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We apply well-known results of the econometric learning literature to the Mortensen and Pissarides real business cycle model. Agents can always learn the unique rational expectations 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.
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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+ (1ut)
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 (1nt)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>b0. 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(1nt) + Dt,(8)
with wtntand b(1nt)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(1nt) +
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=1ut. 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)(1nt).(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
=wtb+
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(yt1) +
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+swt+snt+scvt+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
vt0(ytntwtntcvt) +
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
=ytwt+(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+1wt+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+1b) + (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=ytntwtntcvt.
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
yt1+
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+Byt1+C
ε
t,(23)
since the only predetermined variable in the above system is yt1. 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
t1= ( ˆ
At,ˆ
Bt), then the general recursive
updating algorithm can be represented by
xt=xt1+gtQ(
θ
tˆ
Atˆ
Btyt1),(26)
which shows that agents update their previous parameter estimates xt1by 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
yt1
+ (
ψ
1
ρ
+
ψ
2ˆ
Bt)
ε
t.(27)
This defines the following T-mapping from the PLM,
θ
t=A+Byt1+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:RNRNmaps 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
t1= (1,yt1),x
t1= ( ˆ
At,ˆ
Bt)and
θ
t=z
t1xt1+
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=t1t
i=1zi1z
i1, which allows us to write the RLS estimator as
Rt=Rt1+t1(zt1z
t1Rt1),(29)
xt=xt1+t1R1
tzt1(
θ
tz
t1xt1),(30)
and thus
xt=xt1+t1R1
tzt1z
t1Tˆ
At,ˆ
Btxt1+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
Bˆ
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
bjytj+
r
j=1
cj
θ
tj+
q
j=1
dj
ε
tj+
l
j=1
fj
η
tj+d0
ε
t+f0
η
t.(34)
Accordingly, expectations of
θ
t+1take the form:
θ
e
t+1=a+
s
j=1
bjyt+1j+
r
j=1
cj
θ
t+1j+
q
j=1
dj
ε
t+1j+
l
j=1
fj
η
t+1j,(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
(1ut+1) + 1
µ
z
tT(xt) + V(xt)
ε
t+11
α
ut+1,
(II)Rt+1=Rt+1
t+1(ztz
tRt),
(III)xt+1=xt+1
t+1R1
t+1ztz
t[T(xt)xt] +V(xt)
ε
t+1,
(IV )yt+1= (1
ρ
) +
ρ
yt+
ε
t+1,
(V)
ε
t+2i.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 t02. 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+Byt1+
κ
1
ζ
t1+
κ
2
ζ
t2+
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
21, 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
1nt1+
ϕ
2vt1,(39)
yt= (1
ρ
) +
ρ
yt1+
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
Ctnt1.(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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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+1b) + (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
)( ¯yb) + (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
)( ¯yb) + (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
)( ¯yb)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
1yt1+˜
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
1yt1
+˜
ψ
1
2
θ
t1+d1
ε
t+d2
η
t,(A.4)
with d1and d2being arbitrary parameters, and
η
t:=Et[
θ
t+1]Et1[
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)yt1d1˜
ψ
2
(L˜
ψ
2)
ε
td2
(L˜
ψ
2)
η
t,(A.5)
with Ldenoting the lag operator such that Lxt=xt1. The parameters d1and d2can be
chosen arbitrarily. In particular, to obtain the MSV solution
θ
t=A+Byt1+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
1yt1d1˜
ψ
2
ε
t)
i=1
˜
ψ
i
2Li1.
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
2yt1
i=0
(
ρ
˜
ψ
)i
+
ε
t1(
ρ
1˜
ψ
1˜
ψ
1
2
i=1
(
i
j=1
ρ
jLij)˜
ψ
i
2+d1
i=1
˜
ψ
i
2Li1) + 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
=MzR,(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=xt1+t1u(Rt)zt1z
t1Tˆ
At,ˆ
Btxt1+
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) = R1in 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
21)(( 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
yt1+˜
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+1j+˜
ψ
2
1˜
ψ
2c1
r
j=2
cj
θ
t+1j
+˜
ψ
2
1˜
ψ
2c1
q
j=2
dj
ε
t+1j+˜
ψ
2
1˜
ψ
2c1
l
j=2
fj
η
t+1j.(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,..., s1,bs=0,(A.17)
cj=˜
ψ
2
1˜
ψ
2c1
cj+1,j=1,..., r1,cr=0,(A.18)
dj=˜
ψ
2
1˜
ψ
2c1
dj+1,j=1,..., q1,dq=0,(A.19)
fj=˜
ψ
2
1˜
ψ
2c1
fj+1,j=0,..., l1,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
21<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
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This paper examines implications of incorporating labor market search and matching frictions into a sticky price model for determinacy and E-stability of rational expectations equilibrium (REE) under interest rate policy. When labor adjustment takes place solely at the extensive margin, forecast-based policy that meets the Taylor principle is likely to induce indeterminacy and E-instability, regardless of whether it is strictly or flexibly inflation targeting. When labor adjustment takes place at both the extensive and intensive margins, the strictly inflation-forecast targeting policy remains likely to induce indeterminacy, but it generates a unique E-stable fundamental REE as long as the Taylor principle is satisfied. These results suggest that introducing the search and matching frictions alter determinacy properties of the strictly inflation-forecast targeting policy, but not its E-stability properties in the presence of the intensive margin of lab.
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Standard matching models of unemployment generate far too little volatility in unemployment and vacancies relative to the variation in the shock variables. Shimer (2005) showed that in US data the vacancy-to-unemployment ratio is about 26 times more volatile than the standard model predicts. He identified the flexibility of wages as the key issue and triggered a heated debate on possible improvements of the core model to accommodate these empirical facts. In this paper, we first document Shimer's facts for the UK and find them to be qualitatively similar to US facts. We then develop and calibrate a model based on the Mortensen and Pissarides approach that increases the volatility of the v/u ratio 20-fold compared to the standard framework. The key features of our model relate to the job creation decision by firms and the search options of workers. We allow these to search whilst employed, and firms to re-advertise jobs that have been quit from. This leads us to use a different job creation process, whereby potential vacancies, or job ‘ideas’, arise at a finite rate per period over a range of idiosyncratic productivities. Calibrating the model to UK data, we show that it delivers volatility in unemployment and vacancies much closer to, though still not as large as, that observed for the UK, whilst retaining the standard wage determination process.