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The Time-Varying Beveridge Curve

WP 13-12

Luca Benati
University of Bern
Thomas A. Lubik
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

The Time-Varying Beveridge Curve
Thomas A. Lubik
Federal Reserve Bank
of Richmondz

Luca Benati
University of Berny

August 2013
Working Paper No. 13-12

Abstract
We use a Bayesian time-varying parameter structural VAR with stochastic volatility
to investigate changes in both the reduced-form relationship between vacancies and the
unemployment rate, and in their relationship conditional on permanent and transitory
output shocks, in the post-WWII United States. Evidence points towards similarities
and di¤erences between the Great Recession and the Volcker disin‡ation, and widespread time variation along two key dimensions. First, the slope of the Beveridge curve
exhibits a large extent of variation from the mid-1960s on. It is also notably pro-cyclical,
whereby the gain is positively correlated with the transitory component of output. The
evolution of the slope of the Beveridge curve during the Great Recession is very similar to its evolution during the Volcker recession in terms of both its magnitude and
its time pro…le. Second, both the Great In‡ation episode and the subsequent Volcker
disin‡ation are characterized by a signi…cantly larger negative correlation between the
reduced-form innovations to vacancies and the unemployment rate than the rest of the
sample period. Those years also exhibit a greater cross-spectral coherence between the
two series at business-cycle frequencies. This suggests that they are driven by common
shocks.

JEL Classification:
Keywords:

C11, C32, E20, E24,
Bayesian VARs; stochastic volatility; time-varying parameters;
Great Recession; Great In‡ation; long-run restrictions

The views in this paper are those of the authors and should not be interpreted as those of the Federal
Reserve Bank of Richmond, the Board of Governors, or the Federal Reserve System. We are grateful to
participants at the Applied Time Series Econometrics Workshop at the Federal Reserve Bank of St. Louis
and the Midwest Macroeconomics Meetings at the University of Colorado Boulder for useful comments and
suggestions.
y
Department of Economics, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland. Email:
luca.benati@vwi.unibe.ch
z
Research Department, Federal Reserve Bank of Richmond, 701 E Byrd Street, Richmond, VA 23218,
USA. E-mail: Thomas.Lubik@rich.frb.org

Introduction

The Beveridge curve describes the relationship between the unemployment rate and open
positions, that is, vacancies, in the labor market. Plotting the former against the latter in a
scatter diagram reveals a downward-sloping relationship that appears to be clustered around
a concave curve (see Figure 1). The curve re‡ects the highly negative correlation between
unemployment and vacancies that is a hallmark of labor markets in market economies.
Empirical work on the Beveridge curve has explored the relationship between vacancies
and the unemployment rate under the maintained assumption that it can be regarded, as
a …rst approximation, as time-invariant. The behavior of the two series during the Great
Recession, with the unemployment rate seemingly stuck at high levels, even in the presence
of a vacancy rate that has been progressively improving, has, however, raised doubts about
the validity of the assumption of time-invariance. This suggests exploring the relationship
between the two series allowing for the possibility that it may have evolved over time.
Our paper builds directly on the seminal contribution of Blanchard and Diamond (1989).
These authors reintroduced the concept of the Beveridge curve as one of the key relationships in macroeconomic data. They conducted a vector autoregression (VAR) analysis of
unemployment, vacancies, and the labor force in order to identify the driving forces behind
movements in the Beveridge curve. We build upon their analysis by identifying both permanent and transitory structural shocks in a time-varying VAR context. By doing so, we
are able to trace out the sources of movements, shifts, and tilts in the Beveridge curve over
time.
The theoretical background for our study, and one that we use for identifying the structural shocks, is the simple search and matching approach to modeling labor markets (see
Shimer, 2005). The Beveridge curve encapsulates the logic of this model. In times of
economic expansions, unemployment is low and vacancies, that is, open positions o¤ered
by …rms, are plentiful. Firms want to expand their workforce, but they are unable to do
so since the pool of potential employees (that is, the unemployed) is small. As economic
conditions slow down and demand slackens, …rms post fewer vacancies and unemployment
rises, consistent with a downward move along the Beveridge curve. At the trough of the
business cycle, …rms may have expectations of a future uptick in demand and start posting
open positions. This decision is ampli…ed by the large pool of unemployed, which guarantees …rms high chances of …nding suitable candidates and thus outweighs the incurred
search costs. As the economy improves, unemployment falls and vacancy postings rise in

an upward move along the Beveridge curve.
We approach the Beveridge curve empirically by specifying a time-varying parameter
VAR with stochastic volatility. Our choice is informed by the observation that there are
patterns in the Beveridge-curve relationship that are ill described by a linear framework.
Speci…cally, the data suggest that the slope of the Beveridge curve is di¤erent for each
business cycle episode, that the curve shifts over time, and that the pattern of driving
forces change in a nonlinear fashion as well. Naturally, nonlinearity can take many forms,
as do the framework to capture this. We utilize a time-varying parameter framework since
it is a reasonably straightforward extension of linear VARs. Moreover, and perhaps more
importantly, it can capture and approximate a wide range of underlying nonlinear behavior.
To this point, we will introduce time variation in a nonlinear theoretical model of the labor
market in order to relate it to the results from the VAR.
Our empirical analysis starts by documenting the presence of time variation in the
relationship between vacancies and the unemployment rate by means of Stock and Watson’s
(1996, 1998) time-varying parameter median-unbiased estimation, which allows us to test
for the presence of random-walk time variation in the data. Having detected evidence of
time variation in the bivariate relationship between vacancies and the unemployment rate,
we then use a Bayesian time-varying parameter structural VAR with stochastic volatility to
characterize changes over time in the relationship. Evidence points towards both similarities
and di¤erences between the Great Recession and the Volcker disin‡ation, and widespread
time variation along two key dimensions.
First, the slope of the Beveridge curve, which we capture by the average gain of the
unemployment rate onto vacancies at business cycle frequencies is strongly negatively correlated with the Congressional Budget O¢ ce’s (CBO) estimate of the output gap. The
evolution of the slope of the Beveridge curve during the Great Recession is very similar
to its evolution during the Volcker recession in terms of both its magnitude and its time
pro…le. This suggests that the seemingly anomalous behavior of the Beveridge curve during
the Great Recession, which has attracted much attention in the literature, may not have
been that unusual. Second, both the Great In‡ation episode and the subsequent Volcker
disin‡ation, are characterized by a signi…cantly larger (in absolute value) negative correlation between the reduced-form innovations to vacancies and the unemployment rate than
the rest of the sample period. These years also show a greater cross-spectral coherence
between the two series at business-cycle frequencies. This suggests that they are driven, to
a larger extent than the rest of the sample, by common shocks.

Having characterized changes over time in the relationship between vacancies and the
unemployment rate, we then proceed to interpret these stylized facts based on an estimated
search and matching model. Speci…cally, we explore within a simple theoretical model how
changes in individual parameter values a¤ect the relationship between vacancies and the
unemployment rate in order to gauge the origin of the variation in the Beveridge relationship.
The paper is organized as follows. The next section presents preliminary evidence on the
presence of (random-walk) time variation in the bivariate relationship between the vacancy
rate and the unemployment rate. Section 3 describes the Bayesian methodology we use
to estimate the time-varying parameter VAR with stochastic volatility, whereas Section
4 discusses the evidence of changes over time in the Beveridge relationship. Section 5
details our structural identi…cation procedure based on insights from a simple search and
matching model, where we discuss implementation of both long-run and sign restrictions.
We present the results of the structural identi…cation procedure in Section 6. We also explore
how changes in individual structural parameters of the search and matching model map
into corresponding changes in the relationship between vacancies and the unemployment
rate. Section 7 concludes. An appendix contains a detailed description of the econometric
methods and the theoretical model we use for identi…cation purposes.

