Solving the DMP Model Accurately
Nicolas Petrosky-Nadeau∗
Carnegie Mellon University
Lu Zhang†
The Ohio State University
and NBER
December 2013‡
Abstract
An accurate global algorithm is crucial for quantifying the dynamics of the DiamondMortensen-Pissarides model. Loglinearization understates the mean and volatility of
unemployment, overstates the unemployment-vacancy correlation, and ignores impulse
responses that are an order of magnitude larger in recessions than in booms. Although
improving on loglinearization, the second-order perturbation in logs also induces large
approximation errors. We demonstrate these insights within the context of Hagedorn
and Manovskii (2008). Our quantitative results highlight the extreme importance of
accurately accounting for nonlinear dynamics in quantitative macroeconomic studies.
JEL Classification: E24, E32, J63, J64.
Keywords: Search and matching, the unemployment volatility puzzle, global approximation methods, projection, loglinearization, nonlinear impulse response functions
∗
Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh PA 15213. Tel:
(412) 268-4198 and e-mail: npn@cmu.edu.
†
Fisher College of Business, The Ohio State University, 760A Fisher Hall, 2100 Neil Avenue, Columbus
OH 43210; and NBER. Tel: (614) 292-8644 and e-mail: zhanglu@fisher.osu.edu.
‡
We are grateful to Hang Bai, Andrew Chen, Daniele Coen-Pirani, Steven Davis, Wouter Den Haan, Paul
Evans, Lars-Alexander Kuehn, Dale Mortensen, Paulina Restrepo-Echavarria, Etienne Wasmer, Randall
Wright, and other seminar participants at The Ohio State University and the 2013 North American Summer
Meeting of the Econometric Society for helpful comments. Nicolas Petrosky-Nadeau thanks Stanford Institute for Economic Policy Research and the Hoover Institution at Stanford University for their hospitality.
The Matlab programs are available from the authors upon request. All remaining errors are our own.
1
Introduction
Since the foundational work of Diamond (1982), Mortensen (1982), and Pissarides (1985), the
search model of unemployment has become the dominant framework for studying the labor
market. Shimer (2005) argues that the unemployment volatility in the baseline DiamondMortensen-Pissarides (DMP) model is too low relative to that in the data. A large subsequent
literature has developed to address this quantitative puzzle.1 Given its quantitative focus, an
accurate approximation to the model’s equilibrium is of paramount importance. However,
this literature has mostly relied on loglinearization to solve the model.
Our key message is that an accurate global approximation is crucial for characterizing
the quantitative properties of the DMP model. We demonstrate the importance of nonlinearity within the context of Hagedorn and Manovskii (2008). They argue that under their
calibration, the DMP model produces realistic labor market volatilities. The volatility of
labor market tightness is 0.292 in the model, which is not far from 0.259 in the data. The
unemployment volatility is 0.145 in the model, which is close to 0.125 in the data.
However, using an accurate projection algorithm and the Hagedorn-Manovskii (2008)
calibration, we calculate the unemployment volatility to be 0.258, which is about twice
as large as 0.125 in the data. The unemployment-vacancy correlation is also lower in
magnitude, −0.567, versus −0.724 from loglinearization. Finally, the stochastic mean of the
unemployment rate using projection, 6.17%, is almost one percentage point higher than its
deterministic steady state rate, 5.28%. These results cast doubt on the validity of calibration
1
Hall (2005) uses sticky wages and Mortensen and Nagypál (2007) and Pissarides (2009) use fixed
matching costs to help explain the volatility puzzle. Gertler and Trigari (2009) propose a staggered wage
bargaining framework to generate wage rigidity. Hagedorn and Manovskii (2008) show that a calibration
with small profits and a low bargaining power for workers can produce realistic volatilities. Hall and Milgrom
(2008) replace the Nash bargaining wage with a credible bargaining wage. Finally, Petrosky-Nadeau and
Wasmer (2013) use financial frictions to increase labor market volatilities.
1
that relies exclusively on steady state relations. More generally, these results question
loglinearization as a solution method for the DMP model (despite its simplicity and speed).2
What explains the differences between loglinearization and projection? The crux is that
the unemployment dynamics in the DMP model with the Hagedorn-Manovskii calibration are
highly nonlinear. In recessions unemployment rises rapidly, whereas in booms it declines only
gradually. The projection algorithm fully captures the nonlinearity. In contrast, by focusing
only on local dynamics around the deterministic steady state, loglinearization ignores completely the unemployment spikes in recessions. Because of this omission, loglinearization underestimates the volatility of unemployment but overestimates its correlation with vacancies.
The two algorithms also differ dramatically in impulse responses. With projection, impulse responses are stronger in recessions than in booms. In response to a negative onestandard-deviation shock to log labor productivity, the unemployment rate increases only
by 0.08% if the economy starts at the 95 percentile of the model’s stationary distribution
of employment and productivity (with an unemployment rate of 4%). In contrast, the unemployment rate shoots up by 0.83% (an order of magnitude larger) if the economy instead
starts at the 5 percentile of the bivariate distribution (with an unemployment rate of 10.5%).
In contrast, such nonlinearity in impulse responses is entirely missed by loglinearization.
Since Merz (1995) and Andolfatto (1996), the DMP model has been embedded into
the dynamic stochastic general equilibrium framework to study business cycles. While the
Hagedorn-Manovskii calibration calls for a quarterly persistence of 0.88 for the log productivity, business cycle studies typically calibrate the persistence to be 0.95 as in, for example,
Cooley and Prescott (1995). We show that the higher persistence induces even larger differ2
It should be noted that Hagedorn and Manovskii (2008) use a global approximation algorithm, not loglinearization, to solve their model. However, due to their Markov-chain approximation to the highly persistent
log labor productivity, their results are quantitatively similar to those from loglinearization (see Section 5.1).
2
ences between loglinearization and projection (even with a lower unconditional volatility for
the log productivity). The unemployment volatility is only 0.108 from loglinearization but
0.285 from the projection algorithm. Intuitively, because job creation is forward looking, a
low realization of productivity reduces job creation more if it is expected to last longer. As
such, the high persistence strengthens nonlinear unemployment dynamics, thereby exacerbating the deviation of loglinearization from the projection algorithm.
Although improving on loglinearization, the second-order perturbation in logs also fails to
deliver accurate results. The unemployment volatility is 0.164, which, although higher than
0.133 from loglinearization, is still substantially lower than 0.251 from projection. Similarly,
the unemployment-vacancy correlation is −0.791, although lower in magnitude than −0.848
from loglinearization, is still substantially higher than −0.564 from projection.3
We also perform accuracy tests on the different numerical algorithms. The results show
that the projection algorithm provides an extremely accurate solution, whereas loglinearization and the second-order perturbation in logs induce large approximation errors. In particular, dynamic Euler equation errors in consumption are in the order of 10−14 for the projection
algorithm, but are around 70% for the other two algorithms.
Our work suggests that many prior conclusions in the macro labor literature might need
to be reexamined with an accurate global solution. Although the Hagedorn-Manovskii (2008)
calibration seems somewhat extreme in its implications on the magnitude of profits, it provides a parsimonious setting with realistic labor market volatilities in loglinear approxi3
We have also experimented with the third-order perturbation in logs. However, as discussed in Kim,
Kim, Schaumburg, and Sims (2008) and Den Haan and Wind (2012), high-order perturbations often
produce explosive simulated sample paths. In our practice, after confirming this phenomenon in the DMP
model, we opt not to pursue the third- or higher-order perturbations. As such, although casting doubt on
lower-order perturbations, our results should not be interpreted as a critique on high-order perturbations.
On the contrary, our results indicate that the higher order terms are in fact quantitatively important.
3
mations. As such, their setup offers a natural starting point for our analysis. Many other
economic mechanisms have been proposed to generate realistic labor market volatilities. Our
quantitative insights should apply as a consequence of nonlinear unemployment dynamics.
However, we emphasize that our results should not be interpreted as a critique on Hagedorn
and Manovskii. On the contrary, we show that once their model is solved accurately, a less
extreme calibration is sufficient to match the unemployment volatility (albeit with a lower
volatility for the labor market tightness than that in the data).
