Inference of Seasonal Cointegration: Gaussian Reduced Rank Estimation and Tests for Various Types of Cointegration*

Abstract

An extension of Gaussian reduced rank estimation of Ahn and Reinsel (Journal of Econometrics, Vol. 62, pp. 317-350, 1994) to seasonal periods other than four is presented. Simple adjustments for estimation that are necessary because of complex-valued seasonal unit roots are presented in detail and the asymptotic distribution of the estimators that takes the same form as that in Ahn and Reinsel (1994) is derived. Tests for contemporaneous cointegration and common polynomial cointegrating vectors (PCIVs) for different seasonal unit roots are presented. Finite sample properties are briefly examined through a small Monte Carlo simulation study and a numerical example is presented to illustrate the methods.

Semiparametric Seasonal Cointegrating Rank Selection

“…We assume that the initial value X0 is fixed and, for brevity, Xt is observed on a quarterly basis. Models with the other seasonal periods, e.g., monthly, can be easily implemented as in Ahn et al (2004). Note that r01, r02, and r03(r04) denote the CI ranks at seasonal unit roots 1, −1, and i(−i), respectively, (i.e., frequencies 0, π, and π/2(3π/2), respectively), and B1Ut, B2Vt, (B3 + B4L)Wt, and (B4 − B3L)Wt are stationary processes, i.e., CI relationships.…”

GMM Estimation for Seasonal Cointegration

“…Models with other seasonal periods, e.g., monthly, can be easily implemented as in Ahn et al (2004). Then, as in AR1994, if we expand (2.1) by Lagrange expansion at seasonal unit roots z = 1, −1, i and −i (i.e., frequencies 0, π, π/2 and 3π/2, respectively), we obtain the following VECM:…”

“…Since then, many approaches for analyzing seasonal cointegration have been developed. Among others, Ahn and Reinsel (1994) (AR1994) and Ahn et al (2004) used an iterative method considering all the frequencies of seasonal unit roots simultaneously. Johansen and Schaumburg (1999) considered a switching algorithm based on partial regression, to avoid the complexity generated with the simultaneous estimation.…”

Joint Test for Seasonal Cointegrating Ranks

“…Models with the other seasonal periods, e.g., monthly, can be easily implemented as in Ahn et al (2004) and models with deterministic terms, ΦD t = 0, as in Cubadda (2001) and JS. Then, as in Ahn and Reinsel (1994), if the series are cointegrated of order (1, 1) at seasonal unit roots z = 1, −1, i and −i (i.e., frequencies 0, π, π/2 and 3π/2, respectively), model (2.1) can be rewritten in the following error correction model (ECM):…”