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Unemployment and econometric learningI

Daniel Schaefera, Carl Singletona,1,∗

aSchool of Economics, The Univeristiy of Edinburgh, 31 Buccleuch Place, EH8 9JT, UK

Abstract

We apply well-known results of the econometric learning literature to the Mortensen-

Pissarides real business cycle model. Agents can always learn the unique rational ex-

pectations equilibrium (REE), for all possible well-deﬁned sets of parameter values, by

using the minimum-state-variable solution to the model and decreasing gain learning.

From this perspective the assumption of rational expectations in the model could be

seen as reasonable. But using a parametrisation with UK data, simulations show that the

speed of convergence to the REE is slow. This type of learning dampens the cyclical

response of unemployment to small structural shocks.

Keywords: real business cycle; unemployment; adaptive learning; expectational

stability

JEL codes: E24; E32; J64

1. Introduction

The Mortensen and Pissarides (1994) model of search and matching frictions has be-

come the foundation for studying the cyclical behaviour of labour markets (see Rogerson

and Shimer, 2011 for a survey). The existence of search frictions in labour markets is

usually motivated by decentralisation, due to geography and other differences between

ﬁrms, and because each worker has characteristics which make them more or less suit-

able for the available jobs. While they search and decide whether or not to accept a

match at some agreed upon wage, unemployed workers and ﬁrms must form expecta-

tions of future variables relevant to their choices, including aggregate conditions. The

IWe would like to thank participants at the 2014 SGPE Residential Methodology conference, in addition

to Jan Grobovšek, Philipp Kircher and Andy Snell for their comments and advice.

∗Corresponding author: E-mail: carl.singleton@ed.ac.uk

1This work was supported by the Economic and Social Research Council (UK) [ES/J500136/1].

Preprint submitted to Elsevier 7th October 2017

decentralised nature of labour markets makes it a priori not obvious that workers and

ﬁrms are able to correctly forecast these variables at all times. Nonetheless the rational

expectations (RE) assumption is usually made in search and matching models. This is

likely to pose overly strong requirements on the cognitive abilities of economic agents,

and is unrealistic in the presence of potentially frequent structural or policy shocks. Even

small departures from RE might alter the qualitative or quantitative predictions of these

models.

Here we analyse the equilibrium properties and dynamics of the textbook real busi-

ness cycle (RBC) Mortensen-Pissarides model (see Hagedorn and Manovskii, 2008 for

the standard discrete-time treatment), while representing agents as ‘good econometri-

cians,’ who form forecasts according to their estimates of some structural model para-

meters (Evans and Honkapohja, 2001).2We assume that agents employ a recursive least

squares (RLS) algorithm to update their parameter estimates when new data becomes

available. Econometric (or adaptive) learning can provide a behavioural foundation for

RE if the rational expectations equilibrium (REE) is shown to be learnable or E-stable

(Evans and Honkapohja, 2001); i.e. small deviations from this equilibrium are reversed

over time, with asymptotic convergence. We show that the model’s unique REE is E-

stable. No parameter restrictions are required to ensure this is the case, beyond those

which make the model well-formulated, when agents use a minimum-state-variable rule

to form and update their forecasts of so-called labour market tightness. Furthermore, we

conﬁrm that this equilibrium is globally stable and satisﬁes the properties of Strong E-

stability (i.e. being robust to over-parametrisation of the econometric relationship by the

agents). And so from this perspective, the assumption of RE when studying or applying

this model would seem to be reasonable.

This article contributes to a signiﬁcant literature on the more realistic representation

of agents as behaving like econometricians in macroeconomic models. Mankiw et al.

(2004) offer empirical evidence against RE. Their analysis of surveys of professional

forecasters and households ﬁnds signiﬁcant autocorrelation in forecast errors, which is

compatible with econometric learning, but not with RE. Milani (2007, 2011) also ar-

gues for the presence of adaptive learning in the New Keynesian model. He shows that

2Strictly, the characterisation of agents who behave in this way as ‘good econometricians’ is used by

Branch and McGough (2016). One should think of these as being agents who make conditional forecasts,

which are pertinent to their decisions and based on a simple model such as linear regression, and who update

this model based on forecast errors.

2

learning by agents is capable of replacing the other ‘mechanical’ sources of persistence

in these models, such as habits, whilst at the same time increasing the ﬁt to the data as

compared to assuming RE. Pfajfar and Santoro (2010) examine a survey of households’

inﬂation forecasts over several decades and conclude that the hypothesis of RE can be

rejected, and that there is evidence in support of adaptive learning dynamics. This view

is further supported by Berardi and Galimberti (2012), who examine post-WWII US

inﬂation and output growth. Comparing the performance of different adaptive learning

algorithms in matching survey forecasts, their results suggest that economic agents form

these according to RLS.

The formulation of learning we use here is such that agents need only make one-

step-ahead forecasts of the labour market’s condition. Ours is not the only recent study

to apply the principles of econometric learning to this class of model. From a similar set-

up, Di Pace et al. (2016) consider an approach where agents must make inﬁnite horizon

forecasts about the future paths of wages, unemployment and proﬁts in order to make

choices today. This latter type of learning ﬁts into the anticipated utility approach (Kreps,

1998), and has notably been applied to the RBC model by Eusepi and Preston (2011).

Di Pace et al. focus on results with constant gain learning, for which there are no equi-

valent analytical results to the E-stability conditions we consider. The authors use the

model to address the ‘unemployment volatility puzzle’: the inability of the Mortensen-

Pissarides model to generate a realistically large ampliﬁcation of unemployment for a

given change in wages or productivity (Shimer, 2005). Under inﬁnite horizon learning,

they not only match US professional forecast errors, but also ﬁnd a greater cyclical un-

employment rate response relative to the baseline model. This is driven by persistence

or inertia in agents’ expectations of the future path of wages, which implies that ﬁrms

are over-optimistic about future proﬁts, post more vacancies, and thus unemployment is

more volatile relative to the REE baseline case. They also ﬁnd some, but signiﬁcantly

less, propagation of the unemployment response when the model is reformulated in a

one-step-ahead forecast guise. Kurozumi and Van Zandweghe (2012) also apply econo-

metric learning to an extended model of the business cycle, which includes sticky prices

and monetary policy, as well as labour market search frictions. They analyse determin-

acy and E-stability conditions, ﬁnding that these depend on model parameters. However,

both this set-up and that of Di Pace et al. (2016) differ form our own in so far as they

move beyond the textbook Mortensen-Pissarides model, and in both cases the agents

must forecast several aggregate variables, which do not appear in the reduced form of

3

the REE characterisation.

We also present illustrative simulations and analyse the dynamics of the unemploy-

ment model, and show that the REE could be a poor approximation to an economy in

which agents are econometricians. Convergence to the REE is very slow, even when

agents have a short memory or give greater weight to more recent data. If agents must

learn the REE, then aggregate variables may be persistently some distance away from

their REE equivalents. We also demonstrate how structural shocks generate a more

gradual cyclical adjustment of wages after the introduction of learning, and thus pre-

dicted unemployment volatility when such shocks occur frequently would be reduced

relative to the REE. Therefore we ﬁnd some different results to Di Pace et al. (2016).

The same ampliﬁcation mechanism described therein is not present here. By keeping

closer to the spirit of the most standard RBC variant of the model, in which the only

relevant choice is the number of vacancies that ﬁrms post,3agents need only estimate a

relationship between labour market tightness and productivity to form expectations and

close the model, and so wage determination is absorbed. Econometric learning in our

set-up then generates inertia in expectations of tightness (and wages) following shocks.4

The REE of the model describes a choice of labour market tightness which is in-

dependent of the state of unemployment. Therefore, agents’ learning of the minimum-

state-variable solution implicitly assumes their complete understanding of the economy’s

dynamics of unemployment and vacancy creation. We consider an alternative decision

rule, which relaxes this latter implied assumption. We consider whether or not agents

can learn how many vacancies to create in response to an expanded state of the world,

which includes the state of unemployment. In other words we also ask if agents can

learn the Beveridge curve. This alternative is not E-stable for the complete range of pos-

sible model parameters. And where the economy does converge to the REE, it does so

more slowly than under the minimum-state-variable representation. In this respect, the

approximation of the REE for this model would be even further weakened.

3With linear production technology this is also equivalent to the decision of vacancy creation or destruc-

tion when ﬁrms consist of a single worker. Given worker homogeneity, we will rule out states of the world

whereby workers would choose not to work.