More broadly, our insights apply to a wide body of academic and policy research as
the DMP model has been adopted throughout macroeconomics.4 Characterizing dynamics
accurately is crucial for deriving quantitative properties. The capability to perform policy
counterfactuals requires a macroeconomic model to be estimated (e.g., Smets and Wouter
(2003, 2007)). However, inferences can be biased if an approximate equilibrium ignores
nonlinear dynamics (e.g., Fernández-Villaverde and Rubio-Ramı́rez (2007)).
The projection method for solving dynamic equilibrium models is proposed by Judd
(1992). Christiano and Fisher (2000) show how to incorporate occasionally binding constraints into a projection algorithm. Den Haan (2010a) compares perturbations with projection in solving the incomplete markets model. Finally, we adapt the projection algorithm
in Petrosky-Nadeau, Zhang, and Kuehn (2013), who study asset prices in the DMP model.
We instead examine how different algorithms affect labor market moments.
The rest of the paper is organized as follows. Section 2 sets up the model. Section 3
describes the projection algorithm. Section 4 shows how the labor market moments with
loglinearization deviate quantitatively from with the projection algorithm. Section 5 recon4
Recent examples include Merz (1995) and Andolfatto (1996) in the real business cycle literature, Gertler
and Trigari (2009) in the New Keynesian literature, Blanchard and Gali (2010) on monetary policy, and
Monacelli, Perotti, and Trigari (2010) on fiscal policy.
4
ciles our global results with those in Hagedorn and Manovskii (2008). Section 6 reports the
results with the Cobb-Douglas matching function. Finally, Section 7 concludes.
2
The Model
We follow closely the setup in Hagedorn and Manovskii (2008, HM). There exist a representative household and a representative firm that uses labor as the single productive input.
Following Merz (1995), we use the representative family construct, which implies perfect
consumption insurance. The household has a continuum with a unit mass of members who
are, at any point in time, either employed or unemployed. The fractions of employed and
unemployed workers are representative of the population at large. The household pools
the income of all the members together before choosing per capita consumption and asset
holdings. The household is risk neutral with a time discount factor β.
The representative firm posts a number of job vacancies, Vt , to attract unemployed workers, Ut . Vacancies are filled via a constant returns to scale matching function, G(Ut , Vt ):
G(Ut , Vt ) =
Ut Vt
(Utι
+ Vtι )1/ι
,
(1)
in which ι > 0 is a constant parameter. This matching function, from Den Haan, Ramey,
and Watson (2000), implies that matching probabilities fall between zero and one.
Define θ t ≡ Vt /Ut as the vacancy-unemployment (V /U) ratio. The probability for an
unemployed worker to find a job per unit of time (the job finding rate), f (θt ), is:
ft = f (θt ) =
G(Ut , Vt )
1
=
1/ι .
Ut
1 + θ−ι
t
5
(2)
The probability for a vacancy to be filled per unit of time (the vacancy filling rate), q(θt ), is:
qt = q(θt ) =
G(Ut , Vt )
1
=
.
Vt
(1 + θιt )1/ι
(3)
An increase in the scarcity of unemployed workers relative to vacancies makes it harder to
fill a vacancy, q (θt ) < 0. As such, θt is labor market tightness from the firm’s perspective.
The firm takes aggregate labor productivity, Xt , as given. We specify xt ≡ log(Xt ) as:
xt+1 = ρxt + σt+1 ,
(4)
in which ρ ∈ (0, 1) is the persistence, σ > 0 is the conditional volatility, and t+1 is an independently and identically distributed (i.i.d.) standard normal shock. The firm uses labor to
produce output, Yt , with a constant returns to scale production technology,
Yt = Xt Nt .
(5)
As in HM, the representative firm incurs costs in posting vacancies with the unit cost:
κt = κK Xt + κW Xtξ ,
(6)
in which κK , κW , and ξ are positive parameters. Once matched, jobs are destroyed at a
constant rate of s per period. Employment, Nt , evolves as:
Nt+1 = (1 − s)Nt + q(θt )Vt ,
(7)
in which q(θt )Vt is the number of new hires. The population is normalized to be unity, Ut =
1−Nt . As such, Nt and Ut are also the rates of employment and unemployment, respectively.
6
The dividends to the firm’s shareholders are given by:
Dt = Xt Nt − Wt Nt − κt Vt ,
(8)
in which Wt is the wage rate. Taking q(θt ) and Wt as given, the firm posts an optimal
number of job vacancies to maximize the cum-dividend market value of equity, denoted St :
St ≡
max
{Vt+τ ,Nt+τ +1 }∞
τ t=0
Et
∞
β τ [Xt+τ Nt+τ − Wt+τ Nt+τ − κt+τ Vt+τ ] ,
(9)
τ =0
subject to equation (7) and a nonnegativity constraint on vacancies:
Vt ≥ 0.
(10)
Because q(θt ) > 0, this constraint is equivalent to q(θt )Vt ≥ 0. As such, the only source of
job destruction in the model is the exogenous separation of employed workers from the firm.
Let λt denote the multiplier on the constraint q(θt )Vt ≥ 0. From the first-order conditions
with respect to Vt and Nt+1 , we obtain the intertemporal job creation condition:
κt+1
κt
.
− λt = Et β Xt+1 − Wt+1 + (1 − s)
− λt+1
q(θt )
q(θt+1 )
(11)
Intuitively, the marginal costs of hiring at time t (with the nonnegativity constraint accounted
for) equal the marginal value of employment to the firm, which in turn equals the marginal
benefit of hiring at period t + 1, discounted to t with the discount factor, β. The marginal
benefit at t + 1 includes the marginal product of labor, Xt+1 , net of the wage rate, Wt+1 , plus
the marginal value of employment, which equals the marginal costs of hiring at t + 1, net of
separation. Finally, the optimal vacancy policy also satisfies the Kuhn-Tucker conditions:
q(θt )Vt ≥ 0,
λt ≥ 0,
7
and λt q(θt )Vt = 0.
(12)
The wage rate is from the sharing rule per the outcome of a generalized Nash bargaining
process between the employed workers and the firm. Let η ∈ (0, 1) be the workers’ relative
bargaining weight and b the workers’ flow value of unemployment activities. The wage rate is:
Wt = η (Xt + κt θt ) + (1 − η)b.
(13)
Let Ct denote consumption. In equilibrium, the goods market clearing condition says:
Ct + κt Vt = Xt Nt .
3
(14)
A Projection Algorithm
We approximate the model solution with a projection algorithm. The state space of the
model consists of employment and log productivity, (Nt , xt ). The goal is to solve for the optimal vacancy function, Vt = V (Nt , xt ), and the multiplier function, λt = λ(Nt , xt ) from the
intertemporal job creation condition given by equation (11). We work with the job creation
condition because the competitive equilibrium is in general not Pareto optimal. In addition,
V (Nt , xt ) and λ(Nt , xt ) must also satisfy the Kuhn-Tucker condition (12).
The standard projection method would approximate V (Nt , xt ) and λ(Nt , xt ) directly to
solve the job creation condition, while obeying the Kuhn-Tucker condition. However, with
the vacancy nonnegativity constraint, these kinked functions can cause problems in the approximation with smooth functions. To deal with this issue, we follow Christiano and Fisher
(2000) to approximate the conditional expectation in the right-hand side of equation (11)
as Et ≡ E(Nt , xt ). A mapping from Et to policy and multiplier functions then eliminates the
need to parameterize the multiplier function separately. Specifically, after obtaining Et , we
first calculate q̃(θt ) ≡ κt /Et . If q̃(θt ) < 1, the nonnegativity constraint is not binding, we
8
set λt = 0 and q(θt ) = q̃(θt ), and then solve θt = q −1 (q̃(θt )), in which q −1 (·) is the inverse
function of q(·) from equation (3). We then set Vt = θ t (1 − Nt ). If q̃(θt ) ≥ 1, the constraint
is binding, we set Vt = 0, θ t = 0, q(θt ) = 1, and λt = κt − Et .
