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Law of Iterated Expectations

ConÖdence Intervals for Regression Coe¢ cients The OLS estimator b is a point estimator for a coe¢ cient . A broader concept is a set or interval estimator which takes the form Cb = [L; b Ub]. The goal of an interval estimator Cb is to contain the true value, e.g. 2 C; b with high probability. The interval estimator Cb is a function of the data and hence is random. An interval estimator Cb is called a 1 conÖdence interval when P( 2 Cb) = 1 for a selected value of . The value 1 is called the coverage probability: Typical choices for the coverage probability 1 are 0.95 or 0.90. The probability calculation P( 2 Cb) is easily mis-interpreted as treating as random and Cb as Öxed. (The probability that is in Cb.) This is not the appropriate interpretation. Instead, the correct interpretation is that the probability P( 2 Cb) treats the point as Öxed and the set Cb as random. It is the probability that the random set Cb covers (or contains) the Öxed true coe¢ cient . There is not a unique method to construct conÖdence intervals. For example, one simple (yet silly) interval is Cb = ( R with probability 1 n b o with probability : If b has a continuous distribution, then by construction P( 2 Cb) = 1 ; so this conÖdence interval has perfect coverage. However, Cb is uninformative about b and is therefore not useful. Instead, a good choice for a conÖdence interval for the regression coe¢ cient is obtained by adding and subtracting from the estimator b a Öxed multiple of its standard error: Cb = h b c  s( b); b + c  s( b) i (5.11) where c > 0 is a pre-speciÖed constant. This conÖdence interval is symmetric about the point estimator ; b and its length is proportional to the standard error s( b): Equivalently, Cb 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 (c  T( )  c): (5.12) Since the t-statistic T( ) has the tnk distribution, (5.12) equals F(c) F(c), where F(u) is the student t distribution function with nk degrees of freedom. Since F(c) = 1F(c) (see Exercise 5.20) we can write (5.12) as P  2 Cb  = 2F(c) 1: This is the coverage probability of the interval Cb, and only depends on the constant c. As we mentioned before, a conÖdence interval has the coverage probability 1 . This requires selecting the constant c so that F(c) = 1 =2. This holds if c equals the 1 =2 quantile of the tnk distribution. As there is no closed form expression for these quantiles, we compute their values numerically. For example, by tinv(1-alpha/2,n-k) in MATLAB. With this choice the conÖdence interval (5.11) has exact coverage probability 1 . By default, Stata reports 95% conÖdence intervals Cb for each estimated regression coe¢ cient u