Discretizing a Process with
Non-zero Skewness and High Kurtosis ∗
Simone Civale† Luis Díez-Catalán‡ Fatih Fazilet§
November 6, 2015
Abstract
We develop and test two discretization methods to calibrate a Markov chain
that features non-zero skewness and high kurtosis. (i) The Extended Tauchen
method applies the logic of Tauchen (1986) to a first-order autoregressive process
with normal mixture innovations, which, as we discuss, can be calibrated to feature non-zero skewness and high kurtosis. (ii) The Empirical Calibration method
imposes no structure on the data generating process and maps the data directly
into a discrete Markov chain. Furthermore, we document that commonly used
discretization methods, when applied to a persistent Gaussian AR(1), calibrate
the innovations to have erroneously high kurtosis. This issue motivates the use of
a method such as Extended Tauchen, which can be used to discretize a Gaussian
AR(1) process and is free from this bias. Finally, we illustrate an application of
the Extended Tauchen method in an Aiyagari economy; we find that an idiosyncratic shock with higher kurtosis decreases the general equilibrium interest rate,
where as higher left skewness increases the equilibrium interest rate.
∗
We are especially thankful for the guidance and feedback of Fatih Guvenen. We thank Alessandro
Peri, Georgios Stefanidis, Sergio Salgado, Matt Shapiro, and Guillaume Sublet for helpful comments
and discussions.
†
University of Minnesota. E-mail: [email protected]
‡
University of Minnesota. E-mail: [email protected]
§
University of Minnesota. E-mail: [email protected]
1
1
Introduction
For many economic applications, researchers must specify a stochastic process that
governs the evolution of a key variable. Because the analytical and numerical solutions
of a model depend crucially on this choice, calibrating the process and its discrete
approximation demands great care.
Thanks to their tractability, AR(1) processes with Gaussian innovations are a popular choice for many problems in the literature, as in Aiyagari (1994). Consequently,
several discretization methods, such as Tauchen (1986) or Rouwenhorst (1995), work
well with this type of process.
A growing body of literature, including Bloom et al. (2012) and Guvenen et al.
(2015), shows that higher order moments display interesting empirical patterns, with
major economic implications. The analysis of these patterns poses a challenge for the
economic practitioner who needs to choose (i) a process that is tractable but flexible
enough to capture these patterns and (ii) a discrete approximation of the process.
In this paper we develop and test two discretization methods to calibrate a firstorder Markov process that features non-zero skewness and excess kurtosis—both in
levels and in differences.
The Extended Tauchen (ET) method, the first of these methods, is based on the
simple method of Tauchen (1986). We depart from this method in two important
directions. (i) The innovations to the autoregressive process are distributed as a mixture
of normals. This assumption makes the process flexible enough to feature the desired
level of skewness and kurtosis. (ii) The choice of the state space is optimal with respect
to the targeted moments. The latter change entails a sizable gain in the precision of
the approximation.
The Empirical Calibration (EC) method maps the data directly into the Markov
process, without requiring any assumption on the data generating process. Under this
method, the frequency of the transitions observed in the data identifies a transition
probability.
The literature has often evaluated the performance of a discretization method based
solely on how well it can match the moments of the levels of the process and its autocorrelation. The moments of the differences and the innovations, however, have been
largely overlooked. We document that standard discretization methods perform poorly
2
in this respect. More specifically, when an AR(1) process with Gaussian innovations
is highly persistent, commonly used discretization methods calibrate a Markov chain
whose innovations and differences feature unreasonably high kurtosis. As we illustrate,
this calibration error can have important economic implications. This finding is relevant to our line of research, as it motivates the adoption of a discretization method
such as ET or EC—which do not suffer from this problem—even when calibrating a
process with no skewness and excess kurtosis.
Furthermore, to apply the ET method, the practitioner needs to calibrate a firstorder autoregressive process with normal mixture innovations. For this reason, we study
the properties of this process and further generalize it in order to capture skewness and
kurtosis of the changes over multiple periods in a process.
Finally, we present an economic application of the Extended Tauchen method within
a standard Aiyagari economy. We find that an increase in the kurtosis of the idiosyncratic shocks decreases the general equilibrium interest rate, while an increase in the
left skewness of the idiosyncratic shocks increases the equilibrium interest rate.
The rest of the paper is organized as follows. Section 2 illustrates the connection
between this paper and the existing literature. In Section 3 we document that standard
discretization methods, when applied to an AR(1) process with Gaussian innovations
and high persistence, often calibrate the innovations to have unreasonably high kurtosis.
We then illustrate how this calibration error can have economic implications within a
standard income fluctuation problem.
In Section 4 we discuss the choice of a continuous process that is parsimonious and
yet flexible enough to feature non-zero skewness and high kurtosis. In particular, we
show that an AR(1) process with normal mixture innovations (NMAR) can be calibrated to display non-zero skewness and high kurtosis in the levels and the innovations.
To capture the skewness and kurtosis of the changes in a process over multiple years, a
more flexible process is necessary; we find that an AR(1) process with normal mixture
innovations and a transitory component (NMART) is a suitable choice for this purpose.
Section 5 introduces the Extended Tauchen (ET) method. We show that ET can
be used to calibrate a discrete Markov chain that features non-Gaussian skewness and
kurtosis. We then illustrate two applications of this method. The first application
allows us to compare our method with the existing methods by discretizing an AR(1)
process with Gaussian innovations. In the second application, we discretize an AR(1)
3
process with normal mixture innovations.
In Section 6 we introduce the Empirical Calibration (EC) method and discuss two
applications. The first application uses EC to discretize an AR(1) process with normal
mixture innovations and admits a comparison with the ET method. We then apply EC
to calibrate a process whose changes are characterized by skewness and kurtosis that
are persistent over time. Section 7 presents an economic application of the ET method
in an Aiyagari economy. Section 8 concludes.
2
Related Literature
Because of their tractability, autoregressive processes with Gaussian innovations are
widely used to model key economic variables. Aiyagari (1994), Hubbard et al. (1995)
and Storesletten et al. (2004) use an AR(1) process to model earning dynamics. Cooper
and Haltiwanger (2006) use it to model the evolution of firm profits, while Arellano et al.
(2012) employ it to model the dynamics of volatility.
To implement these frameworks numerically, several discretization methods exist to
approximate the continuous process with a Markov chain. Tauchen (1986), given an
arbitrary rule for placing the state space, proposes calibrating the transition probabilities with the conditional distribution implied by the AR(1) process. Improving on
this method, Hussey and Tauchen (1991) introduce the idea of placing the state space
optimally. In fact, as they argue, the discretization should allow accurately approximating the integral equation that characterizes an economic problem. They show that
a quadrature rule describes the optimal placement.
The method of Rouwenhorst (1995) uses a construction to obtain a discrete Markov
process that exactly matches the conditional and unconditional mean and variance and
the autocorrelation of an AR(1) process. This calibration offers a dramatic improvement, especially in those applications where the persistence of the autoregressive process
is high.
Our study also relates to several other papers comparing and establishing the properties of these competing methods, often introducing some improvements as in Adda and
Cooper (2003). Flodén (2008) compares the relative performance of Tauchen (1986),
Hussey and Tauchen (1991), and Adda and Cooper (2003) in approximating an AR(1)
process and proposes a version of Hussey and Tauchen (1991) that is more accurate than
4
the original. Terry and Knotek II (2011) introduce a method based on Tauchen (1986)
that is suitable for vector autoregressive processes. Galindev and Lkhagvasuren (2010)
develop an approximation method for vector autoregressive processes with correlated
error terms. Kopecky and Suen (2010) consider the impact of different discretization
procedures on the solutions of a macroeconomic model. Gospodinov and Lkhagvasuren
(2014) develop a moment-matching method for approximating a persistent vector autoregression with a finite-state Markov chain. Our method is similar because it is based
on a moment-matching procedure, although in this paper we focus on unconditional
higher order moments. Finally, Farmer and Toda (2015) develop a computationally
tractable method to match conditional moments.
More recently, a large body of literature has shown that higher order moments display interesting empirical patterns, which can have major economic implications. In the
literature of earning dynamics, an early example is Geweke and Keane (1997), who fit a
normal mixture model to earnings innovations using Panel Study of Income Dynamics
(PSID) data. Guvenen et al. (2015), using US Social Security data, report that the innovations to income are characterized by sizable skewness and high kurtosis. Bonhomme
and Robin (2010) use PSID data to document excess kurtosis of the changes in earnings.
Bloom et al. (2012) show that both the distribution of total factor productivity shocks
and sales growth in the United States display negative skewness and excess kurtosis.
Higson et al. (2002) study the correlation between aggregate business cycle fluctuations
and higher moments of the cross-sectional distribution of growth sales. They report
a negative correlation between the GDP growth rate and the cross-sectional variance
and skewness of sales growth rates. They observe the opposite relationship for kurtosis
because it displays a positive correlation at business cycle frequencies. Bachmann et al.
(2015) look at the higher order moments of investment innovations. They find sizable
excess kurtosis but no significant skewness.
In the finance literature, several studies have documented that the returns on financial assets are leptokurtic—feature fat tails—and, in many cases, feature non-zero
skewness. An example is the early work of Mandelbrot (1963) and Fama (1965). More
recent studies are, among others, Liu and Brorsen (1995) and Chiang and Li (2015).
In light of these findings, our study sheds light on how to specify a process that
features non-zero skewness and excess kurtosis, and provides two discretization tools to
implement the process numerically.
5
3
Kurtosis of the Innovations
In this section, we discuss the importance of testing the higher order moments of a
calibrated Markov chain when discretizing an AR(1) process with Gaussian innovations.
In fact, we uncover a calibration error that is common to most familiar discretization
methods, and we illustrate how it can have economic implications. This motivates the
use of a method that is free from this error such as the Extended Tauchen method
proposed in Section 5.
First we introduce some notation. Let y denote a continuous process, with an AR(1)
representation,
yt = ρyt−1 + t ,
(1)
where |ρ| < 1 and t ∼ N (0, σ 2 ). The first differences of the process are denoted by
∆yt = yt − yt−1 . We will denote the approximating Markov chain by (z, T ), where
z = (z1 , . . . , zN ) is a vector of states and T is the transition matrix with typical element
Tij . Let x denote the random variable distributed according to (z, T ) and e the residual
obtained as et = xt − ρ̂xt−1 , where ρ̂ is the first-order autocorrelation of x. S and K
denote skewness and kurtosis, respectively.
Now we consider how some of the most commonly used discretization methods
perform in matching the moments of y and . In this paper, all the tables reporting the
performance of a discretization method use a format that is standard in this literature.
For each method and for each moment of interest, the ratio of the moment associated
with the approximating Markov chain over the true value of the moment evaluates the
performance of the method. A value of one indicates that the approximating Markov
chain perfectly matches the moment.
Table 1 shows that for a highly persistent AR(1) process, standard discretization
methods feature a very high kurtosis in the innovation. For example, if we consider a
9-state Markov chain with an autocorrelation of 0.99 and use Rouwenhorst’s method,
the kurtosis of the innovation implied by the discretization method is about 27, eight
times higher than the kurtosis of the Gaussian innovation, which equals 3. The method
of Hussey and Tauchen (1991) performs much better with respect to the kurtosis of the
innovations, but it misses the variance of the levels entirely. A notable exception is the
method of Flodén (2008), which is more accurate and reasonably matches the moments
of the innovations and the levels, though it becomes less precise as ρ approaches 1.