4Although we do not expand on this point later, it is straightforward to see that the minimum-state-

variable solution of the model we apply, which agents learn and use to form expectations, could be re-

written in terms of wages and productivity by substituting for the standard ‘wage curve’ derived from Nash

bargaining, (19).

4

2. The search and matching model of unemployment

We outline a discrete-time search and matching model of the labour market, follow-

ing Merz (1995) and Andolfatto (1996) by assuming that members of a representative

household perfectly insure each other against income ﬂuctuations. We derive a differ-

ence equation for so-called labour market tightness which summarises its equilibrium,

thus analogous to the treatment in Hagedorn and Manovskii (2008).5This can be re-

garded as a textbook model, which has been applied, critiqued and extended exhaustively

in the literature, not least in attempts to solve the unemployment volatility puzzle.

2.1. The labour market

There is a continuum of identical, risk-neutral workers with total measure one, and

an inﬁnite horizon. The matching function M(ut,vt)provides the number of successful

matches in period t. It is increasing and concave in both of its arguments, utand vt,

which represent the share of the total workforce currently unemployed and the level of

vacancies relative to the size of the workforce respectively. Matches and separations

occur after agents in the economy have made decisions, i.e. at the end of each period. A

Cobb-Douglas, constant returns to scale matching function is chosen, due to its simpli-

city, well-known features and being most common in the literature:

M(ut,vt) =

µ

u

α

tv1−

α

t,

µ

>0,

α

∈(0,1),(1)

where

µ

gives a measure of matching efﬁciency and

α

the elasticity of the number of

matches with respect to unemployment. We deﬁne the level of labour market tightness

as

θ

t=vt

ut

.(2)

Unemployed workers and vacancies are matched randomly, and so the probability of a

ﬁrm ﬁlling an open vacancy each period is

q(

θ

t) = M(ut,vt)

vt

=

µθ

−

α

t.(3)

5Our set-up of the model only differs in so far as we describe a representative household and ﬁrm, rather

than a continuum of the latter. The characterisation of the equilibrium is approximately identical, though

we hope our exposition is more familiar as an extended RBC model with aggregate uncertainty.

5

The corresponding probability that an unemployed worker gets matched to an open va-

cancy is

θ

tq(

θ

t) = M(ut,vt)

ut

=

µθ

1−

α

t.(4)

Matches are destroyed with probability

λ

∈[0,1]. Unemployment changes between peri-

ods due to inﬂows, caused by exogenous separations, and outﬂows from new matches.

The resulting law of motion for the share of workers unemployed at the beginning of

period t+1 is

ut+1=ut+ (1−ut)

λ

−

θ

tq(

θ

t)ut,(5)

whereby the ﬁnal two terms measure these inﬂows and outﬂows respectively. The

steady-state level of unemployment, where these ﬂows are equal, for given

θ

tis

u∗

t=

λ

λ

+

θ

tq(

θ

t).(6)

2.2. The household

For expositional simplicity we consider an economy comprising a single inﬁnitely-

lived representative household of size one, in which all workers are identical and risk

neutral. There is perfect consumption insurance across its members. In period t,ntof

the household’s members are employed and (1−nt)are waiting for a match. In other

words, labour supply is inelastic, which can be ensured by assuming that the equilibrium

wage is always strictly greater than the per-period utility value from non-employment b;

formally we assume wt>b≥0. The household discounts each additional period’s utility

by a factor

δ

∈(0,1). In period tthe household has risk-neutral preferences over income

atand has expected lifetime utility

E∗

t∞

∑

s=0

δ

t+sat+s,(7)

where E∗

tdenotes expectations (not necessarily rational). Income is given by

at=wtnt+b(1−nt) + Dt,(8)

with wtntand b(1−nt)denoting total income from labour and non-employment respect-

ively, and Dtare dividends from a representative ﬁrm, which is owned by the household.

6

This household can also be represented by the Bellman equation

W(nt) = wtnt+Dt+b(1−nt) +

δ

E∗

t[W(nt+1)],(9)

where W(nt)represents the household’s current value function, with their state of the

world given by the employment level nt=1−ut. The household takes as given wages

wt, dividends from the representative ﬁrm Dtand labour market tightness

θ

t. The house-

hold’s expected continuation value is E∗

t[W(nt+1)]. The law of motion for employment

follows directly from (5), and is given by

nt+1= (1−

λ

)nt+

θ

tq(

θ

t)(1−nt).(10)

Applying the envelope theorem to (9) and using the law of motion (10), the marginal

value of household employment is given by

∂

W(nt)

∂

nt

=wt−b+

δ

(1−

λ

−

θ

tq(

θ

t))

∂

E∗

t[W(nt+1)]

∂

nt+1

,(11)

i.e. the net utility value from wages exceeding non-employment income plus the dis-

counted expected continuation value from additional employment.

2.3. The ﬁrm

The production side of the economy consists of a representative ﬁrm. This ﬁrm

employs workers and produces output ytnt, where ytis the marginal product of labour.

We assume that the process of worker productivity is a stationary AR(1) process in logs:

log(yt) =

ρ

log(yt−1) +

ε

t,

ρ

∈[0,1),(12)

with some initial condition y0and

ε

tbeing drawn as an iid zero-mean shock. Because

labour is the only input into production, the ﬁrm maximises proﬁts by choosing the level

of employment, subject to the law of motion for the labour market. However, due to

the exogeneity of separations, the optimal choice for the level of employment and the

optimal quantity of vacancies to open coincide. For each vacancy held open the ﬁrm has

to pay per-period unit cost c>0. The objective of the ﬁrm is to maximise the expected

value of current and future proﬁts, given by

E∗

t∞

∑

s=0

δ

t+s(yt+snt+s−wt+snt+s−cvt+s).(13)

7

Because the ﬁrm is owned by the household, the ﬁrm discounts future proﬁts using the

same discount factor as the household. The ﬁrm takes wages wtand labour market

tightness

θ

tas given. Employment at the ﬁrm follows the law of motion

nt+1= (1−

λ

)nt+q(

θ

t)vt,(14)

which shows that the more vacancies the ﬁrm creates, the higher aggregate employment

will be in the next period. Maximising (13), subject to (14), can be represented as the

Bellman equation

Π(vt;nt,yt) = max

vt≥0(ytnt−wtnt−cvt) +

δ

E∗

t[Π(vt+1;nt+1,yt+1)],(15)

where Π(vt;nt,yt)represents the ﬁrm’s current value function. Proﬁt maximisation in

this case implies that the representative ﬁrm will open or close vacancies until the mar-

ginal cost and beneﬁt of doing so are equal:

∂

E∗

t[Π(vt+1;nt+1,yt+1)]

∂

nt+1

=c

δ

q(

θ

t).(16)

Applying the envelope theorem and using the above ﬁrst order condition gives the sur-

plus to the ﬁrm from employing an additional worker,

∂

Π(vt;nt,yt)

∂

nt

=yt−wt+(1−

λ

)c

q(

θ

t),(17)

i.e. the net proﬁt from employing an additional worker plus the discounted expected

continuation value, taking matching frictions into account. The optimal choices of the

ﬁrm (16) and (17) imply that labour market tightness evolves according to the non-linear

difference equation

c

δ

q(

θ

t)=E∗

tyt+1−wt+1+(1−

λ

)c

q(

θ

t+1).(18)

In other words, the representative ﬁrm must form expectations about the right hand side

of (18) to optimally choose the number of vacancies to open in the current period. In

particular, the ﬁrm forecasts labour productivity yt+1, the real wage which will be real-

ised next period wt+1, and the value of labour market tightness in the next period

θ

t+1.

Note that this problem does not depend on the type of expectations formation; we have

not speciﬁed how forecasts of the right hand side of (18) are formed.

8

2.4. Wage determination

Wages are determined by generalised Nash bargaining between the ﬁrm and workers

over the additional surpluses (11) and (17), with worker bargaining power

β

∈[0,1]:6

wt=argmax

∂

W(nt)

∂

nt

β

∂

Π(vt;nt,yt)

∂

nt1−

β

.

Combining the surplus sharing rules which form the solution of this problem, iterating

forwards, and using (11), (16) and (17) gives what is referred to in the textbook model

as the ‘wage curve’:

wt= (1−

β

)b+

β

(yt+c

θ

t).(19)

To ensure employment is always preferred and a wage successfully negotiated we also

restrict yt>b. Again, we do not have to specify rational expectations to obtain the wage

curve.

2.5. The rational expectations equilibrium

Our search and matching framework consists of the goods and the labour market.