Searching for Time Variation in the Beveridge Relationship

Figure 1 presents a time series plot of the unemployment rate and vacancies from 1949 to
2011. The negative relationship between the two series is readily apparent. At the peak
of the business cycle unemployment is low and vacancies are high. Over the course of a
downturn the former rises and the latter declines as fewer and fewer workers are employed
and …rms have fewer and fewer open positions. Volatility and serial correlation of both
series appear of similar magnitude. The second panel in Figure 1 depicts the same series in
a scatter plot of vacancies against unemployment, resulting in the well-known downwardsloping relationship that has come to be known as the Beveridge curve. In the graph, we
plot individual Beveridge curves for each NBER business cycle. Each episode starts at
the business cycle peak and ends with the period before the next peak. Visual inspection
reveals two observations. First, all curves are downward-sloping, but with di¤erent slopes.
Second, there is substantial lateral movement in the individual Beveridge curves, ranging
from the innermost cycle, the 1953-1957 episode, to the outermost, 1982-1990. We take
these observations as motivating evidence that the relationship between unemployment
4

and vacancies exhibits substantial variation over time, which a focus on a single aggregate
Beveridge curve obscures.
Time variation in data and in theoretical models can take many forms, from continuous
variations in unit-root time-varying parameter models to discrete parameter shifts such as
regime-switching. We regard both discrete and continuous changes as a priori plausible. In
this paper, we focus on the latter. We thus provide evidence of time variation in the bivariate
relationship between vacancies and unemployment. We apply the methodology developed
by Stock and Watson (1996, 1998) to test for the presence of random-walk time-variation
in the two-equation VAR representation for the two variables.1
The regression model we consider is:
xt =

+ (L)Vt

+ (L)Ut

Zt + t ;

(1)

where xt = Vt , Ut , with Vt and Ut being the vacancy rate and the unemployment rate,
respectively.
..., Ut

]0 .

(L) and

(L) are lag polynomials;

= [ ; (L); (L)]0 and Zt = [1, Vt

We select the lag order as the maximum of the lag orders individually chosen

by the Akaike, Schwartz, and Hannan-Quinn criteria. Letting

= [ t;

t (L);

0
t (L)] ,

the

time-varying parameter version of (1) is given by:
xt =

0
t Zt

+ t;

(2)

t 1

(3)

with
2

iid N (04p+1 ;

is the variance of

2 2
Q),

where 04p+1 is a (4p + 1)-dimensional vector of zeros.

Q a covariance matrix, and E[

standard random walk, where

= 0. Following Stock and

Under this normalization, the coe¢ cients

[E(Zt Zt0 )] 1=2 Zt ,
2

evolve according to a (4p + 1)-dimensional

Watson (1996, 1998), we set Q = [E(Zt Zt0 )]
on the transformed regressors,

t t]

is the ratio between the variance of each transformed
h
i 1
^ = T 1 PT zt z 0
.
innovation and the variance of t . We estimate the matrix Q as Q
t
t=1

We estimate the speci…cation (1) by OLS, from which we obtain an estimate of the

innovation variance, ^ 2 . We then perform an exp- and a sup-Wald joint test for a single
unknown break in

and in the sums of the ’s and ’s, using either the Newey and West

1
From an empirical perspective, we prefer their methodology over, for instance, structural break tests for
reasons of robustness to uncertainty regarding the speci…c form of time-variation present in the data. While
time-varying parameter models can successfully track processes subject to structural breaks, Cogley and
Sargent (2005) and Benati (2007) show that break tests possess low power when the true data-generating
process (DGP) is characterized by random walk time variation. Generally speaking, break tests perform
well if the DGP is subject to discrete structural breaks, while time-varying parameter models perform well
under both scenarios.

(1987) or the Andrews (1991) HAC covariance matrix estimator to control for possible
autocorrelation and/or heteroskedasticity in the residuals. Following Stock and Watson
(1996), we compute the empirical distribution of the test statistic by considering a 100point grid of values for

over the interval [0, 0.1]. For each element of the grid we compute
^ j = 2 ^ 2 Q;
^ conditional on Q
^j
the corresponding estimate of the covariance matrix of t as Q
j
we simulate the model (2)-(3) 10,000 times, drawing the pseudo innovations from pseudorandom iid N (0; ^ 2 ). We compute the median-unbiased estimate of

as that particular

value for which the median of the simulated empirical distribution of the test is closest to the
test statistic previously computed based on the actual data. Finally, we compute the p-value
based on the empirical distribution of the test conditional on

= 0, which we compute

based on Benati’s (2007) extension of the Stock and Watson (1996, 1998) methodology.
We report the estimation results in Table 1. We …nd strong evidence of random-walk
time variation in the equation for the vacancy rate. The p-values for the null of no time
variation range from 0:0028 to 0:0195, depending on the speci…c test statistic. The medianunbiased estimates of

are comparatively large, between 0:0235 and 0:0327. On the other

hand, the corresponding p-values for the unemployment rate are much larger, ranging from
0:1661 to 0:2594, which suggests time-invariance. However, the densities of the medianunbiased estimates of

in Figure 2 paint a more complex picture. A substantial fraction

of the probability mass are clearly above zero, whereas median-unbiased estimates of
range between 0:0122 and 0:0153. Although, strictly speaking, the null hypothesis of no
time variation cannot be rejected at conventional signi…cance levels in a frequentist sense,
the evidence reported in Figure 2 suggests more caution. In what follows, we will proceed
under the assumption that both equations feature random-walk time variation. We now
investigate the changing relationship between the vacancy rate and the unemployment rate
based on a Bayesian time-varying parameter VAR.

A Bayesian Time-Varying Parameter VAR with Stochastic
Volatility

We de…ne the data vector Yt

[ yt , Vt , Ut ]0 , where

yt is real GDP growth, computed as

the log-di¤erence of real GDP; Vt is the vacancy rate based the Conference Board’s HelpWanted Index and Barnichon’s (2010) extension; and Ut is the unemployment rate. The
data are all quarterly. The vacancies and unemployment series are both normalized by the
labor force, seasonally adjusted, and converted from the original monthly series by simple
averaging. The overall sample period is 1951Q1-2011Q4. We use the …rst 15 years of data
6

to compute the Bayesian priors, which makes the e¤ective sample period 1965Q1-2011Q4.
Appendix A contains a complete description of the data and of their sources.
We specify the time-varying parameter V AR(p) model:
Yt = B0;t + B1;t Yt

+ ::: + Bp;t Yt

+ t:

(4)

The notation is standard. As is customary in the literature on Bayesian time-varying
parameter VARs, we set the lag order to p = 2. The time-varying lag coe¢ cients, collected
in the vector

are postulated to evolve according to:
p(

t 1,

Q) = I( t ) f (

t 1,

Q);

(5)

where I( t ) is an indicator function that rejects unstable draws and thereby enforces stationarity on the VAR. The transition f (

with

t 1,
t 1

Q) is given by:

(6)

in (4) are normally distributed with

iid N (0; Q).

We assume that the reduced-form innovations

zero mean, where we factor the time-varying covariance matrix
V ar( t )
The time-varying matrices Ht and
2
h1;t
0
h2;t
Ht 4 0
0
0

as:

= At 1 Ht (At 1 )0 :

At are de…ned as:
3
2
0
0 5;
At 4
h3;t

(7)

1
21;t

0
1

31;t

32;t

3
0
0 5:
1

We assume that the hi;t evolve as geometric random walks:
ln hi;t = ln hi;t

For future reference, we de…ne ht

i;t ; i

(8)

= 1; 2; 3:

[h1;t , h2;t , h3;t ]0 and

(9)
1;t ,

2;t ,

3;t ]

with

iid N (0; Z) and Z diagonal. We assume, as in Primiceri (2005), that the non-zero and
non-unity elements of the matrix At , which we collect in the vector

[

21;t ,

31;t ,

0
32;t ] ,

evolve as random walks:
t

where

t 1

iid N (0; S). Finally, we assume that the innovations vector [u0t ,

distributed as:

2
6
6
4

ut
t
t
t

3
7
7
5

3
I3 0 0 0
6 0 Q 0 0 7
7
N (0; V ) , with V = 6
4 0 0 S 0 5 ;
0 0 0 Z
7

(10)
0,
t

0,
t

0 ]0
t

(11)

where ut is such that

At 1 Ht2 ut .