6
The results in Table 1 highlight the significant implications that a discretization
method has on the moments of the innovations. While a discretization method may
well match the moments of the levels, it may also feature incorrect moments of the
innovations. To the best of our knowledge, the literature has overlooked this issue, and
in Section 3.1 we illustrate how it can have major economic consequences in the framework of a typical income fluctuation problem. For a comparison of these discretization
methods with the Extended Tauchen method, we refer the reader to Section 5.1.
3.1
The Income Fluctuation Problem
In this section we consider a typical income fluctuation problem, as in Aiyagari (1994),
to illustrate that our findings in Table 1 are economically consequential. Specifically,
we show that the excess kurtosis arising from a calibration error affects the agent’s
optimal behavior. The household problem is
∞
1−µ
X
t ct
max. E0
β
{ct ,at+1 }
1−µ
t=0
s.t.
(2)
ct + at+1 = (1 + r)at + wlt
where the utility function exhibits constant relative risk aversion and at+1 ≥ 0. The
interest and wage rate are set to their equilibrium values. The log-labor endowment
follows a Gaussian AR(1) process:
log(lt+1 ) = ρ log(lt ) + σ(1 − ρ2 )1/2 t , t ∼ N (0, 1),
(3)
where ρ = 0.99 and σ = 0.4. To solve this problem numerically, we discretize the loglabor endowment process using the method of Rouwenhorst (1995). We calibrate three
Markov chains: with 9, 19, and 49 states. The only relevant difference among these
discrete processes is how well they match the kurtosis of the innovation. As Table 2
shows, how well the processes match the kurtosis of the levels is negligible.
The lower the cardinality of the state space, the higher the kurtosis of the innovations. The agents in the economy with a 9-state Markov process are effectively hit by
shocks with almost four times as much kurtosis as the agents in the economy with a
49-state Markov process.
To understand the implications of the shocks’ kurtosis, we inspect the policy function
for capital under the different Markov processes. To ensure that the policy functions
are comparable under different specifications, we consider the policy function associated
7
Tauchen (1986)
ρ
0.5
0.9
0.99
0.999
0.9999
N =9
N =19
N =49
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
0.998
0.998
1.008
NaN
NaN
1.057
1.219
1.651
NaN
NaN
0.976
0.948
0.876
NaN
NaN
1.058
1.238
0.227
NaN
NaN
0.984
1.007
41.42
NaN
NaN
0.998
0.999
1.000
1.001
NaN
1.006
1.033
1.329
1.636
NaN
0.974
0.960
0.900
0.842
NaN
1.008
1.045
1.330
0.011
NaN
0.983
0.997
1.416
1727
NaN
0.998
0.999
1.000
1.000
NaN
0.995
0.993
1.037
1.374
NaN
0.973
0.962
0.942
0.874
NaN
0.997
1.004
1.064
1.266
NaN
0.982
0.998
0.999
2.060
NaN
Rouwenhorst (1995)
ρ
0.5
0.9
0.99
0.999
0.9999
N =9
N =19
N =49
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.917
0.917
0.917
0.917
0.917
1.000
1.000
1.000
1.000
1.000
0.972
1.627
9.125
84.12
834.1
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.963
0.963
0.963
0.963
0.963
1.000
1.000
1.000
1.000
1.000
0.988
1.279
4.611
37.94
371.2
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.986
0.986
0.986
0.986
0.986
1.000
1.000
1.000
1.000
1.000
0.995
1.105
2.354
14.85
139.8
Adda-Cooper (2003)
ρ
0.5
0.9
0.99
0.999
0.9999
N =9
N =19
N =49
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
0.959
0.974
0.985
0.993
0.998
0.953
0.953
0.953
0.953
0.953
0.773
0.773
0.773
0.773
0.773
0.979
1.160
2.390
7.497
23.76
0.872
1.090
2.161
6.865
21.80
0.984
0.990
0.994
0.997
0.999
0.982
0.982
0.982
0.982
0.982
0.875
0.875
0.875
0.875
0.875
0.993
1.066
1.560
4.169
13.19
0.932
1.052
1.832
6.505
20.62
0.995
0.997
0.998
0.999
1.000
0.995
0.995
0.995
0.995
0.995
0.945
0.945
0.945
0.945
0.945
0.998
1.022
1.172
2.048
5.945
0.971
1.023
1.378
6.721
24.58
Hussey & Tauchen (1991)
ρ
0.5
0.9
0.99
0.999
0.9999
N =9
N =19
N =49
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
1.000
0.985
0.953
0.948
0.948
1.000
0.860
0.159
0.017
0.002
1.000
0.833
0.622
0.601
0.599
1.000
0.972
0.881
0.866
0.864
1.000
0.997
1.049
1.060
1.061
1.000
0.999
0.982
0.977
0.977
1.000
0.994
0.343
0.040
0.004
1.000
0.981
0.660
0.609
0.604
1.000
0.999
0.947
0.926
0.924
1.000
0.999
1.019
1.030
1.032
1.000
1.000
0.995
0.991
0.991
1.000
1.000
0.676
0.100
0.010
1.000
1.000
0.753
0.621
0.608
1.000
1.000
0.988
0.971
0.969
1.000
1.000
1.001
1.005
1.005
Flodén (2008)
ρ
0.5
0.9
0.99
0.999
0.9999
N =9
N =19
N =49
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
1.000
0.999
1.006
NaN
NaN
1.000
0.989
0.821
NaN
NaN
1.000
0.962
0.718
NaN
NaN
1.000
0.994
0.300
NaN
NaN
1.000
1.014
8.163
NaN
NaN
1.000
1.000
1.001
NaN
NaN
1.000
1.000
1.001
NaN
NaN
1.000
1.000
0.940
NaN
NaN
1.000
1.000
0.886
NaN
NaN
1.000
1.000
1.430
NaN
NaN
1.000
1.000
1.000
NaN
NaN
1.000
1.000
1.000
NaN
NaN
1.000
1.000
0.999
NaN
NaN
1.000
1.000
1.000
NaN
NaN
1.000
1.000
1.000
NaN
NaN
Table 1: Moments of a discretized Gaussian AR(1) process
This table shows the performance of the methods of Tauchen (1986), Rouwenhorst (1995), Adda and
Cooper (2003), Hussey and Tauchen (1991), and Flodén (2008) in matching the variance and kurtosis
of the levels and the innovations of a Gaussian AR(1) process. For each method and for each moment
of interest, the performance is evaluated as a ratio of the moment associated with the approximating
Markov chain over the value of the moment of the AR(1) process. A value of one indicates that the
approximating Markov chain perfectly matches the moment. We display the results for different values
of ρ and for different cardinalities of the state space, denoted by N . The variance σ 2 of the AR(1)
process is set to 1 without loss of generality. Notice that the mean and skewness of the process are
equal to zero in this application, and all methods match them by construction.
8
Table 2: Rouwenhorst calibrations
Rouwenhorst (1995)
ρ = 0.99
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
N =9
N =19
N =49
1.000
1.000
1.000
1.000
1.000
1.000
0.917
0.963
0.986
1.000
1.000
1.000
9.125
4.611
2.354
with the shock at the center of the state space under the three specifications.1 In
Figure 1 we plot the percentage difference between the policy functions.
Figure 1 shows two important facts: i) the kurtosis of the innovation changes the
optimal behavior of the agents who are close to hitting the borrowing constraint, and ii)
the higher the coefficient of risk aversion, the greater the difference between the policy
functions. Since higher kurtosis means that more extreme realizations of the shocks
are more likely, we interpret Figure 1 as an increase in precautionary savings to avoid
hitting the borrowing constraint in comparison with the 19-state or 49-state economies.
Our interpretation is consistent with the second finding. As the risk aversion of the
agents increases, precautionary saving increases. Finally, the percentage difference is
sizable, and it amounts to up to 20% when µ = 5.
To better understand how important these findings are, we inspect agents’ asset
holdings in the economies with 9-state and 49-state shocks. In Figure 2 we report the
stationary distribution of agents’ capital. Since most agents hold less than one unit of
capital, the difference in the policy functions is relevant for a sizable mass of agents.
This simple example illustrates how a calibration error—common to several familiar
discretization methods—can have important economic consequences, and it motivates
the adoption of a discretization method that correctly calibrates higher order moments.
4
More Flexible Processes
Several empirical studies have recently shown that a process with non-Gaussian innovations governs key economic variables; see section Section 2 for a review of this
literature. More specifically, the levels and the differences of some processes display
non-zero skewness and high kurtosis, which are impossible to reproduce using an au1
Prices do not change across specifications, which makes these policy functions comparable.
9
Figure 1: Policy function differences.
This figure shows the difference between the policy functions for capital in an Aiyagari economy. Three different policy functions are computed under a discrete process for log-labor
endowment with 9, 19, and 49 states, obtained using the method of Rouwenhorst (1995). We
then compute the percentage difference between them using a middle point rule.
=
=
-
-
-
%
-
Figure 2: Stationary distribution.
This figure shows two stationary distributions of agents over capital in an Aiyagari economy.
The two distributions are computed under a discrete processes for log-labor endowment with
9 and 49 states, obtained using the method of Rouwenhorst (1995). The density of the agents
within each bin is reported as a percentage.
Density
30
9 states
49 states
20
10
1
2
3
Capital
10
4
toregressive process with Gaussian innovations. Furthermore, as shown in Guvenen
et al. (2015), skewness and kurtosis can display substantial persistence; that is, the
differences in a process several periods apart can display remarkably non-Gaussian
skewness and kurtosis.
In this section, we study two continuous processes that are flexible enough to capture
these patterns. In particular, we focus on a first-order autoregressive process with
normal mixture innovations (NMAR) and an NMAR process with transitory shocks
(NMART). To understand how flexible these processes are, we study their moment
structure, and we illustrate their calibration by targeting the moment structure of the
income process found in Guvenen et al. (2015).
We show that NMAR is flexible enough to reproduce non-zero skewness and high
kurtosis for both the levels and the innovations of the process. However, for generating
persistent skewness and kurtosis, we find that NMART is a better choice. The findings
we discuss in this section are auxiliary to Section 5, since the Extended Tauchen method
takes a calibrated NMAR process as an input.
4.1
NMAR Process
A first-order autoregressive process with normal mixture innovations (NMAR) has the
following representation:
yt = ρyt−1 + ηt ,
where
(
ηt ∼
N (µ1 , σ12 )
with probability p1 ,
N (µ2 , σ22 )
with probability p2 ,
(4)
and p2 = 1 − p1 . Geweke and Keane (1997) and Kon (1984) among others have used
this type of process previously. If necessary, ηt can be generalized to be a mixture of
more than two normals. We will denote ∆yt = yt − yt−1 and ∆k yt = yt − yt−k . In
Appendix A2 we report the mean, variance, skewness, and kurtosis of η, y, ∆y, and
∆5 y.
Inspecting this structure yields an interesting fact: the raw moments of η, together
with ρ, uniquely pin down the mean, variance, skewness, and kurtosis of η, y, ∆y,
and ∆5 y. This mapping establishes a calibration trade-off between the moments; for
example, matching the moments of y can imply incorrectly calibrating ∆y and vice
11
versa.
Furthermore, under NMAR ρ determines the persistence of the skewness and kurtosis together with the raw moments of the process. To see this fact, in Appendix A2 we
write S(∆5 y) as a linear function of S(∆y) and K(∆5 y) as a linear function of K(∆y):
B
S(∆yt ),
A
D
K(∆5 y) = 3 + [K(∆, y) − 3] ,
C
S(∆5 yt ) =
where A,B,C, and D are nonlinear functions of ρ, as reported in Equation 20 and 21.
Clearly, the choice of ρ and S(∆yt ) uniquely pins down S(∆5 yt ), establishing a
trade-off in the calibration of the process between matching the moments of the first
differences and the moments of the fifth differences. The same happens with K(∆5 yt ),
which is uniquely pinned down by ρ and K(∆yt ).
