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A Survey about Smooth Transition Panel Data Analysis
SCIENTIFIC LETTERS
SCIENTIFIC LETTERSSCIENTIFIC LETTERS
SCIENTIFIC LETTERS
Econometrics Letters Volume (1), 1, 2014, June
.
A Survey about Smooth Transition Panel Data Analysis
Tolga Omay
1
Abstract
In this study we are introducing a literature survey about the panel smooth transition
regression models. This type of modeling has been emerged from two different strand
of literature where the first one is nonlinear time series the other is the panel data
analysis. Both of these fields have tackled with different type of biases in estimation
process. Therefore, combining these two fields constitutes different problems in
estimation. Hence, instead of giving the studies in chronological order, we preferred to
explain papers with respect to problems which they have solved. In this order, first we
analyze the categories of different models. For example, there are several
categorizations in the panel data estimation with respect to time and cross-section
dimension. Therefore every category has its own biases depending on the time and
cross-section dimension. On the other hand the dynamic structure of the panel data is
another important determinant in which we can classify the biases. Hence, the static
and dynamic panel smooth transition models are also discussed separately in this
study. Finally, smooth transition models has its’ own categories, hence we are giving
these categories under the panel categorization as well.
Keywords:
Panel smooth transition data; Bias; Large-moderate-small panel data;
static panel data; dynamic panel data, Logistic, exponential, time varying
Jel Classification:
1
Cankaya University Banking and Insurance Program, Eskişehir yolu 29. km. P.code: 06790, Etimesgut,
Ankara, Turkey. tel: +90 312 233 11 91- fax: +90 312 233 11 90, e-mail: omayt@cankaya.edu.tr
1. Introduction
In the last two decades the availability of large data set leads to researchers to analyze
economical problems in panel data settings. The initiation of panel data analysis first started
with small time dimension where the data availability is very restricted at those times. The
technological improvements and the regularities in collecting data leads to large data sets with
respect to country, firms and other entities. On the other hand the sustainability of this process
increases the availability of data through the large time dimensions. Therefore, now
researchers have ability to apply old empirical studies such as cross-sectional and time series
studies in panel data settings. But the panel data studies have its’ own type of problems. The
cross-section and time series analysis biases are mostly solved in the literature for the linear
case and can be found in the text books. The linear panel data analysis problems are also
discussed in the text books, but the point which we have mentioned above that the data
availability is increasing and the new category of panel data analysis emerging then the new
type of biases introduced in to the panel data analysis such as cross section dependency. Cross
section dependency is mainly a problem of large time dimension in the panel data analysis.
Therefore it is easy to understand that this problem is a recent problem of panel data analysis.
The first cross section dependency test is proposed at 2004 by Pesaran (2004). On the other
hand an other important strand of literature is occurred about the nonlinearity in parameter
estimation. The most efficient type of nonlinearities the researchers are extensively used in
their studies are, threshold autoregressive models (TAR), smooth transition autoregressive
models (STAR) and Markow switching autoregressive models (MS-AR). In this study we are
concentrated on STAR models. There is plenty of good literature survey about the STAR
models such as van dijk et al. (2003). Here we are focused on the STR models where the
panel literature considers. Therefore, still the panel models further enhanced by introducing
new typology of STR nonlinearity in to panel settings.
In this study we discuss the different categories of panel data analysis. The first
categorization depending on N and T dimension; as Large N and Small T (Type 1), Small N
and Large T (Type 2) and Large N and Large T (Type 3). The second categorization is the
time structure of the models which are static and dynamic panel models. First we deal with
the static case and the three types of panel data models then we introduce the dynamic
structure and deal with the biases in this stetting. In panel data models the biases and the
remedies are changed due the model type and time structure. On the other hand, STR models
can be categorized with respect to transition function and transition (or state) variable. In any
modeling one can use one or more transition function with respect to the data generating
process in hand. If more than one transition function is used the STR model classified as
multiple regime STR model (MR-STR). If we use time as transition variable then the STR
model called as time varying smooth transition model (TV-STR). Therefore, with these two
categories and the panel categories which we have mentioned above, one can obtained various
type of nonlinear panel data model.
