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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