We approximate the log productivity process, xt , in equation (4) based on the discrete
state space method of Rouwenhorst (1995). We use 17 grid points to cover the values of xt ,
which are precisely within four unconditional standard deviations from the unconditional
mean of zero. For the Nt grid, we set the minimum value to be 0.01 and the maximum 0.99
(Nt never hits the bounds in simulations). For each grid point of xt , we use piecewise linear
splines with 1,000 nodes on the Nt space to approximate E(Nt , xt ). To obtain an initial
guess, we use the model’s loglinear solution.
We implement the loglinear solution (and the second-order perturbation in logs) using
Dynare (e.g., Adjemian et al. (2011)). Because Dynare is well known, we do not discuss
the details. Instead, we report our Dynare program in Appendix A. Two comments on the
implementation are in order. First, in the loglinear solution, we ignore the nonnegativity
constraint of vacancy by setting the multiplier, λt , to be zero for all t. Doing so is consistent
with the common practice in the literature. Second, following Den Haan’s (2011) recommendation, we substitute out as many variables as we can and use only a minimum number of
equations in the Dynare program. We use only three equations (the employment accumulation equation, the job creation condition, and the law of motion for log productivity) with
three primitive variables (employment, log productivity, and consumption). The solutions
to all the other variables are obtained using the actual nonlinear equations of the model,
equations that connect them to the three variables in the perturbation system.
9
4
How Loglinearization Deviates from Projection
In this section, we quantify in depth how the results based on loglinearization deviate significantly from those based on the global projection algorithm.
4.1
Labor Market Volatilities
It is customary to detrend variables in log-deviations from the HP-trend with a smoothing
parameter of 1,600. In contrast, we use the HP-filtered cyclical component of proportional
deviations from the mean with the same smoothing parameter. We cannot take logs because
vacancies can be zero in simulations when the Vt ≥ 0 constraint is binding.5
4.1.1
The HM Calibration
To solve and simulate from the model, we use exactly the same parameter values from the
HM’s weekly calibration. Specifically, the time discount factor, β, is 0.991/12 . The persistence
of log labor productivity, ρ, is 0.9895, and its conditional volatility, σ, is 0.0034. The workers’
bargaining weight, η, is 0.052, and their flow value of unemployment activities, b, is 0.955.
The job separation rate, s, is 0.0081. The elasticity of the matching function, ι, is 0.407.
Finally, for the vacancy cost function, the capital cost parameter, κK , is 0.474, the labor
cost parameter, κW , is 0.11, and the exponential parameter in the labor cost, ξ, is 0.449.
With these parameter values, we first reach the model’s stationary distribution by simulating the economy for 500 × 12 × 4 weekly periods from the initial condition of zero for log
productivity and 0.947 for employment (the deterministic steady state). From the station5
In the data, the two detrending methods yield quantitatively similar results. We use the same data
sources as HM (2008) and calculate the standard deviations of unemployment, vacancy, and the labor
market tightness to be 0.119, 0.134, and 0.255, respectively, with our detrending method (untabulated).
Also, unemployment and vacancy have a correlation of −0.913, indicating a downward-sloping Beveridge
curve. These moments are close to those in HM’s Table 3 based on log-deviations.
10
ary distribution, we repeatedly simulate 5,000 artificial samples, each with 648 × 4 weekly
periods. We take the quarterly averages of the weekly unemployment, vacancy, and labor
productivity to obtain 216 quarterly observations, matching HM’s sample length from the
first quarter in 1951 to the fourth quarter in 2004. We then calculate the model moments
for each artificial sample and report the cross-simulation averages.
Table 1 reports the labor market moments from the model. Panel A is borrowed from
Table 4 in HM. In Panel B, we attempt to replicate their results with loglinearization (see the
Dynare code in Appendix A.1). Although the moments are not identical, the results in Panel
B are close to those in Panel A. We detrend all the variables in Panel B as log-deviations
from the HP-trend as in HM (The Vt ≥ 0 constraint is never binding with loglinearization).
The standard deviation for the labor market tightness is 0.284, which is close to 0.292 in
HM. The unemployment volatility is 0.146, which is close to 0.145 in HM. Finally, the
unemployment-vacancy correlation is −0.777, which is not far from −0.724 in HM’s Table 4.
As noted, because vacancies can hit the zero lower bound (albeit infrequently) in the
model solved with the projection algorithm, we cannot take logs. Panel C reports the results
from loglinearization while detrending the variables as HP-filtered proportional deviations
from the mean. Comparing Panels B and C shows that the detrending method does not materially affect the results. The standard deviation of the labor market tightness becomes 0.327,
which is not far from 0.284 with the log-deviations. The unemployment volatility is 0.133,
which is close to 0.146 with the log-deviations. However, the unemployment-vacancy correlation is −0.848, which is somewhat higher in magnitude than −0.777 with the log-deviations.
Panel D reports the key message of the paper. The projection results differ from the loglinearization results in quantitatively important ways. Most important, the unemployment
11
volatility is 0.251, which is almost twice as large as that from loglinearization, 0.133. Also, the
unemployment-vacancy correlation is −0.564 in the projection solution and is substantially
lower in magnitude than −0.848 with loglinearization. As such, loglinearization understates
the unemployment volatility but overstates the unemployment-vacancy correlation.
4.1.2
The Impact of Higher Persistence in Labor Productivity
A higher persistence of the log productivity gives rise to even larger quantitative differences
between loglinearization and projection. This result holds even when the unconditional standard deviation of productivity is lower. We recalibrate the model by setting the persistence
of log productivity to 0.9957 and the conditional volatility to 0.0021. These weekly values
correspond to the quarterly values of 0.95 and 0.007, respectively, which are standard in the
business cycles literature (e.g., Cooley and Prescott (1995)). From Panels E and F of Table
1, the volatility of labor productivity, 0.009, is lower than 0.013 in the HM calibration.
Despite the lower productivity volatility, the differences between loglinearization and
projection increase in magnitude with the high persistence calibration. The unemployment
volatility from projection, 0.285, is almost three times as large as 0.108 from loglinearization.
The unemployment-vacancy correlation is −0.502 from projection in contrast to −0.860 from
loglinearization. While the volatility of the market tightness from loglinearization is (relatively) close to that from projection with the HM calibration, 0.327 versus 0.271, the gap
widens to 0.290 versus 0.215 in the high persistence calibration.
4.1.3
Second-order Perturbations in Logs
Loglinearization is the first-order perturbation in logs. Can the second-order perturbation
in logs (also popular in the business cycle literature) eliminate the quantitative differences
between projection and loglinearization? The answer is not affirmative, unfortunately.
12
From Panel G of Table 1, although improving on loglinearization, the labor market moments from the second-order perturbation still deviate significantly from those from the
projection algorithm. Relative to loglinearization, the unemployment volatility from the
second-order perturbation increases slightly from 0.133 to 0.164, which still differs greatly
from 0.251 with projection. Similarly, the unemployment-vacancy correlation drops in magnitude from −0.848 to −0.791, once we move from loglinearization to the second-order perturbation. However, −0.791 still deviates significantly from −0.564 with projection. Panel
H shows that the deviations are further exacerbated with the high persistence calibration.
4.2
Intuition: Nonlinearity Matters
Why does loglinearization differ so much from projection? The crux is that the unemployment dynamics in the DMP model are highly nonlinear. In recessions unemployment rises
rapidly, whereas in booms unemployment declines only gradually. The distribution of unemployment is highly skewed with a long right tail. The projection solution fully captures these
nonlinear dynamics. In contrast, by focusing only on local dynamics around the deterministic steady state, loglinearization ignores the large unemployment dynamics in recessions
altogether, thereby understating its volatility but overstating its correlation with vacancies.
4.2.1
Stationary Distribution
Figure 1 plots unemployment against labor productivity using one million weekly periods
simulated from the model’s stationary distribution. From Panel A, the projection-based
unemployment-productivity relation is highly nonlinear under the HM calibration. When
the labor productivity is high, unemployment falls gradually, whereas when the productivity
is low, unemployment rises drastically. The unemployment rate can reach above 65% in
simulations. The correlation between unemployment and labor productivity is −0.69. Panel
13
B plots the empirical cumulative distribution function for unemployment in the simulated
data. Unemployment is positively skewed with a long right tail. The mean unemployment
rate is 6.12%, the median is 5.34%, and the skewness is 5.82. The 2.5 percentile, 3.8%, is
close to the median, whereas the 97.5 percentile is far away, 13.43%.