Since our focus is to study the labour market under econometric learning, we abstract

from the goods market. This can be justiﬁed by Walras’ Law, which states that equi-

librium in the labour market implies that the goods market clears. The representative

household supplies labour inelastically, and thus the REE of the model can be summar-

ised and determined uniquely by the value of labour market tightness at which point the

representative ﬁrm is indifferent between opening an additional vacancy or not. In other

words, the ﬁrm has to form expectations about the future state of labour market tightness

as this affects the current discounted value of a match. The non-linear difference equa-

tion determining this value of

θ

t, substituting the outcome of the wage bargaining (19)

into (18), is given by

c

δ

q(

θ

t)=E∗

t(1−

β

)(yt+1−b) + (1−

λ

)c

q(

θ

t+1)−

θ

t+1

β

c.(20)

To provide intuition for this expression we stress the similarities to (16). Aggregate

labour market tightness

θ

twill adjust immediately to deviations from this equality via the

6Note, it is crucial here that both workers and ﬁrms are assumed to form expectations in the same way,

using the same rule, as otherwise the Nash bargaining solution would be signiﬁcantly complicated. In the

sense of the model here, since workers own the ﬁrm, this is not an unreasonable assumption.

9

ﬁrm instantaneously opening or closing vacancies. Thus today’s labour market tightness

is determined by expectations of the value of a ﬁlled vacancy in the next period. In

equilibrium it must also be that Dt=ytnt−wtnt−cvt.

Given the process for productivity (12), the equilibrium

θ

tis the solution of the non-

linear difference equation (20). With this and initial condition u1, the remainder of the

interesting endogenous variables in the equilibrium, {at,wt,vt,ut+1}t>0, can be obtained

using (2), (5), (8) and (19). In the next section we linearise around steady-state values to

obtain an analytical solution to (20) and discuss the rational expectations equilibrium.

3. Linearisation and the rational expectation equilibrium

To solve the system consisting of (12) and (20), we linearise around deterministic

steady-state values ¯

θ

and ¯y=1:7

θ

t=

ψ

0+

ψ

1E∗

tyt+1+

ψ

2E∗

t

θ

t+1,(21)

yt= (1−

ρ

) +

ρ

yt−1+

ε

t,(22)

where the coefﬁcients are functions of the model’s parameters and steady state values

ψ

0= [1−

ψ

2]¯

θ

−

ψ

1¯y,

ψ

1= (1−

β

)

δ

¯

θ

q(¯

θ

)(c

α

)−1,

ψ

2=

δ

(1−

λ

)−

β

¯

θ

q(¯

θ

)

α

−1.

We now assume that expectations are rational, that is, the ﬁrm and the household take

all available information into account and forecast

θ

without systematic errors. We

denote the rational expectations operator by Et. Linear RE models where agents form

expectations regarding an endogenous variable can have multiple equilibria (or bubble

solutions, not related to economic fundamentals).8If there are multiple stable REEs

then a model is said to be indeterminate. In Appendix A.2 we show that a unique stable

equilibrium exists so long as |

ψ

2|<1. Intuitively, this condition requires that the shocks

7See Appendix A.1 for derivation and all subsequently deﬁned parameters, such as

ψ

’s, expressed in

terms of the model parameters and steady-state values.

8The literature on bubbles and the related concept of indeterminacy is reviewed in Benhabib and Farmer

(1999). The classic reference on bubble solutions is Blanchard and Watson (1983); see also Bullard and

Mitra (2002) for an analysis of indeterminacy in a New Keynesian framework.

10

ε

are transitory and

θ

returns to its steady state value, analogous to a stationary ﬁrst-

order autoregressive process. The solution in this case to the system (21) and (22) is

obtained by using the method of undetermined coefﬁcients. After substituting Etyt+1=

(1−

ρ

) +

ρ

ytin (21), the solution can be guessed to have the form

θ

t=A+Byt−1+C

ε

t,(23)

since the only predetermined variable in the above system is yt−1. The parameters A,

B, and Cof the reduced form are functions of the parameters in (21) and (22).9Using

equation (23), RE about next period’s labour market tightness are given by

Et

θ

t+1=A+Byt,(24)

since Et

ε

t+1=0. The parameters used thereby to estimate

θ

t+1are true values, that is the

ﬁrm and the household know the true underlying functional forms and their associated

parameter values. When

ε

tis realised in period t, it becomes part of the information

set of ﬁrms and households and the resulting forecast Et

θ

t+1leads to an immediate

adjustment of vacancies, such that the difference between Et

θ

t+1and

θ

t+1is only from

the next period’s shock C

ε

t+1. In the absence of new shocks the ﬁrm and the household

forecast the response of market tightness to productivity correctly using (24).

4. Adding econometric learning to the model

We now depart from RE and apply the concept of econometric learning to this model.

Unlike the application of Di Pace et al. (2016), who suggest that agents might need to

form inﬁnite horizon forecasts of multiple variables, such as wages or ﬁrm proﬁts, we

depart from the REE summarised above by the single choice variable

θ

t, which just

requires a one-step-ahead forecast of

θ

t+1. Kurozumi and Van Zandweghe (2012) have

also considered the role of learning when agents need to make one-step-ahead forecasts

in the presence of labour search frictions. However, their model also includes sticky

prices and monetary policy, involves forecasting over several aggregate variables, and

the determinacy of the REE is not always certain. In what follows the expectational

stability results become more clear-cut.

9See Appendix A.3 for a description of how (23) can be obtained from (21) and (22), and also A,Band

Cin terms of the model parameters.

11

4.1. E-Stability of the MSV solution

We relax the assumption of RE by modelling agents as econometricians attempting

to estimate the parameters A,B, and C, which underpin the true motion of the economy

under uncertainty. Agents are endowed with a perceived law of motion (PLM) in the

economy of the MSV form (23), because we derived its functional form without having

to impose rational expectations.10 In other words, agents know the structure of the eco-

nomy as expressed in the system (21) and (22), but the parameter values are unknown to

them. They make corresponding estimates of the true coefﬁcients in period t, given by

ˆ

At,ˆ

Btand ˆ

Ct, and update these each period when new data becomes available. Therefore,

the household and ﬁrm forecast as under RE in (24), but instead of the true parameter

values they use estimates ˆ

Atand ˆ

Btto forecast labour market tightness

E∗

t

θ

t+1=ˆ

At+ˆ

Btyt.(25)

The main difference is that forecasts with rational expectations coincide with the true

realisation of next period’s

θ

on average, whereas this is not the case for econometric

learners. They possess less information than rational agents, since they do not know the

parameters of the model. We assume econometric learners perform the task of estimating

parameters using recursive least squares (RLS). This is the most widely used estimation

technique in the learning literature and Berardi and Galimberti (2012) provide evidence

that this estimator matches surveys of forecasts of US time series closely.11 Let the

vector of parameter estimates be denoted by x′

t−1= ( ˆ

At,ˆ

Bt), then the general recursive

updating algorithm can be represented by

xt=xt−1+gtQ(

θ

t−ˆ

At−ˆ

Btyt−1),(26)

which shows that agents update their previous parameter estimates xt−1by a function

of the observed forecasting error. The function Qand the so-called “gain parameter”

gtare further described below. There are potential problems of simultaneity in forward

looking models. Therefore, it is assumed that although agents forecast

θ

t+1using yt, the

variable ytis not in the information set for the estimation of ˆ

Atand ˆ

Bt. As proved by

10If agents are not able to learn the simplest representation (as few state variables as possible), they

cannot be expected to learn equilibria containing more state variables and to coordinate behaviour towards

them.

11The presented algorithm is comparable to a restricted form of the Kalman ﬁlter. For further discussion

see Berardi and Galimberti (2013).