To explore these trade-offs more clearly, in Figure 3 we plot S(∆5 y) as a function of
S(∆y) and K(∆5 y) as a function of K(∆y). To put this graph in context, we consider the
data moments reported in Guvenen et al. (2015). They document that for log-income
S(∆y) = −1.35 and S(∆5 y) = −1.01, whereas K(∆y) = 17.80 and K(∆5 y) = 11.55.
As is clear in the graph, the NMAR process cannot be calibrated in a manner that is
consistent with these moments.
We conclude that NMAR is not suitable for generating persistent skewness and
kurtosis. In Section 4.2 we discuss an NMART process, which generalizes an NMAR
process and can feature persistent skewness and kurtosis. Since NMAR is a special case
of NMART, we refer the reader to Section 4.2 for further discussion of the properties
of an NMAR process and what combinations of skewness and kurtosis are attainable
under NMAR.
In the remaining part of this section, we consider an application using the Generalized Method of Moments (GMM) to calibrate an NMAR process targeting the data
moments found in Guvenen et al. (2015). Table 3 reports the results. Six parameters (ρ, p1 , µ1 , µ2 , σ1 , σ2 ) govern NMAR with two normals. The strategy we follow sets
ρ = 0.99, p1 = 0.9, and imposes µ2 = −p1 µ1 / (1 − p1 ) to ensure that E (η) = 0. This
leaves 3 parameters to calibrate, (µ1 , σ1 , σ2 ). Table 4 reports the parametrization of the
NMAR that yields exactly the moments targeted in Table 3, and in Table 5 we report
the complete moment structure of the calibrated process.
12
Figure 3: S(∆5 y) and K(∆5 y)
This figure shows S(∆5 y) and K(∆5 y) as a function of ρ, given that S(∆y) = −1.35 and
K(∆y) = 17.8.
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Table 3: Targeted data moments
Var(∆y)
0.23
E(∆y)
0
S(∆y)
-1.35
K(∆y)
17.8
Table 4: NMAR calibration
µ1
0.0336
µ2
-0.3021
σ12
0.0574
σ22
1.6749
p1
0.9000
Table 5: Moments of the calibrated process
E(ηt ) = 0
Var(ηt ) = 0.23
S(ηt ) = −1.36
K(ηt ) = 17.95
E(yt ) = 0
Var(yt ) = 11.50
S(yt ) = −0.12
K(yt ) = 3.15
13
E(∆yt ) = 0
Var(∆yt ) = 0.23
S(∆yt ) = −1.35
K(∆yt ) = 17.80
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4.2
NMART Process
In this section, we introduce the NMART process, an autoregressive process with normal
mixture shocks and a transitory shock. In fact, NMART is an NMAR process plus a
transitory Gaussian shock. As we discuss in this section, including the transitory shock
is crucial to modeling persistent skewness and kurtosis.
The NMART process has the following representation:
yt = zt + t
(5)
zt = ρzt−1 + ηt ,
where
(
ηt ∼
N (µ1 , σ12 )
with probability p1 ,
N (µ2 , σ22 )
with probability p2 ,
t ∼ N (0, σ2 ).
To simplify the exposition, we write the variance of the transitory shock as a constant α
of the variance of the permanent shock, that is, Var (t ) = αVar (ηt ). Clearly, if α = 0,
then yt is NMAR.
To appreciate why the transitory component of the process plays a central role in
modeling persistent skewness and kurtosis, in Appendix A3 we obtain analytical formulas of variance, skewness, and kurtosis of ∆t y. Our main findings can be summarized
as follows:
(a) Var (∆t y) is a linear function of Var(η). For a given value of Var(η), Var (∆t y) is
increasing in α.
To study the persistence of variance, one can rewrite Var (∆t y) as a function of
Var (∆y): the relationship is linear, and it is a function of α, ρ, and t. As expected,
for a given value of Var (∆y), Var (∆t y) is decreasing in α. In fact, as α increases,
the magnitude of the temporary shock increases relative to the permanent shock.
(b) For the case S(η) < 0, S (∆t y) is linear in S(η), and it also depends on α, ρ, and t:
S (∆t y) is increasing in α for a given S(η).2
To study the persistence of skewness, one can rewrite S (∆t y) as a linear function
of S (∆y), which also depends on α, ρ, and t. For a given value of S (∆y), S (∆t y) is
2
If S(η) > 0, then S (∆t y) is decreasing in α and S (∆t y) /S (∆y) is increasing in α.
14
decreasing in α. This last fact is particularly important because it allows the process
to capture the persistence in skewness; increasing α increases S (∆t y) /S (∆y).
(c) K (∆t y) is linear in K(η), and it varies with α, ρ, and t. K (∆t y) is decreasing in α
for a given K(η).
To study the persistence of kurtosis, one can rewrite K (∆t y) as a linear function
of K (∆y), which also depends on α, ρ, and t. For a given value of K (∆y), K (∆t y)
is increasing in α. This feature allows the process to capture the persistence in
kurtosis, as increasing α increases K (∆t y) /K (∆y).
In Figure 4 we plot how the persistence of variance, skewness, and kurtosis varies
as a function of α. We repeat this exercise for selected values of ρ and assume that
Var(∆y), S(∆y), and K(∆y) take on the values in Guvenen et al. (2015). To illustrate
the pattern of persistence, we plot the relationships reported in Equation 27, which
allow us to express Var(∆t y) as a function of Var(∆y), S(∆t y) as a function of S(∆y),
and K(∆t y) as a function of K(∆y).
These findings have shown that varying α can increase the persistence of skewness
and kurtosis, but we also explore what combinations of variance, skewness, and kurtosis
are achievable. In fact, two issues affect what combinations of moments are attainable:
(i) since ηt is a mixture of two normals, it may not admit a calibration featuring some
combinations of {Var(η), S(η), K(η)}, and (ii) since the moments of ∆k y are a function
of α, ρ, and the moments of η and , some combinations of {Var(∆k y), S(∆k y), K(∆k y)}
might not be possible to attain.
We address this issue with a numerical exercise that explores what combinations of {Var(η), S(η), K(η)} are achievable.
Using Equation 26, we map the at-
tainable combinations of {Var(η), S(η), K(η)} into the attainable combinations of
{Var(∆k y), S(∆k y), K(∆k y)} for selected values of ρ, α, and k. This numerical exercise is also important for understanding the properties of an NMAR process, since
NMART equals an NMAR process when α = 0.
In the first step of this numerical exercise, we calibrate η, targeting combinations
of Var(η) ∈ [0.1, 1], S(η) ∈ [−5, 0], and K(η) ∈ [3, 21]. The first important finding
is that the attainability of a {S(η), K(η)} combination is independent of Var(η). In
light of this fact, we only discuss the feasible combinations of {S(η), K(η)}. In Figure 5
the feasible combinations of {S(η), K(η)} lie northeast (NE) of the blue line. In other
15
Figure 4: Persistence of Variance, Skewness, and Kurtosis as a function of α
�
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In this figure we fix Var(∆y) = 0.23, S(∆y) = −1.35, and K(∆y) = 17.8, as reported in
Guvenen et al. (2015). We then plot Var(∆k y), S(∆k y), and K(∆k y) as a function of α, for
k = 5, 10. Increasing α makes variance less persistent, whereas skewness and kurtosis become
more persistent. When α = 0, the NMART process boils down to an NMAR process.
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Figure 5: Feasible combinations of {S(η), K(η)}
In this figure we show what combinations of kurtosis, K(η), and skewness, S(η), are feasible
when calibrating a normal mixture η. The combinations NE of the blue line are exactly
attainable; there is a calibration of η that delivers exactly any combination {S(η), K(η)} in
the NE region. For each {S(η), K(η)} in the SW region, we calibrate η using a GMM procedure
and report the average percentage absolute deviation from {S(η), K(η)} using a grayscale.
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words, there is a calibration of η that delivers exactly any combination {S(η), K(η)} in
the NE region. The combinations southwest (SW) of the blue line can only be obtained
with some error.
To understand the magnitude of this error, for each {S(η), K(η)} in the SW region
we calibrate η using a GMM procedure, and we compute the average percentage discrepancy of the calibrated skewness and kurtosis from the target {S(η), K(η)}. In Figure 5
we report the error associated with each calibration in the SW region using a grayscale.
A darker color signifies a bigger discrepancy. For example, the error associated with
K(η) = 5 and S(η) = −3 is 20, meaning that the GMM calibration misses the targets
of S and K by an average of 20%.
To put Figure 5 in context, we report the combinations of {S(η), K(η)} found by
Bloom et al. (2012) and Bachmann et al. (2015), which we label B12 and B15, respectively. Both papers find combinations that are feasible under a normal mixture
specification.3
In the second step of this numerical exercise, we investigate what combinations of
{S(∆y), K(∆y)} are feasible under a NMART process by mapping the feasible combi3
In Figure 17, we repeat this first step of the numerical exercise to understand what is to be gained
from using a mixture of three normals rather than two normals.
17
Figure 6: Feasible combinations of {S(∆y), K(∆y)}
In this figure we show what combinations of kurtosis, K(∆y), and skewness, S(∆y), are attainable exactly when calibrating NMART. Each quadrant of the plot is characterized by a
different value of α. Within each quadrant we draw the frontier for three different values of ρ.
The feasible combinations lie NE of each frontier.
(
)
=
=
=
=
-
-
-
-
-
-
=
-
)
(
=
=
=
=
=
=
=
=
-
-
-
-
-
=
-
=
-
=
-
(
)
(
)
nations {S(η), K(η)} into {S(∆y), K(∆y)} using Equation 26. In Figure 6 the feasible
combinations of skewness and kurtosis lie NE of each frontier. In other words, there is an
NMART process that delivers exactly any combination of skewness and kurtosis NE of
each frontier. We report the feasible frontier for α = 0, 0.5, 1, 1.5 and ρ = 0.8, 0.9, 0.99.
To put Figure 6 in context, we report the combinations of {S(∆y), K(∆y)} found
by Bloom et al. (2012) and Guvenen et al. (2015), which we label B12 and G15,
respectively.
Both papers find combinations that are feasible under NMART. Fi-
nally, we repeat this exercise to study what combinations of {S(∆5 y), K(∆5 y)} and
{S(∆10 y), K(∆10 y)} are feasible under an NMART. We report the results in Figure 18
of Appendix A5.
We conclude this section with an application that allows us to compare the persistent
pattern of skewness and kurtosis of NMAR with that of NMART. In Figure 7 we report
the lag-profile of skewness and kurtosis of NMAR, in black, and of NMART, in blue.
Both processes are calibrated to feature S(∆y) = −1.35 and K(∆y) = 17.8 as in
Guvenen et al. (2015). For the NMART process, α is calibrated to 0.94 in order to
attain skewness and kurtosis at lag 5, consistent with Guvenen et al. (2015).
The NMART process is governed by 8 parameters (µ1 , µ2 , σ12 , σ22 , σ2 , ρ, α, p1 ),
18
which we calibrate using GMM. Table 7 reports the parametrization of the NMART
that yields the moments targeted in Table 6 and in Table 8 we report the complete
moment structure of the calibrated process.
Figure 7 shows that a higher value of α makes the process more persistent, allowing
it to attain S(∆5 yt ) = −0.95 and K(∆5 yt ) = 11.30.
5
Extended Tauchen Method
In this section we propose and discuss the Extended Tauchen method, which consists
of a procedure to discretize an autoregressive process with normal mixture innovations.
