπ{n˘´

ď2on

An. Notice that since ∥pp

πp

Bπ{nq´pr

πr

Bπ{nq∥“oPp1q(Lemma B.2) and, by assumption,

∥r

πr

Bπ{n´IK∥“oPp1q, then PpAc

nq “ op1q. On Anthen

|I| À ζK,nλ2

K,n 

`p

Bπ´Bπ˘1Y{n

“ζK,nλ2

K,n 

R1

nY{n

.

From R1

nY“řn

t“1bK

πpW2tqppϵ1t´ϵ1tqYt“ pΠ1´p

Π1q1W1

1Yit follows that



R1

nY{n

ď

Π1´p

Π1



W1

1Y{n



on An, meaning

|I| “ OP`ζK,nλ2

K,n{?n˘

as ∥W1

1Y{n∥“OPp1qand PpAc

nq “ op1q.

(2) Again we proceed by uniformly bounding II according to

|II| ď ζK,nλK,n 

`p

πp

Bπ{n˘´´`r

πr

Bπ{n˘´



r

πY{n

.

The last factor has order ∥r

πY{n∥“OPp?Kqsince r

Bπis growing in row dimension with

K. For the middle term, introduce

∆B:“p

πp

Bπ{n´r

πr

Bπ{n

and event

Bn:“!

`r

πr

Bπ{n˘´∆B

ď1{2)X!

r

πr

Bπ{n´IK

ď1{2).

On Bn, we can apply the bound (Horn and Johnson,2012)



`p

πp

Bπ{n˘´´`r

πr

Bπ{n˘´

ď∥pr

πr

Bπ{nq´∥2∥∆B∥

1´∥pr

πr

Bπ{nq´∆B∥À

p

πp

Bπ{n´r

πr

Bπ{n

.

Since 

p

πp

Bπ{n´r

πr

Bπ{n

“OPpaK{nqby Lemma B.2, we get

|II| “ OPˆζK,nλK,n

?n˙

on Bn. Finally, using PppAXBqcq ď PpAcq ` PpBcqwe note that PpBc

nq “ op1qso that

the bound asymptotically holds irrespective of event Bn.

Thus, we have shown that

|pπ2pxq ´ pπ˚

2pxq| ď OPˆζK,nλ2

K,n

?n˙`OPˆζK,nλK,n

?n˙

“OPˆζK,nλK,n

?n˙

as clearly ?n´1“opK{?nqand, as discussed in the proof of Lemma B.2,λ2

K,n{n“oPpaK{nq.

This bound is uniform in xand holds for each of the (ﬁnite number of) components of p

Π2,

therefore the proof is complete.

B.4 Theorem 4.1

Before proving impulse response consistency, I show that compositions of the model’s autore-

gressive nonlinear maps are also consistently estimated at any ﬁxed horizon. This means that

the “functional moving average" coeﬃcient matrices Γjinvolved in Proposition 4.1 can be con-

sistently estimated with p

Π1and p

Π2.

Lemma B.3. Under the assumptions of Theorem 3.2 and for any ﬁxed integer jě0it holds

∥p

Γj´Γj∥8“oPp1q.

Proof. By deﬁnition, recall that ΓpLq “ ΨpLqGpLqwhere Ψ“ pId´ApLqLq´1. Since ΨpLqis

an MA(8) lag polynomial, we have that

ΓpLq “ ˜8

k“0

ΨkLk¸pG0`G1L`. . . `GpLpq,

where Ψ0“Id,tΨku8

k“1are purely real matrices and G0is a functional vector that may also

contain linear components (i.e. allow linear functions of Xt). This means that Γjis a convolution

of real and functional matrices,

Γj“

mintj, pu

k“1

Ψj´kGk.

The linear coeﬃcients of ApLqcan be consistently estimated by p

Π1and p

Π2, and thus plug-in

estimate p

Ψjis consistent for Ψj(Lütkepohl,2005). Therefore,

∥p

Γj´Γj∥8ď

mintj, pu

k“1



Ψj´kGk´p

Ψj´kp



8

mintj, pu

k“1



Ψj´k´p

Ψj´k



8∥Gk∥8`



p

Ψj´k



8



Gk´p



8

mintj, pu

k“1

opp1qCG,k `OPp1qopp1q

“opp1q,

where CG,k is a constant and ∥Gk´p

Gk∥8“opp1qas a direct consequence of Proposition 3.2.

Note. Since we assume that the model respects either contractivity or stability conditions,

the impulse responses must decay (eventually) exponentially fast to zero. This means that by

“stitching” bounds appropriately, one should also be able to achieve convergence uniformly over

h“0,1,...,8.