12

Marcet and Sargent (1989), this does not alter the asymptotic stability results obtained in

the following, as compared to an algorithm allowing for simultaneity, so long as agents

are assumed to ignore outliers, deﬁned as being observations outside of some prede-

termined range. This timing assumption is usually thought of as realistic, since robust

macroeconomic data is only available to decision makers with a substantial lag.12

Since the current value of labour market tightness depends on the prediction of next

period’s value, agents estimates have the potential to affect the path of labour market

tightness. To see this, we substitute the stochastic process of labour productivity and

the econometric forecast (25) into (21), which gives the actual law of motion (ALM) for

labour market tightness:

θ

t=

ψ

0+

ψ

1(1−

ρ

)(1+

ρ

) +

ψ

2ˆ

At+

ψ

2ˆ

Bt(1−

ρ

)

+ (

ψ

1

ρ

+

ψ

2ˆ

Bt)

ρ

yt−1

+ (

ψ

1

ρ

+

ψ

2ˆ

Bt)

ε

t.(27)

This deﬁnes the following T-mapping from the PLM,

θ

t=A+Byt−1+C

ε

t, to the ALM:

T(ˆ

At) =

ψ

0+

ψ

1(1−

ρ

)(1+

ρ

) +

ψ

2ˆ

At+

ψ

2ˆ

Bt(1−

ρ

),

T(ˆ

Bt) = (

ψ

1

ρ

+

ψ

2ˆ

Bt)

ρ

,

T(ˆ

Ct) =

ψ

1

ρ

+

ψ

2ˆ

Bt,

where the function T:RN→RNmaps the estimated coefﬁcients into the actual para-

meters, which are in turn determined by the estimates. There is a self-referential feature

inherent in all learning models which can be seen in equation (27). Although the es-

timated parameters are non-stationary during their transition to REE values, learners

neglect this fact, since a least squares method assumes the ‘true’ A,Band Cto be con-

stants. Intuitively, if the coefﬁcient which determines the responsiveness to expectations

is sufﬁciently small, then this speciﬁcation error becomes asymptotically negligible and

the economy converges to the REE (Evans and Honkapohja, 2001). The T-mapping to

ˆ

Ctis determined by the other coefﬁcients, and the estimate ˆ

Ctis independent of Cand

does not inﬂuence stability results. Therefore, in what follows we refer to the mappings

Tˆ

At,ˆ

Btand for ˆ

Ct:V(ˆ

Bt).

12The assumption also plausibly implies that under subjective expectations agents would only ever enter

the wage bargaining process with pre-determined valuations. Otherwise, there would be simultaneity

between the bargaining result and subsequent expectations formation.

13

Let z′

t−1= (1,yt−1),x′

t−1= ( ˆ

At,ˆ

Bt)and

θ

t=z′

t−1xt−1+

η

t.(28)

The estimation error

η

tis perceived by the agents to be independently and identically

distributed iid. However, due to the self-referential nature of the model there is an en-

dogeneity bias which agents are unaware of, and thus

η

tis not truly iid. We deﬁne

Rt=t−1∑t

i=1zi−1z′

i−1, which allows us to write the RLS estimator as

Rt=Rt−1+t−1(zt−1z′

t−1−Rt−1),(29)

xt=xt−1+t−1R−1

tzt−1(

θ

t−z′

t−1xt−1),(30)

and thus

xt=xt−1+t−1R−1

tzt−1z′

t−1Tˆ

At,ˆ

Bt−xt−1+V(ˆ

Bt)

ε

t,(31)

with the gain sequence 1/t, often referred to as decreasing gain learning.13 This gain

guarantees that asymptotically new information is disregarded by agents.

The stability of the system in (29) and (31) with decreasing gain is governed by the

following ordinary differential equation (ODE), where

τ

denotes ‘notional’ time:

d

d

τ

ˆ

A,ˆ

B=Tˆ

A,ˆ

B−ˆ

A,ˆ

B.(32)

The REE is E-stable if (32) is asymptotically locally stable under learning (Evans and

Honkapohja, 2001). This is the case, if all the eigenvalues of the Jacobian of T(ˆ

A,ˆ

B)−

(ˆ

A,ˆ

B)have negative real parts. Here the necessary condition for E-stability is

ψ

2

ρ

<1,

with sufﬁcient condition

ψ

2=

δ

1−

λ

+

β

¯

θ

q(¯

θ

)

α

<1.(33)

This holds for all possible well-deﬁned sets of parameter values, and there is also global

convergence to the REE (see Appendix A.4):

δ

∈[0,1),

λ

∈[0,1],

β

∈[0,1],

µ

>

0,

α

∈(0,1),c>0, and which all imply ¯

θ

≥0. As explained in the previous section,

the model is determinate if |

ψ

2|<1. We can therefore state the following:

Proposition 4.1. If the economy described by the system (21) and (22) exhibits determ-

13In the case of constant gain learning the weight given each observation is geometrically declining with

the time since it was observed, and the gain sequence would be 0 <

γ

<1.

14

inacy and the PLM is of the MSV form, and if agents learn using least squares updating,

then so long as

ψ

2<1the unique REE is E-stable.

In other words, the textbook linearised model of labour market search and matching

frictions, with homogeneous agents and no-on-the-job search (Pissarides, 2000: Chapter

1), has a unique E-stable equilibrium. Sets of parameter values which move

ψ

2closer

to one will imply slower convergence to the REE. It is intuitive and clear from (33)

that these will be parameters which lessen the magnitude of the dynamics in the labour

market, such as a small separation probability or low worker bargaining power.

4.2. Strong E-stability of the MSV solution

One potential criticism of the econometric learning literature is that it is not clear

how agents could settle upon a particular law of motion for the economy. Strong E-

stability of a system is deﬁned if the previous result is robust to over-parametrisation of

the PLM (Evans and Honkapohja, 2001). Assume instead that agents are forming their

expectations of

θ

t+1according to the general ARMA representation (A.4), and are not

endowed with a PLM of the MSV form. Moreover, due to econometric considerations

they start with an arbitrarily over-parametrised version,

θ

t=a+

s

∑

j=1

bjyt−j+

r

∑

j=1

cj

θ

t−j+

q

∑

j=1

dj

ε

t−j+

l

∑

j=1

fj

η

t−j+d0

ε

t+f0

η

t.(34)

Accordingly, expectations of

θ

t+1take the form:

θ

e

t+1=a+

s

∑

j=1

bjyt+1−j+

r

∑

j=1

cj

θ

t+1−j+

q

∑

j=1

dj

ε

t+1−j+

l

∑

j=1

fj

η

t+1−j,(35)

which can be substituted into equation (A.3) to obtain the new ALM and a corresponding

T-mapping in the same way as before (see Appendix A.5). Let b′= (b1, ...,bs),c′=

(c1,..., cr),d′= (d0, ...,dq), and also f′= ( f0, ..., fl). Further, deﬁne

ϕ

′= (a,b′,c′,d′,f′).

According to the E-stability principle, the ODE governing the stability of the above

system is given by

d

ϕ

d

τ

=T(

ϕ

)−

ϕ

.(36)

To investigate whether agents will detect the over-parametrisation and converge towards

the MSV solution, the stability of (36) at the REE must be studied. In Appendix A.5 we

show the following:

15

Proposition 4.2. If the economy described by the system (21) and (22) exhibits determ-

inacy and the PLM is of the over-parametrised ARMA form, and if agents learn using

least squares updating, then so long as

ψ

2<1the unique REE is Strongly E-stable.

5. Analysis

We present a brief analysis of the unemployment model with econometric learning

described above. We consider two illustrative simulations to demonstrate the implied

speed of convergence and dynamics of the model. First, we demonstrate E-stability when

starting ‘realistically’ far away from the REE. Second, with agents initially assumed to

have learned the REE, we consider the impact of a structural shift implied by an arbitrary

change in some parameter value. We then discuss the speed of convergence and results

with constant gain learning. We also consider the implications if we relax an implicit

assumption that agents understand the joint dynamics of unemployment and vacancy

creation.

5.1. Simulations

We follow an illustrative parametrisation strategy, using seasonally adjusted UK

quarterly14 data for the period 1998-2013 (see Appendix B for a brief discussion of

this strategy).15 Table 1 gives the complete list of parameters and implied values of the

endogenous variables for the deterministic steady-state equilibrium. Summary statistics

of some UK labour market variables are described in Table 2, which are consistent with

the parametrisation here.

For completeness we write out in full the stochastic recursive sequence that repres-

14As pointed out by a referee, the timing structure of the model implies an average time between hiring

and production for workers of one and a half months, and when calibrating a model with labour market

search frictions it would be more generally preferable to use a monthly periodicity. But when we wish

to capture the role of aggregate uncertainty affecting agents’ decisions, since UK National Statistics are

generally released quarterly, we believe our timing is justiﬁed. What matters for plausibly estimating the

role of learning dynamics is the frequency at which it is assumed new aggregate data becomes available to

the agents, since between times the agents’ model parameters will remain unchanged.

15All data used and described are from the Ofﬁce for National Statistics, accessed 01/08/2014. Labour

market data are for those aged 16 and over. For a more complete calibration of the unemployment model

using UK data see Burgess and Turon (2010).