We follow the literature in imposing a block-diagonal structure for V , mainly for parsimony, since the model is already quite heavily parameterized. Allowing for a completely
generic correlation structure among di¤erent sources of uncertainty would also preclude any
structural interpretation of the innovations. Finally, following Primiceri (2005) we adopt
the additional simplifying assumption of a block-diagonal structure for S:
S
with S1

V ar (

21;t ),

V ar ( t ) = V ar ( t ) =
and S2

V ar ([

S1
02 1

0
31;t ; 32;t ] ).

01 2
S2

;

(12)

This implies that the non-zero

and non-one elements of At , which belong to di¤erent rows, evolve independently. This
assumption simpli…es inference substantially, since it allows Gibbs sampling on the nonzero and non-one elements of At equation by equation. We estimate (4)-(12) using standard
Bayesian methods. Appendix B discusses our choices for the priors, and the Markov-Chain
Monte Carlo algorithm we use to simulate the posterior distribution of the hyperparameters
and the states conditional on the data.

Reduced-Form Evidence

Figure 3 presents the …rst set of reduced-form results. It shows statistics of the estimated
time-varying innovations in the VAR (4). The …rst panel depicts the median posterior estimate of the correlation coe¢ cient of the innovations to vacancies and the unemployment
rate and associated 68% and 90% coverage regions. The plot shows substantial time variation in this statistic. From the late 1960s to the early 1980s the correlation strengthens from
0:4 to

0:85 before rising (in absolute value) to a low of

0:25. Over the course of the last

decade, the correlation has strengthened again, settling close to the average median value
of

0:55. This suggests that the unemployment-vacancy correlation strengthens during pe-

riods of broad downturns and high volatility, whereas it weakens in general upswings with
low economic turbulence. The evidence over the last decade also supports the impression
that the U.S. economy is in a period of a prolonged downswing.2
The impression of substantial time variation is strongly supported by the second panel,
which shows the fraction of draws from the posterior distribution for which the correlation
coe¢ cient is greater than the average median value over the sample. The fraction of draws
2
We also note that at the same time the coverage regions are tightly clustered around the median estimate
during the period of highest instability, namely the last 1970s and the Volcker disin‡ation, whereas they are
more spread out in the beginning and towards the end of the sample.

sinks toward zero at the end of the Volcker disin‡ation, while it oscillates for much of the
Great Moderation between 0:6 and 0:9. Similarly to the results in the …rst panel, the period
since the beginning of the …nancial crisis in August 2007 is characterized by a substantial
decrease in the fraction of draws.
In the third panel of Figure 3, we highlight the ratio of the estimated standard deviations of the unemployment and vacancy innovations. The graph shows substantial time
variation in this ratio, although overall both innovation variances are of roughly equal size.
While the innovation variance of the vacancy rate appears overall dominant, unemployment
innovations play a relatively larger role at the end of the Great In‡ation, the Volcker disin‡ation, and the Great Recession. All of these are periods during which the unemployment
rate shot up sharply. This suggests a dominant role of speci…c shocks, namely those tied
closely to reduced-form innovations to the unemployment rate, at the onset of an economic
downturn. We attempt to identify the sources for this behavior in the following section.
We now narrow our focus to the behavior of unemployment and vacancies at the businesscycle frequencies between six quarters and eight years. We report these results using statistics from the frequency domain. Figure 4 shows median posterior estimates (and associated
coverage regions) of the average cross-spectral gain and coherence between the two variables. The gain of a variable xt onto another variable yt at the frequency ! is de…ned as
the absolute value of the OLS-coe¢ cient in the regression of yt on xt at that frequency,
whereas the coherence is the R2 in that regression. Consequently, the gain has a natural
interpretation in terms of the slope of the Beveridge curve, while the coherence measures
the fraction of the vacancy-rate’s variance at given frequencies that is accounted for by
the variation in the unemployment rate. We …nd it convenient to express time variation
in the Beveridge curve in terms of the frequency domain since it allows us to isolate the
‡uctuations of interest, namely policy-relevant business cycles, and therefore abstract from
secular movements.
Overall, evidence of time variation is signi…cantly stronger for the gain than for the coherence. The coherence between the two series appears to have remained broadly unchanged
since the second half of the 1960s, except for a brief run-up during the Great In‡ation of
the 1970s, culminating in the tight posterior distribution during the Volcker disin‡ation of
the early 1980s. Moreover, average coherence is always above 0.8, with 0.9 contained in
the 68% coverage region. The high explanatory power of one variable for the other at the
business cycle frequencies thus suggests that unemployment and vacancies are driven by a
set of common shocks over the sample period.

The gain is large during the same periods in which the relative innovation variance of
reduced-form shocks to the unemployment rate is large, namely during the …rst oil shock,
the Volcker recession, and the Great Recession; that is, during these recessionary episodes
movements in the unemployment rate are relatively larger than those in the vacancy rate.
This points towards a ‡attening of the Beveridge curve in downturns, when small movements
in vacancies are accompanied by large movements in unemployment. Time variation in the
gain thus captures the shifts and tilts in the individual Beveridge curves highlighted in
Figure 1 in one simple statistic.
As a side note, our evidence does not indicate fundamental di¤erences between the
Volcker disin‡ation and the Great Recession, that is, between the two deepest recessions in
the post-war era. This is especially apparent from the estimated gain in Figure 4, which
shows a similar time pro…le during both episodes. The relationship between vacancies and
unemployment, although clearly di¤erent from the years leading up to the …nancial crisis,
is broadly in line with that of the early 1980s.
We can now summarize our …ndings from the reduced-form evidence as follows. The
correlation pattern between unemployment and vacancies shows a signi…cant degree of time
variation. It strengthens during downturns and weakens in upswings. This is consistent
with the idea that over the course of a business cycle, as the economy shifts from a peak
to a trough, labor market equilibrium moves downward along the Beveridge curve. This
movement creates a tight negative relationship between unemployment and vacancies. As
the economy recovers, however, vacancies start rising without much movement in unemployment. Hence, the correlation weakens. The economy thus goes o¤ the existing Beveridge
curve, in the manner of a counter-clockwise loop, as identi…ed by Blanchard and Diamond
(1989), or it moves to a new Beveridge curve, as suggested by the recent literature on
mismatch, e.g. Furlanetto and Groshenny (2012), Lubik (2012), and Sahin et al. (2012).
Evidence from the frequency domain suggests that the same shocks underlie movements
in the labor market, but that over the course of the business cycle shocks change in their
importance. During recessions movements in unemployment dominate, while in upswings
vacancies play a more important role. We now try to identify the structural factors determining this reduced-form behavior.

Identi…cation

A key focus of our analysis is to identify the underlying sources of the movements in the
Beveridge curve. In order to do so we need to identify the structural shocks behind the
10

behavior of unemployment and vacancies. Our data set contains a nonstationary variable,
GDP, and two stationary variables, namely the unemployment and vacancy rates. This
allows us to identify one permanent and two transitory shocks from the reduced-form innovation covariance matrix. While the permanent shock has no e¤ect on the two labor
market variables in the long run, it can still lead to persistent movements in these variables,
and thus in the Beveridge curve, in the short to medium run.3 More speci…cally, we are
interested in which shocks can be tied to the changing slope and the shifts in the Beveridge
curve. We let our identi…cation strategy be guided by the implications of the simple search
and matching model, which o¤ers predictions for the e¤ects of permanent and transitory
productivity shocks as well as for other transitory labor market disturbances.