In contrast, Panel C shows that the unemployment-productivity relation from loglinearization is linear. Their correlation is nearly perfect, −0.96. The maximum unemployment
rate in simulations is only 10.38%. Panel D also shows that the empirical cumulative distribution for unemployment from loglinearization is largely symmetric. The mean unemployment
rate is 5.28%, which is close to the median of 5.29%. The skewness is almost zero, −0.05. The
2.5 and 97.5 percentiles, 2.72% and 7.77%, respectively, are distributed symmetrically around
the median of 5.29%. Also, the stochastic mean of the unemployment rate from projection,
6.12%, is almost one percentage point higher than its deterministic steady state rate, 5.28%.
From Panels E and F, unlike loglinearization, the second-order perturbation in logs captures some nonlinear dynamics. However, the nonlinearity is not nearly as strong as that
from the projection algorithm. As such, although improving on loglinearization, the labor
market moments from the second-order perturbation are closer to those from loglinearization
than those from the projection algorithm (Panel G in Table 1).
Panels G and H in Figure 1 report the unemployment-productivity relation and the empirical distribution of unemployment under the high persistence calibration. Comparing with
Panels A and B shows that this calibration exhibits stronger nonlinear dynamics than the
HM calibration. The unemployment rate can reach above 90% in simulations in contrast
to 65% under the HM calibration. Intuitively, job creation is a forward looking decision,
in which the persistence of labor productivity plays a determining role. A low realization
14
reduces job creation more if it is expected to last a longer time. As such, nonlinear dynamics are stronger under the high persistence calibration, and loglinearization deviates further
from the projection algorithm (Panels E and F in Table 1).
4.2.2
An Illustrative Example
To further illustrate the differences between projection and loglinearization, we contrast
simulated sample paths of unemployment from the algorithms. The underlying labor productivity series is identical. As such, the only difference is the algorithm.
Figure 2 shows that the two sample paths differ in two important ways. First, in recessions the unemployment rates from the projection algorithm spike up frequently to more
than 10% (the blue solid line). In contrast, no such spikes are visible from the sample path
based on loglinearization (the red broken line). For instance, toward the end of the sample
path, the unemployment rate from the projection algorithm spikes up to more than 25%,
whereas the corresponding unemployment rate from loglinearization is only about 8.5%. Second, in booms the unemployment rates from the projection algorithm are often higher than
those from loglinearization. In particular, around the 3,000th weekly period, the unemployment rate from the projection algorithm reaches a lower level of about 3.2%. However, the
corresponding unemployment rate from loglinearization is about 1%.
Intuitively, these nonlinear dynamics are driven by a combination of matching frictions
and small profits due to high and inelastic wages. The matching frictions induce a congestion externality in the labor market. In booms, many vacancies compete for a small pool of
unemployed workers. An extra vacancy can cause a large drop in the vacancy filling rate,
causing the marginal costs of hiring to increase rapidly. The increasing marginal costs of
hiring mitigate the impact of rising profits on the incentives of hiring, slowing down job
15
creation flows. As such, the unemployment rates fall gradually in booms.
In contrast, many unemployed workers compete for a small pool of vacancies in recessions. An extra vacancy is quickly filled and hardly reduces the vacancy filling rate. As such,
the marginal costs of hiring hardly decline in recessions (downward rigidity), and a given
decline in vacancies leads to a disproportionately large drop in hiring. Consider a large, negative shock to labor productivity. Profits drop disproportionately more due to the high and
inelastic wages. The downward rigidity of the marginal costs of hiring further exacerbates
the impact of falling profits on the incentives of hiring, stifling job creation flows. As such,
as vacancies become scarce, the unemployment rates spike up in recessions (e.g., PetroskyNadeau, Zhang, and Kuehn (2013)). As noted, loglinearization misses these dynamics.
4.2.3
Nonlinear Impulse Response Functions
We consider three different initial points, bad, median, and good economies. The bad
economy is the 5 percentile of the model’s bivariate stationary distribution of employment
and log productivity under the projection algorithm, the median economy is the 50 percentile,
and the good economy is the 95 percentile. (The unemployment rates are 10.54%, 5.34%,
and 3.96%, and the log productivity levels are −3.82, 0, and 3.82, across the bad, median,
and the good economies, respectively.) We calculate two sets of responses. The first set is the
responses to a one-standard-deviation shock to log productivity, both positive and negative,
starting from all three initial points. The second set is the responses to a two- and a threestandard-deviation negative shock to log productivity, starting from the median economy.
Figure 3 reports the first set of impulse responses. Several nonlinear patterns emerge.
First, the responses from the projection algorithm are clearly stronger in recessions than those
in booms. For instance, in response to a negative impulse, the unemployment rate shoots up
16
by 0.83% in the bad economy (Panel A). In contrast, the response is only 0.08%, which is
an order of magnitude smaller, in the good economy (Panel C). The response in the median
economy is 0.18%, which is closer to that in the good economy than in the bad economy.
Second, the responses to a negative impulse from the projection algorithm are somewhat
larger in magnitude than those to a positive impulse. The response to a positive impulse is
−0.72% in the bad economy, in contrast to the response to a negative impulse, 0.83%.
Third, and more important, although close in the median economy, the responses in
unemployment from loglinearization diverge significantly from the projection responses. In
particular, the responses from loglinearization are substantially weaker in the bad economy.
For instance, the response to a negative impulse under loglinearization is 0.15%, which is less
than 20% of the response under the projection solution, 0.83%. However, in the good economy, the loglinearization responses are somewhat stronger than the projection responses.
The loglinearization response to a negative impulse is 0.16%, which is twice as large as the
projection response, 0.08%. The intuition is that loglinearization misses the gradual nature of economic expansions captured by projection as illustrated in Figure 2. Panels D to F
present quantitatively similar results for the impulse responses in the labor market tightness.
From Panels B and E in Figure 3, the projection-based responses and the loglinearizationbased responses to one-standard-deviation shocks to log productivity are quantitatively similar in the median economy. It is natural to ask to what extend this finding holds up quantitatively in the face of larger impulses. To this end, Figure 4 reports the responses to negative two- and three-standard-deviation shocks to log productivity starting from the median
economy. Panel A shows that, for the unemployment rate, the projection-based responses
are larger than the loglinearization-based responses: 0.39% versus 0.32% as the maximum
17
response to a two-standard-deviation impulse and 0.61% versus 0.47% to a three-standarddeviation impulse. However, for the labor market tightness (Panel B), the two algorithms
produce quantitatively similar responses even to large shocks in the median economy.
4.3
Accuracy Tests
While it is natural to expect that projection would be more accurate than loglinearization, we
are not aware of any prior attempt to quantify the approximation errors for loglinearization
relative to an accurate global solution in the DMP model. In this subsection, we fill this gap.
4.3.1
Static Euler Equation Errors
Following Judd (1992) and Den Haan (2010b), we calculate Euler equation errors, both static
and dynamic. The static Euler equation error is defined as:
eSt
κt
κt+1
−
≡ Et β Xt+1 − Wt+1 + (1 − s)
− λt+1
− λt .
qt+1
qt
(15)
If an algorithm is accurate, eSt should be zero on all points in the state space. We calculate the static error on a fine employment-log productivity grid. To compute the conditional
expectation in equation (15), we discretize xt via the Rouwenhorst (1995) method and calculate the conditional expectation via matrix multiplication.6 The grid size remains at 17. We
create an evenly spaced grid of Nt with 10,000 points (in contrast to the original 1,000-point
grid). We use linear interpolation for Nt values that are not on the original grid.
