An Efficient Estimation for Switching Regression Models: A Monte Carlo Study

Abstract

This article investigates an efficient estimation method for a class of switching regressions based on the characteristic function (CF). We show that with the exponential weighting function, the CF-based estimator can be achieved from minimizing a closed form distance measure. Due to the availability of the analytical structure of the asymptotic covariance, an iterative estimation procedure is developed involving the minimization of a precision measure of the asymptotic covariance matrix. Numerical examples are illustrated via a set of Monte Carlo experiments examining the implementation, finite sample property and the efficiency of the proposed estimator.

Keywords:

• Characteristic function
• Gaussian mixtures
• Integrated squared error
• Switching regression

Mathematics Subject Classification:

• 62F10
• 62F12

Acknowledgments

The author thanks an anonymous referee, Associate Editor, and the Editor for their constructive suggestions and comments. The author is also grateful for the suggestions and comments from participants in the Far East and South Asia Meeting of the Econometric Society (FEMES) at University of Tokyo and Econometrics Workshop at Southwestern University of Finance and Economics and University of Guelph.

Notes

If is a set of different order of lagged values of y _i , (3) assumes the similar set up as the mixture autoregressions (MAR) structure; see Wong and Li (2000).

We will provide more discussions on the failure of the ordinary MLE procedure in the next section.

We thank an anonymous referee for pointing out the related literature.

Quandt and Ramsey (1978) and Quandt (1988) also pointed out that the problem is essentially due to the singularity of the matrix of second partial derivatives of the log-likelihood function, which is equivalent to a vanishing Jacobian for Gaussian mixtures model from the conventional MLE approach.

(6) is essentially analog to z _i in the MGF application to the SWR, see p. 723 in Quandt and Ramsey (1978) and p. 510 in Schmidt (1982).

As pointed out by an anonymous referee, in (7), if we allow the mean of each mixture component to be constant, i.e., μ _k , then this distance measure collapses to the objective function achieved in Heathcote (1977) in the i.i.d. context because is the associated ECF.

While the distance function includes a constant term, which does not depend on the unknown parameters, consequently, this constant term does not affect the optimization process in practice. For convenience, the constant term is omitted in (8).

Xu and Knight (2010) applied a similar iterated procedure for estimating the MN parameters. See the article for more details.

Provided that the influence function of is bounded, the optimal b theoretically exists, which has been shown in Besbeas (1999).

The parameter space tends to expand rapidly as the number of the mixture component increases. For example, in a simple SWR environment with two regressors in each regime, when K = 3, the number of the unknown parameters to estimate is 11, and when K = 5, the number of the unknown parameters increases to 19. This can lead to numerical convergence problems in practice and sometimes result in a prohibitive computational cost. This has been a major impediment to the attempts to extend the model to multivariate settings.

Some other experiments with different parameter designs have also been examined, for example, x is generated from a normal distribution or student-t distribution. We found similar results and patterns. To save space, those simulation results are not reported in this article.

*Normality is not rejected at 5% significance level (cut-off value is 0.0428); **Normality is not rejected at 1% significance level (cut-off value is 0.0513).

These numbers are calculated using Tables 3 and 4 in Quandt (1972), pp. 308–309.

The duration of the iterative estimation may depend on other factors, such as the initial values, the characteristics of the sampling data, the stopping rules for the convergence both in the parameters and bandwidth, optimization software as well as computer speed, etc. Here, we just provide an average approximation time on a computer with CPU 6400 at 2.13 G Hz (0.99 GB of RAM) using Matlab 7.1. We also carried out some experiments for testing the sensitivity of the initial values. In general, the results show that the convergent estimates are close to the true parameter values. Consequently, the initial values for the program in the Monte Carlo simulation are set to be close to the true parameter values.

The results of RMSEs are reported in Table 3 (p. 308) in Quandt (1972). For instance, for Case 7 in this article and Case 3 in Quandt (1972), the RMSEs of (p, β₁, β₂, γ₁, γ₂) are [0.0600, 1.3336, 0.0859, 2.3752, 0.1508] and [0.0648, 1.0366, 0.0632, 3.8029, 0.2379], respectively.

The RMSE found to be slightly larger in this article is mainly due to the existence of some outliers in the estimates. One can observe some outliers in the QQ-plot in Fig. 2.

To make a “fair” comparison, we randomly sample 200 subsets of size 50 from 1,000 ISE estimates in the simulations. As a result, we find the RMSEs are, in general, smaller than what were reported in both Quandt (1972) and Quandt and Ramsey (1978). To save space, the comparison results are not reported in this article. But, they are available upon request.

To save space, the QQ-plots for the first five parameters are reported for each simulation case. The variance estimates are generally found to be significantly deviated from the normality (except for Case 4). Those graphs are available upon request.