5.1

A Simple Theoretical Framework

We organize the interpretation of our empirical …ndings around the predictions of the standard search and matching model of the model labor market as described in Shimer (2005).
The model is a data-generating process for unemployment and vacancies that is driven
by a variety of fundamental shocks. The speci…c model is taken from Lubik (2012). The
speci…cation and derivation is described in more detail in the appendix.
The model can be reduced to three key equations that will guide our thinking about the
empirics. The …rst equation describes the law of motion for employment:
Nt+1 = (1

h
1
)
t+1 Nt + mt Ut Vt

(13)

The stock of existing workers Nt is augmented by new hires mt Ut Vt1

, which are the re-

sult of a matching process between open positions Vt and job seekers Ut via the matching
function. The matching process is subject to exogenous variation in the match e¢ ciency
mt , which a¤ects the size of the workforce with a one-period lag. Similarly, employment
is subject to exogenous variations in the separation rate

which a¤ects employment con-

temporaneously. It is this timing convention that gives rise to an identifying restriction.
The second equation is the job-creation condition, which describes the optimal vacancy
posting decision by a …rm:
t

= Et (1

t+1 )

) (At+1

bt+1 )

t+1 t+1

t+1

mt+1

t+1

(14)

3
While this rules out strict hysteresis e¤ects, in the sense that temporary shocks can have permanent
e¤ects, it can still lead to behavior that looks over typical sample periods as hysteresis-induced. Moreover,
the empircial evidence concerning hysteresis is decidedly mixed.

= Vt =Ut is labor market tightness and a crucial statistic in the search and matching

model. E¤ective vacancy creation costs

mt t

are increasing in tightness since …rms have to

compete with other …rm’s hiring e¤orts given the size of the applicant pool. Hiring costs
are subject to exogenous variations in the component

and have to be balanced against

the expected bene…ts, namely the right-hand side of the above equation. This consists of
surplus of a worker’s marginal product At over his outside options (unemployment bene…ts
bt ), net of a hold-up term,

t t,

accruing to workers and extracted from the …rm on account

of the latter’s costly participation in the labor market, and the …rm’s implicit future cost
savings

mt t

when having already hired someone.

is a parameter indicating the strength of

worker’s bargaining power. Finally, production is assumed to be linear in employment, but
subject to both permanent and temporary productivity shocks, APt and ATt , respectively:
Yt = At Nt = APt ATt Nt :

(15)

The permanent shock is a pure random walk, while the temporary shock is serially correlated
but stationary.
Figure 5 depicts the theoretical impulse response functions of the unemployment and
vacancy rate to each of the shocks. We can categorize the shocks in two groups, namely
into shocks that move unemployment and vacancies in the same direction, and those that
imply opposite movements of these variables. This classi…cation underlies the identi…cation
by sign restrictions that we use later on. Both productivity shocks increase vacancies on
impact and lower unemployment over the course of the adjustment period. The e¤ect of the
temporary shock is much more pronounced since it is calibrated at a much higher level of
persistence than the productivity growth rate shock. Persistent productivity shocks increase
vacancy posting because they raise the expected value of a …lled position. As more vacancies
get posted, new employment relationships are established and the unemployment rate falls.
We note that permanent shocks have a temporary e¤ect on the labor market because they
tilt the expected pro…t pro…le in a manner similar to temporary shocks. However, they are
identi…ed by their long-run e¤ect on output, which by de…nition no other shock can muster.
Shocks to match e¢ ciency, vacancy posting costs and unemployment bene…ts lead to
negative comovement between unemployment and bene…ts. Increases in match e¢ ciency
and decreases in the vacancy costs both lower e¤ective vacancy creation cost

mt t

ceteris

paribus and thereby stimulate initial vacancy creation. These vacancies then lead to lower
unemployment over time. In the case of
since the hold-up term

t t

there is additional feedback from wage setting

can rise or fall. Similarly, increases in match e¢ ciency have an

additional e¤ect via the matching function as the higher level of vacancies is now turned
12

into even more new hires, so that employment rises. Movements in bene…ts also produce
negative comovements between the key labor market variables, but the channel is via wage
setting. Higher bene…ts increase the outside option of the worker in bargaining which leads
to higher wages. This reduces the expected pro…t stream to the …rm and fewer vacancy
postings and higher unemployment.
On the other hand, a persistent increase in the separation rate drives both unemployment
and vacancy postings higher. There is an immediate e¤ect on unemployment, which ceteris
paribus lowers labor market tightness, thereby reducing e¤ective vacancy posting cost. In
isolation, this e¤ect stimulates vacancy creation. At the same time, persistent increases
in separations reduce expected pro…t streams from …lled positions which has a dampening
e¤ect on desired vacancies. This is balanced, however, by persistent declines in tightness
because of increased separations. The resulting overall e¤ect is that …rms take advantage
of the larger pool of potential hires and increase vacancy postings to return to the previous
long-run level over time.

5.2

Disentangling Permanent and Transitory Shocks

We now describe how we implement identi…cation of a single permanent shock and two transitory shocks in our time-varying parameter VAR model, based on the theoretical insights
derived in the previous section. The permanent shock is identi…ed from a long-run restriction as originally proposed by Blanchard and Quah (1989). We label a shock as permanent
if it a¤ects only GDP in the long run, but not the labor market variables. The short- and
medium-run e¤ects on all variables is left unrestricted. In terms of the simple model, the
identi…ed permanent shock is consistent with the permanent productivity shock APt which
underlies the stochastic trend in output. We follow the procedure proposed by Galí and
Gambetti (2009) for imposing long-run restrictions within a time-varying parameter VAR
model.
Let

= Pt Dt Pt0 be the eigenvalue-eigenvector decomposition of the VAR’s time-varying

covariance matrix

in each time period and for each draw from the ergodic distribution.

We compute a local approximation to the matrix of the cumulative impulse-response functions (IRFs) to the VAR’s structural shocks as:
Ct;1 = [IN
|

B1;t

:::
{z

Bp;t ]

1
A0;t ;
}

(16)

where IN is the N N identity matrix. The matrix of the cumulative impulse-response functions is then rotated via an appropriate Householder matrix H in order to introduce zeros in
13

the …rst row of Ct;1 , which corresponds to GDP, except for the (1,1) entry. Consequently,
the …rst row of the cumulative impulse-response functions,
P
Ct;1
= Ct;1 H = C0 A0;t H = C0 AP0;t ;

(17)

is given by [x 0 0], with x being a non-zero entry. By de…nition, the …rst shock identi…ed
by AP0;t is the only one exerting a long-run impact on the level of GDP. We therefore label
it the permanent output shock.

5.3

Identifying the Transitory Shocks Based on Sign Restrictions

We identify the two transitory shocks by assuming that they induce a di¤erent impact
pattern on vacancies and the unemployment rate. Our theoretical discussion of the search
and matching model has shown that a host of shocks, e.g., temporary productivity, vacancy
cost, or match e¢ ciency shocks, imply negative comovement for the two variables, while
separation rate shocks increase vacancies and unemployment on impact. We transfer these
insights to the structural VAR identi…cation scheme.
Let ut

[uPt ; uTt 1 ; uTt 2 ]0 be the vector of the structural shocks in the VAR: uPt is the

permanent output shock, uTt 1 and uTt 2 are the two transitory shocks; let ut = A0;t1 t , with A0;t
being the VAR’s structural impact matrix. Our sign restriction approach postulates that uTt 1
induces the opposite sign on vacancies and the unemployment rate contemporaneously, while
uTt 2 induces an impact response of the same sign. We compute the time-varying structural
impact matrix A0;t by combining the methodology proposed by Rubio-Ramirez et al. (2005)
for imposing sign restrictions, and the procedure proposed by Galí and Gambetti (2009) for
imposing long-run restrictions in time-varying parameter VARs.
= Pt Dt Pt0 be the eigenvalue-eigenvector decomposition of the VAR’s time-varying
1
covariance matrix t , and let A~0;t Pt D 2 . We draw an N N matrix K from a standardLet

normal distribution and compute the QR decomposition of K; that is, we …nd matrices Q
and R such that K = Q R. A proposal estimate of the time-varying structural impact matrix
can then be computed as A0;t = A~0;t Q0 . We then compute the local approximation to the
matrix of the cumulative IRFs to the VAR’s structural shocks, Ct;1 , from (16). In order
to introduce zeros in the …rst row of Ct;1 , we rotate the matrix of the cumulative IRFs via
an appropriate Householder matrix H. The …rst row of the matrix of the cumulative IRFs,
P
Ct;1
as in (17), is given by [x 0 0]. If the resulting structural impact matrix A0;t = A0;t H

satis…es the sign restrictions, we store it; otherwise it is discarded. We then repeat the
procedure until we obtain an impact matrix that satis…es both the sign restrictions and the
long-run restriction at the same time.
14