In the rest of the paper, section 2 gives the biases and the remedies of static panel smooth
transition models and these biases and remedies are discussed in this section as well. In
section 3, the biases and the remedies of dynamic panel smooth transition model are
discussed. Section 4 concludes.
2. Static panel data models biases and remedies
Panel estimators utilize the concept that by combining the data from different groups, one
can improve efficiency in parameter estimation, if there are similarities in the process
A Survey about Smooth Transition Panel Data Analysis
generating the data in the different groups (Smith and Fuertes, 2010). Smith and Fuertes,
(2010) state also that panel data estimation establishes flexibility in how one identifies
parameter heterogeneity, e.g. over time and over units. We can consider the parameters as
fixed or random, homogenous or heterogenous, and we can permit for parameter variation
over time periods or both. Nevertheless, the relative magnitudes of N and T are one of the
major categorization of panel data estimation. The categories can be classified as Large N and
Small T (Type 1), Small N and Large T (Type 2) and Large N and Large T (Type 3). All these
categories have its’ own estimation strategies and working assumptions.
Type 1 category is most traditional approach in panel data estimation depending on
data availability. In the static case, where the regressors are strictly exogenous and the
coefficients differ randomly and distributed independently of the regressors across groups,
any type of averaging such as pooling, aggregating averaging group estimates, and cross-
section regression provide consistent (and unbiased) estimates of the coefficient means
(Pesaran and Smith, 1995)
2
. Pesaran and Smith (1995) have proven that the parameter
heterogeneity is critical for estimating Dynamic panel estimation. Hereafter, we called this
problem as “Heterogeneity Bias”. However, increasing the time periods lead to Heterogeneity
Bias, but this bias is not a problem in static case as we mentioned above
3
.
Furthermore in the presence of cross-sectionally correlated error terms, traditional
OLS-based estimation are inefficient and invalidates much inferential theory of panel data
models which we mentioned before. The traditional remedy, SURE-GLS (Seemingly
Unrelated Regression Equations and Generalized Least Squares) is feasible when the cross-
section dimension N is smaller than the time series dimension T. The standard approach is to
treat the equations from the different cross-section units as a system of seemingly unrelated
regression equations and then estimate the system by Generalized Least Squares technique. If
both of these dimensions are same the disturbance covariance matrix will be rank deficient.
However, when the non-zero covariance’s between the errors of different cross-section units
due to common omitted variables, it is not apparent that SURE-GLS is always the correct
response (Coakley et al., 2002). For this reason, researcher should have searched for a more
efficient method which is not subject to these kinds of problems. Therefore, it is apparent to
use Pesaran (2006) model which suggested a method that makes use of cross-sectional
averages to provide valid inference for stationary panel regressions with multifactor error
structure. CCE procedure is applicable to panels with single or multiple unobserved factors so
long as the numbers of unobserved factors is fixed. Groote and Everaert (2011) suggest that
the PCCE estimator is quite useful for estimating cross-sectional dependent panel data models
provided T is not too small. Furthermore, Pesaran (2006) have shown by Monte Carlo
experiment that the simulation results favor the CCEP estimator for small to moderate sample
sizes and slightly favor CCEMG when N and T are relatively large. Omay and Kan (2010)
suggest a new approach by nothing that the linear and non-linear combinations of the
observed factors can be well approximated by cross-section averages of the dependent,
independent and state dependent variable) and proposed nonlinear mean group common
correlated effect estimator (CCEMG) for static panel. The estimation procedure has the
advantage that it can be computed by least squares to auxiliary regression where the observed
2
Zellner (1962) has proven this issue by using asymptotic theory.
3
In Pesaran and Smith (1995) heterogeneity comes from the random coefficient models. Nonlinear estimation
procedure Gonzales et al. (2005) may capture these random coefficients in two different regimes. But we know
that the nonlinear estimation which we explained above can be interpreted as heterogeneous panel estimation.
Gonzales et al. (2005) interpreted nonlinear model as heterogeneous and linear model as homogenous. In this
respect this type of estimation limits or totally eliminate the heterogeneity in estimation procedure in dynamic
panel as well.
regressors augmented with cross-sectional averages of dependent variable, individual specific
regressors and state dependent variable. This leads to a new set of estimators, which is
mentioned in Pesaran (2006), referred to as the Common Correlated Effects (CCE) estimators,
that can be computed by running smooth transition panel regressions augmented with cross-
section averages of the dependent, independent and state dependent variables (Omay and Kan,
2010).