Recall now that the sample estimate for the relaxed-shock impulse response is

IRFh,ℓpδq “ Θh,¨1δ n´1

t“1

ρppϵ1tq `

j“0p

Vj,ℓpδq

where

Vj,ℓpδq “ 1

n´j

t“1pvj,ℓ`Xt`j:t;p

δt˘“1

n´j

t“1„p

ΓjpγjpXt`j:t;p

δtq ´ p

ΓjXt`jȷ.

Therefore, the estimated horizon himpulse response of the ℓth variable is

IRFh,ℓpδq:“p

Θh,ℓ1δ n´1

t“1

ρpp

ϵ1tq `

j“0«1

n´j

t“1p

vj,ℓ`Xt`j:t;p

δt˘ﬀ.

Lemma B.4. Under the assumptions of Theorem 4.1 , let xj:0 “ pxj, . . . , x0q P Xjand εPE1

be nonrandom quantities. Let r

δbe the relaxed shock determined by δ,ρand ε. Then

(i) supxj:0,ε |pγjpxj:0;r

δq ´ γjpxj:0;r

δq|“oPp1q,

(ii) supxj:0,ε |pvj,ℓ`xj:0;r

δ˘´vj,ℓ`xj:0 ;r

δ˘|“oPp1q,

for any ﬁxed integers jě0and ℓP t1, . . . , du.

Proof.

(i) From Proposition 4.1, we have that

pγjpxj:0;δq “ xj`Θj,11 δρpεq `

k“1pΓk,11xj´kpr

δq ´ Γk,11xj´kq,

thus

γjpxj:0;δq ´ γjpxj:0 ;δq|“



k“1”pp

Γk,11xj´kpr

δq ´ p

Γk,11xj´kq´pΓk,11xj´kpr

δq ´ Γk,11xj´kqı



k“1



p

Γk,11xj´kpr

δq ´ Γk,11xj´kpr

δq



`

k“1



p

Γk,11xj´k´Γk,11xj´k



.

This yields

sup

xj:0,ε |pγjpxj:0 ;r

δq ´ γjpxj:0;r

δq|ď2jsup

xPX



p

Γk,11x´Γk,11x



.

Since jis ﬁnite and ﬁxed and the uniform consistency bound of Lemma B.3 holds, a fortiori

supxPX



p

Γk,11x´Γk,11x



“oPp1q.

(ii) Similarly to above,

vj,ℓ`xj:0 ;r

δ˘´vj,ℓ`xj:0 ;r

δ˘|“



´p

Γj,ℓp

γjpxj:0;r

δq ´ Γj,ℓγjpxj:0 ;r

δq¯´´p

Γj,ℓxj´Γj,ℓ xj¯



ď∥p

Γj,ℓ ´Γj,ℓ∥8`∥Γj,ℓ ∥8|pγjpxj:0;δq ´ γjpxj:0;δq|

`|p

Γj,ℓxj´Γj,ℓ xj|

ď2∥p

Γj,ℓ ´Γj,ℓ∥8`CΓ,j,l |pγjpxj:0 ;δq ´ γjpxj:0;δq|,

where we have used that γjpxj:0;r

δq P Xto derive the ﬁrst term in the second line. In the

last line, CΓ,j,l is a constant such that

∥Γj,ℓ∥8ď

mintj, pu

k“1

∥Ψj´k∥8∥Gk∥8ďCΓ,j,l.

The claim then follows thanks to Lemma B.3 and (i).

In what follows, deﬁne pvj,ℓ`Xt`j:t;r

δt˘to be a version of vj,ℓ that is constructed using coef-

ﬁcient estimates from tp

Π1,p

Π2ubut evaluated on the true innovations ϵt.

Proof of Theorem 4.1.If we introduce

IRFh,ℓpδq˚:“p

Θh,ℓ1δ n´1

t“1

ρpϵ1tq `

j“0«1

n´j

t“1p

vj,ℓ`Xt`j:t;r

δt˘ﬀ,

then clearly



y

IRFh,ℓpδq ´ Ą

IRFh,ℓpδq



ď



y

IRFh,ℓpδq ´ Ą

IRF˚

h,ℓpδq



`



Ą

IRF˚

h,ℓpδq ´ Ą

IRFh,ℓpδq



“I`II.

