SUPPLEMENT TO TESTING FOR REGIME SWITCHING: A COMMENT (Econometrica, Vol. 80, No. 4, July 2012, )

Transcription

1 Econometrica Supplementary Material SUPPLEMENT TO TESTING FOR REGIME SWITCHING: A COMMENT Econometrica, Vol. 80, No. 4, July 01, BY ANDREW V. CARTER ANDDOUGLAS G. STEIGERWALD We present proof of the inconsistency of the QMLE defined by Cho and White 007. Inconsistency arises from the fact that the quasi-log-likelihood does not attain a maximum at the population parameter values. S1. PROOF OF INCONSISTENCY OF A QMLE IN THIS SUPPLEMENTAL section we prove that, for an autoregressive process, the gradient of the quasi-log-likelihood does not equal zero when evaluated at the population parameter values. Recall the mixture model for an autoregressive process S1 1 Nθ 1 + αx t Nθ + αx t 1 1 on which the quasi-likelihood is based. For θ 1 = μ and θ = μ + γ, wehave 1 n E[l t αμγ=e[l t αμγ:=mαμγ n t=1 where [ Mαμγ = E log exp [γx t αx t 1 μ γ 1 [log pi + EX t αx t 1 μ Remaining variable definitions are given in the main paper. The key is to calculate the first derivative of Mαμγ with respect to the autoregressive coefficient α evaluated at the population values of the parameters.tobegin,letz t = X t α X t 1 μ. The derivative for α is α Mαμγ S γ X t 1 exp [γ Z t γ = E [γ Z t γ + EX t 1Z t 01 The Econometric Society DOI: /ECTA96

2 A. V. CARTER AND D. G. STEIGERWALD To calculate these expectations, we must account for the correlation between Z t and X t 1. The distribution of Z t conditional on X t 1 depends on this previous observation through S t 1. The conditional density is fz X t 1 = φz γ PS t = X t 1 + φzps t = 1 X t 1 [ = ζ t exp [γ Z t γ φz where φz is the density of a standard Gaussian random variable and ζ t := PS t = σx t 1. ThefirstexpectationinSis γ X t 1 exp [γ Z t γ E S3 [γ Z t γ = γ E X t 1 exp [γ z γ ζ t exp [γ z γ [γ z γ φzdz Because exp[γ z γ φz = φz γ, ζ exp [γ z γ t exp [γ z γ [γ z γ φzdz [γ z γ = [γ z γ φz γdz [γ v + γ = [γ v + γ φvdv

3 TESTING FOR REGIME SWITCHING: A COMMENT 3 and it follows that this integral is [γ v + γ [γ v + γ φvdv = ζ [ t + ζ t [γ v + γ ζ t / [ [γ v + γ [γ v + γ = ζ [ t + ζ t + ζ t 1 [γ v + γ φvdv Therefore, substituting this expression back into S3, α Mαμγ 1 φvdv = γ EX t 1 ζ t + γ E[X t 1 ζ t C γ + EX t 1Z t where C γ is the expectation C γ [ := E exp [γ ξ + γ 1 for ξ a standard Gaussian random variable. Clearly, C γ does not depend on X t 1 or S t and is a bounded positive quantity: 0 <C γ 1 1. Furthermore, we can use that EZ t X t 1 = γ PS t = X t 1 = γ ζ t to simplify the expression α Mαμγ = γ C γ E[X t 1ζ t The last factor in this expression is E[X t 1 ζ t =EX t 1 ζ t EX t 1 ES t = CovX t 1 S t The last equality follows from EX t 1 S t = EX t 1 ES t X t 1 and ES t X t 1 = ζ t. To obtain an expression for the covariance of X t 1 and S t, we use the following lemma.

4 4 A. V. CARTER AND D. G. STEIGERWALD LEMMA A: Under model S1 CovX t 1 S t = γ1 p 1 α p 1 Therefore, the expression for the partial derivative of the M function with respect to α becomes α Mαμγ = γ C γ p 1 1 α p 1 PROOF OF LEMMA A: We first use the recursive expression X t = μ + αx t 1 + γs t + ξ t = α k μ + γs t k + ξ t k where ξ t iidn0 1. This implies that the covariance is CovX t 1 S t = γ α k CovS t 1 k S t The covariance of the binary state variables is CovS t k S t = PS t = S t k = where = PS t = = PS t k = is the stationary probability in the Markov chain. It can be shown that the conditional probability is k p1 PS t = S t k = = Thus k p1 CovS t k S t = 1 and p1 CovX t 1 S t = γ1 α k p1 = γ1 p 1 α p 1 k Q.E.D.

5 TESTING FOR REGIME SWITCHING: A COMMENT 5 REFERENCE CHO, J., AND H. WHITE 007: Testing for Regime-Switching, Econometrica, 75, [1 Dept. of Statistics, University of California Santa Barbara, Santa Barbara, CA 93106, U.S.A.; carter@pstat.ucsb.edu and Dept. of Economics, University of California Santa Barbara, Santa Barbara, CA 93106, U.S.A.; doug@econ.ucsb.edu. Manuscript received October, 010; final revision received November, 011.