6
We have experimented with the Gauss-Hermite quadrature in the continuous state space of xt when
calculating the conditional expectation. However, when xt is extreme, implementing this method involves
large extrapolation errors in the policy functions. For instance, to calculate the expectation conditional on
the minimum value of the xt grid, which is precisely four unconditional standard deviations below the mean
of zero, we need to put another grid of xt around this extreme xt value. As such, the lower half of the
second grid lies outside the original xt grid, on which we solve the policy functions. Extrapolating the policy
functions outside the original xt grid can induce large errors. However, these errors are irrelevant because the
original grid is large enough to cover xt values within four unconditional standard deviations from the mean.
18
To calculate interpretable Euler equation errors per Den Haan (2010), we compute two
consumption (employment) values on each grid point. The first consumption (employment)
value, Ct (Nt ), is computed from a given numerical solution method. The second consumption (employment) value, CtS (NtS ), is the implied consumption (employment) value from
an accurately calculated conditional expectation, which is the right-hand side of the Euler
equation. Denoting the deterministic steady state levels of consumption and employment as
Css and Nss , we define the static consumption and employment errors, respectively, as:
eSCt ≡
Ct − CtS
,
Css
and eSN t ≡
Nt − NtS
.
Nss
(16)
Figure 5 reports these errors. From Panels A to C, the projection algorithm offers an extremely accurate solution to the model. In particular, eSt is in the magnitude of 10−13 , and the
two interpretable errors are in the magnitude of 10−14 . Panels D to F show that loglinearization exhibits large approximation errors: eSt is in the magnitude of 10% and can potentially go
up to 20%. Even at the deterministic steady state (employment = 0.947 and log productivity
= zero), eSt is −3.63%. The consumption error, eSCt , ranges from −8% to 2% and remains at
−2.13% at the steady state. The employment error, eSN t , ranging from slightly below −1.31%
to 3.64%, is the smallest among the three errors, and its level is 0.79% at the steady state.
Finally, Panels G to I show that the second-order perturbation in logs can be even more
inaccurate than loglinearization. Although at the deterministic steady state, the three errors
are close to those under loglinearization, the errors can be larger once the economy wonders
away from the steady state. The static Euler equation error varies from −75% to 25%, the
consumption error from −20% to 11%, and the employment error from −0.5% to 8.5%.
19
4.3.2
Dynamic Euler Equation Errors
The static errors are only one-period ahead forecast errors, which ignore the accumulation of
small errors over time. To allow for this possibility, we follow Den Haan (2010) to calculate
dynamic errors. We simulate two series for Ct and Nt . First, we use a given algorithm
in simulations. Second, we simulate alternative series using the following steps in each period: (i) Use a given algorithm to calculate the conditional expectation accurately; (ii) use
this conditional expectation to calculate implied consumption value, CtD ; and (iii) compute
employment, NtD , from this implied consumption value and the resource constraint. The
dynamic consumption and employment errors are defined, respectively, as:
eD
Ct
Ct − CtD
≡
Css
and
eD
Nt
Nt − NtD
≡
.
Nss
(17)
Figure 6 shows that the differences in accuracy between projection and perturbations
are even starker in dynamic Euler equation errors. We simulate one million weekly periods
from the model’s stationary distribution under a given algorithm and report the empirical
D
cumulative distribution functions of eD
Ct and eN t . From Panels A and B, the dynamic errors
from the projection algorithm are extremely small (in the magnitude of 10−14 ). In contrast,
Panels C and D report very large dynamic errors for loglinearization. For the consumption
errors, the mean is 73.78%, and the median is 73.51%. For the employment errors, the mean
is 74.20%, and the median is 73.79%. Finally, the dynamic errors from the second-order
perturbation in logs are only slightly smaller (Panels E and F).
20
5
Reconciling with the HM Results
As noted, HM in fact use a global algorithm to solve their model. In this section, we address
two issues. First, why the results based on their global method differ from those from our
projection algorithm (Section 5.1). Second, we ask whether there exists a different small
surplus calibration (but in the same spirit of HM) that can explain the Shimer puzzle when
solved accurately with our projection algorithm (Section 5.2).
5.1
The Importance of Choosing Markov-chain Approximations
to the Highly Persistent Productivity Process
Our projection algorithm differs from the HM algorithm in a crucial aspect. While we follow
Kopecky and Suen (2010) in using the Rouwenhorst (1995) method with a grid of 17 points
to approximate the log productivity, HM use the Tauchen (1986) method with 35 grid points.
We show that this difference goes a long way in reconciling our results with theirs.
In Table 2, we report labor market volatilities under alternative Markov-chain approximations to the log productivity, xt . Panel A shows the results under the Rouwenhorst
approximation but with a small grid of only five points. The five-point grid covers a range of
xt that is precisely two unconditional standard deviations above and below the unconditional
mean of zero. (The range is four standard deviations with the 17-point grid.) We choose the
five-point grid to be consistent with the HM implementation of the Tauchen method that
also covers a range of two standard deviations above and below the mean.
Panel A shows that the quantitative results from the Rouwenhorst method are (relatively)
robust with respect to the range of the grid. Even with only five points, the unemployment
volatility is 0.220, which is not far from 0.251 with the 17-point grid (Panel D in Table 1).
21
The volatility of the market tightness is 0.267, which is close to 0.271 from the large grid.
Finally, the unemployment-vacancy correlation is −0.606, which is not far from −0.564 from
the large grid. These results on the DMP model echo those reported in Kopecky and Suen
(2010) in the context of the stochastic growth model.
Panels B to D in Table 2 show that the quantitative results from the Tauchen method
are extremely sensitive to the range of the grid chosen. From Panel B, when the range of the
grid covers two standard deviations above and below the conditional mean as in HM, the
results from the Tauchen method are largely similar to those reported in their Table 4. For
instance, the unemployment volatility is 0.154, which is close to 0.145 in their Table 4. We
calculate the unemployment-vacancy correlation as −0.698, which is close to their estimate
of −0.724. The volatility of the market tightness is 0.246, which is not far from their 0.292.
Panels C and D shows that enlarging the range of the Tauchen grid raises the unemployment volatility but dampens the unemployment-vacancy correlation. As we vary the
grid range from two to three and four unconditional standard deviations above and below
the mean, the unemployment volatility increases from 0.154 to 0.267 and 0.319, and the
unemployment-vacancy correlation falls in magnitude from −0.698 to −0.562 and −0.521,
respectively. The only difference across Panels B to D is the range of the Tauchen grid, as all
three panels use our projection algorithm. As such, these results cast doubt on the Tauchen
method in approximating highly persistent autoregressive processes.7
7
Kopecky and Suen (2010) also note that the performance of the Tauchen (1986) method is extremely
sensitive to the choice of a free parameter that determines the range of the discrete state space. Their results
are based on the stochastic growth model. We reinforce their conclusion in the context of the DMP model.
22
5.2
Restoring the Small Surplus Calibration
Table 1 shows that when solved accurately, the HM small surplus calibration matches the
volatility of the labor market tightness but overshoots the unemployment volatility by almost
50%. It is natural to ask whether there exists a different small surplus calibration (but in the
same spirit of HM) that can explain the Shimer puzzle, when we solve the model accurately
with our projection algorithm. The glass is half full, at least.
When we reduce the flow value of unemployment slightly from 0.955 in HM to 0.94,
the unemployment volatility falls to 0.125, which is identical to that in the data (see HM’s
Table 3). As such, the key insight that small profits increase the unemployment volatility
survives our computational scrutiny. The volatility of the market tightness is only 0.191 in
the model, which is about 36% lower than that in the data, 0.259. However, to the extent
that the quality of the unemployment data might be higher than the quality of the available
vacancy data, the succuss of the HM calibration in matching the unemployment volatility
seems more important than its failure in matching the volatility of the market tightness.
6
Cobb-Douglas Matching Function
HM use the matching function first introduced by Den Haan, Ramey, and Watson (2000)
with the property that the job finding rate, ft , and the vacancy filling rate, qt , both fall
between zero and one. As such, no additional constraints are necessary to ensure that the
two matching probabilities are properly bounded. In contrast, the Cobb-Douglas matching
function does not have this property. We show that loglinearization still diverges from the
projection algorithm in terms of labor market moments in the DMP model with the CobbDouglas matching function. However, the differences are smaller than those with the Den
23
Haan-Ramey-Watson matching function, at least in some specifications.
