Markov Regime-Switching Tests: Asymptotic Critical Values
-
Andrew V. Carter
and
Douglas G. Steigerwald
Abstract
Empirical research with Markov regime-switching models often requires the researcher not only to estimate the model but also to test for the presence of more than one regime. Despite the need for both estimation and testing, methods of estimation are better understood than are methods of testing. We bridge this gap by explaining, in detail, how to apply the newest results in the theory of regime testing, developed by Cho and White [Cho, J. S., and H. White 2007. “Testing for Regime Switching.” Econometrica 75 (6): 1671–1720.]. A key insight in Cho and White is to expand the null region to guard against false rejection of the null hypothesis due to a small group of extremal values. Because the resulting asymptotic null distribution is a function of a Gaussian process, the critical values are not obtained from a closed-form distribution such as the χ². Moreover, the critical values depend on the covariance of the Gaussian process and so depend both on the specification of the model and the specification of the parameter space. To ease the task of calculating critical values, we describe the limit theory and detail how the covariance of the Gaussian process is linked to the specification of both the model and the parameter space. Further, we show that for linear models with Gaussian errors, the relevant parameter space governs a standardized index of regime separation, so one need only refer to the tabulated critical values we present. While the test statistic under study is designed to detect regime switching in the intercept, the test can be used to detect broader alternatives in which slope coefficients and error variances may also switch over regimes.
- 1
As Carter and Steigerwald (2012) note, inconsistency of a QMLE does not necessarily imply inconsistency of a QLR test.
- 2
This is similar to the behavior of a one-sided likelihood ratio test (van der Vaart 1998, p. 235).
- 3
The element of the gradient corresponding to θ0 is identically zero when evaluated at π=1 and so is deleted from the vector that forms ℐ (θ0) (Cho and White 2007, assumption A.6, 1678).
- 4
Cho and White (2007, 1693) show
- 5
Cho and White select J=150 and consider a maximal value of η=5, so η2/J≤1/6.
- 6
If the errors are homoskedastic, so that ν1=ν2, then the covariance contains an additional term, see Carter and Steigerwald (2011) for details.
6 Appendix
6.1 Formal Conditions
We present the assumptions that define a class of processes to which the asymptotic theory presented in Section 2 applies. The two assumptions presented here combine A1–A2(i) from Cho and White with A2(ii) from Carter and Steigerwald (2012)
Assumption 1
The observable random variables
d∈ℕ, are generated as a sequence of strictly stationary β-mixing random variables such that for some c>0 andρ∈[0,1) the beta-mixing coefficient, gτ, is at most cρτ.The sequence of unobserved state variables that indicate regimes,
, is generated as a first-order Markov process such that ℙ(St=1|St−−1=0)=p0and ℙ(St=0|St−−1=1)=p1with pi ∈ [0,1] (i=0,1).The given
= is a Markov regime-switching process. That is, for some 
,
where ℱt−1:= σ (Yt−1, Zt, St) is the smallest σ-algebra generated by
r0∈ℕ; and the conditional cumulative distribution function of Yt|ℱ, F (·|Zt;γ,θj) has a probability density function f (·|Zt;γ,θj) (j=0,1). Further, for (p0, p1) ∈ (0,1]×(0,1]\{(1,1)}, (γ, θ0, θ1) is unique in
The vector γ captures all parameters of F(·), including the scale parameter, that do not vary across regimes. The point p0=p1=1 is excluded from the parameter space to rule out a deterministically periodic process for {St}, which would imply that {Yt} is not strictly stationary.
The model for the data generating process specifies a compact parameter space.
Assumption 2
A model for f (·|Zt;γ,θj) is
where 
, and Γ and Θ are compact convex sets in
and ℝ respectively. Further, for each (γ,θj)∈
, f(·|Zt;γ,θj) is a measurable probability density function, where the support of f (·|Zt;γ,θj) is the same for all 
, with cumulative distribution function F (·|Zt;γ,θj) (j=0,1).The covariates are exogenous in the sense that ℙ (St=j|ℱt−1) is independent ofZt for (j=0,1).
Additional conditions that imply a uniform bound on the first eight partial derivatives of the quasi-log-likelihood and an invertible information matrix are needed to establish (7) [see Cho and White 2007, Theorem 6(b) and Assumptions A.3, A.4, A.5 (ii), (iii), A.6 (iv)]. These conditions are satisfied for a Gaussian density.
6.2 Gaussian Process Covariance
6.2.1 Single Equation – Derivative Calculations
For the process given by (3) with Ut ~ i.i.d.N(0,ν), the quasi-log-likelihood for observation t, lt, equals
where c=2 · pi (where pi=3.14 …).
The gradient of lt evaluated at (1,γ,θ0,θ⁎) contains
where 
. The remaining elements of the gradient are
We analyze the behavior of 
in detail, as this forms the heart of the calculations for ℐ(θ0)=ℐ(θ0,θ0). Further detail, covering the remaining calculations, can be found in Carter and Steigerwald (2011).
To determine the behavior of 
first note that because 
the definition of a moment generating function yields 
for any real number s. Let 
, so 
. Hence
In similar fashion, 
and 
, hence
We also need to calculate 
and 
For the first quantity,
where 
. Note 
so
For the second quantity,
Because 
With these calculations in hand, the elements of the first row of ℐ(θ0) are
(1, 1) 
(1, 2) 
(1, 3) 
(1, 4) 
6.2.2 Simultaneous Equations – Covariance Calculations
From the reduced form, the coefficient on the state variable, St, is d=δA−1 (1 0)T and the covariance matrix of the errors is Ω−1=A−1Σ(A−1)T with 
As detailed in Carter and Steigerwald (2011), the covariance of the Gaussian process is
where 
.4 The quantity 
simplifies as
6.3 Pseudo-Code
Prior to the first iteration, the researcher must select the set H that contains η, the resolution of the grid of values in H (we recommend 0.001) and the number of normal random variables, J, used to approximate the Gaussian process covariance. (We detail how to select J on page 12. For many applications J=150 is sufficient.) For each of r=1,…,R iterations:
Generate
For each value of η in the grid mesh, construct
(the equation for ℊA(η) appears at the top of page 12)
Obtain mr=max {[max (0, ε4)]2, maxη [min (0, ℊA (η))]2} [use of ε4 is described at the top of page 11; the formula for mr corresponds to the right side of (7)]
This yields 
. Let 
be the ordered values from which the critical value for a test with size 5% is m[.95R].
References
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©2013 by Walter de Gruyter Berlin Boston
Articles in the same Issue
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- Research Articles
- A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model
- Markov Regime-Switching Tests: Asymptotic Critical Values
- Finite Mixture for Panels with Fixed Effects
- Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks
- A Control Function Approach to Estimating Dynamic Probit Models with Endogenous Regressorsa
- Practitioner’s Corner
- Reproducible Econometric Simulations
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Articles in the same Issue
- Masthead
- Masthead
- Research Articles
- A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model
- Markov Regime-Switching Tests: Asymptotic Critical Values
- Finite Mixture for Panels with Fixed Effects
- Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks
- A Control Function Approach to Estimating Dynamic Probit Models with Endogenous Regressorsa
- Practitioner’s Corner
- Reproducible Econometric Simulations
- Teaching Corner
- Understanding and teaching unequal probability of selection1)