16

Table 1: Assumed/estimated parameter values and steady-state equilibrium

Parameter Assumed value

y- labour productivity 1

b- non-employment ﬂow value 0.8

c- vacancy ﬂow cost 0.25

λ

- separation rate 0.023

µ

- matching efﬁciency 0.56

α

- matching elasticity 0.67

β

- worker bargaining power 0.67

δ

- discount factor 0.99

ρ

- persistence of y 0.84

σ

- std dev. of innovations to y0.006

Endogenous variable Steady-state eq. value

θ

- tightness 0.35

u- unemployment 0.055

v- vacancy rate 0.019

w- wage 0.99

Source: authors’ calculations.

ents the adaptive learning model, stating from an initial period t0:

(I)ut+2=

λ

(1−ut+1) + 1−

µ

z′

tT(xt) + V(xt)

ε

t+11−

α

ut+1,

(II)Rt+1=Rt+1

t+1(ztz′

t−Rt),

(III)xt+1=xt+1

t+1R−1

t+1ztz′

t[T(xt)−xt] +V(xt)

ε

t+1,

(IV )yt+1= (1−

ρ

) +

ρ

yt+

ε

t+1,

(V)

ε

t+2∼i.i.d.N(0,

σ

2).

When written out in sequence order, the simultaneity which requires us to exclude yt

from the information set used to estimate xtbecomes clearer. The adaptive learning

process, which takes place at the beginning of each period, can also be described by

Figure 1. To initiate the sequence from t0we must choose initial values u1,x0,z0R0

and

ε

1. The asymptotic properties of decreasing or constant gain least squares recursion

will hold irrespective of the initial conditions. As suggested by Carceles-Poveda and

Giannitsarou (2007), the approach to setting initial values z0and R0should depend on the

particular model in question and the empirical purpose of the researchers. One approach

could be to use historic or randomly generated data, with t0set sufﬁciently large such

17

Figure 1: Timeline of the labour market & agents’ learning

Table 2: Summary statistics of labour market states & quarterly transition rates: consistent with the model’s

parametrisation, 2002q1-13q2

Mean Std err.

Tightness -

θ

t=vt

ut0.35 0.022

Job ﬁnding rate -

θ

tq(

θ

t)0.39 0.011

Job separation rate -

λ

t0.023 0.00093

‘Steady-state unemployment rate’ - u∗

t=

λ

t

λ

t+

θ

tq(

θ

t)0.56 0.0041

Unemployment rate 0.57 0.0022

Source: authors’ calculations using UK Labour Force Survey and Labour Market Statistics. The unemploy-

ment rate is the share of the economically active population ILO unemployed. The job ﬁnding and separation

rates are consistent with in reality a three-state system, which includes inactivity; i.e. the job separation rate is

not only the direct ﬂow rate from employment to unemployment but in addition the indirect ﬂow via inactivity

(see Smith (2011). See Appendix B for more details.

that R0is invertible; in this case t0≥2. This would be most appropriate when comparing

the performance of models which assume that agents are ‘good econometricians’ against

18

real data. However, this gives few clues as to how large t0should be, and the subsequent

simulation is likely to be sensitive to this assumed level of agents’ memory, particularly

for decreasing gain least squares. Another attractive option is to choose initial values

from an assumed distribution around the REE.

To set initial conditions here, using the same data used to parametrise the model, we

estimate using least squares

θ

t=

κ

0+cubtrt+Byt−1+

κ

1

ζ

t−1+

κ

2

ζ

t−2+

ζ

t,t=2001q3...2013q3,(37)

where cubtrtrepresents a cubic time trend to address the possibility that agents could

recognise low frequency structural breaks in the relationship, output per worker is nor-

malised but not de-trended, and we include signiﬁcant MA terms, to account for auto-

correlation when the MSV is applied to real world data, which the good econometrician

may in practice account for by what we have referred to before as an over-parametrised

PLM.16 Given an estimate of ˆ

Bfrom (37), we choose an initial value for ˆ

Asuch that the

economy is initially at ¯

θ

, the deterministic steady-state equilibrium. t0=49 is the max-

imum number of UK observations available. Using this approach, we set R0=1 1

1 1.0014 ,

x′

0= (−1.42,1.77),z′

0= (1,1),

ε

1=0 and

u1=

λ

λ

+

µ

z′

0T(x0)1−

α

(= 0.265).

To analyse the impact of adaptive learning we focus on the simulated time paths of

wages and the tightness parameter, which are independent of the choice of u1. With the

parametrisation described above, the REE parameters of the MSV solution are given by

x′

REE = (−0.70,1.055). The elasticity of

θ

to productivity at the long-run average level

is then around three, which is signiﬁcantly lower than observed in the data.

Figure 2 demonstrates a simulation over a hundred quarters of wages, unemployment

and labour market tightness for the baseline case of agents with RE.17 Unsurprisingly,

as is common with this class of models, and as described in Table 3 when compared

with Table 2, the generated sample path under the REE signiﬁcantly underestimates

16In determining initial conditions, one could also consider the class of GARCH, error correction, or even

VAR models, however we believe this would be an unnecessarily signiﬁcant leap from the straightforward

least squares updating we assume that a ‘good econometrician’ carries out in practice, and which constitutes

the learning algorithm we study here.

17See Appendix Figure C1 for the simulated paths of output per worker and shocks used in all simulations

here. These were generated using the random number seed 42 in the Python Numpy application.

19

the variance of tightness in the UK labour market; i.e. the model does not generate

a realistic magnitude of unemployment ﬂuctuations over the business cycle, with the

standard deviation being approximately a quarter of that observed in the data.

Table 3: Simulation results under the REE, decreasing and constant gain learning

Number of qtrs after init. val. 20 100

Std dev. Std dev. Min. Max.

REE

w0.011 0.0085 0.97 1.01

u0.00054 0.00053 0.054 0.056

θ

0.015 0.012 0.032 0.037

Decreasing gain

w0.012 0.0092 0.97 1.01

u0.00072 0.0070 0.054 0.057

θ

0.0020 0.0016 0.031 0.038

Constant gain (

γ

=0.05)

w0.012 0.090 0.97 1.01

u0.00073 0.00069 0.054 0.057

θ

0.021 0.016 0.31 0.38

Source: authors’ calculations.

Figure 2 also shows the equivalent simulation results when agents learn the REE

with decreasing gain, with initial estimates of the PLM parameters as described above.

It shows the path of these parameter estimates as agents learn from their forecast errors.

The key result is that convergence is very slow, when agents are given a relatively small

amount of historical data (12.5 years) and with initial estimates of the model parameters

not unrealistically far from the true REE values. As shown in Appendix Figure C2, this

takes thousands of years despite being exponential. This indicates that under adaptive

learning, an economy could be persistently away from its REE level of unemployment,

on the high or low side, even though agents are behaving rationally in the limited sense

prescribed by the ‘good econometrician.’ In this sense, RE can be a poor approximation

in terms of levels to a model with learning. One recommendation from this result is

that when calibrating the Mortensen-Pissarides model, targeting second moments of the

data should always be preferable, whereas not exactly hitting levels of the endogenous

variables may not be too concerning.

As a further example, in Appendix C3 we simulate the model with no memory, and

allow the agents to have guessed the correct initial parameter estimates, x0=xREE , but

20

Figure 2: Simulations of the labour market model and agents’ parameter estimates: a comparison of the

REE, decreasing gain and constant gain learning

Note.- initial parameter estimates of the PLM, and ˆ

B0is assumed to be ‘realistically’ far away from the true

REE values, whereas ˆ

A0is chosen such that under learning the economy begins at ¯

θ

.

suppose that there is an immediate negative twenty percent shock to the ﬂow value of

unemployment b. In the REE, due to the rise in the surplus of a match, ﬁrms immedi-

ately open more vacancies, and the unemployment rate falls. Under learning, the initial

increase in

θ

is smaller. Therefore, unemployment falls more slowly as agents attempt to

disentangle the effects of the structural shock from the stochastic process. In this sense,

21

the response to the shock under learning leads to a less volatile path for unemployment.

If actual labour market data contain the effects of frequent structural shocks of this kind,

then econometric learning will not improve the ability of the standard search model to

match their cyclical properties.

5.2. Speed of convergence

As shown theoretically in Benveniste et al. (2012), the learning of the agents results

in root-t convergence to the true REE parameter estimates if all the eigenvalues of the

system’s Jacobian have a real part strictly less than a half.18 Here this requires

ψ

2<1/2.

In the example parametrisation above this is not ensured, with

ψ

2=0.57. More gener-

ally, it can be shown with simulations that the speed of convergence decreases substan-

tially as

ψ

2→1, the threshold for E-Stability. To illustrate a decrease in the speed of

convergence, in Appendix Figure C4 we consider a value of

ψ

=0.91 by decreasing

worker bargaining power to

β

=0.1, keeping all other parameters except cconstant,

which is always used to match the mean value of

θ

from the UK data. As expected, the

rate of convergence decreases, and the economy remains more persistently away from

the REE. As such, choosing parameter values which guarantee a higher speed of conver-

gence is one way in which the REE model could become an improved approximation of

an alternative with econometric learning.

