This paper suggests a generalized method of moments (GMM) based estimation for dynamic panel data models with individual specific fixed effects and threshold effects simultaneously. We extend Hansen’s (Hansen, 1999) original setup to models including endogenous regressors, specifically, lagged dependent variables. To address the problem of endogeneity of these nonlinear dynamic panel data models, we prove that the orthogonality conditions proposed by Arellano and Bond (1991) are valid. The threshold and slope parameters are estimated by GMM, and asymptotic distribution of the slope parameters is derived. Finite sample performance of the estimation is investigated through Monte Carlo simulations. It shows that the threshold and slope parameter can be estimated accurately and also the finite sample distribution of slope parameters is well approximated by the asymptotic distribution.
Since many economic relationships are dynamic and nonlinear, nonlinear/dynamic panel data models could obtain more information from data sources than traditional models [
Many results exist in the theoretical literature concerning the estimation and inference for dynamic panel data models. Since the lagged dependent variables and the disturbance term are correlated due to the unobserved effects, standard least square methods could not obtain consistent estimators when the model is dynamic. To overcome this problem, Anderson and Hsiao [
Several models could be chosen to describe the nonlinear relationship such as mixture models, switching models, smooth transition threshold models, and threshold models. In this paper threshold model is used because of wide applications in empirical researches. This model splits the sample into classes based on an observed variable—whether or not it exceeds some thresholds. In most situations, the complexity of the problem increases because the exact threshold is unknown and needed to be estimated. The estimation and inference are fairly well developed for linear models with exogenous regressors [
The dynamic panel threshold models have been used in empirical literature. Cheng et al. [
The remainder of the paper proceeds as follows. Section
Consider a simple AR
One can also write (
In this section, we first consider a simple model without exogenous covariates and derive GMM based estimator for the threshold parameter
In traditional dynamic panel data model, two methods are commonly used to remove individual effect
First, we take first-difference for model (
For any given
Define
Stacking over individuals, (
In fact, this estimator is infeasible in empirical studies, since it depends on an unknown parameter
Once
According to Hansen [
Hansen [
Now we extend the results in the previous subsection to cases with strictly exogenous variables. Consider additional regressors
In this section Monte Carlo experiments are implemented to examine the finite sample performance of our estimator. For this purpose we consider the following design.
The data generating process (DGP) is given by
The computation of the threshold
Tables
Quantiles of distribution,
Quantiles |
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0.05 | 0.5 | 0.95 | 0.05 | 0.5 | 0.95 | 0.05 | 0.5 | 0.95 | |
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0.10 | 2.09 | 4.12 | 0.91 | 2.05 | 3.17 | 1.39 | 2.00 | 2.34 |
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0.89 | 2.00 | 3.40 | 1.16 | 2.00 | 2.84 | 1.60 | 2.00 | 2.33 |
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−0.22 | 1.88 | 3.26 | 1.54 | 2.00 | 2.58 | 1.72 | 2.00 | 2.36 |
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0.25 | 0.44 | 0.61 | 0.37 | 0.46 | 0.55 | 0.38 | 0.46 | 0.52 |
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0.33 | 0.45 | 0.57 | 0.40 | 0.48 | 0.54 | 0.43 | 0.48 | 0.53 |
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0.19 | 0.46 | 0.56 | 0.43 | 0.49 | 0.54 | 0.45 | 0.49 | 0.53 |
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0.37 | 0.55 | 0.84 | 0.47 | 0.56 | 0.65 | 0.50 | 0.57 | 0.63 |
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0.46 | 0.57 | 0.70 | 0.50 | 0.57 | 0.64 | 0.53 | 0.58 | 0.63 |
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0.45 | 0.56 | 0.66 | 0.53 | 0.59 | 0.64 | 0.54 | 0.58 | 0.63 |
Quantiles of distribution,
Quantiles |
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0.05 | 0.5 | 0.95 | 0.05 | 0.5 | 0.95 | 0.05 | 0.5 | 0.95 | |
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1.84 | 2.02 | 2.17 | 1.90 | 2.00 | 2.02 | 1.98 | 1.99 | 2.00 |
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1.93 | 2.00 | 2.06 | 1.94 | 2.00 | 2.03 | 1.98 | 2.00 | 2.02 |
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1.93 | 2.01 | 2.04 | 1.96 | 2.00 | 2.04 | 2.00 | 2.00 | 2.02 |
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0.26 | 0.42 | 0.57 | 0.37 | 0.46 | 0.54 | 0.40 | 0.47 | 0.53 |
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0.32 | 0.45 | 0.58 | 0.40 | 0.47 | 0.54 | 0.43 | 0.48 | 0.53 |
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0.37 | 0.47 | 0.58 | 0.42 | 0.48 | 0.54 | 0.44 | 0.49 | 0.53 |
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0.54 | 0.71 | 0.86 | 0.65 | 0.74 | 0.83 | 0.70 | 0.76 | 0.82 |
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0.62 | 0.74 | 0.85 | 0.70 | 0.77 | 0.83 | 0.73 | 0.78 | 0.83 |
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0.64 | 0.76 | 0.87 | 0.70 | 0.77 | 0.84 | 0.73 | 0.79 | 0.84 |
Table
Table
Figure
Density distribution of slope parameters (small threshold).
Figure
Density distribution of slope parameters (big threshold).
This paper extends the estimation of threshold models in nondynamic panels to dynamic panels and presents practical estimation methods for these econometric models with individual-specific effects and threshold effects. The foremost feature of these models is that they allow the econometrician to consider the dynamic and threshold relationships in economic system simultaneously. As mentioned in the introduction, many applications may have such relationships. Using the first-difference to eliminate the individual-specific effects, we prove that the orthogonality conditions proposed by Arellano and Bond [
There are several possible extensions to this work. The asymptotic properties of the threshold parameter would be an interesting topic. Also, testing for one or multiple thresholds is also worth studying, which is saved for future research.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the editor and three anonymous referees for many constructive and helpful comments. This work was partially supported by the National Natural Science Foundation of China (Grant nos. 71301160 and 71301173), China Postdoctoral Science Foundation funded project (Grant nos. 2012M520419, 2012M520420, and 2013T60186), Beijing Planning Office of Philosophy and Social Science (13JGB018), and Program for Innovation Research in Central University of Finance and Economics.