University of Vienna, Dept. of Economics Master in Economics Vienna 2010
The Model (1) Westerlund (2007) consider the following DGP: y it = φ 1i + φ 2i t + z it (1) x it = x it 1 + υ it (2) where the stochastic term z it is modelled as follows: α i (L)υ z it = α i (z it 1 β i x it 1 ) + γ i (L)υ it + e it, (3) where α i (L) = 1 j=1 α ijl j and γ i (L) = j=0 γ ijl j are scalar and k-dimensional polynomials in the lag operator.
The Model (2) Note that equation (3) is basically the conditional model for z it given x it in a standard vector error correction setup, with equation (2) being the associated marginal model for x it. By substituting equation (1) into (3), we get the following conditional error correction model for y it α i (L)y it = δ 1i + δ 2i t + α i (y it 1 β i x it 1 ) + γ i (L)υ it + e it, (4) where δ 1i = α i (1)φ 2i α i φ 1i + α i φ 2i and δ 2i = α i φ 2i now represent the deterministic components. Typical deterministic elements include a constant and a linear time trend. To this end, three cases are considered.
The Model (3) In case 1, φ 1i and φ 2i are both restricted to zero so y it has no deterministic terms, while in case 2, φ 1i is unrestricted but φ 2i is zero so y it is generated with a constant. In case 3, there are no restrictions on either φ 1i or φ 2i such that y it is generated with both a constant and a trend.
The Model (4) In all three cases, note that the error correction model in equation (4) can only be stable if the variables it comprises are all stationary. Thus, as y it 1 β i x it 1 must be stationary, the vector β i defines a long-run equilibrium relationship between x it and y it provided that the errors υ it and e it are also stationary.
The Model (5) Any deviation from this equilibrium relationship lead to a correction by a proportion 2 < α i < 0, which is henceforth referred to as the error correction parameter. If α i < 0, then there is error correction, which implies that y it and x it are cointegrated, whereas if α i = 0, the error correction will be absent and there is no cointegration. This suggests that the null hypothesis of no cointegration for cross-sectional unit i can be implemented as a test of H 0 : α i = 0 vs. H 1 : α i < 0.
The Model (6) Westerlund proposes four new panel statistics that are based on this idea. Two of the statistics are based on pooling the information regarding the error correction along the cross-sectional dimension of the panel. These are referred to as panel statistics. The second pair does not exploit this information and are referred to as group mean statistics. The relevance of this distinction lies in the formulation of the alternative hypothesis.
The Model (7) For the panel statistics, the null and alternative hypotheses are formulated as H 0 : α i = 0 for all i vs. H p 1 1 : α i < 0 for all i, which indicates that a rejection should be taken as evidence of cointegration for the panel as a whole. By contrast, for the group mean statistics, H 0 is tested vs. H g 1 : α i < 0 for at least some i, suggesting that a rejection should be taken as evidence of cointegration for at least one of the cross-sectional units.
Test construction (1) In constructing the new statistics, it is useful to rewrite equation (4) as y it = δ i d t +α i (y it 1 β i x it 1 )+ α ij y it j + γ ij x it j +e it j=1 j=0 where d t = (1, t) now holds the deterministic components, with δ i = (δ 1i, δ 2i ) being the associated vector of parameters. (5)
Test construction (2) The problem is how to estimate the error correction parameter α i, for the new tests. One way is to assume that β i is known, and to estimate α u by OLS. However, as shown by Boswijk (1994) and Zivot (2000), tests based on a prespecified β i are generally not similar and depend on nuisance parameters, even asymptotically.