5.3. Constant gain learning

In Figure 2 we also compare the results of our ﬁrst simulation with decreasing gain

learning to an equivalent example with constant gain parameter

γ

=0.05.19 When agents

weight recent data more, convergence to the REE is faster, and agents’ parameter estim-

ates are more volatile. This faster convergence results in more volatile series of labour

market tightness, wages and unemployment. However, the gain parameter generating

this faster convergence roughly implies that agents only use data over the past twenty

quarters to update their beliefs, and is notably outside the range suggested by the adapt-

ive learning literature (see Di Pace et al. (2016) for a discussion). Our simulation results

with constant gain learning and more reasonable levels of memory weighting are not

dissimilar to those obtained with decreasing gain.

18I.e. the rate of convergence at which in classical econometrics the mean of the least squares parameter

estimate converges to the true value.

19For constant gain learning there is no analytic solution for expectational stability and so we must select

a reasonably small gain parameter to ensure convergence.

22

5.4. An alternative non-steady-state perceived law of motion

So far we have described a model of econometric learning in which agents endeavour

to forecast labour market tightness

θ

. However this is a construct of the model and

its assumptions. It is an attractive feature of the search and matching models that the

equilibrium can be described by this single choice variable, determined by the state of

the productivity process, but independent of unemployment. But ﬁrms in the model

are described as choosing the number of vacancies to post, or analogously whether or

not to enter the labour market. And for given levels of

θ

and productivity, this choice

does depend on the state of the labour market. In the REE, if we consider the economy

as initially being at some steady state (i.e. unemployment and vacancy rates are on

the Beveridge curve), then in moving to any new steady state the vacancy rate changes

non-monotonically. In characterising agents as learning how to choose and forecast

θ

, we imply that they fully understand the non-steady-state dynamics of the model.

Here we consider the implications of relaxing this assumption (for what follows, see

Appendix A.6 for complete derivations and descriptions of parameter values).

Linearising (14), (20) and (12) around the steady-state deterministic values of va-

cancies, employment and output per worker, we derive an alternative system deﬁning

the economy:

vt=

κ

0+

κ

1ye

t+1+

κ

2ve

t+1+

κ

3ne

t+1+

κ

4nt,(38)

nt=

ϕ

0+

ϕ

1nt−1+

ϕ

2vt−1,(39)

yt= (1−

ρ

) +

ρ

yt−1+

ε

t.(40)

We endow agents with a PLM in which they use both the output per worker and employ-

ment states to forecast vacancy creation,

vt=ˆ

At+ˆ

Btyt+ˆ

Ctnt−1.(41)

Given (38)-(41), we can then derive the ALM for this version of the economy, and

subsequently a ˜

T-mapping

˜

Tˆ

At,ˆ

Bt,ˆ

Ct=˜

κ

0+˜

κ

2ˆ

At+ (1−

ρ

)ˆ

Bt,˜

κ

1+˜

κ

2

ρ

ˆ

Bt,˜

κ

3+˜

κ

2ˆ

Ct.(42)

Assuming agents update their parameter estimates for the PLM using RLS as previously,

and applying the same E-stability principle, it can be shown that the sufﬁcient condition

to guarantee local convergence to the REE is given by ˜

κ

2<1. Comparing this with

23

the condition for stability of the MSV-PLM, ˜

κ

2≥

ψ

2. Hence, convergence to the REE

is slower when agents do not implicitly know the out of steady-state dynamics of em-

ployment and vacancy creation. The REE model is then a poorer approximation to an

economy with econometric learning. What is more, for a subset of parameter values

we cannot claim that the model is E-stable. For example, it is less likely to be E-stable

in the circumstance of inefﬁciently high vacancy creation, departing from the Hosios

(1990) condition (i.e.

α

>

β

). Though for the parametrisation we have used here the

model would still certainly converge to the REE.

6. Conclusion

We take the textbook linearised RBC version of the model of search and match-

ing frictions for the labour market and show that the unique REE is not only always

E-stable, for all well-deﬁned sets of parameter values, but this result is robust to over-

parametrisation of the MSV-PLM used by agents (Strong E-stability) with decreasing

gain learning. These local convergence conditions also extend trivially to global con-

vergence. Because the economy will eventually move to the REE when agents use

econometric learning, the potentially unrealistic RE assumption in this class of model

is nonetheless reasonable. We use recent UK data to parametrise the model, and show

that although the model is E-stable, implied convergence can be very slow. Therefore,

the RE model of unemployment ﬂuctuations could in fact be a poor approximation to

an economy in which agents more realistically learn as econometricians, especially in

the presence of frequent structural or permanent shocks. The MSV-PLM implicitly as-

sumes that agents understand the out of steady-state paths of employment and vacancy

creation in the model. When we consider a version of the PLM which relaxes this as-

sumption, we see that convergence is further slowed, and local E-stability of the model

is not guaranteed, making the approximation of the RE model even weaker.

24

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26

Appendix A. Methodology

Appendix A.1. Linearisation

We take a ﬁrst order Taylor approximation around the deterministic steady-state val-

ues of

θ

and y,¯

θ

and ¯y=1 respectively, approximating the right and the left hand side

of equation (20) which is stated here again for convenience,

c

δ

q(

θ

t)=(1−

β

)(yt+1−b) + (1−

λ

)c

q(

θ

t+1)−

θ

t+1

β

ce

.

This results in

c

δ

q(¯

θ

)−cq′(¯

θ

)

δ

[q(¯

θ

)]2(

θ

t−¯

θ

) =(1−

β

)( ¯y−b) + (1−

β

)(ye

t+1−¯y)

−¯

θβ

c−

β

c(

θ

e

t+1−¯

θ

) + (1−

λ

)c

q(¯

θ

)

−(1−

λ

)cq′(¯

θ

)

[q(¯

θ

)]2(

θ

e

t+1−¯

θ

).

(A.1)

By noting that

c

δ

q(¯

θ

)= (1−

β

)( ¯y−b) + (1−

λ

)c

q(¯

θ

)−¯

θβ

c

must hold in equilibrium according to (20), this steady-state condition can be subtracted

from both sides of the approximated equation. Then solving explicitly for

θ

tand deﬁning

the functional form q(¯

θ

) =

µ

¯

θ

−

α

, (A.1) becomes

θ

t=1+

β δ µ

2(¯

θ

)−2

α

αµ

(¯

θ

)−

α

−1−(1−

λ

)

δ

¯

θ

−(1−

β

)

δ µ

2(¯

θ

)−2

α

c

αµ

(¯

θ

)−

α

−1¯y

+(1−

β

)

δ µ

2(¯

θ

)−2

α

c

αµ

(¯

θ

)−

α

−1ye

t+1

+−

β δ µ

2(¯

θ

)−2

α

αµ

(¯

θ

)−

α

−1+ (1−

λ

)

δ

θ

e

t+1,

(A.2)

27

which can be simpliﬁed to the form given in the text, (21), with coefﬁcients

ψ

0= [1−

ψ

2]¯

θ

−

ψ

1¯y,

ψ

1=(1−

β

)

δ

¯

θ

q(¯

θ

)

c

α

,

ψ

2=

δ

(1−

λ

)−

β

¯

θ

q(¯

θ

)

α

,

and with the steady-state value for labour market tightness the solution to

(1−

β

)( ¯y−b)−c(1−

δ

δ

+

λ

)

q(¯

θ

)−

β

c¯

θ

=0.

Appendix A.2. Determinacy of the REE

The operator Etdenotes mathematical expectations formed at period t. The lin-

earised dynamics of output (22) can be substituted into (21) by noting under RE that

Etyt+1= (1−

ρ

) +

ρ

yt;

θ

t=˜

ψ

0+˜

ψ

1yt−1+˜

ψ

2E∗

t

θ

t+1+˜

ψ

1

ρ

−1

ε

t,(A.3)

with

˜

ψ

0=

ψ

0+

ψ

1(1−

ρ

)(1+

ρ

),

˜

ψ

1=

ψ

1

ρ

2,

˜

ψ

2=

ψ

2.

