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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.

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