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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