A REE of the system (22) and (A.3) is a stochastic process for

θ

tthat satisﬁes this system

with Et

θ

t+1=

θ

e

t+1. To see this possibility, note that (A.3) can be written in ARMA(1,1)

form by iterating (A.3) forward by one period, and subsequently comparing this to the

result one obtains by solving (A.3) for

θ

e

t+1. This gives

θ

t=˜

ψ

−1

2

ρ

−1˜

ψ

1(1−

ρ

)−˜

ψ

0−˜

ψ

1˜

ψ

−1

2

ρ

−1yt−1

+˜

ψ

−1

2

θ

t−1+d1

ε

t+d2

η

t,(A.4)

with d1and d2being arbitrary parameters, and

η

t:=Et[

θ

t+1]−Et−1[

θ

t+1]being a mar-

tingale difference sequence with Et[

η

t+1] = 0 by the law of iterated expectations. No

restrictions are imposed on d1or d2, since RE formed according to (A.4) regarding

θ

t+1

are unaffected by those parameters. Therefore there is a continuum of possible solutions

28

to (A.4). Evans and Honkapohja (1986) have shown that any ﬁnite degree ARMA solu-

tion of an equation in the form of (A.3) can at most be ARMA(1,1), and the particular

form of (A.4) nests all possible ARMA solutions of ﬁnite degree. The ARMA class of

solutions is stable if |

ψ

2|>1, and is unstable for |

ψ

2|<1.

. In this case the solution to (21) and (22) is the fundamental or minimal-state-

variable (MSV) solution; it is impossible to delete any state variable from the minimum

set and still obtain solutions to (22) and (A.3) for all permitted parameter values (Mc-

Callum, 1983). The MSV solution here is guessed to be

Appendix A.3. ARMA(1,1) and the MSV solution

Derivation of MSV solution: (A.4) can be re-written as

θ

t=

ρ

˜

ψ

0−˜

ψ

1(1−

ρ

)

ρ

(1−˜

ψ

2)+˜

ψ

1

ρ

(L−˜

ψ

2)yt−1−d1˜

ψ

2

(L−˜

ψ

2)

ε

t−d2

(L−˜

ψ

2)

η

t,(A.5)

with Ldenoting the lag operator such that Lxt=xt−1. The parameters d1and d2can be

chosen arbitrarily. In particular, to obtain the MSV solution

θ

t=A+Byt−1+C

ε

tone

must ﬁrst set d2=0. (A.5) can be re-written as:

θ

t=

ρ

˜

ψ

0−˜

ψ

1(1−

ρ

)

ρ

(1−˜

ψ

2)−(

ρ

−1˜

ψ

1yt−1−d1˜

ψ

2

ε

t)

∞

∑

i=1

˜

ψ

−i

2Li−1.

θ

t=

ρ

˜

ψ

0−˜

ψ

1(1−

ρ

)

ρ

(1−˜

ψ

2)+

ρ

−1˜

ψ

1˜

ψ

−1

2(1−

ρ

)

∞

∑

i=1

(

i

∑

j=1

ρ

−j)˜

ψ

−i

2−

ρ

−1˜

ψ

1˜

ψ

−1

2yt−1

∞

∑

i=0

(

ρ

˜

ψ

)−i

+

ε

t−1(

ρ

−1˜

ψ

1˜

ψ

−1

2

∞

∑

i=1

(

i

∑

j=1

ρ

−jLi−j)˜

ψ

−i

2+d1

∞

∑

i=1

˜

ψ

−i

2Li−1) + d1

ε

t.

(A.6)

Therefore, to derive an MSV solution from a broader the class of ARMA(1,1) solutions,

in which no lags of

ε

tcan remain, we therefore see from (A.6) that

d1=−˜

ψ

1

ρ

˜

ψ

21

ρ

˜

ψ

2

+ ( 1

ρ

˜

ψ

2

)2+ ( 1

ρ

˜

ψ

2

)3+...,(A.7)

=˜

ψ

1

ρ

˜

ψ

2(1−

ρ

˜

ψ

2)i f ˜

ψ

2>1

ρ

>1,(A.8)

which corresponds to the condition for stable ARMA(1,1) solutions. Otherwise, the

MSV solution cannot be derived from the class of unstable ARMA(1,1) solutions, and

is instead the only stable solution.

29

The REE values of the parameters A,B, and Care found using the method of un-

determined coefﬁcients:

A=˜

ψ

0

1−˜

ψ

2

+˜

ψ

1˜

ψ

2(1−

ρ

)

(1−˜

ψ

2)(1−˜

ψ

2

ρ

),

B=˜

ψ

1

1−˜

ψ

2

ρ

,

C=B

ρ

−1,

where we have assumed that ˜

ψ

2̸=1 and ˜

ψ

2

ρ

̸=1.

Appendix A.4. Global convergence

Given the model discussed here has a unique equilibrium, and satisﬁes the assump-

tions of Evans and Honkapohja (1998) that guarantee global convergence, we simply

apply their Theorem 2 to the recursive learning algorithm given by (29) and (31).

For Rt, using Eztz′

t=Mz, where Mzis some positive deﬁnite matrix, taking expecta-

tions we have the ODE,

dR

d

τ

=Mz−R,(A.9)

which is globally asymptotically stable and independent of xt.

It is possible that for some t Rtmay not be invertible, though this will happen only a

ﬁnite number of times with probability 1. We modify the algorithm for xtto

xt=xt−1+t−1u(Rt)zt−1z′

t−1Tˆ

At,ˆ

Bt−xt−1+

η

t,(A.10)

where u(R)is a bounded regular function from the space of 2x2 matrices to the subspace

of positive deﬁnite matrices such that u(R) = R−1in the neighbourhood of Mz. Then

taking expectations the ODE is given by

dx

d

τ

=u(R)Mz(T(ˆ

A,ˆ

B)−(A,B))′(A.11)

=u(R)Mz(

ψ

2−1)(( ˆ

A,ˆ

B)−(A,B))′.(A.12)

Given that the other requirements of the theorem are trivially satisﬁed, then it applies,

and this differential equation is clearly globally asymptotically stable for

ψ

2<1, and

this stability is exponential; (ˆ

A,ˆ

B)→(A,B)globally almost surely.

30

Appendix A.5. ALM and T-mapping ARMA solution and E-stability

θ

t=˜

ψ

0+˜

ψ

2(a+b1(1−

ρ

))

1−˜

ψ

2c1

+˜

ψ

1+˜

ψ

2(b2+b1

ρ

)

1−˜

ψ

2c1

yt−1+˜

ψ

2(b1+d1) + ˜

ψ

1

ρ

−1

1−˜

ψ

2c1

ε

t

+˜

ψ

2f1

1−˜

ψ

2c1

η

t+˜

ψ

2

1−˜

ψ

2c1

s

∑

j=3

bjyt+1−j+˜

ψ

2

1−˜

ψ

2c1

r

∑

j=2

cj

θ

t+1−j

+˜

ψ

2

1−˜

ψ

2c1

q

∑

j=2

dj

ε

t+1−j+˜

ψ

2

1−˜

ψ

2c1

l

∑

j=2

fj

η

t+1−j.(A.13)

This deﬁnes again a T-mapping from the PLM to the ALM with corresponding elements:

a=˜

ψ

0+˜

ψ

2(a+b1(1−

ρ

))

1−˜

ψ

2c1

,(A.14)

b1=˜

ψ

1+˜

ψ

2(b1

ρ

+b2)

1−˜

ψ

2c1

,(A.15)

d0=˜

ψ

1

ρ

−1+˜

ψ

2(b1+d1)

1−˜

ψ

2c1

,(A.16)

bj=˜

ψ

2

1−˜

ψ

2c1

bj+1,j=2,..., s−1,bs=0,(A.17)

cj=˜

ψ

2

1−˜

ψ

2c1

cj+1,j=1,..., r−1,cr=0,(A.18)

dj=˜

ψ

2

1−˜

ψ

2c1

dj+1,j=1,..., q−1,dq=0,(A.19)

fj=˜

ψ

2

1−˜

ψ

2c1

fj+1,j=0,..., l−1,fl=0.(A.20)

Since (A.14) - (A.20) describes a non-linear system of differential equations, we ﬁrst

have to linearise (36) to study stability properties. However, the subsystem (A.18) is

independent of the other equations and can be analysed separately. The eigenvalues of

the Jacobian of T(c)−cat the REE values cj=0 for j=1,...,rare found to be rtimes

repeatedly equal to −1 and therefore the subsystem (A.18) will converge towards the

REE values. Due to the convergence of cit is apparent that d(apart from d0) and fwill

also converge to their REE values of vectors of zeros. Moreover, bj=0 for j=2, ..., s

is easily veriﬁed to be the values towards which the economy under learning converges.

