# alternative of l breaks, where l<=ub, and ub is the upper bound in the number of breaks imposed

a priori. Another version of this statistic is what BP call the Wdmax which applies weights to supF_{T}(l) so that the marginal p-values are equal across values of l. Alternatively, the number of breaks can be determined based on the values of F-statistics that explicitly test the null of l breaks against the alternative of l+1 breaks. In other words, this statistic tests whether further structural change is present in the data, given that some break points have already been identified. As in the previous case, the tests used have non-standard asymptotic distributions. The critical value tables are provided by BP.

Table 2 presents results for the individual and joint Lc tests applied to the ten economic sectors. For each of the 10 sectors, constancy in the variance of the error terms is rejected, with the individual Lc statistic significant at the 1% level. This result is consistent with the extensive literature on time-varying variances in financial and economic time-series. The individual test results are less clear for the _{k }parameters in equation (2). Specifically, the null of constant

exposure to changes in the dividend yield is not accepted at the 5% level for the Financials or Utilities sectors. Similarly, the null hypothesis is rejected at the same significance level for constancy of exposure to the yield spread for the Health Care sector and for the change in the price of oil for the Technology sector. The null of parameter stability is also rejected for the intercept of the Technology and Telecommunications sectors.

# (Insert Table 2 here)

An important result in Table 2 is that the joint Lc statistic rejects stability at the 1% significance level for all sectors with the exception of Consumer Staples, for which the null is rejected at the 5% level. Given these results, it is relevant to consider Hansen’s own observation that joint significance tests may be more reliable than single parameter tests, especially when “…the shifting error variance induces too much noise into the series for the test to be able to distinguish parameter variation from sampling variation”.

a conditional factor model.

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