In the nonlinear panel literature, we have seen different type’s of panel smooth
transition models depending on sample size and dynamic structure. Gorgens et al. (2009)
propose nonlinear dynamic panel model in the STR context by using the Type 1 category
approach. Their model is Arrelano-Bond type of panel extension of STR modelling. Hence
their model has some shortcomings which are inherited in linear model like not taking in
consideration of cross section dependency and the one which we mentioned above.
Furthermore they find that estimation of the parameters in the transition function can be
problematic but that there may be significant benefits in terms of forecast performance. On
the other hand, van Dijk et al. (2005) proposed a dynamic PSTR model that can be positioned
in between a fully pooled model, which imposes such common features, and a fully
heterogeneous model, which might render estimation problems for some of the panel
members. Moreover, van Dijk et al. (2005) claims that fully pooling is not effective in
understanding possible common non-linear features across the series, and that a fully
heterogeneous model introduces estimation problems which they have faced in their empirical
research that for 4 out of the 18 series are not convergent by means of STR estimation.
Furthermore, they document that a partially heterogeneous panel STAR model outperforms a
fully pooled model, leading to subtle differences across sectors in leads and lags for business
cycle recessions and expansions. However Omay and Kan (2010) have proposed PSTAR
model estimation which remedies cross section dependency. In their modeling strategy first
they have estimated the nonlinear panel by using Gonzales et al. (2005) type PSTR model. By
using this methodology they have found that they cannot handle Cross Section dependency
problem. Hence, they have used the Pesaran (2006) Common Correlated Effects (CCE)
estimation in nonlinear context in order to handle cross section dependency. The CCE
estimator is a fully heterogeneous panel time series model which augments means of
dependent and independent variables in order to remove factors from error term. This factor
removing remedies the cross section dependency problem from panel estimation.
Consequently, Omay and Kan (2010) employ a method which is mentioned by van Dijk et al.
(2005). However, they have not encounter a convergence problem in time series estimations
which hinders fully heterogeneous nonlinear panel estimation. Fully heterogeneous panel
estimation is more suitable for Type 2 panel models and smaller the N, larger the time series
dimension feasibility increases. Increasing time series dimension prevent convergence
problems in time series models. van Dijk et al. (2005)) has used a sample period 1972Q1-
2002Q4 which leads to 120 data points. The dimension of T for their empirical research is
satisfactorily fulfills large T condition. However, they used 18 sectors which can cause a
problem with respect to small N criteria. In this study, we have used 8 countries covering the
period 1993 to 2008 which satisfies the Small N but not large T conditions. Therefore,
estimating fully heterogeneous panel time series model is not appropriate depending on the
arguments mentioned above and sample structure. Furthermore, Omay and Kan (2010) claim
that the CCE estimation removes the endogeneity bias depending on the model structure they
have used in their estimation. On the other hand, Fouquau et al. (2008) estimated static
PSTAR model using the Gonzales et al. (2005) methodology. They have shown from the
parameter estimation the nonlinear model limits the potential endogeneity bias. From these
arguments nonlinear estimation and CCE estimator both reduce the potential endogeneity bias
A Survey about Smooth Transition Panel Data Analysis
in estimation procedure. Hence, in our estimation process we are not dealing with the
endogeneity bias in Nonlinear Static Pool Common Correlated Effect (NPCCE) estimation
4
.
Table 2. Biases and Remedies in Static Panel Data
Biases in Static Panel
Heterogeneity CSD* Endogeneity
Remedies
MG, PMG, RCM
Pool FE, RE**
CCE, SURE-GLS*** IV, GMM
For N small T moderate or large
Not
Prevail:
Χ
Prevail: Prevail:
Nonlinear Panel estimation
Χ
↵
****
Nonlinear CCE Panel estimation
Χ
Χ
Χ
*****
*CSD denotes Cross Section Dependency
** We can include Swamy model as well. MG denotes mean group, PMG pooled mean group, RCM denotes
random coefficient model.
