The spurious regression of AR(pp) infinite-variance sequence in the presence of structural breaks

This paper analyzes a spurious regression involving AR(p) infinite-variance processes in the presence of structural breaks by least squares using asymptotic theory. It is found that when regressing two independent infinite-variance sequence with breaks in the level and slope of trend, no matter whether the breaks occur at different points or not, the t-ratios become divergent and spurious phenomenon happens. The intuition behind this is that structural breaks can increase persistency in the level of regression errors, which then leads to spurious regressions. Simulation reveals that the effects of spurious regression not only depend on the autoregressive parameter and tailed index, but are sensitive to the presence of a linear trend in the regression model, and to the relative location of breaks with the sample. As a result, spurious effects might occur more often than we previously believed as they can arise even between AR(p) infinite-variance series with structural breaks. The spurious regression problem, as it is conceived in the time series econometric literature, can be traced back to Yule (1926), who identified the phenomenon by means of a computerless Monte Carlo experiment in which correlation coefficients were obtained from pairs of independent non-stationary variables. Granger and Newbold (1974) identified it again for simple least squares estimates and showed that when unrelated data series are close to the integrated processes of order one or the I(1) processes, then running a regression with this type of data will yield spurious effects, i.e., results suggesting the presence of significant relationships among time series variables when in fact no such relationship is present in the data generating process under study. Phillips (1986) proved that the usual t-ratios in a spurious regression do not have a limiting distribution but diverge as the sample size approaches infinity. Tsay (1999) examined the possibility of spurious relationship between two independent integrated errors processes belonging to the domain of attraction of a stable law with tailed index κ, and showed that the t-ratios diverge at the rate of T 1/2 , which is identical to what Phillips (1986) has obtained for the Gaussian case where κ = 2. The above results served as a springboard to a subsequent long series of investigations of the phenomenon for different types of regression and different types of data generating process. While Granger and Newbold (1974) and Phillips (1986) used driftless random walks, Entorf (1997) analyzed two independent random walks with non-zero …