6.1
The Model with the Cobb-Douglas Matching Function
The Cobb-Douglas matching function is given by:
G(Ut , Vt ) = χUtα Vt1−α ,
(18)
in which χ > 0 is the level parameter, and α ∈ (0, 1) is the constant elasticity of the matching
process to unemployment, Ut . Equation (18) implies that the job finding rate is:
f (θt ) = χθ 1−α
t
(19)
q(θt ) = χθ−α
t .
(20)
and the vacancy filling rate is:
The matching probabilities are bounded below by zero but not bounded above by one.
To impose the upper bound, we define an upper bound for the market tightness, denoted θ,
such that f (θ) = 1. Equation (19) implies θ = χ1/(α−1) . We also define a lower bound for the
market tightness, θ, such that q(θ) = 1. Equation (20) implies θ = χ1/α . The corresponding
lower and upper bounds for vacancy, Vt , are then given by V t ≡ θUt and V t ≡ θUt . As such,
ft ≤ 1 is equivalent to Vt ≤ V t and qt ≤ 1 is equivalent to Vt ≥ V t . Again because qt > 0,
in terms of new hires, these constraints are equivalent to qt Vt ≤ qt V t and qt Vt ≥ qt V t .
Let λt and λt be the Lagrangian multipliers associated with qt Vt ≤ qt V t and qt Vt ≥ qt V t ,
respectively. The intertemporal job creation condition is given by:
κ
κ
− λt + λt = Et β Xt+1 − Wt+1 + (1 − s)
− λt+1 + λt+1
,
q(θt )
q(θt+1 )
24
(21)
in which we simplify HM’s procyclical vacancy costs, κt , to be a constant, κ. The optimal
vacancy policy should also satisfy the Kuhn-Tucker conditions:
Vt ≥ V t ,
λt ≥ 0,
and λt Vt = 0;
(22)
Vt ≤ V t ,
λt ≥ 0,
and λt Vt = 0.
(23)
We continue to approximate the conditional expectation in the right-hand side of equation
(21) as Et = E(Nt , xt ). After obtaining Et , we calculate q̃t = κ/Et . There are three scenarios
in regard to q̃t . First, if q̃t ≤ q(θ), then Vt ≤ V t is binding. We set qt = q(θ), θt = θ, and
Vt = θUt . The lower bound for vacancies is not binding. As such, λt = 0 and λt = Et − κ/qt .
Second, if q(θ) < q̃t < 1, then neither constraint is binding for vacancies. We set λt = λt = 0
and q(θt ) = q̃t . We then solve for θt = q −1 (q̃t ), in which q −1 (·) is the inverse function of q(·)
from equation (20). We then set Vt = θt (1 − Nt ). Finally, if q̃t ≥ 1, then the lower bound
Vt ≥ V t is binding. We set λt = 0, θt = θ, Vt = θUt , qt = 1, and λt = κ − Et .
6.2
Quantitative Results
We calibrate the model with the Cobb-Douglas matching function in weekly frequency as
in HM. The general strategy is to match the mean unemployment rate and labor market
volatilities under loglinearization and then use the projection algorithm with the same parameter values to quantify its differences in quantitative results. As common in business
cycle studies, we calibrate the persistence of the labor productivity process to be ρ = 0.9957
and its conditional volatility to be 0.0021. Following Pissarides (2009), we set the curvature
parameter, α, in the matching function to be 0.5 and the level parameter, χ, to be 0.17.
For the constant vacancy cost, κ, we set it to be 0.584, which is HM’s average procyclical
vacancy cost. All the other parameter values are from the HM calibration.
25
6.2.1
Weekly, Both Vt Constraints
Under these parameter values, Panel A of Table 3 reports that the model solved with loglinearization produces labor market volatilities that come close to those in the data. The
unemployment volatility is 0.101, and the volatility of the market tightness is 0.288. The
mean unemployment rate is 6.46% (untabulated). Comparing Panels A and B shows that,
relative to the projection algorithm, loglinearization still underestimates the unemployment
volatility (0.101 versus 0.150) but overestimates the magnitude of the unemployment-vacancy
correlation (−0.860 versus −0.681). However, the biases are smaller in magnitude than those
with the Den Haan-Ramey-Watson matching function (0.108 versus 0.285 for the unemployment volatility and −0.860 versus −0.502 for the unemployment-vacancy correlation, see
Table 1). In untabulated results, we show that the second-order perturbation in logs brings
the unemployment volatility to 0.124, which is closer to its accurate value of 0.150. However,
the unemployment-vacancy correlations remains largely unchanged at −0.826.
To dig deeper behind these quantitative results, Figure 7 reports the scatter plots of unemployment versus labor productivity in the DMP model with the Cobb-Douglas matching
function. From Panel A, the model exhibits similar but weaker nonlinear dynamics than
those with the Den Haan-Ramey-Watson matching function (see Panel G of Figure 1). The
unemployment rate seems capped at a level slightly higher than 20%, when the Vt ≥ V t
constraint (i.e., qt ≤ 1) starts to bind. The frequency of this constraint binding is 1.01% in
the models’s simulations. The other constraint, Vt ≤ V t (i.e., ft ≤ 1), is never binding.
6.2.2
Daily, Both Vt Constraints
Despite the low frequencies with which the qt ≤ 1 constraint is binding, it might be possible that the effective cap imposed by the constraint on the unemployment rate dampens
26
its volatility from the projection algorithm. The remaining panels in Table 3 confirm this
conjecture. In Panel C, we recalibrate the model in daily frequency. Because the daily qt is
lower on average than the weekly qt , the qt ≤ 1 constraint should be binding less frequently.
The parameters in daily frequency closely match those in weekly frequency. The persistence of log productivity is set to be .99571/5 = .9991. The conditional volatility is 0.00094,
which, given the daily persistence, implies the same unconditional volatility of .0227 as in the
weekly calibration. We set β = .991/60 , χ = .17/5, and s = .0081/5. All the other parameter
values are the same as in the weekly calibration. The results from loglinearization are similar
across daily and weekly frequencies (untabulated).
Comparing Panels B and C shows that as we move from weekly to daily frequency,
the unemployment volatility increases from 0.150 to 0.178, and the unemployment-vacancy
correlation from −0.681 to −0.649. The frequency of the qt ≤ 1 constraint being binding
drops from 1.01% to 0.31%. More important, Panels A and C reveal large differences between loglinearization and projection (the unemployment volatility 0.101 versus 0.178 and
the unemployment-vacancy correlation −0.860 versus −0.649).
Panel B in Figure 7 confirms stronger nonlinear unemployment dynamics in daily frequency. No longer capped at around 20%, the unemployment rate can reach above 50%.
The stronger nonlinear dynamics in turn convert to higher unemployment volatilities.
6.2.3
Weekly, No Vt Constraints
If one interprets the vacancy filling rate, qt , not as a probability of a vacancy being filled with
an unemployed worker but as the number of workers hired per vacancy, then it is conceivable
that qt can rise above one. As such, in Panel D of Table 3, we experiment with the model
without imposing ft ≤ 1 or qt ≤ 1. (As noted, ft ≤ 1 is never binding in simulations. Panel D
27
shows that relaxing the qt ≤ 1 constraint in weekly frequency delivers labor market volatilities
that are close to those from the daily frequency but with the constraint imposed. As such,
the differences between loglinearization and projection remain important. Panel C in Figure
7 further confirms the nonlinear dynamics in that the unemployment rate can literally reach
100%.8 In all, it seems safe to conclude that the Cobb-Douglas matching function also
exhibits strong nonlinear dynamics, which impact on the labor market moments.
7
Conclusion
An accurate global algorithm is crucial for quantifying the theoretical properties of the
Diamond-Mortensen-Pissarides model. In the context of Hagedorn and Manovskii (2008),
we show that the standard loglinearization understates the unemployment volatility by about
50%, overstates the magnitude of the unemployment-vacancy correlation also by about 50%,
and ignores impulse responses that are an order of magnitude larger in recessions than in
booms. Our quantitative results highlight the extreme importance of accurately accounting
for nonlinear dynamics in quantitative macroeconomic studies.