Test construction (3) As an alternative approach, note that equation (5) can be reparameterized as y it = δ i d t + α i y it 1 λ ix it 1 + α ij y it j + γ ij x it j + e it j=1 j=0 In this regression, the parameter α i is unaffected by imposing an arbitrary β i, which suggests that the least squares estimate of α i can be used to provide a valid test of H 0 vs. H 1. (6)
Test construction (4) Indeed, because λ i is unrestricted, and because the cointegration vector is implicitly estimated under the alternative hypothesis (λ i = α i β i ), this means that it is possible to construct a test based on α i that is asymptotically similar, and whose distribution is free of nuisance parameters. The four new tests proposed by Westerlund are based on the least squares estimate of α i in equation (6) and its t ratio.
The group mean test statistics (1) The construction of the group mean statistics is carried out in three steps. The first stes to estimate equation (6) by least squares for each individual i, which yields y it = ˆδ i d t + ˆα i y it 1 ˆλ ix it 1 + ˆα ij y it j + ˆγ ij x it j + e it j=1 j=0 The lag order is permitted to vary across individuals, and can be determined preferably by using a data-dependent rule. Another possibility is to use an information criterion, such as the Akaike criterion. Alternatively, the number of lags can be set as a fixed function of T. (7)
The group mean test statistics (2) The second stenvolves estimating α i (1) = 1 α ij. A natural way of doing this is to use a parametric approach and to estimate α i (1) using j=1 ˆα i (1) = 1 ˆα ij. Unfortunately, tests based on ˆα i (1) are known to suffer from poor small-sample performance because of the uncertainty inherent in the estimation of the autoregressive parameters, especially when is large. j=1
The group mean test statistics (3) As an alternative approach, consider the following kernel estimator ˆω 2 yi = 1 T 1 M i j= M i ( ) j T 1 Y it Y it j M i + 1 t=j+1 where M i is a bandwidth parameter that determines the number of covariances to be estimated in the kernel. The relevance of ω 2 yi is evident by noting that under the null hypothesis, the long-run variance ω 2 yi of y it is given by ω 2 ui α i (1) 2 where ω 2 ui is the corresponding long-run variance of the composite error term u it = γ i (L)υ it + e it.
The group mean test statistics (4) This suggests that α i (1) can be estimated alternatively using ω ui ω yi, where ω ui may be obtained as above using kernel estimation with y it replaced by û it = ˆγ ij x it j + ê it j=0 where γ ij and ê it are obtained from equation (6). The resulting semiparametric kernel estimator of α i (1) will henceforth be denoted α i (1).
The group mean test statistics (5) The third stes to compute the test statistics as follows G τ = 1 N N ˆα i SE ˆα i i and G α = 1 N N i T ˆα i ˆα i (1) where SE is the conventional standard error.
The panel statistics (1) The panel statistics are complicated by the fact that the both the parameters and dimension of equation (6) are allowed to differ between the cross-sectional units, and therefore a three-step procedure to implement these tests is suggested. The first stes the same as for the group mean statistics and involves determining, the individual lag order. Once has been determined, one regresses y it and y it 1 onto d t, the lags of y it as well as the contemporaneous and lagged values of x it.
The panel statistics (2) This yields the projection errors ỹ it = y it ˆδ i d t ˆλ ix it 1 ˆα ij y it j ˆγ ij x it j + e it and j=1 j=0 ỹ it = y it 1 ˆδ i d t ˆλ ix it 1 ˆα ij y it j ˆγ ij x it j + e it j=1 j=0
The panel statistics (3) The second stenvolves using ỹ it and ỹ it 1 to estimate the common error correction parameter α and its standard error. In particular, one computes ( N T ˆα = i=1 t=2 ) 1 N ỹit 1 2 i=1 The standard error of ˆα is given by: T t=2 i=1 t=2 1 ˆα i (1)ỹit 1 ỹ it ( N T ) 1/2 SE(ˆα) = (Ŝ2 N ) 1 ỹit 1 2 where Ŝ 2 N = 1 N N i=1 Ŝ2 i, with Ŝ i = ˆσ i /ˆα i (1) and σ i the SE in equation (7).
The panel statistics (4) The third stes to compute the panel statistics as P τ = ˆα SE(ˆα) P α = T ˆα