Structural Evidence

Our identi…cation strategy discussed in section 5 allows us distinguish between one permanent and two transitory shocks. The permanent shock is identi…ed as having a long-run
e¤ect on GDP, while the transitory shocks are identi…ed from sign restrictions derived from
a simple search and matching model. A side product of our strategy is that we can identify
the natural rate of output as its permanent component. Figure 6 shows real GDP in logs
together with the median of the posterior distribution of the estimated permanent component and the 68% coverage region. We also report the corresponding transitory component
together with the output gap estimate from the CBO.
Our estimate of the transitory component is most of the time quite close to the CBO
output gap, which is produced from a production function approach to potential output,
whereas our estimate is largely atheoretical. The main discrepancy between the two estimates is in the wake of the Great Recession, particularly the quarters following the collapse
of Lehman Brothers. Whereas the CBO estimate implies a dramatic output shortfall of
around 7.5% of potential output in the …rst half of 2009, our estimated gap is much less
at between 3-4% with little change since then. The reason behind our smaller estimate of
the current gap is a comparatively large role played by permanent output shocks in the
Great Recession. As the …rst panel shows, the time pro…le of the permanent component
of log real GDP is estimated to have been negatively a¤ected in a signi…cant way by the
Great Recession, with a downward shift in the trend path; that is, natural output is now
permanently lower. The question we now investigate is whether and to what extent these
trend shifts due to permanent output shocks seep into the Beveridge curve.

6.1

Impulse Response Functions

As a …rst pass, we report IRFs to unemployment and vacancies for each of the three shocks
in Figures 7-9. Because of the nature of the time-varying parameter VAR, there is not
a single IRF for each shock-variable combination.

We therefore represent the IRFs by

collecting the time-varying coe¢ cients on impact, two quarters ahead, one year ahead, and
…ve years ahead in individual graphs in order to track how the dynamic behavior of the labor
market variables changes over time. An IRF for a speci…c period can then be extracted by
following the impulse response coe¢ cient over the four panels. The IRFs are normalized
such that the long-run e¤ect is attained at a value of one, while transitory shocks eventually
return the responses to zero.
In Figure 7, an innovation to the permanent component of output raises GDP on impact
15

by one-half of the long-run e¤ect, which is obtained fairly quickly after around one year
in most periods. A permanent shock tends to raise the vacancy rate on impact, after
which it rises for a few quarters before falling to its long-run level. The unemployment rate
rises on impact, but then quickly settles around zero. The initial, seemingly counterfactual
response is reminiscent of the …nding by Galí (1999) that positive productivity shocks have
negative employment consequences, which in our model translates into an initial rise in
the unemployment rate. Furthermore, the behavior of the estimated impulse responses is
broadly consistent with the results from the calibrated theoretical model, both in terms of
direction and size of the responses. As we will see below, compared to the transitory shocks
the permanent productivity shock, which in the theoretical model takes the form of a growth
rate variation, exerts only a small e¤ect on unemployment and vacancy rates. Notably, the
coverage regions for both variables include zero at all horizons. Overall, the extent of time
variation in the IRFs appears small. It is more pronounced at shorter horizons than in the
long run.
We report the IRFs to the …rst transitory shock in Figure 8. This shock is identi…ed as
inducing an opposite response of the vacancy and the unemployment rate on impact. In the
theoretical model, this identi…ed empirical shock is associated with a transitory productivity
shock, variations in match e¢ ciency, hiring costs, or bene…t movements. The IRFs of all
three variables in the VAR are hump-shaped, with a peak response after one year. Moreover,
the amplitudes of the responses are much more pronounced than in the previous case. The
vacancy rate is back at its long-run level after 5 years, while there is much more persistence
in the unemployment rate and GDP. We also note that our simple theoretical framework
cannot replicate this degree of persistence.
The vacancy rate exhibits the highest degree of time variation. What stands out is
that its response is asymmetric over the business cycle, but only in the pre-1984 period.
During the recessions of the early and mid-1970s, and the deep recession of the early 1980s
culminating in the Volcker disin‡ation, the initial vacancy response declines (in absolute
value) over the course of the downturn before increasing in the recovery phase. That is, the
vacancy rate responds less elastically to the …rst transitory shock during downturns than
in expansions - which is not the case for the unemployment rate. This pattern is visible
at all horizons. Between the Volcker disin‡ation and shortly before the onset of the Great
Recession the impact response of the vacancy rate declines gradually from -1% to almost
-2% before rising again sharply during a recession.
The second transitory shock is identi…ed by imposing the same sign response on un-

employment and vacancies. In the context of the theoretical model, such a pattern is due
to movements in the separation rate. The IRFs in Figure 9 show that the vacancy rate
rises on impact, then reaches a peak four quarters out before returning gradually over the
long run. The unemployment rate follows the same pattern, while the shock induces a
large negative response of GDP. None of the responses exhibits much time variation, at
best there are slow-moving changes in the IRF-coe¢ cients towards less elastic responses.
Interestingly, the impact behavior of the vacancy rate declines over the course of the Great
Recession. We note, however, that the coverage regions are very wide and include zero for
the unemployment rate and GDP at all horizons.

6.2

Variance Decompositions

Figure 10 provides evidence on the relative importance of permanent and transitory shocks
for ‡uctuations in vacancies and the unemployment rate. We report the median of the posterior distributions of the respective fractions of innovation variance due to the permanent
shock and the associated coverage regions. For the vacancy rate, permanent shocks appear
to play a minor role, with a median estimate of between 10% and 20%. The median estimate
for the unemployment rate exhibits a greater degree of variation, oscillating between 10%
and 40%. Despite this large extent of time variation, it is di¢ cult to relate ‡uctuations in
the relative importance of permanent shocks to key macroeconomic events. Possible candidates are the period after the …rst oil shock, when the contribution of permanent shocks
increased temporarily, and the long expansion of the 1980s until the late 1990s, which was
temporarily punctured by the recession in 1991. Moreover, there is no consistent behavior
of the permanent shock contribution over the business cycle. Their importance rises both
in downturns and in upswings. On the other hand, this observation gives rise to the idea
that all business cycles, at least in the labor market, are di¤erent along this dimension.
We now turn to the relative contribution of the two transitory shocks identi…ed by sign
restrictions. The evidence is fairly clear-cut. Given the strongly negative unconditional relationship between vacancies and the unemployment rate, we would expect the contribution
of uTt 2 , that is, the shock that induces positive contemporaneous comovement between the
two variables, to be small. This is, in fact, borne out by the second column of the graph in
Figure 11. The median estimate of the fraction of innovation variance of the two series due
to uTt 2 is well below 20%. Correspondingly, the …rst transitory shock appears clearly to be
dominant for both variables. Based on the theoretical model, we can associate this shock
with either temporary productivity disturbances or with stochastic movements in hiring

costs, match e¢ ciency, or unemployment bene…ts. Given the parsimonious nature of both
the theoretical and empirical model, however, we cannot further disentangle this.

