**Unformatted text preview:**

Optimal Tests for Reduced Rank Time Variation in Regression Coefficients and Level Variation in the Multivariate Local Level Model November 2003 (Revised October 16, 2004) Piotr Eliasz Department of Economics, Princeton University James H. Stock Department of Economics, Harvard University and the National Bureau of Economic Research and Mark W. Watson* Woodrow Wilson School and Department of Economics, Princeton University and the National Bureau of Economic Research * This research was funded in by NSF grant SBR-0214131. We thank Zongwu Cai and Ulrich Müller for useful discussions.1 Abstract This paper constructs tests for martingale time variation in regression coefficients in the regression model yt = xt′βt + ut, where βt is k×1, and Σβ is the covariance matrix of Δβt. Under the null there is no time variation, so Ho: Σβ = 0; under the alternative there is time variation in r linear combinations of the coefficients, so Ha: rank(Σβ ) = r, where r may be less than k. The Gaussian point optimal invariant test for this reduced rank testing problem is derived, and the test’s asymptotic behavior is studied under local alternatives. The paper also considers the analogous testing problem in the multivariate local level model Zt = μt + at, where Zt is a k×1 vector, μt is a level process that is constant under the null but is subject to reduced rank martingale variation under the alternative, and at is an I(0) process. The test is used to investigate possible common trend variation in the growth rate of per-capita GDP in France, Germany and Italy. Keywords: TVP tests, multivariate local level model, POI tests JEL Numbers: C12, C22, C322 1. Introduction A long-standing problem in econometrics involves testing for stability of regression coefficients in the linear regression model. Using standard notation, the model is yt = xt′βt + ut (1.1) where yt is a scalar and xt is a k×1 vector. Under the null hypothesis the regression coefficients are stable, while under the alternative they are time varying. When k is large, standard tests for time variation have low power because they look for time variation in k different dimensions. However, in many empirical applications it is plausible to assume that time variation in the coefficients will be restricted to a relatively small number of linear combinations of the regression coefficients. For example, it might be assumed that any time variation is concentrated in the linear combinations Rβt where R is an r×k matrix. When R is known, the regression model can be transformed to isolate the coefficients Rβt, and the problem involves testing whether a subset of the regression coefficients are unstable (see Leybourne (1993)). In many applications a researcher might not know the value of R, and test power will deteriorate if the wrong value is used. This concern leads us to consider the testing problem when R is unknown. That is, we suppose that r linear combinations of the regression coefficients are unstable under the alternative hypothesis, but that these linear combinations are unknown. In our leading case r = 1, so there is only one dimension of time variation in the regression coefficients. We are concerned with three related questions. First, what are the power gains that can be attained using this rank information3relative to tests that look for time variation in all of the regression coefficients? Second, what are the power losses associated with using only the rank information relative to tests that use the value of R? Finally, what are the power losses from using a pre-specified but incorrect value of R? We carry out the analysis using an otherwise standard framework. We assume that Δβt is an I(0) process with covariance matrix ΣΔβ. Under the null hypothesis ΣΔβ = 0, while under the alternative ΣΔβ ≠ 0 with rank(ΣΔβ ) = r. As usual, we consider tests that are invariant to transformations yt → yt + xt′b. However, to capture the notion that R is unknown, we also restrict attention to tests that are invariant to transformations xt → Axt. As in Shively (1988), Stock and Watson (1998), and Elliott and Muller (2002) we consider versions of Gaussian point optimal invariant tests. As it turns out, a closely related problem involves testing whether a k×1 vector process Zt is I(0) against the alternative that is I(1). We write this model as Zt = μt + at (1.2) where at is I(0) and μt is I(1). This is a version of what Harvey (1989) calls the “local level model,” because μt represents the local level of the process. If ΣΔμ = 0, then μt is constant and Zt is I(0); when ΣΔμ ≠ 0, then Zt is I(1). In many applications, shifts in μt are a function of a small number of factors or common trends, so that the elements of Zt are cointegrated, ΣΔμ has reduced rank, and (1.2) is a reduced rank local level model. This leads us to consider testing the null that ΣΔμ = 0 against the alternative that ΣΔμ ≠ 0, but with rank(ΣΔμ) = r. This testing problem is also carried out in an otherwise standard framework. Stock (1994) surveys the large literature concerned with testing ΣΔμ = 0 in (1.2) when k4=1; we utilize multivariate versions of the Gaussian point optimal invariant tests derived from King (1980) that have been used for the univariate testing problem. These tests are invariant to transformations of the form Zt → Zt + b. We further restrict the tests so that they are invariant to transformations of the form Zt → AZt to capture the notion that the I(1) linear combinations of Zt are unknown. Jansson (2002) considers a multivariate problem closely related to ours. He supposes that r = 1, but that the I(1) linear combination of Zt is known. Our analysis can then be viewed as an extension to the case of an unknown linear combination. The paper is organized as follows. Section 2 considers the reduced rank multivariate local level model (1.2), and presents exact results using a benchmark Gaussian version of the model, and then extends the results to more general stochastic processes using asymptotic approximations. Section 3 shows the asymptotic equivalence of the testing problems for the regression model (1.1) and the multivariate local level model (1.2). This implies that the testing results derived in section 2 carry over to regression model. Section 4 presents asymptotic power results and answers