The method of Tauchen (1986) discretizes a Gaussian AR(1) process. It relies on the
fact that once n states have been fixed, there is a straightforward way to calibrate a
transition probability using the conditional distribution implied by the AR(1) process.
The Extended Tauchen (ET) method we propose differs from Tauchen in two important ways. First, we discretize an autoregressive process with normal mixture innovations (NMAR). The NMAR allows us to calibrate a process with non-zero skewness
and high kurtosis. With the exception that innovations are distributed as a normal
mixture, the transition probability calibration follows Tauchen. Second, the placement
of the state space is chosen optimally with respect to a set of targeted moments; this
allows a sizable gain in the precision of the approximation.
Section 4.1 demonstrated that an NMAR process is flexible enough to feature nonzero skewness and high kurtosis. We also illustrate how to calibrate NMAR using
income data moments. In this section, we explore how to discretize an NMAR process
specified as in Equation 4 and parametrized by θ = (ρ, p1 , µ1 , µ2 , σ1 , σ2 ).
The objective of the procedure is to calibrate a Markov chain, which we denote by
(z, T ), where z is a state vector and T is a transition matrix. To apply this method, the
practitioner also needs to choose a set of moments to be targeted; the choice is based
on what moments of the process are relevant for the specific application.
Let m (θ) denote a mapping from the continuous process into this set of relevant
moments and m̂ (z, T ) a mapping of (z, T ) into the same set of moments. The practitioner also needs to choose a notion of distance between the targets and the moments
of the Markov chain. We denote this distance by |m (θ) − m̂ (z, T )|, shorthand for
[m (θ) − m̂ (z, T )]0 W [m (θ) − m̂ (z, T )], where W is a weighting matrix. Finally, we
19
Figure 7: Lag-profile of Skewness and Kurtosis
This figure plots the lag-profile of skewness and kurtosis of an NMAR process and of an
NMART process with α = 0.94 and ρ = 0.85.
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Table 6: Targeted data moments
E(∆y)
0
Var(∆y)
0.23
S(∆y)
-1.35
E(∆5 y) Var(∆5 y) S(∆5 y)
0
0.46
-1.01
K(∆y)
17.8
K(∆5 y)
11.55
Table 7: NMART calibration
µ1
0.0069
µ2
-3.99
σ12
0.0468
σ22
2.07
σ2
0.0733
ρ
0.85
α
0.9434
Table 8: Moments of the calibrated process
E(yt ) = 0
Var(yt ) = 0.35
S(yt ) = −1.86
K(yt ) = 16.15
E(∆yt ) = 0
Var(∆yt ) = 0.23
S(∆yt ) = −1.35
K(∆yt ) = 17.80
20
E(∆5 yt ) = 0
Var(∆5 yt ) = 0.46
S(∆5 yt ) = −0.95
K(∆5 yt ) = 11.30
p1
0.9983
will denote by F the cumulative distribution function of η.
The Extended Tauchen method we propose has the following steps:
1. Choose the number of states n for the discrete process.
2. Choose a grid of states z = z1 , . . . , zn of dimension n.
3. Compute n + 1 nodes d = d1 , . . . , dn+1 as
di =



 −∞
if i = 1
∞
if i = n + 1


 (z + z ) /2 otherwise.
i−1
i
4. For any two states i and j, calibrate the probability of transition between the two
states Tij as
Tij = Pr {y 0 = zj | y = zi }
= Pr {dj ≤ ρzi + ηt ≤ dj+1 }
= Pr {dj − ρzi ≤ ηt ≤ dj+1 − ρzi }
= F (dj+1 − ρzi ) − F (dj − ρzi ) .
5. Compute the distance |m (θ) − m̂ (z, T )|.
6. Repeat steps from (2) to (5), choosing z that minimizes the distance
|m (θ) − m̂ (z, T )|.
7. Repeat steps from (1) to (6), choosing the minimum n that delivers a distance
|m (θ) − m̂ (z, T )| < δ, where δ is a predetermined threshold.
This method maps NMAR into a discrete process and is for the practitioner who
has in mind a specific representation of a process in the form of an NMAR process, and
for those applications in which no raw data are available and one relies on the findings
of other researchers who have calibrated an NMAR process based on data.
ET has the advantage of being computationally manageable and fast. At each step
of the procedure, mapping the continuous process and the discrete Markov process into
the relevant set of moments follows the formulas we report in Appendix A1 and A2.
Consequently, the procedure consists of a GMM application.
21
When raw data are available, one can follow a two-step procedure. First calibrate
an NMAR process based on the available data, as shown in Section 4.1, and then use
ET to calibrate a Markov chain. On the one hand, ET is fast and easy enough to
justify using this two-step procedure over the more computationally onerous procedure
proposed in Section 6. On the other hand, using ET requires looking at the data
through the lense of an NMAR process, which a practitioner could find restrictive or
unsuitable for her application. For this reason, in Section 6 we introduce the Empirical
Calibration method, which calibrates a Markov chain using the available raw data,
without requiring any assumption on the nature of the data generating process.
To apply this method, the practitioner needs to make a number of choices. The
most important choice is the set of moments to be targeted. This selection is entirely
dependent on the application at hand, but some guidance is valuable. When following
the two-step procedure just described, the set of moments targeted by ET should be a
subset of the moments used to calibrate the NMAR process. Other choices involve the
selection of the weighting matrix W and the threshold δ that identifies an acceptable
discrepancy between m (θ) and m̂ (z, T ). We provide more details on these and other
issues in Section 5.1 and 5.2, where we illustrate the method with two applications.
The illustration in Section 5.1 applies the ET method to an AR(1) process with
Gaussian innovations. Even though in this application the process does not feature any
skewness and excess kurtosis, it establishes a comparison between ET and the existing
discretization methods. The illustration in Section 5.2 puts the method to the test
with an NMAR process that displays skewness and kurtosis that is consistent with
those found for the income process in Guvenen et al. (2015).
5.1
Discretizing an AR(1) process with a Gaussian innovation
This first application establishes a connection with the existing literature and allows a
comparison of the ET method with other existing methods. We discretize an AR(1)
process with Gaussian innovations.
As explained in Section 5, the ET method differs from Tauchen in two ways: (i)
the innovations are distributed as a normal mixture, and (ii) the states are placed
optimally with respect to a set of targeted moments. Because this application only
departs from Tauchen on the second point, it is the ideal experiment to understand
22
what improvement in accuracy we can gain over other methods by placing the states
optimally.
In this application we assume
t ∼ N (0, 1),
yt = ρyt−1 + t ,
and we denote the Markov chain by (z, T ), with xt ∼ (z, T ) and et = xt − ρ̂xt−1 , where
ρ̂ is the first-order autocorrelation associated with xt .
Since the innovations are symmetrically distributed around 0, in this application
we impose that z is symmetric around 0. Since E (x) = E (e) = S (x) = S (e) = 0 by
construction, in our application we target ρ, Var (y) , Var () , K (y) , K (). In this case
Var () = 1, K (y) = K () = 3 whereas the value of Var (y) changes with ρ.
Our choice of the distance function |m (y) − m̂ (z, T )| is the following:
2 2 2
ρ (x) − ρ (y)
Var (x) − Var (y)
Var (e) − Var ()
δ(z, T ) =
+
+
ρ (y)
Var (y)
Var ()
2 2
K (x) − K (y)
K (e) − K ()
+
+
,
K (y)
K ()
(6)
where each squared percentage deviation is weighted equally. We choose the sum of
squared percentage deviations because it is scale-independent and easy to minimize.
In Table 9 we report the performance of our calibration. The results are reported in
standard format; the ratio of the moment associated with the approximating Markov
chain over the true value of the moment evaluates the performance of the method.
A value of one indicates that the approximating Markov chain perfectly matches the
moment. The calibration is reported for ρ = 0.95, 0.99, and N = 9, 15, 19. In Appendix A4 we also report the results for N = 2, 5, 29. Since the criterion function
used in the minimization, as reported in Equation 6, can be difficult to interpret, in the
tables we report the following measure of goodness of approximation:
1
δ̂(z, T ) =
5
(
ρ (x) − ρ (y) Var (x) − Var (y) Var (e) − Var () +
+
ρ (y)
Var (y)
Var ()
)
K (x) − K (y) K (e) − K () +
,
+ K (y)
K ()
23
(7)
Table 9: Extended Tauchen, Gaussian AR(1)
This table compares the performance of the Extended Tauchen method with the other discretization
methods common in the literature. A more comprehensive version of this table, with N =2, 5, and 29,
is in Appendix A4. The results are reported in standard format; the ratio of the moment associated
with the approximating Markov chain over the true value of the moment evaluates the performance
of the method. A value of one indicates that the approximating Markov chain perfectly matches the
moment.
Method
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
ρ
N
0.95
9
0.95
15
0.95
19
0.99
9
0.99
15
0.99
19
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
1.000
1.001
0.978
0.970
1.000
0.986
0.989
1.000
0.999
0.989
0.990
1.000
0.994
0.995
1.000
0.999
0.992
0.995
1.000
0.996
0.996
1.000
1.009
0.985
0.953
1.006
0.985
0.996
1.000
1.002
0.992
0.975
1.003
0.997
0.997
1.000
1.000
0.994
0.982
1.001
0.998
0.998
1.000
1.374
0.953
0.602
0.950
0.934
0.983
1.000
1.129
0.976
0.827
0.996
0.974
0.994
1.000
1.072
0.982
0.905
0.999
0.985
0.996
1.000
1.651
0.953
0.159
0.821
0.488
0.719
1.000
1.442
0.976
0.272
0.983
0.873
0.940
1.000
1.329
0.982
0.343
1.001
0.926
0.985
0.917
0.923
0.773
0.720
0.890
0.928
0.982
0.952
0.944
0.848
0.818
0.984
0.987
0.997
0.963
0.950
0.875
0.871
0.996
0.993
0.998
0.917
0.876
0.773
0.622
0.718
0.661
0.827
0.952
0.886
0.848
0.646
0.875
0.964
0.968
0.963
0.900
0.875
0.660
0.940
0.970
1.001
1.000
1.340
1.334
0.935
0.954
1.168
1.189
1.000
1.153
1.179
0.979
0.998
1.084
1.092
1.000
1.092
1.134
0.990
1.000
1.057
1.061
1.000
0.227
2.390
0.881
0.300
1.207
0.987
1.000
1.190
1.751
0.930
0.738
1.135
1.182
1.000
1.330
1.560
0.947
0.886
1.142
1.136
2.459
1.405
1.208
1.014
1.143
1.081
1.125
1.834
0.994
1.140
1.002
1.005
1.017
1.014
1.649
0.997
1.114
1.000
1.000
1.008
1.007
9.125
41.42
2.161
1.049
8.163
1.262
1.876
5.643
2.562
1.944
1.026
2.080
1.242
1.272
4.611
1.416
1.832
1.019
1.430
1.116
1.162
24
Av. % Dev.
30.86
23.95
16.75
15.76
6.99
8.03
7.20
17.63
6.90
10.14
7.77
0.55
2.90
2.42
13.71
4.36
8.00
4.80
0.10
1.84
1.55
164.1
839.5
56.82
28.68
166.6
26.69
26.93
93.81
46.07
37.57
24.03
29.74
10.87
10.96
72.97
23.52
30.81
21.75
12.13
7.31
6.30
which is easily interpreted as the average percentage absolute deviation of the calibrated
moments from the targets.