6.3

Structural Shocks and Beveridge Curve Shifts

We now turn to one of the main results of the paper, namely the structural sources of time
variation in the Beveridge curve. We …rst discuss the relationship between the business cycle,
as identi…ed by the transitory component in GDP and measures of the Beveridge curve. We
then decompose the estimated gain and coherence of unemployment and vacancies into their
structural components based on the identi…cation scheme discussed above.
Figure 12 reports two key pieces of evidence on the cyclical behavior of the slope of the
Beveridge curve. The left panel shows the fraction of draws from the posterior distribution
for which the transitory component of output is positive. This is plotted against the fraction
of draws for which the average cross-spectral gain between vacancies and the unemployment
rate at the business-cycle frequencies is greater than one. The graph thus gives an indication
of how the slope of the Beveridge curve moves with aggregate activity over the business
cycle.
We can di¤erentiate two separate time periods. During the 1970s and the early 1980s,
that is, during the Great In‡ation, the slope of the Beveridge curve systematically comoves
contemporaneously with the state of the business cycle. It is comparatively larger (in
absolute value) during business-cycle upswings, and comparatively smaller during periods
of weak economic activity. Similarly, the Great Recession is characterized by very strong
comovement between the slope of the Beveridge curve and the transitory component of
output, but this time the slope slightly leads the business cycle.
On the other hand, in the long expansion period from 1982 to 2008, labelled the Great
Moderation that was only marred by two minor recessions, the slope of the Beveridge
curve comoves less clearly with the business cycle, a pattern that is especially apparent
during the 1990s. In the early and late part of this period the Beveridge curve appears
to lag the cycle. This is consistent with the notion of jobless recoveries after the two mild
recessions. Despite upticks in economic activity, the labor market did not recover quickly
after 1992 and, especially, after 2001. In the data, this manifests itself in a large gain
between unemployment and vacancies (see Figure 13). Moreover, this is also consistent
with the changing impulse response patterns to structural shocks discussed above. In a
sense, the outlier is the Great Recession, which resembles more the recessions of the Great
In‡ation rather than those of the Great Moderation.

The second panel reports additional evidence on the extent of cyclicality of the slope of
the Beveridge curve. It shows the distribution of the slope coe¢ cient in the LAD (Least
Absolute Deviations) regression of the cross-spectral gain on a constant and the transitory
component of output. Overall, the LAD coe¢ cient is greater than zero for 82.5% of the
draws from the posterior distribution, which points towards the pro-cyclicality of the slope
of the Beveridge curve.
Figure 13 shows how the two types of shocks shape the evolution of the Beveridge
curve. We plot the average gain and coherence between vacancies and the unemployment
rate at business-cycle frequencies over time together with the fraction of draws for which
the average gain is greater than one. The upper row of the panel reports the statistics
conditional on the permanent shock, the lower panel contains those conditional on the
two transitory shocks. Whereas the coherence conditional on the permanent shock does
not show much time variation, conditioning on transitory shocks reveals a pattern that is
broadly similar to the reduced-form representation. This suggests that the comparatively
greater coherence between the two series around the time of the Great In‡ation and of the
Volcker disin‡ation is mostly due to transitory shocks.
Time variation in the gain, on the other hand, appears to be due to both types of shocks.
Although the middle column suggests that the extent of statistical signi…cance of the ‡uctuations in the gain is similar, the …rst column shows a di¤erent magnitude. In particular,
‡uctuations in the gain conditional on permanent output shocks, which accounted for a
comparatively minor fraction of the innovation variance of the two series, is signi…cantly
wider than the corresponding ‡uctuations conditional on transitory shocks. Moreover, and
unsurprisingly in the light of the previously discussed evidence on the relative importance
of the two types of shocks, both the magnitude and the time-pro…le of the ‡uctuations of
the gain conditional on transitory shocks are very close to the reduced-form evidence.

6.4

Interpreting Changes in the Beveridge Curve Based on an Estimated
DSGE Model

One of our key contributions is to have demonstrated the presence and the extent of nonlinearity in the Beveridge-curve relationship over time in U.S. data. We did so in a structural
VAR, where the identi…cation restrictions were derived from the behavior of a simple search
and matching model for the labor market. Nevertheless, VARs are by their very nature
largely atheoretical, in the sense that they represent the reduced form of a potentially much
richer underlying dynamic stochastic general equilibrium (DSGE) model. The best that we

can do is to identify structural innovations, but they do not necessarily reveal much about
the structure of the theoretical model. However, researchers may want to go further than
this. One particular point of interest is the source of the non-linearity in the data. We now
make some forays in this direction.
Following Fernández-Villaverde and Rubio-Ramirez (2007) we assume that all parameters in the DSGE model in Section 4 are …rst-order autoregressive processes. The set of
parameters thus includes the original primitive parameters, their …rst-order autocorrelation
coe¢ cients, and their innovation variances. We then estimate the model using Bayesian
methods.4 We draw from the posterior distribution for each individual parameter and
compute the associated gain of the unemployment rate onto vacancies at business cycle frequencies. Figure 14 contains a graph that shows, for parameter intervals around the modal
estimates generated by the Random Walk Metropolis algorithm, the average gain of the
unemployment rate onto vacancies at the business-cycle frequencies, as a function of each
individual parameter.
The parameters with the largest impact on the gain, as a measure of the slope of the
Beveridge curve, are the separation rate , the match e¢ ciency m, and the match elasticity
. The estimated gain is independent of the vacancy creation cost

and the bargaining share

. It is also largely inelastic to variations in the parameters’ autocorrelation coe¢ cients,
with the exception of the match e¢ ciency and separation rate parameters and the serial
correlation of the permanent productivity shock. The innovation variances do not a¤ect the
gain, the exception being the separation rate. These results are in line with the observation
of Lubik (2012) that the key driver of shifts in the Beveridge curve are variations in the
matching function parameters. While productivity shocks can generate movements along
the Beveridge curve, movements of the Beveridge curve have to come through changes in
the law of motion for employment. In contrast to his …ndings, our exercise puts additional
weight on variations in the separation rate. As Figure 14 suggests, variations in the level,
the persistence and the volatility of the separation rate are the main factor underlying the
non-linearity and the time variation in the Beveridge curve relationship. We regard this as
a crucial starting point for future research.
4

The speci…cation of the prior follows Lubik (2012). Posterior estimates and additional results are
available from the authors upon request.

Conclusion

We have used a Bayesian time-varying parameter structural VAR with stochastic volatility
to investigate changes in both the reduced-form relationship between vacancies and the
unemployment rate, and in their relationship conditional on permanent and transitory output shocks, for the post-WWII United States. Evidence points towards both similarities
and di¤erences between the Great Recession and the Volcker disin‡ation, and a widespread
time-variation along two key dimensions. First, the slope of the Beveridge curve, as captured by the average cross-spectral gain between vacancies and the unemployment rate at
business-cycle frequencies, exhibits a large extent of variation since the second half of the
1960s. Moreover, it is broadly pro-cyclical, with the gain being positively correlated with
the transitory component of output. The evolution of the slope of the Beveridge curve
during the Great Recession appears to be very similar to its evolution during the Volcker
recession in terms of its magnitude and its time pro…le. Second, both the Great In‡ation
episode, and the subsequent Volcker disin‡ation, are characterized by a signi…cantly larger
(in absolute value) negative correlation between the reduced-form innovations to vacancies
and the unemployment rate than the rest of the sample period. Those years also exhibit
a greater cross-spectral coherence between the two series at the business-cycle frequencies,
thus pointing towards them being driven, to a larger extent than the rest of the sample, by
common shocks.

Appendix
A

The Data

The series for real GDP (‘GDPC96, Real Gross Domestic Product, 3 Decimal, Seasonally
Adjusted Annual Rate, Quarterly, Billions of Chained 2005 Dollars’) is from the U.S. Department of Commerce: Bureau of Economic Analysis. It is collected at quarterly frequency
and seasonally adjusted. A quarterly seasonally adjusted series for the unemployment rate
has been computed by converting the series UNRATE (‘Civilian Unemployment Rate, Seasonally Adjusted, Monthly, Percent, Persons 16 years of age and older’) from the U.S.
Department of Labor: Bureau of Labor Statistics to quarterly frequency (by taking averages within the quarter). A monthly seasonally adjusted series for the vacancy rate has
been computed as the ratio between the ‘Help Wanted Index’(HWI) and the civilian labor
force. The HWI index is from the Conference Board up until 1994Q4, and from Barnichon
(2010) after that. The labor force series is from the U.S. Department of Labor: Bureau
of Labor Statistics (‘CLF16OV, Civilian Labor Force, Persons 16 years of age and older,
Seasonally Adjusted, Monthly, Thousands of Persons’). The monthly seasonally adjusted
series for the vacancy rate has been converted to the quarterly frequency by taking averages
within the quarter.

