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# Residual Regression

t-statistic Let  = r( ) : R k ! R be a parameter of interest, b its estimator and s(b) its asymptotic standard error. Consider the statistic T() = b  s(b) : (7.33) Di§erent writers have called (7.33) a t-statistic, a t-ratio, a z-statistic or a studentized statistic, sometimes using the di§erent labels to distinguish between Önite-sample and asymptotic inference. As the statistics themselves are always (7.33) we wonít make this distinction, and will simply refer to T() as a t-statistic or a t-ratio. We also often suppress the parameter dependence, writing it as T: The t-statistic is a simple function of the estimate, its standard error, and the parameter. By Theorems 7.9 and 7.10, p n  b   d ! N (0; V) and Vb p ! V: Thus T() = b  s(b) = p n  b   q Vb d ! N (0; V) p V = Z  N (0; 1): The last equality is by the property that a¢ ne functions of normal distributions are normal (Theorem 5.4). This calculation also requires that V > 0, otherwise the continuous mapping theorem cannot be employed. This seems like an innocuous requirement, as it only excludes degenerate sampling distributions. Formally we add the following assumption. Assumption 7.4 V  = R0V R > 0: Assumption 7.4 states that V  is positive deÖnite. Since R is full rank under Assumption 7.3, a su¢ cient condition is that V > 0, and since Qxx > 0 a su¢ cient condition is > 0. Thus Assumption 7.4 could be replaced by the assumption > 0. Assumption 7.4 is weaker so this is what we use. Thus the asymptotic distribution of the t-ratio T() is the standard normal. Since this distribution does not depend on the parameters, we say that T() is asymptotically pivotal. In Önite samples T() is not necessarily pivotal (as in the normal regression model) but the property means that the dependence on unknowns diminishes as n inc CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 240 As we will see in the next section, it is also useful to consider the distribution of the absolute t-ratio jT()j : Since T() d ! Z, the continuous mapping theorem yields jT()j d ! jZj : Letting (u) = P (Z  u) denote the standard normal distribution function, we can calculate that the distribution function of jZj is P (jZj  u) = P (u  Z  u) = P (Z  u) P (Z < u) = (u) (u) = 2(u) 1: (7.34) Theorem 7.11 Under Assumptions 7.2, 7.3 and 7.4, T() d ! Z  N (0; 1) and jtn()j d ! jZj : The asymptotic normality of Theorem 7.11 is used to justify conÖdence intervals and tests for the