To control II , we can observe

II ď



p

Θh,ℓ1δ n´1

t“1

ρpϵ1tq ´ Θh,ℓ1δErρpϵ1tqs



j“0



n´j

t“1p

vj,ℓ`Xt`j:t;r

δt˘´Ervj,ℓ`Xt`j:t;r

δ˘s



ďδ



p

Θh,ℓ1´Θh,ℓ1



n´1

t“1

ρpϵ1tq



`δ



p

Θh,ℓ1



n´1

t“1

ρpϵ1tq ´ Erρpϵ1tqs



j“0



n´j

t“1p

vj,ℓ`Xt`j:t;r

δt˘´Ervj,ℓ`Xt`j:t;r

δ˘s



ďδ



p

Θh,ℓ1´Θh,ℓ1



n´1

t“1

ρpϵ1tq



`δ



p

Θh,ℓ1



n´1

t“1

ρpϵ1tq ´ Erρpϵ1tqs



j“0



n´j

t“1p

vj,ℓ`Xt`j:t;r

δt˘´vj,ℓ`Xt`j:t;r

δt˘



j“0



n´j

t“1

vj,ℓ`Xt`j:t;r

δt˘´Ervj,ℓ`Xt`j:t;r

δ˘s



The ﬁrst two terms in the last bound are oPp1qsince 



p

Θh,ℓ1´Θh,ℓ1



“oPp1q, as discussed in

Lemma B.3, and n´1řn

t“1ρpϵ1tqp

ÑErρpϵ1tqs by a WLLN. For the other terms in the last sum

above, we similarly note that



n´j

t“1p

vj,ℓ`Xt`j:t;r

δt˘´vj,ℓ`Xt`j:t;r

δt˘



“oPp1q

from Lemma B.4, while thanks again to a WLLN it holds



n´j

t“1

vj,ℓ`Xt`j:t;r

δt˘´Ervj,ℓ`Xt`j:t;r

δ˘s



“oPp1q.

Since his ﬁxed ﬁnite, this implies that II “oPp1q.

Considering now I, we can write

Iďδ



p

Θh,ℓ1



n´1

t“1

ρpp

ϵ1tq ´ ρpϵ1tq



`

j“0



n´j

t“1p

vj,ℓ`Xt`j:t;p

δt˘´p

vj,ℓ`Xt`j:t;r

δt˘



“I1`I2.

Since by assumption ρis a bump function, thus continuously diﬀerentiable over the range of ϵt,

by the mean value theorem



n´1

t“1

ρppϵ1tq ´ ρpϵ1tq



ďn´1

t“1

|ρ1

t|

p

ϵ1t´ϵ1t



for a sequence tρ1

tun

t“1of evaluations of ﬁrst-order derivative ρ1at values ϵtin the interval with

endpoint ϵtand p

ϵt. One can use |ρ1

t| ď Cρ1with a ﬁnite positive constant Cρ1, and by recalling

that pϵ1t´ϵ1t“ pΠ1´p

Π1q1W1tone thus gets



n´1

t“1

ρppϵ1tq ´ ρpϵ1tq



ďCρ1

t“1



pΠ1´p

Π1q1W1t



ďCρ1∥Π1´p

Π1∥2

t“1

∥W1t∥2“oPp1q.

This proves that term I1is itself oPp1q. Finally, to control I2, we use that by construction

estimator p

Π2is composed of suﬃciently regular functional elements i.e. B-spline estimates of

order 1 or greater. Thanks again to the mean value theorem



n´j

t“1pvj,ℓ`Xt`j:t;p

δt˘´pvj,ℓ`Xt`j:t;r

δt˘



ď1

n´j

t“1



pvj,ℓ`Xt`j:t;p

δt˘´pvj,ℓ`Xt`j:t;r

δt˘



ďCpv1,j,ℓ

n´j

t“1



p

ϵ1t´ϵ1t



for any ﬁxed jand some Cpv1,j,ℓ ą0. This holds since pvj,ℓ is uniformly continuous by construc-

tion. Note that we have assumed that the nonlinear part of Π2belongs to a Hölder class with

smoothness są1(for simplicity, assume here that sis integer, otherwise a similar argument can

be made). Then, even though Cpv1,j,ℓ depends on the sample, it is bounded above in probability

for nsuﬃciently large. Following the discussion of term I1, we deduce that the last line in the

display above is opp1q. As his ﬁnite and independent of n, it follows that also I2is of order

oPp1q.

C Additional Plots

0 5 10 15 20 25

0.005

0.01

0.015 n = 2400

Sieve OLS

0 5 10 15 20 25

-0.02

-0.01

0.01

0.02

0.03

0.04 n = 2400

Sieve OLS

(a) δ“ `2

0 5 10 15 20 25

0.005

0.01

0.015 n = 2400

Sieve OLS

0 5 10 15 20 25

-0.01

0.01

0.02

0.03

0.04 n = 2400

Sieve OLS

(b) δ“ ´2

Figure 8: Simulation results for DGP 21when considering rφin place of φ.

-2 -1 0 1 2

-4

-2

Fed Funds Rate

-2 -1 0 1 2

-2

-1

Fed Funds Rate

-2 -1 0 1 2

-2

-1

Log Real GDP

-2 -1 0 1 2

-4

-2

Log Real GDP

-2 -1 0 1 2

-2

-1

PCE Inflation

-2 -1 0 1 2

-2

-1

PCE Inflation

Figure 9: Estimated nonlinear regression functions for the narrative U.S. monetary policy vari-

able. Contemporaneous (left side) and one-period lag (right side) eﬀects are shown, linear and

nonlinear functions. For comparison, linear VAR coeﬃcients (dark gray) and the identity map

(light gray, dashed) are shown as lines.