Deconvoluting the Probability Density Function of ^

This appendix describes the procedure we use in section 2 to deconvolute the probability
density function of ^ . We consider the construction of a (1
)% con…dence interval for
L
U
^, [^
;^
]. We assume for simplicity that j and ^ can take any value over [0; 1).
(1

)

Given the duality between hypothesis testing and the construction of con…dence intervals,
the (1
)% con…dence set for ^ comprises all the values of j that cannot be rejected

based on a two-sided test at the % level. Given that an increase in j automatically shifts
^U
^ j conditional on j upwards, ^ L
the probability density function (pdf) of L
(1
) and (1
)
are therefore such that:
^j > L
^j
P L

L
= ^ (1

)

= =2;

(B1)

^j < L
^j
P L

U
= ^ (1

)

= =2:

(B2)

and
be the pdf and, respectively, the cumulative pdf of ^ , de…ned over the
L
U
domain of j . The fact that [ ^ (1 ) ; ^ (1 ) ] is a (1
)% con…dence interval automatically
Let

^( j)

and

^( j)

implies that (1

)% of the probability mass of
L
This, in turn, implies that ^ ( ^ (1 ) ) = =2 and
holds for any 0 <

^( j)
^U
^ ( (1

L
lies between ^ (1
))

)

U
and ^ (1

=2. Given that this

< 1, we therefore have that:
^( j)

^j > L
^j
=P L

(B3)

^ and on the simulated distributions of the L
^ j ’s
Based on the exp-Wald test statistic, L,
conditional on the j ’s in , we thus obtain an estimate of the cumulative pdf of ^ over the
, ^ ^ ( j ). Finally, we …t a logistic function to ^ ^ ( j ) via nonlinear least squares and
we compute the implied estimate of ^ ( j ), ^ ^ ( j ), whereby we scale its elements so that
grid

they sum to one.

Details of the Markov-Chain Monte Carlo Procedure

We estimate (4)-(12) using Bayesian methods. The next two subsections describe our choices
for the priors, and the Markov-Chain Monte Carlo algorithm we use to simulate the posterior distribution of the hyperparameters and the states conditional on the data. The
third section discusses how we check for convergence of the Markov chain to the ergodic
distribution.

C.1

Priors

The prior distributions for the initial values of the states,

and h0 , which we postulate to

be normally distributed, are assumed to be independent both from each other and from the
distribution of the hyperparameters. In order to calibrate the prior distributions for

and

h0 we estimate a time-invariant version of (4) based on the …rst 15 years of data. We set:
0

h
i
N ^OLS ; 4 V^ (^OLS ) ;

(C1)

where V^ (^OLS ) is the estimated asymptotic variance of ^OLS . As for h0 , we proceed as
follows. Let ^ OLS be the estimated covariance matrix of t from the time-invariant VAR,
and let C be its lower-triangular Cholesky factor, CC 0 = ^ OLS . We set:
ln h0
where

N (ln

0 ; 10

IN );

(C2)

is a vector collecting the logarithms of the squared elements on the diagonal of C.

As stressed by Cogley and Sargent (2002), this prior is weakly informative for h0 .

Turning to the hyperparameters, we postulate independence between the parameters
corresponding to the two matrices Q and A for convenience. Further, we make the following standard assumptions. The matrix Q is postulated to follow an inverted Wishart
distribution:
Q

IW Q

; T0 ;

(C3)

with prior degrees of freedom T0 and scale matrix T0 Q. In order to minimize the impact
of the prior, we set T0 equal to the minimum value allowed, the length of t plus one. As
^ OLS , setting = 1:0 10 4 , as in Cogley and Sargent
for Q, we calibrate it as Q =
(2002). This is a comparatively conservative prior in the sense of allowing little randomwalk drift. We note, however, that it is smaller than the median-unbiased estimates of
the extent of random-walk drift discussed in section 2, ranging between 0.0235 and 0.0327
for the equation for the vacancy rate, and between 0.0122 and 0.0153 for the equation for
the unemployment rate. As for

, we postulate it to be normally distributed with a large

variance:
f ( ) = N (0; 10000 IN (N

1)=2 ):

(C4)

Finally, we follow Cogley and Sargent (2002, 2005) and postulate an inverse-Gamma distribution for

2
i

V ar(

i;t )

for the variances of the stochastic volatility innovations:
2
i

C.2

10 4 1
;
2 2

(C5)

Simulating the Posterior Distribution

We simulate the posterior distribution of the hyperparameters and the states conditional on
the data using the following MCMC algorithm (see Cogley and Sargent, 2002). xt denotes
the entire history of the vector x up to time t, that is, xt

[x01 , x02 ,

, x0t ]0 , while T is the

sample length.
1. Drawing the elements of

Conditional on Y T , , and H T , the observation equation

(4) is linear with Gaussian innovations and a known covariance matrix. Following
Carter and Kohn (1994), the density p(
p(

Conditional on

jY ; ; H ) = p(

T jY

jY T ; ; H T ) can be factored as:
T

; ;H )

TY1
t=1

p( t j

t+1 ; Y

; ; H T ):

(C6)

and H T , the standard Kalman …lter recursions determine the …rst

element on the right hand side of (C6), p(
24

T jY

; HT ) = N (

T ; PT ),

with PT being

the precision matrix of

produced by the Kalman …lter. The remaining elements in

the factorization can then be computed via the backward recursion algorithm found
in Cogley and Sargent (2005). Given the conditional normality of
tjt+1

tjt

1
+ Ptjt Pt+1jt
(

we have:

t) ;

t+1

(C7)

and
1
Ptjt Pt+1jt
Ptjt ;

Ptjt+1 = Ptjt
which provides, for each t from T
; HT ) = N (

(C8)

1 to 1, the remaining elements in (4), p( t j

tjt+1 ; Ptjt+1 ).

t+1 ; Y

Speci…cally, the backward recursion starts with a draw
from N ( T ; PT ), call it ~T . Conditional on ~T , (C7)-(C8) give us T 1jT and PT 1jT ,
thus allowing us to draw ~T 1 from N ( T 1jT ; PT 1jT ), and so on until t = 1.
T

2. Drawing the elements of Ht : Conditional on Y T ,
novations ut

A(Yt

, and

, the orthogonalized in-

Xt t ), with V ar(ut ) = Ht , are observable. Following Cogley

and Sargent (2002), we then sample the hi;t ’s by applying the univariate algorithm of
Jacquier et al. (1994) element by element.
3. Drawing the hyperparameters: Conditional on Y T ,
to

, H T , and

, the innovations

and to the hi;t ’s are observable, which allows us to draw the hyperparameters,

namely the elements of Q and the
4. Drawing the elements of

2,
i

from their respective distributions.

: Finally, conditional on Y T and

, the t ’s are observable.

They satisfy:
A

= ut ;

(C9)

with the ut being a vector of orthogonalized residuals with known time-variying variance Ht . Following Primiceri (2005), we interpret (C9) as a system of unrelated regressions. The …rst equation in the system is given by

1;t

u1;t , while the following

equations can be expressed as transformed regressions:
1

h2;t2

2;t

2;1

h2;t2

3;t

3;1

h3;t2

1;t

+ h2;t2 u2;t

(C10)

1;t

3;2

h3;t2

2;t

+ h3;t2 u3;t ;

where the residuals are independently standard normally distributed. Assuming normal priors for each equation’s regression coe¢ cients the posterior is also normal and
can be computed as in Cogley and Sargent (2005).
25

Summing up, the MCMC algorithm simulates the posterior distribution of the states and
the hyperparameters, conditional on the data, by iterating on (1)-(4). In what follows, we use
a burn-in period of 50,000 iterations to converge to the ergodic distribution. After that, we
run 10,000 more iterations sampling every 10th draw in order to reduce the autocorrelation
across draws.

