《计量经济学导论》ch4.pptVIP

Chapter 4 ;Statistical inference in the regression model Hypothesis tests about population parameters Construction of confidence intervals Sampling distributions of the OLS estimators The OLS estimators are random variables We already know their expected values and their variances However, for hypothesis tests we need to know their distribution In order to derive their distribution we need additional assumptions Assumption about distribution of errors: normal distribution;Assumption MLR.6 (Normality of error terms);Discussion of the normality assumption The error term is the sum of ?many“ different unobserved factors Sums of independent factors are normally distributed (CLT) Problems: How many different factors? Number large enough? Possibly very heterogenuous distributions of individual factors How independent are the different factors? The normality of the error term is an empirical question At least the error distribution should be ?close“ to normal In many cases, normality is questionable or impossible by definition;Discussion of the normality assumption (cont.) Examples where normality cannot hold: Wages (nonnegative; also: minimum wage) Number of arrests (takes on a small number of integer values) Unemployment (indicator variable, takes on only 1 or 0) In some cases, normality can be achieved through transformations of the dependent variable (e.g. use log(wage) instead of wage) Under normality, OLS is the best (even nonlinear) unbiased estimator Important: For the purposes of statistical inference, the assumption of normality can be replaced by a large sample size;Terminology Theorem 4.1 (Normal sampling distributions);Testing hypotheses about a single population parameter Theorem 4.1 (t-distribution for standardized estimators) Null hypothesis (for more general hypotheses, see below);t-statistic (or t-ratio) Distribution of the t-statistic if the null hypothesis is true Goal: Define a rejection rule so that, if it is true, H0 is rejected only with a sm