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30
Table 1 : The Second Moments of the Labor Market in the DMP Model
Panel A is borrowed from HM (2008, Table 4). All the panels use the HM calibration except
for Panels E, F, and H, in which we use the high persistence calibration with ρ = 0.9957 and
σ = 0.0021. For each combination of calibration and solution method, we simulate 5,000
artificial samples with 648 × 4 weekly observations in each sample. We take the quarterly
averages of weekly unemployment U, vacancy, V , and labor productivity, X, to convert to
216 quarterly observations. θ = V /U denotes the labor market tightness. All the variables
in Panel B are in log-deviations from the HP-trend with a smoothing parameter of 1,600.
The variables in the remaining panels are in HP-filtered proportional deviations from the
mean with the same smoothing parameter. We calculate all the moments on these simulated
samples and report the cross-simulation averages.
U
V
θ
X
Panel A: HM (2008, Table 4)
U
V
θ
X
Panel B: HM, loglinearization
(log-deviations)
Standard deviation 0.145
0.169
0.292
0.013
0.146
0.155
0.284
0.013
Autocorrelation
0.830
0.575
0.751
0.765
0.826
0.674
0.787
0.760
Correlation matrix
−0.724 −0.916 −0.892 U
−0.777 −0.939 −0.901
0.940
0.904 V
0.946
0.965
0.967 θ
0.990
Panel C: HM, loglinearization
Panel D: HM, projection
Standard deviation 0.133
0.144
0.327
0.013
0.251
0.177
0.271
0.013
Autocorrelation
0.831
0.681
0.783
0.760
0.823
0.579
0.760
0.760
Correlation matrix
−0.848 −0.864 −0.927 U
−0.564 −0.662 −0.699
0.858
0.985 V
0.887
0.906
0.890 θ
0.996
Panel E: High persistence,
loglinearization
Panel F: High persistence,
projection
Standard deviation 0.108
0.115
0.290
0.009
0.285
0.158
0.215
0.009
Autocorrelation
0.848
0.711
0.808
0.784
0.856
0.610
0.783
0.783
Correlation matrix
−0.860 −0.820 −0.934 U
−0.502 −0.551 −0.603
0.819
0.986 V
0.841
0.874
0.844 θ
0.991
Panel G: HM, 2nd-order
perturbation in logs
Panel H: High persistence,
2nd-order perturbation in logs
Standard deviation 0.164
0.178
0.263
0.013
0.146
0.155
0.212
0.009
Autocorrelation
0.831
0.704
0.788
0.760
0.855
0.739
0.808
0.784
Correlation matrix
−0.791 −0.794 −0.795 U
−0.756 −0.720 −0.747
0.946
0.973 V
0.935
0.962
0.993 θ
0.989
31
Table 2 : Labor Market Volatilities Under Alternative Markov-chain Approximations
to the Highly Persistent Productivity Process in the HM Calibration
We simulate 5,000 samples from each approximation with the projection algorithm. In Panel
A, we discretize the log productivity, xt , using the Rouwenhorst method with five grid points.
In Panels B to D, we discretize xt with the Tauchen method with 35 grid points but vary the
bounds of the grid in terms of the number of unconditional standard deviations, denoted m,
from the unconditional mean of zero. The simulation design is identical to that in Table 1.
U
V
θ
X
Panel A: Rouwenhorst (5 points)
Standard deviation 0.220 0.173 0.267
Autocorrelation
0.831 0.581 0.761
Correlation matrix
−0.606 −0.713
0.901
32
V
θ
X
Panel B: Tauchen (m = 2)
0.013
0.154 0.149 0.246 0.013
0.761
0.814 0.583 0.750 0.748
−0.741 U
−0.698 −0.825 −0.842
0.913 V
0.930 0.936
0.998 θ
0.997
Panel C: Tauchen (m = 3)
Standard deviation 0.267 0.183 0.279
Autocorrelation
0.823 0.581 0.758
Correlation matrix
−0.562 −0.657
0.888
U
Panel D: Tauchen (m = 4)
0.014
0.319 0.198 0.292 0.015
0.758
0.827 0.578 0.759 0.760
−0.695 U
−0.521 −0.609 −0.655
0.908 V
0.869 0.895
0.996 θ
0.994
Table 3 : The Second Moments of the Labor Market in the DMP Model with the
Cobb-Douglas Matching Function
In Panels A, B, and D, we calibrate the model in weekly frequency, with ρ = .9957, σ = .0021,
χ = .17, α = .5, κ = .584, and all the other parameters from the HM calibration. In Panel
C, we calibrate the model in daily frequency, with ρ = .99571/5 = .9991, σ = .00094,
β = .991/60 , χ = .17/5, s = .0081/5, and all the other parameter values from the weekly
calibration. In Panel D, we solve the model without imposing the two Vt constraints. We
simulate 5,000 samples from each combination of calibration and algorithm, with 648 × 4
weeks (in Panels A, B, and D) or 648 × 4 × 5 days (in Panel C) for each sample. We take the
quarterly averages of weekly (or daily) unemployment U, vacancy, V , and labor productivity,
X, to convert to 216 quarters. θ = V /U is the labor market tightness. All the variables are
in HP-filtered proportional deviations from the mean with a smoothing parameter of 1,600.
We calculate the moments on each simulated sample and report cross-simulation averages.
U
V
θ
Panel A: Loglinearization
X
U
V
θ
X
Panel B: Projection
Standard deviation 0.101 0.128 0.288 0.009
0.150 0.144 0.218 0.009
Autocorrelation
0.852 0.719 0.807 0.784
0.857 0.658 0.7812 0.783
Correlation matrix
−0.860 −0.830 −0.928 U
−0.681 −0.686 −0.764
0.848 0.988 V
0.923 0.952
0.858 θ
0.985
Panel C: Daily frequency
Panel D: No Vt constraints
Standard deviation 0.178 0.150 0.222 0.009
0.175 0.150 0.221 0.009
Autocorrelation
0.860 0.663 0.783 0.785
0.858 0.653 0.783 0.783
Correlation matrix
−0.649 −0.642 −0.723 U
−0.648 −0.657 −0.731
0.915 0.950 V
0.911 0.945
0.986 θ
0.988
33
Figure 1 : The Unemployment-labor Productivity Relation and Empirical
Cumulative Distribution Functions of Unemployment
From the model’s stationary distribution under each combination of calibration and
algorithm, we simulate one million weekly periods and plot unemployment against labor
productivity and the empirical cumulative distribution function (ecdf) of unemployment.
Panel A: HM, projection, Ut -Xt
Panel B: HM, projection, ecdf
1
Probability
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Unemployment
Panel C: HM, loglinearization, Ut -Xt
Panel D: HM, loglinearization, ecdf
1
Probability
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Unemployment
Panel E: HM, 2nd-order, Ut -Xt
Panel F: HM, 2nd-order, ecdf
1
Probability
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Unemployment
Panel G: High persistence, projection,
Ut -Xt
Panel H: High persistence, projection,
ecdf
1
Probability
0.8
0.6
0.4
0.2
34
0
0
0.2
0.4
0.6
Unemployment
0.8
1
35
0
0
0.05
0.1
0.15
0.2
0.25
1000
2000
3000
4000
5000
This figure plots two unemployment rate series with 5,000 weekly periods. The blue solid line is from the projection algorithm,
and the red broken line from loglinearization. The underlying labor productivity process is fixed across the two algorithms.
Figure 2 : An Illustrative Example of Simulated Sample Paths of the Unemployment Rate with Identical Shocks,
Projection versus Loglinearization
Unemployment
36
−0.2
0
−0.2
0
400
−0.1
−0.1
300
0
0
200
0.1
0.1
100
0.2
0.2
Panel D: θt , 5 percentile
−1
0
−1
0
400
−0.5
−0.5
300
0
0
200
0.5
0.5
100
1
1
Panel A: Ut , 5 percentile
200
300
100
200
300
Panel E: θt , median
100
Panel B: Ut , median
400
400
0
−0.2
−0.1
0
0.1
0.2
−1
0
−0.5
0
0.5
1
200
300
100
200
300
Panel F: θt , 95 percentile
100
Panel C: Ut , 95 percentile
400
400
Three initial points are the 5, 50, and 95 percentiles of the model’s bivariate distribution of employment and log productivity
under the projection algorithm. The impulse responses are averaged across 5,000 simulations, each with 480 weeks. The blue
solid (red broken) lines are the projection responses to a positive (negative) one-standard-deviation shock to log productivity.
