Chapter 5 Confidence Intervals Introduction Hypothesis testing, which we discussed in the previous chapter, is the foundation for all inference in classical econometrics. It can be used to find out whether restrictions imposed by economic theory are compatible with the data | Chapter 5 Confidence Intervals Introduction Hypothesis testing which we discussed in the previous chapter is the foundation for all inference in classical econometrics. It can be used to find out whether restrictions imposed by economic theory are compatible with the data and whether various aspects of the specification of a model appear to be correct. However once we are confident that a model is correctly specified and incorporates whatever restrictions are appropriate we often want to make inferences about the values of some of the parameters that appear in the model. Although this can be done by performing a battery of hypothesis tests it is usually more convenient to construct confidence intervals for the individual parameters of specific interest. A less frequently used but sometimes more informative approach is to construct confidence regions for two or more parameters jointly. In order to construct a confidence interval we need a suitable family of tests for a set of point null hypotheses. A different test statistic must be calculated for each different null hypothesis that we consider but usually there is just one type of statistic that can be used to test all the different null hypotheses. For instance if we wish to test the hypothesis that a scalar parameter 0 in a regression model equals 0 we can use a t test. But we can also use a t test for the hypothesis that 0 00 for any specified real number 00. Thus in this case we have a family of t statistics indexed by 00. Given a family of tests capable of testing a set of hypotheses about a scalar parameter 0 of a model all with the same level a we can use them to construct a confidence interval for the parameter. By definition a confidence interval is an interval of the real line that contains all values 00 for which the hypothesis that 0 00 is not rejected by the appropriate test in the family. For level a a confidence interval so obtained is said to be a 1 a confidence interval or to be at confidence .