*** There are other methods such as time effects/ demeaning, orthoganilization, PANIC Panel Analysis of Non-
stationarity in the Idionsyncratic and Common components and residual principal component methods
****
↵
means nonlinear model partially correct endogeneity bias (Fouquau et al. 2008).
***** CCE estimation further corrected this bias Omay and Kan (2010).
3. Dynamic panel data models biases and remedies
In the panel data literature, there are several methods to handle estimation process.
Pesaran and Smith (1995) compare these estimation methods extensively for dynamic panels
by using asymptotic theory and conclude that the Mean Group (MG) estimation is the most
efficient with respect to other methods which they consist different versions of taking average
implicitly where the MG estimation did explicitly
5
. The widely used four procedures are
pooling, aggregating, averaging group estimates and cross-section regression.
(MG) and Pooled Mean Group (PMG) estimations are classified as averaging group estimates,
Dynamic Panel GMM estimation of Arrelano and Bond (1991) and the other estimation
procedures which are using this kind of estimation procedure are classified as pooling. In this
study, we do not consider time aggregation and cross-section regression for two purposes. The
first one is Pesaran and Smith (1995) found out that these estimation procedures are
asymptotically inconsistent estimates. These estimators are more efficient in different cases.
The relative magnitudes of N and T are one of the major categorization of panel data
estimation. The categories can be classified as Large N and Small T (Type 1), Small N and
4
For PCCE estimation Sarafidis and Yamagata (2010) investigate the properties of instrumental variable
estimation and compare their estimators with GMM. However, they are dealing with the linear model where this
estimator is more efficient and constitute consistent estimators.
5
The aggregation method which is proposed by Zellner (1969) for random coefficient (with lagged dependent
variable) model is suitable for static models. Pasaran and Smith (1995) proved that this aggregation is not prevail
in dynamic panel models.
Large T (Type 2) and Large N and Large T (Type 3). All these categories have its’ own
estimation strategies and working assumptions.
Type 1 category is most traditional approach in panel data estimation depending on
data availability. In this type the classical estimator for estimation of pooled models
depending on Nickell (1981) bias such as fixed effects instrumental variables or Generalized
Method of Moments (GMM) estimators proposed, for example, by Anderson and Hsiao
(1981, 1982), Arellano (1989), Arrelano and Bover (1995), Keane and Runkle (1992) and
Ahn and Schmidt (1995). Pesaran and Smith (1995) show that these estimators can produce
inconsistent and potentially very misleading estimates in fact the slope coefficients are
identical. Therefore, Pesaran and Smith (1995) have proven that the parameter heterogeneity
is critical for estimating Dynamic panel estimation. Here after, we called this problem as
“Heterogeneity Bias”. Alvarez and Arrelona (2003) have shown that the OLS estimates are
consistent if time dimension grow sufficiently fast relative to N such that /N T
κ
→
where
0
κ
≤ <∞
. However, increasing the time periods lead to Heterogeneity Bias, hence this bias
can prevail in most of the nonlinear panels. In the nonlinear estimation procedure and cross
section dependency part we re-analyze this phenomenon. But we know that the nonlinear
estimation which we explained above can be interpreted as heterogeneous panel estimation.
Gonzales et al. (2005) interpreted nonlinear model as heterogeneous and linear model as
homogenous. In this respect this type of estimation limits or totally eliminate the
heterogeneity in estimation procedure
6
.