Finally, convergence of a,b1and d0are studied by analysing the Jacobian of the system

(A.14)-(A.16). If this Jacobian has eigenvalues strictly less than unity, then the whole

system is E-stable. It can easily be veriﬁed that the eigenvalues are

ψ

2and

ψ

2

ρ

.

31

Appendix A.6. A non-steady-state PLM

The system deﬁned as (38)-(40), linearised around steady-state values ¯v,¯n,¯y=1 has

derived parameter values as follows,

κ

0= (1−

κ

2)¯

θ

−¯y

κ

1,

κ

1=

δ

(1−

β

)(1−¯n)¯

θ

q(¯

θ

)

c

α

,

κ

2=

δ

(1−

λ

)−

β

¯

θ

q(¯

θ

)

α

(=

ψ

2),

κ

3=¯

θκ

2,

κ

4=−¯

θ

,

ϕ

0=

α

q(¯

θ

)( ¯v+¯

θ

¯n),

ϕ

1= (1−

λ

)−

αθ

q(¯

θ

),

ϕ

2=q(¯

θ

)(1−

α

).

Given the PLM (41), agents form expectations according to

ve

t+1=ˆ

At+ˆ

Bt[(1−

ρ

) +

ρ

yt] + ˆ

Ctnt,(A.21)

and the ALM is given by

vt=˜

κ

0+˜

κ

2ˆ

At+ (1−

ρ

)ˆ

Bt+˜

κ

1+˜

κ

2

ρ

ˆ

Btyt+˜

κ

3+˜

κ

2ˆ

Ctnt,(A.22)

where

˜

κ

0=

κ

0+

κ

1(1−

ρ

) +

κ

3

ϕ

0

1−

κ

3

ϕ

2

,

˜

κ

1=

κ

1

ρ

1−

κ

3

ϕ

2

,

˜

κ

2=

κ

2

1−

κ

3

ϕ

2

,

˜

κ

3=

κ

4+

κ

3

ϕ

1

1−

κ

3

ϕ

2

.

Given the mapping ˜

Tdeﬁned in the main text, the REE is E-stable if all the eigenvalues

of the Jacobian of ˜

T(ˆ

A,ˆ

B,ˆ

C)−(ˆ

A,ˆ

B,ˆ

C)have negative real parts. Thus, we must have

˜

κ

2

ρ

−1<0

32

and

˜

κ

2−1<0,

whereby the second condition implies the validity of the ﬁrst. Therefore, we need to

check for what range of parameter values of the model the second condition is true.

Writing out the term ˜

κ

2and rearranging, we see that the required condition is

δ

(1−

λ

)−

β

¯

θ

q(¯

θ

)

α

1+¯

θ

q(¯

θ

)(1−

α

)<1,(A.23)

or

ψ

21+¯

θ

q(¯

θ

)(1−

α

)<1.(A.24)

Given that ˜

κ

2≥

ψ

2, if the E-stability condition holds with this alternative PLM, then

convergence will be slower. For the complete range of possible model parameters, this

condition does not hold. As realistic levels of

λ

are small, the condition would be

sensitive to assumed parameter values of

β

and

α

. For example, given

α

>

β

>0,

which is the case of low worker bargaining power, whereby wages are reduced towards

the value of the outside option, and there is excessive ﬁrm entry, or inefﬁciently high

according to the Hosios (1990) condition, it is more likely E-stability will not hold.

33

Appendix B. Parametrisation of the model

We normalise average productivity to be one. For the productivity process we estim-

ate an AR(1) in log deviations from trend output per worker, dynamically de-trended

using the HP ﬁlter with standard quarterly smoothing parameter, and ﬁnd an auto-

regressive parameter

ρ

for the period of 0.84, and a standard deviation for the shocks

σε

of 0.0063 (assuming them to be normally distributed). For the labour market, we

parametrise the model to the unemployment rate, measured as the fraction of the eco-

nomically active population aged 16 and over who are ILO unemployed. We use ofﬁcial

quarterly time series from Ofﬁce for National Statistics (ONS) Labour Market Statistics.

For transition rates between labour market states we use the ﬂows time series similarly

published by ONS, which are derived from the Two-quarter Longitudinal Labour Force

Survey and are consistent with all stocks series. The economy we describe has two

states. In reality there is a third: economic inactivity. To adhere to our interpretation of

utas the unemployment rate, abstracting from the relative size of the inactive population

over the business cycle, as is common in the literature (Shimer, 2005; Hagedorn and

Manovskii, 2008), we must carefully construct from the raw data measures of job ﬁnd-

ing and separation rates. In the notation of the model, the steady-state unemployment

rate is given by

u∗

t=

λ

λ

+

θ

tq(

θ

t).(B.1)

As per Smith (2011), using three-state ﬂows data between the stocks in employment,

unemployment and inactivity, denoted by {E,U,I}, with transition rates, for example

between inactivity and unemployment, denoted by pIUt, we can re-write (B.1) as

u∗

t=pEUt+pEItpIUt

pIUt+pIEt

pEUt+pEItpIUt

pIUt+pIEt

λ

t

+pUEt+pUItpIEt

pIUt+pIEt

θ

tq(

θ

t)

.(B.2)

As such, the separation rate from real data which is consistent with the model described

here is the sum of the direct transition rate from employment to unemployment and a

term which captures the indirect role of transitions to unemployment via inactivity -

with a similar interpretation for the job ﬁnding rate.

Using this measure of the hiring rate from the transition rates data, we estimate the

parameters of the aggregate matching function using least squares as follows for 2002q1-

34

13q2:

logpUEt+pUItpIEt

pIUt+pIEt=log(

µ

) + (1−

α

)logvt

ut+

ζ

t,(B.3)

where data for vtcome from the quarterly ONS aggregate vacancies series, and utis the

UK national unemployment rate. Following Borowczyk-Martins et al. (2013), we con-

sider time trends in the estimation to account for the endogeneity of unobserved shifts

in the matching efﬁciency with the number of vacancies that ﬁrms open, but these all

drop out. We also carry out tests that the matching function is Cobb-Douglas, and reject

the alternative. In line with the existing literature, we ﬁnd that the data suggests the

matching function has decreasing returns to scale, although we proceed as though it is

constant (see Pissarides and Petrongolo (2001) for a thorough review of estimates of the

aggregate matching function). We ﬁnd estimates of

α

=0.67 and

µ

=0.56. For the con-

stant separation rate parameter in the model, over the same period we choose an average

value of the two-quarter composite hazard rate: pEU +pE I pIU

pIU +pIE =

λ

=0.023. (In practice

we regress the data on a constant and cubic trend to account for low frequency shifts

for the short period in question, then selecting the estimated constant as the parameter

value - we similarly do this when estimating moments of the labour market variables

presented in Table 2). The discount factor is set as

δ

=0.99, and to restrict the number

of free parameters we let the bargaining power adhere to the Hosios (1990) condition,

β

=

α

=0.67. We set the ﬂow value of unemployment to 0.8. How to select or estimate

appropriate values of both the bargaining power and the ﬂow value of unemployment are

open to debate. Shimer (2005) and subsequently Hagedorn and Manovskii (2008) are

often considered in the literature as more extreme examples for parametrisations, and

highlight how this affects the ability of the model to match the observed volatility of

unemployment and vacancy creation. With the relatively arbitrary parametrisation ap-

plied here, we are somewhere in between these two examples. The remaining parameter,

the ﬂow vacancy cost c, is chosen to match the observed level of average labour market

tightness over the period, as displayed in Table 2 and as used to estimate the parameters

of the matching function.

35

Appendix C. Additional ﬁgures

Figure C1: Simulation of the equilibrium of the stochastic model with the assumption of rational expecta-

tions: the REE

Figure C2: Convergence of agents’ parameter estimates under decreasing gain learning to the REE values

Note.- The simulation here is identical to that described under decreasing gain learning for Figure 2. Dashed

lines give the true REE parameter values.

36

Figure C3: Comparison of sample paths for endogenous variables under RE and decreasing gain learning,

and agents’ parameter estimates, following a structural shock

Note.- these simulation paths are the results of a negative 20% shock to the ﬂow value of unemployment b,

with initial parameter estimates assumed to be at the true pre-shock REE values.

37

Figure C4: Comparing the speed of convergence to the REE under decreasing gain learning: changing

worker bargaining power

β

Note.- given

β

=0.1, then

ψ

2=0.91. For

β

=

α

,

ψ

2=0.57, as in Figure 2. The crossing of the time paths

indicates a decreased speed of convergence since, for example, the REE parameter value with

β

=0.1 of

B=0.56 is substantially lower than value with

β

=

α

.

38