Under the label “Adjusted weights”, we report the results obtained by implementing
a different weighting scheme, which puts more weight on ρ(y), Var(y), and Var(); these
are the moments most commonly used in the literature to assess the performance of a
discretization method. The results show that ET performs much better at matching
K(y) and K(), and performs better than most methods at matching ρ(y), Var(y), and
Var(). Our results show that ET performs better than all the other methods except for
that of Flodén (2008) when ρ = 0.95. When ρ = 0.99, ET performs better than all the
other methods we consider. The most remarkable feature of ET is that it matches the
moments of and also matches the moments of y in a satisfactory way. These results
illustrate that placing the state space optimally entails a sizable gain in precision.
Figure 8 concludes this application by plotting the stationary distribution and the
histogram of the innovations of the process calibrated with ET. To put these figures in
context, in Appendix A5 we plot the same objects when the process is calibrated with
the other discretization methods common in the literature.
5.2
Discretizing NMAR
The second application of the ET method discretizes an NMAR process and illustrates
how this method allows us to calibrate a Markov process with non-zero skewness and
excess kurtosis. In this application, we assume an NMAR process as in Equation 4.
The parameters of the process are set so that the moments of the first differences of
the process are consistent with the values found by Guvenen et al. (2015), reported in
Table 3. In this application, we target the moments of the innovations reported in the
first column of Table 5. Because we discretize a process with non-zero skewness, the
state space is not assumed to be symmetric.
We target 7 moments: Var(y), Var(η), S(y), S(η), K(y), K(η), together with ρ. The
conditions E(y) = E(η) = 0 are satisfied by construction and need not be targeted.
The practitioner who uses this method and wants the process to feature a non-zero
mean should explicitly target it. For this application, we define the distance function
similarly as in Equation 6.
Table 10 shows the results from the calibration. For ease of interpretation, we
25
Figure 8: Extended Tauchen, Gaussian AR(1): ρ = 0.99, N=9.
This figure displays the histogram of the innovations, that is, Eij = zj − ρ(x)zi , and the
histogram of the levels x implied by a Markov chain calibrated with the Extended Tauchen
method (equal weights). In this example, there are 9 states, the variance of the shocks equals
1 and ρ = 0.99. In the second panel of this figure, we plot the histogram of the innovations
again, but we rescale the y-axis by applying a cubic root transformation in order to make the
tails conspicuous. For a comparison with other existing methods, see Appendix A5.
Histogram of the innovations
0.08
0.05
0.02
−4
−2
0
E
2
4
ij
Innovations, scaled
0.05
0.01
0.001
−4
−2
0
Eij
2
4
Histogram of the levels
0.12
0.07
0.02
−10
−5
0
zi
26
5
10
Table 10: Extended Tauchen, NMAR
This table shows the performance of the Extended Tauchen method when applied to NMAR.
The results are reported in standard format; the ratio of the moment associated with the
approximating Markov chain over the true value of the moment evaluates the performance of
the method. A value of one indicates that the approximating Markov chain perfectly matches
the moment.
ρ
0.5
0.9
0.95
0.99
N
ρ̂
ρ
Var(x)
Var(y)
S(x)
S(y)
K(x)
K(y)
Var(e)
Var(η)
S(e)
S(η)
K(e)
K(η)
5
9
15
19
5
9
15
19
5
9
15
19
5
9
15
19
0.967
0.999
0.999
1.000
1.029
1.013
1.003
0.999
1.005
1.006
1.003
1.003
0.968
0.994
0.999
1.000
1.005
1.002
1.000
1.000
1.088
1.098
1.035
1.015
0.878
1.072
1.050
1.063
0.176
0.534
0.890
1.021
0.998
1.000
1.000
1.000
1.015
1.016
1.000
0.993
1.023
1.007
1.008
1.008
1.016
1.002
1.001
1.010
1.033
1.008
1.000
1.000
0.621
0.831
0.984
0.992
0.673
0.844
0.920
0.962
0.680
0.706
0.886
0.927
1.027
1.003
1.000
1.000
0.812
0.974
1.012
1.022
0.795
0.945
0.988
1.003
0.717
0.869
0.966
1.001
1.006
1.000
1.000
1.000
1.119
1.064
1.011
0.997
0.921
1.072
1.033
1.030
0.812
0.770
0.838
0.857
0.942
0.991
1.000
1.000
0.985
0.861
0.904
0.924
1.226
1.043
0.946
0.970
1.237
1.224
1.107
1.048
Av. % Dev.
2.341
0.349
0.038
0.000
11.902
7.518
2.468
1.885
14.094
5.884
3.420
2.497
27.143
19.325
7.559
4.240
report the average percentage absolute deviation of the moments from the targets as
in Equation 7. We repeat the exercise for ρ = 0.5, 0.9, 0.95, 0.99, and N = 5, 9, 15,
19. For every specification, the moments of η remain constant at the levels reported in
Table 5, whereas the moments of y change with ρ.
We observe that the performance improves with the number of states and that a
more persistent process is calibrated less accurately. Since we are not aware of any
previous work done in this direction, we do not have a benchmark against which to
compare our method.
Figure 9 concludes this application by plotting the histogram of the innovations and
the stationary distribution of the calibrated process for ρ = 0.95 and for N = 9, 19.
27
Figure 9: Extended Tauchen, NMAR
This figure displays the histogram of the innovations, that is, Eij = zj − ρ(x)zi , and the
histogram of the levels x implied by a Markov chain calibrated with the Extended Tauchen
method. In this example, ρ = 0.95, and we report the results for 9 and 19 states. Notice
that in the first and third panel of this figure, we rescale the y-axis by applying a cubic root
transformation in order to make the tails of the distribution conspicuous.
Innovations, ;=0.95, N=9
Var(2 ) = 0.23
S(2 ) = -1.36
K(2 ) = 17.95
0.5
0.1
0.01
-5
-2.5
0
2.5
5
E ij
Levels, ;=0.95, N=9
0.25
Var(y) = 2.36
S(y) = -0.30
K(y) = 3.77
0.14
0.03
-5
-2.5
0
2.5
5
zi
Innovations, ;=0.95, N=19
0.5
0.1
0.01
-5
-2.5
0
2.5
5
E ij
Levels, ;=0.95, N=19
0.14
0.08
0.02
-5
-2.5
0
z
28
i
2.5
5
6
Empirical Calibration Method
The Empirical Calibration (EC) method follows a nonparametric approach to calibrating a Markov chain. The practitioner who uses this method remains agnostic about the
structure of the data generating process. This method identifies a transition probability
by the frequency of the transitions observed in the data. Clearly, a downside of the
method is the raw data requirements.
Assume we have t realizations of a stochastic process y = {y1 , . . . , yt }, and denote
the Markov chain to be calibrated by (z, T ), where z is a state vector and T is a
transition matrix. As in ET we calibrate (z, T ) to match a specific set of moments of
y. Denote by m (y) a mapping from the available data into the set of relevant moments
and by m̂ (z, T ) a mapping of (z, T ) into the same set of moments.
For applying EC, the practitioner needs to choose a notion of distance between
the targets and the moments of the Markov chain.
We denote this distance by
0
|m (y) − m̂ (z, T )|, shorthand for [m (y) − m̂ (z, T )] W [m (y) − m̂ (z, T )], where W is
a weighting matrix.
The Empirical Calibration method we propose has the following steps:
1. Choose the number of states n for the discrete process.
2. Choose a grid of states z = {z1 , . . . , zn } of dimension n.
3. Having chosen z, map the realizations of the stochastic process y into a sequence
of realizations of a discrete process {x1 , . . . , xt }, according to
xs = arg min |ys − w| .
w∈z
4. For any two states i and j, compute the probability of transition between the two
states Tij as
Tij =
|{h : xh = zi , xh+1 = zj }|
,
|{h : xh = zi }|
where |.| denotes the cardinality of the set.
5. Compute the distance |m (y) − m̂ (z, T )|.
6. Repeat steps from (3) to (5), choosing z that minimizes the distance
|m (y) − m̂ (z, T )|.
29
7. Repeat steps from (1) to (6), choosing the minimum n that delivers a distance
|m (y) − m̂ (z, T )| < δ, where δ is a predetermined threshold.
The greatest advantage in using EC is that no assumption is required on the data
generating process; the practitioner does not need to calibrate any continuous process
or to make any parametric assumption. Furthermore, this approach delivers a Markov
chain that is “as close as possible” to the realized process, as the transition matrix
is determined based only on the observed data. We expect this method to deliver a
discrete process that mimics the data more closely than a method that targets the same
moments but does not use all the information contained in the data.
A drawback of this method compared with ET is its raw data requirements and
significantly slower computational time. Additionally, calibrating a Markov chain with
many states requires a large amount of data in order to identify the transition probabilities with satisfactory precision.
As with the ET method, using EC requires choosing the set of moments to target,
the weighting matrix W , and the threshold δ that identifies an acceptable discrepancy
between m (y) and m̂ (z, T ).
The rest of this section illustrates the EC method with two applications. In Section 6.1, we consider an application that parallels that of Section 5.2. In fact, we
generate data from the NMAR process that we discretize in Section 5.2. We then apply
EC to this data to calibrate a Markov chain. In Section 6.2, we apply EC to data
generated by using the NMART process calibrated in Section 4.2. This application
allows us to illustrate how EC performs when dealing with data that feature persistent
skewness and kurtosis.
6.1
Discretizing NMAR
This application of the EC method is parallel to that in Section 5.2. We generate
100, 000 realizations from the calibrated NMAR process and apply EC to calibrate a
Markov chain.
In this application, differently from before, we target the moments of the differences
of the process rather than the moments of the innovations. In fact, if no specific
assumption is made about the data generating process, the differences of the process
are observable in the data, whereas the innovations are not. Therefore, we target 7
30
moments: Var(y), Var(∆y), S(y), S(∆y), K(y), K(∆y), and ρ. The criterion function
we minimize is the sum of the squared percentage deviations—as in Equation 6—but
for ease of interpretation, we report the performance of the approximation in terms of
the average percentage absolute deviation from the targets—as in Equation 7.
Table 11 reports the performance of EC for ρ = 0.5, 0.9, 0.95, 0.99, and N = 5, 9,
15, 19. The table shows that a more persistent process is calibrated less precisely and
that the performance improves with the cardinality of the state space.
Comparing Table 11 with Table 10, we observe that ET performs better than EC.
This result stems from the fact that EC requires an intermediate simulation step,
whereas the ET method calibrates the transition probabilities analytically using the
density function of the NMAR shocks. This illustrates that to accurately calibrate a
Markov chain with EC, a large amount of data is necessary.
Figure 10 concludes this application by plotting the histogram of the innovations
and of the stationary distribution of the calibrated process, for ρ = 0.95 and for N = 9
and N = 19.
6.2
Discretizing NMART
The second application of the EC method discretizes an NMART process and illustrates
how this method allows us to calibrate a Markov process with persistent non-zero
skewness and excess kurtosis.
In this application, we assume an NMART process as in Equation 5, and we denote
the first-order autocorrelation of yt by ρ∗ —recall that ρ is the first-order autocorrelation
of zt . The parameters of the process are set according to Table 7, so the differences of
the process feature moments that are consistent with Guvenen et al. (2015), reported
in Table 6. In this application, we target the moments of the differences of the process,
which we report in the last two columns of Table 8.
For this application, we generate 100, 000 realizations from the calibrated NMART
process of Section 4.2, and we apply EC to calibrate a Markov chain.
We run two different calibration experiments. In the first illustration, we target the
moments of the levels and the first differences of the process: Var(y), Var(∆y), S(y),
S(∆y), K(y) and K(∆y), together with ρ∗ . In the second illustration, we also target the
moments of the fifth differences of the process: Var(∆5 y), S(∆5 y), and K(∆5 y). The
31
Figure 10: Empirical Calibration, NMAR.
