Nonlinear IV Panel Unit Root Tests

This paper presents the nonlinear IV methodology as an effective inferential basis for nonstationary panels. The nonlinear IV method resolves the inferential difficulties in testing for unit roots arising from the intrinsic heterogeneities and cross-dependencies of panel models. Individual units are allowed to be dependent through correlations among innovations, interrelatedness of short-run dynamics and/or cross-sectional cointegrations. If based on the instrumental variables that are nonlinear transformations of the lagged levels, the usual IV estimation of the augmented Dickey-Fuller type regressions yields asymptotically normal unit root tests for panels with general dependencies and heterogeneities. Moreover, the nonlinear IV estimation allows for the use of covariates to further increase power, and order statistics to test for more flexible forms of hypotheses, which are especially important in heterogeneous panels.

Taking a New Contour: A Novel Approach to Panel Unit Root Tests

The paper introduces a novel approach to testing for unit roots in panels. Following Chang and Park (2004), the approach takes a new contour that is drawn along the line given by the equi-squared-sum instead of the traditional one given by the equi-sample-size. As we show in the paper, the distributions of the unit root tests based on nonlinear IV t-ratios (which includes the Dickey-Fuller test as a special case) are asymptotically normal along the new contour. The normal asymptotics hold under both the null of a unit root and the local-to-unity alternative. Moreover, they are applicable also for the models with intercept, as long as is used the demeaning method relying only on the past information. Subsequently, we demonstrate that this startling finding may be exploited to invent tools and methodologies for the effective inferences in panel unit root models. In particular, our theory implies that the individual tests may be regarded asymptotically as normal samples if they are computed using the samples having the same sum of squares across all cross-sectional units, which may be obtained through the standard bootstrap method. Consequently, we may conveniently use various functionals of the individual tests to do valid inferences in panels.

Nonlinear IV Unit Root Tests in Panels with Cross-Sectional Dependency

We propose a unit root test for panels with cross-sectional dependency. We allow general dependency structure among the innovations that generate data for each of the cross-sectional units. Each unit may have different sample size, and therefore unbalanced panels are also permitted in our framework. Yet, the test is asymptotically normal, and does not require any tabulation of the critical values. Our test is based on nonlinear IV estimation of the usual augmented Dickey-Fuller type regression for each cross-sectional unit, using as instruments nonlinear transformations of the lagged levels. The actual test statistic is simply de2ned as a standardized sum of individual IV t-ratios. We show in the paper that such a standardized sum of individual IV t-ratios has limit normal distribution as long as the panels have large individual time series observations and are asymptotically balanced in a very weak sense. We may have the number of cross-sectional units arbitrarily small or large. In particular, the usual sequential asymptotics, upon which most of the available asymptotic theories for panel unit root models heavily rely, are not required. Finite sample performance of our test is examined via a set of simulations, and compared with those of other commonly used panel unit root tests. Our test generally performs better than the existing tests in terms of both 2nite sample sizes and powers. We apply our nonlinear IV method to test for the purchasing power parity hypothesis in panels.

Bootstrap Unit Root Tests in Panels with Cross-Sectional Dependency

We apply bootstrap methodology to unit root tests for dependent panels with N cross-sectional units and T time series observations. More specifically, we let each panel be driven by a general linear process which may be different across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual panels, we construct our unit root tests from the estimation of the system of the entire N panels. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply the bootstrap method to the approximated autoregressions to obtain the critical values for the panel unit root tests, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions. The proposed bootstrap tests are indeed quite general covering a wide class of panel models. They in particular allow for very general dynamic structures which may vary across individual units, and more importantly for the presence of arbitrary cross-sectional dependency. The finite sample performance of the bootstrap tests is examined via simulations, and compared to that of the t-bar statistics by Im, Pesaran and Shin (1997), which is one of the commonly used unit root tests for panel data. We find that our bootstrap panel unit root tests perform well relative to the t-bar statistics.

Bootstrap Unit Root Tests in Panels with Cross-Sectional Dependency

We apply bootstrap methodology to unit root tests for dependent panels with N cross-sectional units and T time series observations. More specifically, we let each panel be driven by a general linear process which may be different across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual panels, we construct our unit root tests from the estimation of the system of the entire N panels. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply the bootstrap method to the approximated autoregressions to obtain the critical values for the panel unit root tests, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions. The proposed bootstrap tests are indeed quite general covering a wide class of panel models. They in particular allow for very general dynamic structures which may vary across individual units, and more importantly for the presence of arbitrary cross-sectional dependency. The finite sample performance of the bootstrap tests is examined via simulations, and compared to that of the t-bar statistics by Im, Pesaran and Shin (1997), which is one of the commonly used unit root tests for panel data. We find that our bootstrap panel unit root tests perform well relative to the t-bar statistics, especially when N is small.Panels with cross-sectional dependency, unit root tests, sieve bootstrap, AR approximation

