CUSUM Methods for Monitoring Structural Changes in Structural Equations

Abstract

In this article, we introduce a monitoring scheme to (sequentially) detect structural changes in linear models with endogenous explanatory variables and develop asymptotic theories for them. The method is based on cumulative sums (CUSUM) of weighted residuals, in which the unknown in-control parameters have been replaced by its GMM estimate from the training sample. We characterize the limit distributions of the CUSUM process and the corresponding test statistic and results show that these tests are consistent. Moreover, We also obtain and tabulate the asymptotic critical values of the tests when p = 2. We also investigate our distribution theory with a Monte Carlo simulation which indicates the applicability of the methods.

Keywords:

• CUSUM
• GMM
• Instrument variable
• Structural changes
• Weighted residuals

2000 Mathematics Subject Classification:

• Primary 62J05
• Secondary 62L10

Acknowledgments

We would like to thank the Associate Editor and the two anonymous referees for their helpful comments and insightful suggestions. This research was supported jointly by the National Natural Science of China (Grant No. 50846021) and by Shannxi Science Development Funds (Grant No. 2002K08-G15).

Notes

The standard Brownian Motion is approximated using 1000 i.i.d. standard normal random variables on a unit and transformed to Brownian bridge. The number of replications is 2,500. τ = [N/n]

The percentage of rejection for the process observed under H ₀ is reported here. The results are based on 2,500 replications.

The numbers in each cell are rejection rates under alternative hypothesis which are based on 1,000 replications.

The numbers in each cell are mean and standard deviation (in parentheses) which are based on 2,500 replications.