This figure displays the histogram of the innovations, that is, Eij = zj − ρ(x)zi , and the
histogram of the levels x implied by a Markov chain calibrated with the Empirical Calibration
method. In this example, ρ = 0.95, and we report the results for 9 and 19 states. In the first
and third panel of this figure, we rescale the y-axis by applying a cubic root transformation
in order to make the tails of the distribution conspicuous.
First Differences, ;=0.95, N=9
0.5
Var("y) = 0.23
S("y) = -1.35
K("y) = 17.80
0.1
0.01
-5
-2.5
0
2.5
5
E ij
Levels, ;=0.95, N=9
Var(y) = 2.30
S(y) = -0.30
K(y) = 3.80
0.15
0.09
0.03
-5
-2.5
0
2.5
5
zi
First Differences, ;=0.95, N=19
0.5
0.1
0.01
-5
-2.5
0
E
2.5
5
ij
Levels, ;=0.95, N=19
0.12
0.07
0.02
-5
-2.5
0
zi
32
2.5
5
Table 11: Empirical Calibration, NMAR
This table shows the performance of the Empirical Calibration method applied to data generated by an NMAR process. The performance is computed as a ratio of the calibrated moment
over the target moment.
ρ
0.5
0.9
0.95
0.99
N
ρ̂
ρ
Var(x)
Var(y)
S(x)
S(y)
K(x)
K(y)
Var(∆x)
Var(∆y)
S(∆x)
S(∆y)
K(∆x)
K(∆y)
5
9
15
19
5
9
15
19
5
9
15
19
5
9
15
19
0.890
0.968
0.986
0.990
0.947
0.962
0.983
0.989
0.963
0.973
0.988
0.990
0.980
0.989
0.992
0.994
0.914
0.987
0.995
0.991
0.800
0.876
0.939
0.976
0.644
0.864
0.931
0.966
0.559
0.760
0.964
0.983
0.989
1.003
1.003
1.003
1.174
0.984
1.014
1.032
1.119
0.954
1.074
1.040
1.139
1.075
0.791
0.987
1.029
0.990
1.004
0.996
0.680
1.056
1.008
1.017
0.569
1.000
0.980
1.042
0.546
0.666
1.002
1.026
1.014
1.018
1.009
1.001
1.180
1.171
1.081
1.069
1.091
1.297
1.140
1.147
1.597
1.540
1.647
1.533
0.999
1.000
0.998
0.997
0.599
0.777
0.947
0.982
0.536
0.650
0.848
0.891
0.207
0.359
0.418
0.527
0.971
1.003
0.994
1.002
0.543
1.053
1.006
0.954
0.572
1.044
0.979
0.967
0.695
0.596
0.927
0.889
Av. % Dev.
4.006
1.122
0.629
0.466
25.517
9.720
3.421
3.099
27.493
12.859
6.993
5.927
39.268
32.081
22.231
16.852
conditions E(y) = E(∆y) = E(∆5 y) = 0 are satisfied by construction, and therefore we
do not target them. The calibration is repeated for N = 5, 9, 15, 19.
Once again, the criterion function we minimize is the sum of the squared percentage
deviations—as in Equation 6—but for ease of interpretation, we report the performance
of the approximation in terms of the average percentage absolute deviation from the
targets—as in Equation 7.
Table 12 reports the performance of the calibration. The performance relative to
the moments of ∆5 y is also shown when such moments are not targeted. The table
shows that the accuracy of the method increases with the cardinality of the state space.
Two remarks are in order. (i) In this illustration, we target the moments of the
fifth differences because we want to establish a connection with Guvenen et al. (2015),
who discuss the persistency of skewness and kurtosis. However, this method can also
be applied by targeting a completely different set of moments. (ii) Table 12 shows
that targeting the moments of the fifth differences does not considerably change the
33
calibration. In particular, the calibration misses the skewness of the fifth differences by
a wide margin, but it performs reasonably well with the other moments.
Table 12: Empirical Calibration, AR(1) Mixture
This table shows the performance of the Empirical Calibration method when applied to an
NMART process. We report the results for different cardinalities of the state space, N . In
panel (1) of this table, we report the performance of EC when skewness and kurtosis of the fifth
differences are not targeted. In panel (2) we report the performance of EC when skewness and
kurtosis of the fifth differences are targeted. In the last two rows of the table, we report the
average percentage absolute deviation from the targets: (%) denotes the average that excludes
the moments of the fifth differences, and (%∆5 ) denotes the average that includes them.
7
(1)
(2)
∆5 moments not targeted
∆5 moments targeted
N
5
9
15
19
5
9
15
19
ρˆ∗
ρ∗
Var(x)
Var(y)
S(x)
S(y)
K(x)
K(y)
Var(∆x)
Var(∆y)
S(∆x)
S(∆y)
K(∆x)
K(∆y)
Var(∆5 x)
Var(∆5 y)
S(∆5 x)
S(∆5 y)
K(∆5 x)
K(∆5 y)
0.901
0.957
0.979
0.986
0.884
0.957
0.989
0.988
0.970
0.990
0.996
0.992
0.953
0.977
0.999
0.988
1.166
1.031
0.994
0.994
1.138
1.077
1.096
0.998
1.043
1.018
0.996
0.969
1.135
1.042
1.065
0.982
1.179
1.083
1.041
1.023
1.194
1.068
1.023
1.014
0.327
0.820
0.890
0.907
0.585
0.831
0.994
0.927
0.666
0.869
0.902
0.903
0.742
0.891
1.015
0.928
1.375
1.336
1.322
1.312
1.330
1.317
1.283
1.300
0.114
0.398
0.487
0.497
0.271
0.414
0.487
0.525
0.812
0.762
0.734
0.719
0.855
0.774
0.681
0.726
(%)
21.8
7.09
4.06
3.88
18.6
7.59
3.09
2.90
(%∆5 )
29.7
16.7
13.9
13.7
25.1
16.6
13.3
12.5
Aiyagari Calibrated with Extended Tauchen
In this section, we illustrate an economic application of the ET method. In an Aiyagari
economy with a standard household problem as in Equation 2, we calibrate a discrete
process for labor endowment with 15 states to feature non-zero skewness and high
kurtosis, and we observe the implications in general equilibrium.
34
The variance of the idiosyncratic shock is set to 0.1, the autocorrelation of the
process to 0.6, and the relative risk aversion (RRA) coefficient to 3. We then solve
for the general equilibrium interest rate with different combinations of skewness and
kurtosis for the shock process. Figure 11 reports the results of this experiment, by
drawing the level curves of the general equilibrium interest rate.
Figure 11: Level curves of the general equilibrium interest rate
For each combination of skewness and kurtosis of the shocks to labor endowment displayed in
this figure, we compute the interest rate at the general equilibrium of an Aiyagari economy.
In this figure, we report the level curves of such interest rate functions. Notice that since
we use the ET method to calibrate the 15-state Markov chain for labor endowment, only
the combinations northeast of the blue line can be calibrated exactly: for more details, see
Section 4.2.
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The figure shows that the interest rate varies considerably with the skewness and
kurtosis of the shocks. More specifically, a higher kurtosis is associated with a lower
35
interest rate—at the aggregate level, the agents want to save more—whereas a higher
left skewness is associated with a higher interest rate—at the aggregate level, the agents
want to save less. Since higher kurtosis is synonymous with the greater likelihood of
tail events, the result about higher kurtosis is intuitive.
The fact that higher left skewness—increased downturn risk—translates into smaller
aggregate saving is somewhat surprising and requires further explanation. To better
understand the results shown in Figure 11, we compare the stationary asset distribution
and the stationary shock distribution under 4 calibrations. The benchmark calibration
is characterized by 0 skewness and kurtosis of 3. The other three calibrations match
the skewness and kurtosis reported in Guvenen et al. (2015). In the first panel of
Figure 12, we introduce only skewness and we observe that the agents who are close
to the borrowing constraint save more whereas wealthier agents save less. Because the
effect on the upper tail dominates the effect on the left shoulder of the distribution,
aggregate saving decreases and the interest rate increases. By comparing the first and
second panel of Figure 13, we can gain some intuition into why wealthier agents save less
under a process with higher left skewness. In fact, matching negative skewness while
keeping the other moments of the distribution constant means that some probability
mass must move above zero. Therefore, the probability of a positive shock is 0.57, as
opposed to 0.40 under the benchmark case. Since wealthy agents are not sensitive to
left skewness but face a higher probability of receiving a good shock they save less.
In the second panel of Figure 12, we introduce only kurtosis, which results in higher
aggregate saving. In the third panel, we introduce both skewness and kurtosis. Since the
effect of kurtosis dominates the effect of skewness, aggregate saving increases. Because
the two effects tend to cancel each other out, the change in interest rate is only moderate,
even though the change in the asset distributions is sizable.
36
Figure 12: Stationary asset distribution
This figure displays the stationary asset distribution in an Aiyagari economy under three
different calibrations of the process for labor endowment. Each distribution—in gray— is
represented together with a benchmark asset distribution that is derived under a Gaussian
process. In the benchmark economy, the equilibrium interest rate is 3.06 and the Gini coefficient equals 0.38. The values of skewness and kurtosis that we calibrate are taken from
Guvenen et al. (2015).
Skewness=-1.36, Kurtosis=3
Density
Interest rate = 3.21
Gini = 0.32
0
5
10
15
Capital
20
25
30
Skewness=0, Kurtosis=17.95
Density
Interest rate = 2.72
Gini = 0.46
0
5
10
15
Capital
20
25
30
Skewness=-1.36, Kurtosis=17.95
Density
Interest rate = 2.95
Gini = 0.41
0
5
10
15
Capital
37
20
25
30
Figure 13: Stationary distribution of labor endowment
This figure displays the stationary distribution of the levels of labor endowments under four
different calibrations. The values of skewness and kurtosis that we calibrate are taken from
Guvenen et al. (2015).
Skewness=0, Kurtosis=3
0.2
0.1
-2.5
0
2.5
zi
Skewness=-1.36, Kurtosis=3
0.1
-2.5
0
2.5
zi
Skewness=0, Kurtosis=17.95
0.2
0.1
-2.5
0
z
2.5
i
Skewness=-1.36, Kurtosis=17.95
0.2
0.1
-2.5
0
zi
38
2.5
8
Conclusion
The main contribution of this paper is to provide two discretization methods to calibrate
a Markov chain that features non-zero skewness and high kurtosis.
When data are available firsthand to calibrate a Markov chain, the Empirical Calibration method is accurate and requires no assumptions about the data generating
process.
The Extended Tauchen method calibrates a Markov chain using a procedure that
is similar to that of Tauchen. This method is fast and bypasses a common calibration
error with persistent Gaussian AR(1) processes that we uncover in this paper. As
we illustrate, this calibration error can affect the behavior of modeled agents and is
economically consequential.
Additionally, we contribute to the literature with some results that are auxiliary
to the main point of the paper and which are interesting in themselves. In fact, we
characterize the moment structure of an NMART process and its lag-profile of skewness
and kurtosis.
Finally, we illustrate an economic application of the Extended Tauchen method in
an Aiyagari economy. We find that introducing idiosyncratic shocks with skewness
and excess kurtosis affects the aggregate level of savings and its distribution, therefore
changing the general equilibrium interest rate.
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41
A1
Analytical Moments of a Markov Process
This section describes how we analytically compute the moments of a Markov chain.
The formulas we report are a straightforward application of the definition of these
moments, and we prefer them to simulation-based computation of the moments because
they are faster and more accurate.