Panel Unit Root Tests in the Presence of Cross-Sectional Dependency and Heterogeneity

An IV approach, using as instruments nonlinear transformations of the lagged levels, is explored to test for unit roots in panels with general dependency and heterogeneity across cross-sectional units. We allow not only for the cross-sectional dependencies of innovations, but also for the presence of cointegration across cross sectional levels. Unbalanced panels and panels with differing individual short run dynamics and cross-sectionally related dynamics are also permitted. Panels with such cross-sectional dependencies and heterogeneities appear to be quite commonly observed in practical applications. Yet none of the currently available tests can be used to test for unit roots in such general panels. We also more carefully formulate the unit root hypothesis in panels. In particular, using order statistics we make it possible to test for and against the presence of unit roots in some of the individual units for a given panel. The individual IV t-ratios, which are the bases of our tests, are asymptotically normally distributed and cross-sectionally independent. Therefore, the critical values of the order statistics as well as the usual averaged statistic can be easily obtained from simple elementary probability computations. We show via a set of simulations that our tests work well, while other existing tests fail to perform properly. As an illustration, our tests are applied to some of the data sets that were used in earlier studies.Panels with cross-sectional dependency and heterogeneity, unit root test, cointegration, covariate, nonlinear instrument, order statistics

Unit Root Tests for Panels in the Presence of Short-run and Long-run Dependencies: Nonlinear IV Approach with Fixed N and Large T

An IV approach, using as instruments nonlinear transformations of the lagged levels, is explored to test for unit roots in panels with general dependency and heterogeneity across cross-sectional units. We allow not only for the cross-sectional dependencies of innovations, but also for the presence of cointegration across cross-sectional levels. Unbalanced panels and panels with differing individual short-run dynamics and cross-sectionally related dynamics are also permitted. Panels with such cross-sectional dependencies and heterogeneities appear to be quite commonly observed in practical applications. Yet, none of the currently available tests can be used to test for unit roots in such general panels. We also more carefully formulate the unit root hypotheses in panels. In particular, using order statistics we make it possible to test for and against the presence of unit roots in some of the individual units for a given panel. The individual IV t-ratios, which are the bases of our tests, are asymptotically normally distributed and cross-sectionally independent. Therefore, the critical values of the order statistics as well as the usual average statistic can be easily obtained from simple elementary probability computations. We show via a set of simulations that our tests work well, while other existing tests fail to perform properly. As an illustration, we apply our tests to the panels of real exchange rates, and find no evidence for the purchasing power parity hypothesis, which is in sharp contrast with the previous studies.

Electricity Demand Analysis Using Cointegration and Error-Correction Models with Time Varying Parameters: The Mexican Case

We specify and estimate a double-log functional form of the demand equation, using monthly Mexican electricity data for residential, commercial and industrial sectors. Income, prices and a nonparametric temperature measure are used as explanatory variables, and the income elasticity is allowed to evolve slowly over time by employing the time varying coefficient (TVC) cointegrating model. The specification of the proposed TVC cointegrating model is justified by testing it against the spurious regression and the usual fixed coefficient (FC) cointegration regression. The estimated coefficients suggest that the income elasticity has followed a predominantly increasing path for all sectors during the entire sample period, and that electricity prices do not significantly affect in the long-run the residential and commercial demand for electricity in Mexico.

"Nonlinear IV Unit Root Tests in Panels with Cross-Sectional Dependency"

We propose a unit root test for panels with cross-sectional dependency. We allow general dependency structure among the innovations that generate data for each of the cross-sectional units. Each unit may have different sample size, and therefore unbalanced panels are also permitted in our framework. Yet, the test is asymptotically normal, and does not require any tabulation of the critical values. Our test is based on nonlinear IV estimation of the usual ADF type regression for each cross-sectional unit, using as instruments nonlinear transformations of the lagged levels. The actual test statistics is simply defined as a standardized sum of individual IV t-ratios. We show in the paper that such a standardized sum of individual IV t-ratios has limit normal distribution as long as the panels have large individual time series observations and are asymptotically balanced in a very weak sense. We may have the number of cross-sectional units arbitrarily small or large. In particular, the usual sequential asymptotics, upon which most of the available asymptotic theories for panel unit root models heavily rely, are not required. Finite sample performance of our test is examined via a set of simulations, and compared to those of other commonly used panel unit root tests. Our test generally performs better than the existing tests in terms of both finite sample sizes and powers. We apply our nonlinear IV method to test for the purchasing power parity hypothesis in panels.

Bootstrapping Cointegrating Regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. We do not assume any specific data generating process, and employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the firrst differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen (1991) and Stock and Watson (1993) relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do inferences that are asymptotically valid. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.