The blue and red dotted lines are the responses under loglinearization. The responses in the unemployment rate, Ut , are
changes in levels (times 100), and those in the market tightness, θt , are changes in levels scaled by the pre-impulse level.
Figure 3 : Nonlinear Impulse Response Functions, Projection versus Loglinearization
37
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
100
200
300
Panel A: Unemployment
400
−0.25
0
−0.2
−0.15
−0.1
−0.05
0
100
200
300
Panel B: Labor market tightness
400
The responses are averaged across 5,000 simulations, each with 480 weeks. The blue solid lines are the projection-based
responses to negative two- and three-standard-deviation shocks to log productivity. The red dotted lines are the responses
under loglinearization. The initial condition of the simulations is the 50 percentile of the model’s bivariate stationary
distribution of employment and log productivity under the projection solution. The responses in the unemployment rate
are changes in levels (times 100), and those in the labor market tightness are changes in levels scaled by the pre-impulse level.
Figure 4 : Impulse Responses to Large Negative Shocks in the Median Economy
Figure 5 : Static Euler Equation Errors
Static Euler equation errors, static consumption errors, and static employment errors are
defined in equations (15) and (16), respectively. Each line in a panel is for one value of log
labor productivity on its 17-point grid.
Panel A: Euler equation
errors, projection
Panel B: Consumption
errors, projection
−13
3
−14
x 10
4
2
2
1
0
0
−2
−1
−4
−2
0
Panel C: Employment
errors, projection
0.2
0.4
0.6
Employment
0.8
1
Panel D: Euler equation
errors, loglinearization
−14
x 10
2
x 10
1
0
−6
0
0.2
0.4
0.6
Employment
0.8
1
−1
0
Panel E: Consumption
errors, loglinearization
0.2
0.4
0.6
Employment
0.8
1
Panel F: Employment
errors, loglinearization
0.1
0.04
0.04
0.05
0.02
0.03
0
0
−0.05
−0.02
−0.1
−0.04
−0.15
−0.06
−0.2
−0.08
0.02
0.01
−0.25
0
0.2
0.4
0.6
Employment
0.8
1
Panel G: Euler equation
errors, 2nd-order
perturbation in logs
−0.1
0
0
−0.01
0.2
0.4
0.6
Employment
0.8
1
−0.02
0
Panel H: Consumption
errors, 2nd-order
perturbation in logs
0.6
0.15
0.4
0.1
0.2
0.4
0.6
Employment
0.8
1
Panel I: Employment
errors, 2nd-order
perturbation in logs
0.1
0.08
0.05
0.2
0.06
0
0
−0.05
0.04
−0.2
−0.1
−0.4
−0.6
−0.8
0
0.02
−0.15
0
−0.2
0.2
0.4
0.6
Employment
0.8
1
−0.25
0
0.2
0.4
0.6
Employment
38
0.8
1
−0.02
0
0.2
0.4
0.6
Employment
0.8
1
Figure 6 : Dynamic Consumption and Employment Errors
D
Dynamic consumption and employment errors, denoted eD
Ct and eN t , respectively, are defined
in equation (17). We simulate one million weeks from the model’s stationary distribution
and plot the empirical cumulative distribution functions for the dynamic errors.
Panel B: eD
N t , projection
1
1
0.8
0.8
Probability
Probability
Panel A: eD
Ct , projection
0.6
0.4
0.2
0
−4
0.6
0.4
0.2
−2
0
2
Consumption errors
4
0
−2
6
x 10
1
1
0.8
0.8
0.6
0.4
0.2
0.7
0.75
0.8
Consumption errors
0.85
6
−14
x 10
0.6
0.4
0
0.65
0.9
1
1
0.8
0.8
0.6
0.4
0.2
0.7
0.75
0.8
Employment errors
0.85
0.9
Panel F: eD
N t , 2nd-order perturbation in
logs
Probability
Probability
4
0.2
Panel E: eD
Ct , 2nd-order perturbation in logs
0
0.5
2
Employment errors
Panel D: eD
N t , loglinearization
Probability
Probability
Panel C: eD
Ct , loglinearization
0
0.65
0
−14
0.6
0.4
0.2
0.6
0.7
0.8
Consumption errors
0
0.5
0.9
39
0.6
0.7
0.8
Employment errors
0.9
40
Panel A: Weekly, both constraints
Panel B: Daily, both constraints
Panel C: Weekly, no constraints
From each model’s stationary distribution, we simulate one million weeks and plot unemployment against labor productivity.
Figure 7 : The Unemployment-labor Productivity Relation with the Cobb-Douglas Matching Function
For Online Publication
A
Dynare Programs
// HM_loglinear.mod
var N, x, C;
predetermined_variables N;
varexo e;
parameters bet, rhox, stdx, eta, b, s, iota, kappa_K, kappa_W, xi;
bet
= 0.99^(1/12);
rhox
= 0.9895;
stdx
= 0.0034;
eta
= 0.052;
b
= 0.955;
s
= 0.0081;
iota
= 0.407;
kappa_K = 0.474;
kappa_W = 0.11;
xi
= 0.449;
model;
# kappa_t
# V
# theta
# q
# kappa_p
# V_p
# theta_p
# q_p
# W_p
=
=
=
=
=
=
=
=
=
kappa_K*exp(x) + kappa_W*(exp(x)^xi);
(exp(x)*exp(N) - exp(C))/kappa_t;
V/(1 - exp(N));
(1 + theta^iota)^(-1/iota);
kappa_K*exp(x(+1)) + kappa_W*(exp(x(+1))^xi);
( exp(x(+1))*exp(N(+1)) - exp(C(+1)) )/kappa_p;
V_p/(1 - exp(N(+1)));
(1 + theta_p^iota)^(-1/iota);
eta*(exp(x(+1)) + kappa_p*theta_p) + (1 - eta)*b;
exp(N(+1)) = (1 - s)*exp(N) + q*V;
kappa_t/q = bet*( exp(x(+1)) - W_p + (1 - s)*kappa_p/q_p );
x
= rhox*x(-1) + e;
end;
initval;
N = log(1 - 0.1);
x = 0;
e = 0;
end;
steady;
check;
shocks;
var e = stdx^2;
end;
stoch_simul (order = 1, nocorr, nomoments, IRF = 0);
41
// CD_loglinear.mod
var N, x, C;
predetermined_variables N;
varexo e;
parameters bet, rhox, stdx, eta, b, s, alpha, chi, kappa;
bet
= .99^(1/12);
rhox = .9895;
stdx = .0034;
eta
= .052;
b
= .955;
s
= .0081;
alpha = .5;
chi
= .17;
kappa = .584;
model;
# V
# theta
# q
# V_p
# theta_p
# q_p
# W_p
=
=
=
=
=
=
=
(exp(x)*exp(N) - exp(C))/kappa;
V/(1 - exp(N));
chi*theta^(-alpha);
( exp(x(+1))*exp(N(+1)) - exp(C(+1)) )/kappa;
V_p/(1 - exp(N(+1)));
chi*theta_p^(-alpha);
eta*(exp(x(+1)) + kappa*theta_p) + (1 - eta)*b;
exp(N(+1)) = (1 - s)*exp(N) + q*V;
kappa/q
= bet*( exp(x(+1)) - W_p + (1 - s)*kappa/q_p );
x
= rhox*x(-1) + e;
end;
initval;
N = log(.925);
x = 0;
e = 0;
end;
steady;
check;
shocks;
var e = stdx^2;
end;
stoch_simul (order = 1, nocorr, nomoments, IRF = 0);
For the second-order perturbation solution in logs, we change the last command to:
stoch_simul (order = 2, nocorr, nomoments, IRF = 0);
42