-5 -4 -3 -2 -1 0 1 2 3 4 5

(a) δ“ `1

-5 -4 -3 -2 -1 0 1 2 3 4 5

(b) δ“ ´1

Figure 10: Comparison of histograms and shock relaxation function for a positive (left) and

negative (right) shock in monetary policy. Original (blue) versus shocked (orange) distribution

of the sample realization of ϵ1t. The dashed vertical line is the mean of the original distribution,

while the solid vertical line is the mean after the shock.

0 5 10 15 20

Quarters

-0.5

0.5

1.5

2.5 Fed Funds Rate

Linear

Nonlin-par

Sieve

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

(a) δ“ `1, knots at t´1,1u

0 5 10 15 20

Quarters

-2

-1.5

-1

-0.5

0Fed Funds Rate

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

(b) δ“ ´1, knots at t´1,1u

0 5 10 15 20

Quarters

-0.5

0.5

1.5

2.5 Fed Funds Rate

Linear

Nonlin-par

Sieve

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

0 5 10 15 20

Quarters

-2

-1.5

-1

-0.5

0Fed Funds Rate

0 5 10 15 20

Quarters

-0.6

-0.4

-0.2

0.2

0.4

0.6

Log Real GDP

0 5 10 15 20

Quarters

-0.2

-0.1

0.1

0.2 PCE Inflation

(d) δ“ ´1, knot at t0u

Figure 11: Robustness plots for U.S. monetary policy shock when changing knots compared to

those used in Figure 6. Note that linear and parametric nonlinear responses do not change.

GDP

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5 Linear

Nonlin-par

Sieve

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

0 5 10 15 20

Quarters

-3

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

Figure 12: Relative changes in the GDP impulse responses function when the size of the shock is

reduced from that used in Figure 6. The standard deviation of Xt”ϵ1tis σϵ,1«0.5972. Linear

IRFs are re-scaled such that for all values of δthe linear response at h“0is one in absolute

value. Nonlinear IRFs are re-scaled by δtimes the linear response scaling factor.

0 0.1 0.2 0.3 0.4

-1

-0.5

0 0.1 0.2 0.3 0.4

-10

-5

0 0.1 0.2 0.3 0.4

-0.5

0.5

CPI

0 0.1 0.2 0.3 0.4

CPI

0 0.1 0.2 0.3 0.4

-2

-1

0 0.1 0.2 0.3 0.4

-1

-0.5

0 0.1 0.2 0.3 0.4

0.2

0.4

0 0.1 0.2 0.3 0.4

0.2

0.4

Figure 13: Estimated nonlinear regression functions for the 3M3M subjective interest rate un-

certainty measure. One-period (left side) and two-period lag (right side) eﬀects are shown,

combining linear and nonlinear functions. For comparison, linear VAR coeﬃcients (dark gray)

and the identity map (light gray, dashed) are shown as lines.

-0.3 -0.2 -0.1 0 0.1 0.2

(a) δ“σϵ

0 20 40 60

Months

0.2

0.4

0.6

(b) Envelope

Figure 14: [Top] Histograms and shock relaxation function for a one-standard-deviation shock

in interest rate uncertainty. Original (blue) versus shocked (orange) distribution of the sample

realization of ϵ1t. The dashed vertical line is the mean of the original distribution, while the

solid vertical line is the mean after the shock. [Bottom] Envelope (min-max) of shocked paths

for one-standard-deviation impulse response.

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

Linear

Sieve

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

0 20 40 60

Quarters

-12

-10

-8

-6

-4

-2

Figure 15: Relative changes in the industrial production impulse responses function when the

size of the shock is reduced from that used in Figure 7. The standard deviation of ”ϵ1tis

σϵ,1«0.0389. Linear IRFs are re-scaled such that for all values of δthe linear response at h“0

is one in absolute value. Nonlinear IRFs are re-scaled by δtimes the linear response scaling

factor.

CPI

0 20 40 60

Quarters

-3

-2

-1

Linear

Sieve

0 20 40 60

Quarters

-3

-2

-1

0 20 40 60

Quarters

-3

-2

-1

0 20 40 60

Quarters

-3

-2

-1

0 20 40 60

Quarters

-3

-2

-1

0 20 40 60

Quarters

-3

-2

-1

Figure 16: Relative changes in the CPI impulse responses function when the size of the shock

is reduced from that used in Figure 7. The standard deviation of ”ϵ1tis σϵ,1«0.0389. Linear

IRFs are re-scaled such that for all values of δthe linear response at h“0is one in absolute

value. Nonlinear IRFs are re-scaled by δtimes the linear response scaling factor.

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