We consider the cumulative sum (CUSUM) of squares test in a linear regression model with general mixing assumptions on the regressors and the errors. We derive its limit distribution and show how it depends on the nature of the error process. We suggest a corrected version that has a limit distribution free of nuisance parameters. We also discuss how it provides an improvement over the standard approach to testing for a change in the variance in a univariate times series. Simulation evidence is presented to support this.

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This paper considers various asymptotic approximations to the finite sample distribution of the estimate of the break date in a simple one-break model for a linear trend function that exhibits a change in slope, with or without a concurrent change in intercept. The noise component is either stationary or has an autoregressive unit root. Our main focus is on comparing the so-called "bounded-trend" and "unbounded-trend" asymptotic frameworks. Not surprisingly, the "bounded-trend" asymptotic framework is of little use when the noise component is integrated. When the noise component is stationary, we obtain the following results. If the intercept does not change and is not allowed to change in the estimation, both frameworks yield the same approximation. However, when the intercept is allowed to change, whether or not it actually changes in the data, the "bounded-trend" asymptotic framework completely misses important features of the finite sample distribution of the estimate of the break date, especially the pronounced bimodality that was uncovered by Perron and Zhu (2005) and shown to be well captured using the "unbounded-trend" asymptotic framework. Simulation experiments confirm our theoretical findings, which expose the drawbacks of using the " bounded-trend" asymptotic framework in the context of structural change models. Copyright Royal Economic Society 2006

In view of the fact that classic asymptotic theory can not provide satisfactory explanation for Ferson, Sarkissian and Simin's (2003a, 2003b) simulation findings on spurious regression in the context of financial economics, we develop an alternative distributional theory. Closely related is the well-known (nearly) observational equivalence issue in unit root testing literature. This study employs Nabeya-Perron type asymptotics and shows their simulation results can be well predicted. We hence re-enforce the fact that autocorrelation of dependent variable can not be used as an indication of spurious regression bias. Further, a convergent t test based on fix-b asymptotics following Sun (2005) is studied. Our simulation studies reveal further interesting result which explains and generalizes an illustrative simulation finding in FSS (2003a) and shows the asymptotic distribution of the convergent t statistic can be very close to standard normal if one chooses b properly. The interaction between spurious regression effect and data mining is also discussed.