Furthermore in the presence of cross-sectionally correlated error terms, traditional OLS-based
estimation are inefficient and invalidates much inferential theory of panel data models which
we mentioned before. In Omay et al. (2014 a), they have used system SURE-GLS in order to
eliminate cross section depdendency and endogeneity bias together. Depending on Coakley et
al. (2002) system SURE-GLS approach is not seem to be that much feasible with respect to
cross section dependency bias. In the static case we have mentioned that the other estimation
strategy in order to eliminate cross section dependency bias is CCE estimator which is
proposed by Pesaran (2006). Pesaran (2006). model which suggested a method that makes
use of cross-sectional averages to provide valid inference for stationary panel regressions with
multifactor error structure. Omay et al. (2012) extend Omay and Kan (2010) approach to
dynamic nonlinear panel models. Omay and Kan (2010) has proposed nonlinear mean group
common correlated effect estimator (MGCCE) for static panel where they employ this
estimator to dynamic panel. CCE procedure is applicable to panels with single or multiple
unobserved factors so long as the numbers of unobserved factors is fixed. The above
mentioned estimations are dealing with the static case, for dynamic case new studies appear in
the literature. Sarafidis and Robertson (2009) prove that also standard dynamic panel IV and
GMM estimators either in levels or first difference are consistent as N
→∞
for fixed T as
the moment conditions applied by these estimators are invalid under cross sectional
dependence. Groote and Everaert (2010) find that POLS estimator is also inconsistent for
T
→∞
as the temporal dependence in the unobserved factor implies that it is correlated with
lagged dependent variable. Groote and Everaert (2010) show that for dynamic case PCCE
consistent if and only if
,N T
→ ∞
. On the other hand, they have proposed an other estimator
which is called restricted PCCE. The restricted version is to be preferred over its unrestricted
version for N large T fixed version. Groote and Everaert (2010) suggest that the PCCE
estimator is quite useful for estimating cross-sectional dependent dynamic panel data models
provided T is not too small. Furthermore, Pesaran (2006) have shown by Monte Carlo
experiment that the simulation results favor the CCEP estimator for small to moderate sample
6
In Pesaran and Smith (1995) heterogeneity comes from the random coefficient models. Nonlinear estimation
procedure Gonzales et al. (2005) may capture these random coefficients in two different regimes.
A Survey about Smooth Transition Panel Data Analysis
sizes and slightly favor CCEMG when N and T are relatively large. We have mentioned that
Heterogeneity Bias occurred if the sample size of T increases, hence, preference of CCEP
estimator in small samples may be related to this issue
7
. Hence, in our estimation process we
are not dealing with the endogeneity bias in NDPCCE estimation
8
.
Biases
Nickell Heterogeneity CSD* Endogeneity
Remedies
IV, GMM
MG, PMG,
RCM**
CCE, SURE-
GLS*** IV, GMM
For N small T moderate or large
Not prevail:
Χ
Prevail: Prevail: Prevail:
Nonlinear Panel estimation
Χ
Χ
↵
****
Nonlinear CCE Panel estimation
Χ
Χ
Χ
Χ
*****
*CSD denotes Cross Section Dependency
** We can include Swamy model as well. MG denotes mean group, PMG pooled mean group, RCM denotes random
coefficient model.
*** There are other methods such as time effects/ demeaning, orthoganilization, PANIC Panel Analysis of Non-
stationarity in the Idionsyncratic and Common components and residual principal component methods
****
↵
means nonlinear model partially correct endogeneity bias (Fouquau et al. 2008).
***** CCE estimation further corrected this bias Omay and Kan (2010).
4. Concluding Remarks
In this survey we intend to review the nonlinear panel data analysis with respect to biases and
model structures. The flourishing literature of nonlinear panel data models introduces new
remedies and difficulties to the panel data analysis. Fortunately, the solutions are found
hastily as documented in this survey. For the future study this review can be extended by the
new developments in this area.
7
Smith and Fuertes (2010) classified the heterogeneity bias in large T sample structure.
8
For DPCCE estimation Sarafidis and Yamagata (2010) investigate the properties of instrumental variable
estimation and compare their estimators with GMM. However, they are dealing with the linear model where this
estimator is more efficient and constitute consistent estimators.
References
Ahn, S.C., Schmidt, P., 1995. Efficient estimation of dynamic panel models. Journal
of Econometrics 68, 5-27.
Alvarez, J., Arrelona, M., 2003. The time series and cross-section asymptotics of
dynamic panel data estimators. Econometrica 71, 1121-1159.
Anderson, T.W., Hsiao C. 1981. Estimation of dynamic models with error
components. Journal of American Statistical Association 76, 598-606.
Anderson, T.W., Hsiao C. 1982. Formulation and estimation of dynamic models using
panel data. Journal of Econometrics 18, 47-82.
Arellano M., 1989. A note on the Anderson-Hsiao estimator for panel data.
Econometrics Letters 31, 337-341.
Arrelano , M. Bond, S., 1991. Some tests of specification for panel data: Monte Carlo
evidence and an equations. Review of Economic Studies 58, 277-297.
Arrelano M., Bover, O., 1995. Another look at the instrumental variables estimation of error-
component models. journal of Econometrics 68, 29-51.
