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# Conditional Expectation Function

t-statistic An alternative way of writing (5.10) is bj j r  2 h (X0X) 1 i jj  N (0; 1): This is sometimes called a standardized statistic, as the distribution is the standard normal. Now take the standardized statistic and replace the unknown variance  2 with its estimator s 2 . We call this a t-ratio or t-statistic T = bj j r s 2 h (X0X) 1 i jj = bj j s( bj ) where s( bj ) is the classical (homoskedastic) standard error for bj from (4.37). We will sometimes write the t-statistic as T( j ) to explicitly indicate its dependence on the parameter value j , and sometimes will simplify notation and write the t-statistic as T when the dependence is clear from the context. By some algebraic re-scaling we can write the t-statistic as the ratio of the standardized statistic and the square root of the scaled variance estimator. Since the distributions of these two components are normal and chi-square, respectively, and independent, then we can deduce that the t-statistic has the distribution T = bj j r  2 h (X0X) 1 i jj ,s (n k)s 2  2  (n k)  N (0; 1) q  2 nk  (n k)  tnk a student t distribution with n k degrees of freedom. This derivation shows that the t-ratio has a sampling distribution which depends only on the quantity nk. The distribution does not depend on any other features of the data. In this context, we say that the distribution of the t-ratio is pivotal, meaning that it does not depend on unknowns. The trick behind this result is scaling the centered coe¢ cient by its standard error, and recognizing that each depends on the unknown  only through scale. Thus the ratio of the two does not depend on . This trick (scaling to eliminate dependence on unknowns) is known as studentizati