Let a Markov chain be defined by an n-dimensional state vector z and a transition
matrix T . We will denote by Π the stationary distribution of T . Let x be a discrete
random process distributed according to (z, T ). We define the differences and residuals
of the process as follows:
et = xt − ρ(x)xt−1 ,
where ρ(x) is the autocorrelation coefficient that characterizes the Markov chain and e
is the innovation to the process,
∆xt = xt − xt−1 ,
where ∆x is the first difference of the process,
∆h xt = xt − xt−h ,
where ∆h x is the h-th difference of the process.
Expected value, standard deviation, variance, skewness, and kurtosis are denoted
by E, Std, Var, S, and K, respectively.
42
Moments of x
E(x) =
Var(x) =
n
X
Πi zi
i=1
n
X
Πi (zi − E(x))2
i=1
n
X
S(x) =
(8)
i=1
Std(x)3
n
X
K(x) =
Πi (zi − E(x))3
Πi (zi − E(x))4
i=1
Var(x)2
Autocorrelation
n X
n
X
ρ(x) =
Πi Tij (zi − E(x))(zj − E(x))
i=1 j=1
Var(x)
(9)
Moments of e
Let E be an n × n matrix, where each element is defined as follows:
Ei,j = zj − ρ(x)zi .
(10)
Then the moments of e are given by
E(e) =
n X
n
X
Πi Tij Eij
i=1 j=1
Var(e) =
n X
n
X
Πi Tij (Eij − E(e))2
i=1 j=1
n
X
S(e) =
Πi Tij (Eij − E(e))3
i=1
Std(e)3
n X
n
X
Πi Tij (Eij − E(e))4
K(e) =
i=1 j=1
Var(e)2
43
(11)
Moments of ∆x
Let δ be a n × n matrix, where each element is defined as follows:
δi,j = zj − zi
(12)
Then the moments of ∆x are given by:
E(∆x) =
n X
n
X
Πi Tij δij
i=1 j=1
Var(∆x) =
n X
n
X
Πi Tij (δij − E(∆x))2
i=1 j=1
n
X
S(∆x) =
Πi Tij (δij − E(∆x))3
i=1
Std(∆x)3
n X
n
X
K(∆x) =
(13)
Πi Tij (δij − E(∆x))4
i=1 j=1
Var(∆x)2
Moments of ∆h x
To calculate the moments of ∆h x, it is sufficient to replace Tij with (T h )ij in the formulas
for ∆x. Notice that (T h )ij is the element on row i, column j, of a matrix obtained by
multiplying T by itself h times.
44
A2
Moments of NMAR
Consider an NMAR process with the following representation:
yt = ρyt+1 + ηt ,
where
(
ηt =
η1 ∼ N (µ1 , σ12 )
with probability p1 ,
η2 ∼ N (µ2 , σ22 )
with probability p2 .
In order to calculate the moments of this process, we use the following properties.
First, the raw moments of the innovation can be computed as a weighted sum of the
raw moments of each of the normals:
E(η k ) = p1 E(η1k ) + p2 E(η2k ).
Second, to calculate the moments of yt , ∆yt , and ∆5 yt , we use some properties of
cumulants:
• The nth cumulant of a distribution is homogeneous of degree n, which means that
Cn (aX) = an C(X).
• Cumulants are additive, so that if X and Y are independent random variables,
then Cn (X + Y ) = Cn (X) + Cn (Y ).
• The following relationships hold:
C1 (X) = E(X)
C2 (X) = Var(X)
C3 (X) = S(X)Var(X)3/2
C4 (X) = K(X)Var(X)2 − 3Var(X)2
45
(14)
Applying these properties, as an example, we compute the skewness for yt :
yt = ρyt−1 + ηt ⇒ C3 (yt ) = C3 (ρyt−1 + ηt ) ⇒ C3 (yt ) = ρ3 C3 (yt−1 ) + C3 (ηt )
i3
i3
hp
hp
Var(yt ) (1 − ρ3 ) = S(ηt )
Var(ηt )
⇒ (1 − ρ3 )C3 (yt ) = C3 (ηt ) ⇒ S(yt )
hp
i3
S(ηt )
Var(ηt )
⇒ S(yt ) =
hp
i3
3
(1 − ρ )
Var(yt )
Following a similar argument, we obtain the complete moment structure:
E(ηt ) = p1 µ1 + p2 µ2
E(ηt2 ) = p1 (µ21 + σ12 ) + p2 (µ22 + σ22 )
E(ηt3 ) = p1 (µ31 + 3µ1 σ12 ) + p2 (µ32 + 3µ2 σ22 )
(15)
E(ηt4 ) = p1 (µ41 + 6µ21 σ12 + 3σ14 ) + p2 (µ42 + 6µ22 σ22 + 3σ24 )
Var(ηt ) = E(ηt2 ) − E(ηt )2
E(ηt3 ) − 3E(ηt )Var(ηt ) − E(ηt )3
Std(ηt )3
E(ηt4 ) − 3E(ηt )4 + 6E(ηt2 )E(ηt )2 − 4E(ηt )E(ηt3 )
K(ηt ) =
Var(ηt )2
S(ηt ) =
E(ηt )
1−ρ
Var(ηt )
Var(yt ) =
1 − ρ2
(1 − ρ2 )3/2
S(yt ) = S(ηt )
1 − ρ3
(1 − ρ2 )2 (K(ηt ) − 3)
K(yt ) = 3 +
1 − ρ4
(16)
E(yt ) =
46
(17)
E(∆yt ) = 0
(ρ − 1)2
Var(∆yt ) = Var(ηt ) 1 +
1 − ρ2
1+
S(∆yt ) = S(ηt ) h
1+
(ρ−1)3
1−ρ3
(ρ−1)2
(18)
i3/2
1−ρ2
K(∆yt ) = 3 + (K(ηt ) − 3)
(1 + ρ2 + 2ρ3 )
2(1 + ρ2 )
E(∆5 yt ) = 0
(ρ5 − 1)2
Var(∆5 yt ) = Var(ηt ) 1 + ρ + ρ + ρ + ρ +
1 − ρ2
2
4
6
8
(ρ5 −1)3
1−ρ3
i3/2
(ρ5 −1)2
1−ρ2
1 + ρ3 + ρ6 + ρ9 + ρ12 +
S(∆5 yt ) = S(ηt ) h
1 + ρ2 + ρ4 + ρ6 + ρ8 +
(19)
5
4
−1)
1 + ρ4 + ρ8 + ρ12 + ρ16 + (ρ1−ρ
4
K(∆5 yt ) = 3 + [K(ηt ) − 3] h
i2
5
2
−1)
1 + ρ2 + ρ4 + ρ6 + ρ8 + (ρ1−ρ
2
Now we can use Equation 18 and 19 to write S(∆5 y) as a linear function of the
S(∆y) and K(∆5 y) as a linear function of the K(∆y).
S(∆yt ) = S(ηt ) h
|
1+
1+
(ρ−1)3
1−ρ3
(ρ−1)2
1−ρ2
{z
A
i3/2 ,
}
and
S(∆5 yt ) = S(ηt ) h
(ρ5 −1)3
1−ρ3
i3/2 ,
(ρ5 −1)2
1−ρ2
1 + ρ3 + ρ6 + ρ9 + ρ12 +
1 + ρ2 + ρ4 + ρ6 + ρ8 +
|
{z
B
}
so that
S(∆5 yt ) =
B
S(∆yt ).
A
47
(20)
Similarly, for the kurtosis of the process we have
K(∆y) = 3 + [K(ηt ) − 3]
(1 + ρ2 + 2ρ3 )
,
2(1 + ρ2 )
|
{z
}
C
and
5
4
−1)
1 + ρ4 + ρ8 + ρ12 + ρ16 + (ρ1−ρ
4
K(∆5 y) = 3 + [K(ηt ) − 3] h
i2 ,
5
2
−1)
1 + ρ2 + ρ4 + ρ6 + ρ8 + (ρ1−ρ
2
{z
}
|
D
so that
K(∆5 y) = 3 +
D
[K(∆y) − 3] .
C
48
(21)
A3
Moments of NMART
Consider an NMART process with the following representation:
yt = zt + t
zt = ρzt−1 + ηt ,
where
(
ηt ∼
N (µ1 , σ12 )
with probability p1 ,
N (µ2 , σ22 )
with probability p2 ,
t ∼ N (0, σ2 ).
and where ρ < 1. Equivalently,
∞
X
yt = t +
ηt−k ρk ,
k=0
so that
∆h yt = yt − yt−h ,
∞
∞
X
X
= t +
ηt−k ρk − t−h −
ηt−h−k ρk ,
k=0
= (t − t−h ) +
= (t − t−h ) +
= (t − t−h ) +
= (t − t−h ) +
= (t − t−h ) +
= (t − t−h ) +
h−1
X
k=0
h−1
X
k=0
h−1
X
k=0
h−1
X
k=0
h−1
X
k=0
h−1
X
k
k=0
∞
X
k
k=h
∞
X
ηt−k ρ +
ηt−k ρ +
ηt−k ρk +
ηt−k ρk +
ηt−k ρk +
k=0
∞
X
k=0
∞
X
k=h
∞
X
k
ηt−k ρ −
∞
X
ηt−h−k ρk ,
k=0
k+h
ηt−h−k ρ
−
∞
X
k=0
ηt−(h+k) ρk+h − ρk ,
ηt−k ρk − ρk−h ,
ηt−k ρk 1 − ρ−h ,
k=h
k
−h
ηt−k ρ + 1 − ρ
k=0
∞
X
k=h
49
ηt−h−k ρk ,
ηt−k ρk .
(22)
The cumulant i of y is
Ci (yt ) = Ci
t +
∞
X
!
k
ηt−k ρ
,
k=0
= Ci (t ) + Ci
∞
X
!
ηt−k ρk
,
k=0
= Ci (t ) +
∞
X
(23)
Ci ηt−k ρk ,
k=0
= Ci (t ) +
∞
X
ρki Ci (ηt ) ,
k=0
= Ci (t ) +
Ci (ηt )
.
1 − ρi
The cumulant i of the h-th difference of the process is denoted by Ci (∆h yt ) and is
obtained as follows:
"
Ci (∆h yt ) = Ci (t − t−h ) +
h−1
X
#
∞
X
ηt−k ρk + 1 − ρ−h
ηt−k ρk ,
k=0
k=h
= Ci (t ) + Ci (−t−h ) + Ci
" h−1
X
#
"
ηt−k ρk + Ci
1−ρ
∞
X
−h
k=0
= Ci (t ) + Ci (−t ) +
h−1
X
= Ci (t ) + Ci (−t ) +
ρk ηt−k ,
k=h
"
Ci (ηt ) ρki + 1 − ρ
−h i
Ci
∞
X
#
ρk ηt−k ,
k=h
k=0
h−1
X
#
"
i
Ci (ηt ) ρki + 1 − ρ−h
k=0
#
ρki Ci (ηt ) ,
(24)
k=h
hi
= Ci (t ) + Ci (−t ) + Ci (ηt )
∞
X
∞
X
i
1−ρ
+ 1 − ρ−h Ci (ηt )
ρ(k+h)i ,
i
1−ρ
k=0
∞
X
1 − ρhi
−h i
hi
= Ci (t ) + Ci (−t ) + Ci (ηt )
+ 1−ρ
Ci (ηt ) ρ
ρki ,
i
1−ρ
k=0
hi
1 − ρhi
−h i ρ
+
C
(η
)
1
−
ρ
,
i
t
1 − ρi
1 − ρi
"
i #
1 − ρhi + ρhi 1 − ρ−h
= Ci (t ) + Ci (−t ) + Ci (ηt )
.
