Testing for panel cointegration with a level break
Introduction
During the last few years, it has become more common to test cointegration when working with panel data. The most widely applied tests are those of Kao (1999) and Pedroni, 1999, Pedroni, 2004, which are designed to test the null hypothesis of no cointegration vs. the alternative of cointegration. A common feature of all these tests is that they are based on the assumption that the cointegration relation is time-invariant. Under these circumstances, their asymptotic properties are well known and the reader is referred to Kao (1999) and Pedroni (2004) for a detailed treatment. In some empirical exercises, however, the researcher may wish to entertain the possibility that the series are cointegrated but that the relation between them has shifted at some unknown point in the sample. As shown by Hao (1996), this alters the limiting distribution of most conventional tests based on a time-invariant cointegration relation as the estimated regression needs to be modified to collect for the presence of the structural break. It is also likely to result in a loss of power and deceptive inference.
In this paper, we propose four simple tests for the null hypothesis of no cointegration that allow for a time-varying cointegration relation under both the null and alternative hypotheses. The proposed tests are based on those developed in the time series context by Gregory and Hansen (1996) and they are able to accommodate for a single unknown break in the level of each individual regression that may be located in different dates for different individuals. Using sequential limit arguments, it is found that the distributions of the tests are normal and that they are free of nuisance parameter dependencies. Since there are no closed-form solutions for the limit distributions, critical values for up to five regressors are obtained via simulation methods. We also evaluate the small-sample performance of the test in which case our simulation results suggest that the tests have small size distortions and good power.
The paper proceeds as follows. In the next section, we present the test statistics and their limiting distributions. Section 3 is then devoted to the Monte Carlo study and Section 4 concludes the paper. For notational convenience, the Bownian motion Bi(r) defined on r ∈ [0, 1] will be written as Bi and integrals such as ∫01Wi(r)dr will be written as ∫01Wi and ∫01Wi(r)dWi(r) as ∫01WidWi. The symbol ⇒ signifies weak convergence and [z] denotes the integer part of z.
Section snippets
The tests
Let zit = (yit,x′it)′ be an K + 1 dimensioned vector of integrated variables that may be partitioned into a scalar variate yit and a K dimensional vector xit. The data generating process (DGP) of zit is given by
For convenience in deriving the tests and their asymptotic distributions, we assume that vit is cross-sectionally independent and that it follows a linear process whose parameters satisfy the summability conditions of the following assumption. Assumption 1 (Error process.) The vector vit
Monte Carlo experiments
To examine the small-sample properties of the tests, we conduct a small set of Monte Carlo experiments using the following DGPwhere (uit,Δxit)′ ∼ N(0,I2) and xi0 = ei0 = 0. The data is generated for 1000 panels with N cross-sectional and T + 50 time series observations. The first 50 observations for each i are then disregarded to reduce the effect of the initial condition. For simplicity, we assume that there is a single regressor, that μ = δ = β = 1, and that λi = λ
Conclusions
In this paper, we propose four tests for the null hypothesis of no cointegration that allow for a single unknown break in the level of each individual regression. Using sequential limit arguments, it is found that the distributions of the tests are normal and that they are free of nuisance parameter dependencies, other than the number of regressors and the deterministic specification of the individual regressions. Since there are no closed-form solutions for the limit distributions, critical
Acknowledgements
The author thanks David Edgerton and one anonymous referee for valuable comments and suggestions. Financial support from the Jan Wallander and Tom Hedelius Foundation, research grant number P2005-0117:1, is gratefully acknowledged.
References (5)
- et al.
Residual-based tests for cointegration in models with regime shifts
Journal of Econometrics
(1996) Spurious regression and residual-based tests for cointegration in panel data
Journal of Econometrics
(1999)