Coakley, J., Fuertes A.M., Smith, R., 2002. A Principal Components Approach to
Cross-Section Dependence in Panels. 10th International Conference on Panel Data, Berlin,
July 5-6, 2002 B5-3, International Conferences on Panel Data.
van Dijk, D., Fok D., Franses, P.H., 2005. A multi-level panel STAR model for US
manufacturing sectors. Journal of Applied Econometrics 20, 811-827.
Fouquau, J. Hurlin, C., Rabaud, I., 2008. The Feldstein-Horioka puzzle: A panel
smooth transition regression approach. Economic Modelling 25, 284-299.
Gonzalez, A., Teräsvirta, T., Dijk, D. 2005. Panel smooth transition regression
models, Working Paper Series in Economics and Finance No 604, Stockholm School of
Economics, Sweden.
De Groote T., Everaert, G. 2011. Common Correlated Effects Estimation of Dynamic
Panels with Cross-Sectional Dependence. Working Papers of Faculty of Economics and
Business Administration, Ghent University, Belgium 11/723, Ghent University, Faculty of
Economics and Business
Gørgens, T., Skeels, C.L., Würtz, H.A., 2009. Efficient Estimation of Non-Linear
Dynamic Panel Data Models with Application to Smooth Transition Models. CREATES
Research Papers 2009-51, School of Economics and Management, University of Aarhus
A Survey about Smooth Transition Panel Data Analysis
Keane, M.P. Runkle, D.E., 1992. On the estimation of panel-data models with serial
correlation when instruments are not strictly exogenous. Journal of Business and Economic
Statistics 10, 1-9.
Nickell, S., 1981. Biases in dynamic models with fixed effects. Econometrica 49, 1417-
1426.
O’connel, P.G.J., 1998. The overvaluation of purchasing power parity. Journal of
International Economics 44, 1-19.
Omay, T., Kan, Ö. E., 2010. Re-examining the threshold effects in the inflation-growth
nexus with cross-sectionally dependent non-linear panel: Evidence from six industrialized
economies. Economic Modelling 27, 996-1005.
Omay, T., Hasanov, M. Ucar, N., 2014(a). Energy Consumption and Economic Growth:
Evidence from Nonlinear Panel Cointegration and Causality Tests. Applied Econometrics, 34,
Omay, T., Apergis, N., Özleçelebi, H., 2014 (b). Energy Consumption and Growth: New
Evidence from a Non-Linear Panel and a Sample of Developing Countries. Singapore
Economic Review. (Forthcoming)
Pesaran, M.H., Smith, R., 1995. Estimating long-run relationships from dynamic
heterogeneous panels. Journal of Econometrics. 68, 79-113.
Pesaran, M.H., Shin, Y., Smith, R., 1999. Pooled mean group estimation of dynamic
heterogeneous panels. Journal of the American Statistical Association 94, 621-634.
Pesaran, M.H., 2004. General diagnostic tests for cross-section dependence in panels.
Cambridge Working Papers in Economics 0435. Faculty of Economics, University of
Cambridge.
Pesaran, M.H., 2006. Estimation and inference in large heterogeneous panels with a multifactor
error structure. Econometrica 74, 967-1012.
Pesaran, M.H., 2007. A simple panel unit root test in the presence of cross-section dependence.
Journal of Applied Econometrics 22: 265-312.
Ucar, N., Omay, T., 2009. Testing for unit root in nonlinear heterogeneous panels.
Economics Letters 104, 5 –8.
Sarafidis, V., Robertson, D., 2009. On the impact of error cross-sectional dependence in
short dynamic panel estimation. Econometrics Journal12, 62-81.
Sarafidis, V., Takashi, Y., 2010. Instrumental Variable Estimation of Dynamic Linear
Panel Data Models with Defactored Regressors under Cross-sectional Dependence. MPRA
Paper 25182, University Library of Munich.
Smith RP, Fuertes AM. 2010. Panel Time Series. Available online at:
http://www.ems.bbk.ac.uk/faculty/smith/RSpanel.pdf
Zellner, A., 1962. An efficient method of estimating seemingly unrelated regression
and tests for aggregation bias. Journal of the American Statistical Association 57, 348-368.