1 − ρi
= Ci (t ) + Ci (−t ) + Ci (ηt )
Now using Equation 23 and 24, we derive the moments of y and ∆t y as a function of
50
the moments of η and . To make further progress and simplify the exposition, we write
the variance of the transitory shock as a multiple α of the variance of the permanent
shock, that is, Var (t ) = αVar (ηt ). We obtain
1
Var (y) = Var (η) α +
,
1 − ρ2
S (η)
S (y) =
3/2 ,
1
(1 − ρ3 ) α + 1−ρ
2
2
1
3 α + 1−ρ
+ 3−K(η)
2
ρ4 −1
K (y) =
,
2
1
α + 1−ρ2
2Var (η) [α (ρ2 − 1) + ρt − 1]
,
Var (∆t y) =
ρ2 − 1
3S(η)ρt (ρt − 1)
S (∆t y) = √
,
2
t −1 3/2
2 2 (ρ3 − 1) α(ρ −1)+ρ
ρ2 −1
t −1
t −1 ρt +1
2 −1 +ρt −1 2
(K(η)−3)
ρ
2ρ
6
α
ρ
(
)[(
)
]
[
(
)
]
2
(ρ2 − 1)
+
ρ4 −1
(ρ2 −1)2
K (∆t y) =
,
2 [α (ρ2 − 1) + ρt − 1]2
51
(25)
(26)
(αρ2 − α + ρt − 1)
,
(ρ − 1)(αρ + α + 1)
q
αρ+α+1
1
α + ρ+1 ρt−1 (ρt − 1)
(ρ + 1)
ρ+1
q
,
S(∆t y) =S(∆y)
2
t −1
(αρ2 − α + ρt − 1) αρ −α+ρ
ρ2 −1
Var(∆t y) =Var(∆y)
2
(ρ2 + 1) (ρ2 − 1) (αρ + α + 1)2 (ρt − 1)
K(∆t y) =K(∆y)
(2ρ3 + ρ2 + 1) (ρ4 − 1) [α (ρ2 − 1) + ρt − 1]2
× 2ρt − 1 ρt + 1
2
(ρ2 − 1) (ρt − 1) ((2ρt − 1) ρt + 1)
+
2 (ρ4 − 1) (α (ρ2 − 1) + ρt − 1)2
(
3 [2α2 (ρ + 1)2 (ρ2 + 1) + 4α (ρ3 + ρ2 + ρ + 1)]
× −
2ρ3 + ρ2 + 1
−
3 (−2ρ3 + ρ2 + 1)
2ρ3 + ρ2 + 1
)
2
6 (ρ4 − 1) (α (ρ2 − 1) + ρt − 1)
+
−3 .
(ρ2 − 1)2 (ρt − 1) ((2ρt − 1) ρt + 1)
52
(27)
A4
Tables
Table 13: Extended Tauchen, Gaussian AR(1): ρ=0.95
This table compares the performance of the Extended Tauchen method with the other discretization
methods common in the literature.
N
2
5
9
15
19
29
Method
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
1.000
NaN
0.840
0.779
1.034
0.671
0.729
1.000
1.040
0.953
0.922
1.006
0.960
0.975
1.000
1.001
0.978
0.970
1.000
0.986
0.989
1.000
0.999
0.989
0.990
1.000
0.994
0.995
1.000
0.999
0.992
0.995
1.000
0.996
0.996
1.000
0.999
0.995
0.999
1.000
0.996
0.997
1.000
NaN
0.637
0.098
0.243
0.090
0.113
1.000
1.726
0.897
0.349
0.746
0.666
0.810
1.000
1.374
0.953
0.602
0.950
0.934
0.983
1.000
1.129
0.976
0.827
0.996
0.974
0.994
1.000
1.072
0.982
0.905
0.999
0.985
0.996
1.000
1.020
0.990
0.980
1.000
0.973
0.992
0.333
NaN
0.333
0.333
0.333
0.333
0.333
0.833
0.925
0.650
0.630
0.698
0.690
0.777
0.917
0.923
0.773
0.720
0.890
0.928
0.982
0.952
0.944
0.848
0.818
0.984
0.987
0.997
0.963
0.950
0.875
0.871
0.996
0.993
0.998
0.976
0.955
0.913
0.955
1.000
0.986
0.996
1.000
NaN
2.373
0.453
0.086
0.545
0.601
1.000
0.427
1.663
0.834
0.663
1.147
1.175
1.000
1.340
1.334
0.935
0.954
1.168
1.189
1.000
1.153
1.179
0.979
0.998
1.084
1.092
1.000
1.092
1.134
0.990
1.000
1.057
1.061
1.000
1.037
1.080
0.998
1.000
1.038
1.054
12.675
NaN
2.668
1.945
37.518
1.246
1.563
3.919
17.714
1.318
1.063
2.375
1.258
1.591
2.459
1.405
1.208
1.014
1.143
1.081
1.125
1.834
0.994
1.140
1.002
1.005
1.017
1.014
1.649
0.997
1.114
1.000
1.000
1.008
1.007
1.417
0.999
1.077
1.000
1.000
1.016
1.021
53
Ave. % Dev.
246.838
84.637
65.658
777.794
52.135
55.748
61.709
362.547
29.634
26.588
45.464
21.778
24.072
30.855
23.954
16.747
15.759
6.987
8.027
7.201
17.631
6.900
10.138
7.773
0.547
2.903
2.421
13.713
4.360
7.997
4.802
0.104
1.842
1.552
8.816
2.076
5.199
1.368
0.002
1.979
1.806
Table 14: Extended Tauchen, Gaussian AR(1): ρ=0.99
This table compares the performance of the Extended Tauchen method with the other discretization
methods common in the literature.
N
2
5
9
15
19
29
Method
ρ̂
ρ
Var(x)
Var(y)
K(x)
K(y)
Var(e)
Var()
K(e)
K()
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
Rouwenhorst
Tauchen
Adda-Cooper
Hussey-Tauchen
Flodén
ET: Equal Weights
ET: Adjusted Weights
1.000
NaN
0.919
0.765
NaN
0.636
0.676
1.000
NaN
0.969
0.905
1.009
0.933
0.953
1.000
1.009
0.985
0.953
1.006
0.985
0.996
1.000
1.002
0.992
0.975
1.003
0.997
0.997
1.000
1.000
0.994
0.982
1.001
0.998
0.998
1.000
1.000
0.997
0.990
1.000
0.999
0.999
1.000
NaN
0.637
0.020
NaN
0.016
0.019
1.000
NaN
0.897
0.079
0.498
0.136
0.194
1.000
1.651
0.953
0.159
0.821
0.488
0.719
1.000
1.442
0.976
0.272
0.983
0.873
0.940
1.000
1.329
0.982
0.343
1.001
0.926
0.985
1.000
1.146
0.990
0.495
1.001
0.944
0.988
0.333
NaN
0.333
0.333
NaN
0.333
0.333
0.833
NaN
0.650
0.585
0.596
0.591
0.616
0.917
0.876
0.773
0.622
0.718
0.661
0.827
0.952
0.886
0.848
0.646
0.875
0.964
0.968
0.963
0.900
0.875
0.660
0.940
0.970
1.001
0.976
0.926
0.913
0.696
0.994
0.979
0.996
1.000
NaN
5.506
0.426
NaN
0.494
0.533
1.000
NaN
3.560
0.786
0.036
1.003
1.064
1.000
0.227
2.390
0.881
0.300
1.207
0.987
1.000
1.190
1.751
0.930
0.738
1.135
1.182
1.000
1.330
1.560
0.947
0.886
1.142
1.136
1.000
1.189
1.330
0.969
0.989
1.073
1.083
66.002
NaN
6.748
2.127
NaN
1.208
1.417
17.250
NaN
2.653
1.114
113.990
1.244
1.607
9.125
41.420
2.161
1.049
8.163
1.262
1.876
5.643
2.562
1.944
1.026
2.080
1.242
1.272
4.611
1.416
1.832
1.019
1.430
1.116
1.162
3.321
0.998
1.616
1.010
1.049
1.076
1.094
54
Ave. % Dev.
1313.367
227.282
71.646
54.577
57.102
328.342
93.931
35.180
2297
31.733
38.166
164.171
839.533
56.817
28.682
166.595
26.693
26.928
93.812
46.065
37.574
24.032
29.736
10.872
10.958
72.965
23.519
30.811
21.745
12.125
7.310
6.298
46.906
8.240
20.957
17.204
1.355
4.529
3.875
A5
Figures
Figure 14: Histograms of the Levels, Gaussian AR(1): ρ = 0.99, N =9
This figure displays the histogram of the levels (x) implied by the Markov chain of the methods of
Tauchen (1986), Rouwenhorst (1995), Adda and Cooper (2003), Hussey and Tauchen (1991), and
Flodén (2008). In this example, ρ = 0.99, the variance of the shocks equals 1 and we use 9 states. The
x-axis shows the state space and the y-axis the stationary probability of each state.
Tauchen (1986)
Rouwenhorst (1995)
0.2
0.2
0.1
0.1
−15
0
15
−15
Hussey−Tauchen (1991)
0.1
0.05
0.05
0
15
−15
Flodén (2008)
0.1
0.05
0.05
0
0
15
ET: Equal Weights
0.1
−15
15
Adda−Cooper (2003)
0.1
−15
0
15
−15
55
0
15
Figure 15: Histograms of the Innovations, Gaussian AR(1): ρ = 0.99, N =9
This figure displays the histogram of the innovations (Eij = zj −ρ(x)zi ) implied by the Markov chain of
the methods of Tauchen (1986), Rouwenhorst (1995), Adda and Cooper (2003), Hussey and Tauchen
(1991), and Flodén (2008). In this example, ρ = 0.99 and we use 9 states. The x-axis shows the
elements of the matrix E and the y-axis the stationary probability of each innovation. We also report
(in red) the excess of kurtosis of the innovations for each of these methods.
Tauchen (1986)
Rouwenhorst (1995)
41.42
0.2
9.125
0.2
0.1
0.1
−4
0
4
−4
Hussey−Tauchen (1991)
0
Adda−Cooper (2003)
1.049
2.161
0.06
0.08
0.03
0.04
−4
0
4
4
−4
Flodén (2008)
0
4
ET: Equal Weights
8.163
1.262
0.08
0.1
0.04
0.05
−4
0
4
−4
56
0
4
Figure 16: Histograms of the Innovations, Scaled, Gaussian AR(1): ρ = 0.99, N =9
This figure presents the same data as Figure 15, but the y-axis is rescaled by applying a cubic root
transformation that highlights the tail probabilities.
Tauchen (1986)
Rouwenhorst (1995)
41.42
9.125
0.1
0.1
0.01
0.01
−4
0
4
−4
Hussey−Tauchen (1991)
0
4
Adda−Cooper (2003)
1.049
2.161
0.04
0.04
0.001
0.001
−4
0
4
−4
Flodén (2008)
0
4
ET: Equal Weights
8.163
1.262
0.05
0.05
0.01
0.01
−4
0
4
−4
57
0
4
Figure 17: Feasible combinations of {S(η), K(η)} with a mixture of 2 or 3 normals
In this figure, we show what combinations of kurtosis, K(η), and skewness, S(η), are feasible
when calibrating a mixture of two or three normals. The combinations NE of each line are
exactly attainable. This figure shows that using a mixture of three normals expands the set
of feasible combinations only moderately.
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Figure 18: Feasible combinations of {S(∆5 y), K(∆5 y)} and {S(∆10 y), K(∆10 y)}
In this figure, we show what combinations of kurtosis, K(∆k y), and skewness, S(∆k y), are
attainable exactly when calibrating NMART, for k = 5, 10. Each quadrant of the plot is
characterized by a different value of α. Within each quadrant, we draw the frontier for three
different values of ρ. The feasible combinations lie NE of each frontier.
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