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Regression Components

rs. 7.13 ConÖdence Intervals The estimator b is a point estimator for , meaning that b is a single value in R q . A broader concept is a set estimator Cb which is a collection of values in R q : When the parameter  is real-valued then it is common to focus on sets of the form Cb = [L; b Ub] which is called an interval estimator for . An interval estimate Cb is a function of the data and hence is random. The coverage probability of the interval Cb = [L; b Ub] is P( 2 Cb): The randomness comes from Cb as the parameter  is treated as Öxed. In Section 5.13 we introduced conÖdence intervals for the normal regression model, which used the Önite sample distribution of the t-statistic to construct exact conÖdence intervals for the regression coe¢ cients. When we are outside the normal regression model we cannot rely on the exact normal distribution theory, but instead use asymptotic approximations. A beneÖt is that we can construct conÖdence intervals for general parameters of interest , not just regression coe¢ cients. An interval estimator Cb is called a conÖdence interval when the goal is to set the coverage probability to equal a pre-speciÖed target such as 90% or 95%. Cb is called a 1 conÖdence interval if inf P( 2 Cb) = 1 : When b is asymptotically normal with standard error s(b); the conventional conÖdence interval for  takes the form Cb = h b c  s(b); b+ c  s(b) i (7.35) where c equals the 1 quantile of the distribution of jZj. Using (7.34) we calculate that c is equivalently the 1 =2 quantile of the standard normal distribution. Thus, c solves 2(c) 1 = 1 : This can be computed by, for example, norminv(1- =2) in MATLAB. The conÖdence interval (7.35) is symmetric about the point estimator ; b and its length is proportional to the stand CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 241 Equivalently, (7.35) is the set of parameter values for  such that the t-statistic T() is smaller (in absolute value) than c; that is Cb = f : jT()j  cg = (  : c  b  s(b)  c ) : The coverage probability of this conÖdence interval is P   2 Cb  = P (jT()j  c) ! P (jZj  c) = 1