A Simple Search and Matching Model of the Labor Market

The model speci…cation follows Lubik (2012). Time is discrete and the time period is a
quarter. The model economy is populated by a continuum of identical …rms that employ
workers, each of whom inelastically supplies one unit of labor. Output Yt of a typical …rm
is linear in employment Nt :
Yt = At Nt :

(D1)

At is a stochastic aggregate productivity process. It is composed of a permanent productivity shock, APt , which follows a random walk, and a transitory productivity shock, ATt ,
which is an AR(1)-process. Speci…cally, we assume that At = APt ATt .
The labor market matching process combines unemployed job seekers Ut with job openings (vacancies) Vt . This can be represented by a constant returns matching function,
Mt = mt Ut Vt1

, where mt is stochastic match e¢ ciency, and 0 <

< 1 is the match

elasticity. Unemployment is de…ned as those workers who are not currently employed:
Ut = 1

Nt ;

(D2)

where the labor force is normalized to one. In‡ows to unemployment arise from job destruction at rate 0 <

< 1, which can vary over time. The dynamics of employment are

thus governed by the following relationship:
Nt = (1

+ mt

1
1 Ut 1 V t 1

(D3)

This is a stock-‡ow identity that relates the stock of employed workers Nt to the ‡ow of
new hires Mt = mt Ut Vt1

into employment. The timing assumption is such that once a

worker is matched with a …rm, the labor market closes. This implies that if a newly hired
worker and a …rm separate, the worker cannot re-enter the pool of searchers immediately
and has to wait one period before searching again.

The matching function can be used to de…ne the job …nding rate, i.e., the probability
that a worker will be matched with a …rm:
p( t ) =

Mt
= mt
Ut

1
t

;

(D4)

and the job matching rate, i.e., the probability that a …rm is matched with a worker:
q( t ) =
where

Mt
= mt
Vt

;

(D5)

= Vt =Ut is labor market tightness. From the perspective of an individual …rm,

the aggregate match probability q( t ) is exogenous and una¤ected by individual decisions.
Hence, for individual …rms new hires are linear in the number of vacancies posted: Mt =
q( t )Vt .
A …rm chooses the optimal number of vacancies Vt to be posted and its employment
level Nt by maximizing the intertemporal pro…t function:
E0

1
X

[At Nt

Wt Nt

t Vt ] ;

(D6)

t=0

subject to the employment accumulation equation (D3). Pro…ts are discounted at rate
0<

< 1. Wages paid to the workers are Wt , while

> 0 is a …rm’s time-varying cost of

opening a vacancy. The …rst-order conditions are:

where

Nt :

= At

Wt + Et (1

Vt :

= q( t )Et (1

t+1 ) t+1

;

(D7)

;

(D8)

is the multiplier on the employment equation.

Combining these two …rst-order conditions results in the job creation condition (JCC):
t

q( t )

= Et (1

t+1 )

At+1

Wt+1 +

t+1

t+1 )

(D9)

This captures the trade-o¤ faced by the …rm: the marginal e¤ective cost of posting a
vacancy,

q( t ) ,
t

that is, the per-vacancy cost

adjusted for the probability that the position

is …lled, is weighed against the discounted bene…t from the match. The latter consists of
the surplus generated by the production process net of wage payments to the workers, plus
the bene…t of not having to post a vacancy again in the next period.
In order to close the model, we assume in line with the existing literature that wages
are determined based on the Nash bargaining solution: surpluses accruing to the matched

parties are split according to a rule that maximizes their weighted average. Denoting the
workers’weight in the bargaining process as
Wt

Ut =

2 [0; 1], this implies the sharing rule:
(Jt

Vt ) ;

(D10)

where Wt is the asset value of employment, Ut is the value of being unemployed, Jt is the

value of the marginal worker to the …rm, and Vt is the value of a vacant job. By free entry,
Vt is assumed to be driven to zero.

The value of employment to a worker is described by the following Bellman equation:
Wt = Wt + Et [(1

t+1 )Wt+1

t+1 Ut+1 ]:

(D11)

Workers receive the wage Wt , and transition into unemployment next period with probability

t+1 .

The value of searching for a job, when the worker is currently unemployed,

is:
Ut = bt + Et [pt (1

t+1 )Wt+1

+ (1

pt (1

t+1 ))Ut+1 ]:

(D12)

An unemployed searcher receives stochastic bene…ts bt and transitions into employment with
probability pt (1

t+1 ).

Recall that the job …nding rate pt is de…ned as p( t ) = M (Vt ; Ut )=Ut

which is decreasing in tightness

It is adjusted for the probability that a completed match

gets dissolved before production begins next period. The marginal value of a worker Jt is
equivalent to the multiplier on the employment equation, Jt =

so that the respective

…rst-order condition de…nes the Bellman-equation for the value of a job. Substituting the
asset equations into the sharing rule (D10) results in the wage equation:
Wt = (At +

t t)

+ (1

)bt :

(D13)

Wage payments are a weighted average of the worker’s marginal product At , which the
worker can appropriate at a fraction , and the outside option bt , of which the …rm obtains
the portion (1

). Moreover, the presence of …xed vacancy posting costs leads to a hold-up

problem where the worker extracts an additional

t t

from the …rm.

Finally, we can substitute the wage equation (D13) into (D9) to derive an alternative
representation of the job creation condition:
t

= Et (1

t+1 )

) (At+1

bt )

t t+1

mt+1

t+1

(D14)

References
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29

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Table 1 Results based on the Stock-Watson TVP-MUB methodology: exp- and sup-Wald test statistics, simulated p-values,
and median-unbiased estimates of
Equation for:

exp-Wald
(p-value)

vacancy rate
unemployment rate

9.40
4.97

vacancy rate
unemployment rate

7.65
4.68

sup-Wald
^
(p-value)
Newey and West (1987) correction
(0.0053) 0.0286 28.91 (0.0028)
0.0327
(0.1661) 0.0153 16.17 (0.1770)
0.0153
Andrews (1991) correction
(0.0195) 0.0235 25.40 (0.0086)
0.0286
(0.1987) 0.0133 14.61 (0.2594)
0.0122

Figure 1 The unemployment rate and the vacancy rate

Figure 2 Deconvoluted PDFs of λ

Figure 3 Correlation coefficient of reduced-form innovations to vacancies and the unemployment rate,
and ratio between the standard deviations of reduced-form innovations to the two variables

Figure 4 Average gain of the unemployment rate onto vacancies, and average coherence between
vacancies and the unemployment rate, at the business-cycle frequencies

Figure 5 Impulse response functions of the calibrated search and matching model

Figure 6 Logarithm of real GDP and estimated permanent component, estimated transitory component
of log real GDP, and Congressional Budget Office estimate of the output gap

Figure 7 Impulse response functions to a permanent output shock

Figure 8 Impulse response functions to the first transitory shock

Figure 9 Impulse response functions to the second transitory shock

Figure 10 Fractions of innovation variance due to the permanent output shock

Figure 11 Fractions of innovation variance due to the two transitory shocks
identified from sign restrictions
42

Figure 12 Evidence on the pro-cyclicality of the Beveridge curve

Figure 13 Business-cycle frequencies: average gain and coherence between vacancies and the
unemployment rate conditional on the permanent and the transitory output shocks

Figure 14 Average gain of the unemployment rate onto vacancies at the business-cycle frequencies, as
a function of individual parameters of the DSGE model (for parameter intervals around the
modal estimates generated by the Random Walk Metropolis)
45

Full text of Working Papers (Federal Reserve Bank of Richmond) : The Time-Varying Beveridge Curve, Working Paper 13-12

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