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1、Economics 20 - Prof. Anderson,1,Multiple Regression Analysis,y = b0 + b1x1 + b2x2 + . . . bkxk + u 2. Inference,Economics 20 - Prof. Anderson,2,Assumptions of the Classical Linear Model (CLM),So far, we know that given the Gauss-Markov assumptions, OLS is BLUE, In order to do classical hypothesis te
2、sting, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2, xk and u is normally distributed with zero mean and variance s2: u Normal(0,s2),Economics 20 - Prof. Anderson,3,CLM Assumptions (cont),Under CLM, OLS is not only BLUE, but is the mi
3、nimum variance unbiased estimator We can summarize the population assumptions of CLM as follows y|x Normal(b0 + b1x1 + bkxk, s2) While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality,Economics 20 - Prof. Anderson,4,.,.,x1,x2,The homoskedas
4、tic normal distribution with a single explanatory variable,E(y|x) = b0 + b1x,y,f(y|x),Normal distributions,Economics 20 - Prof. Anderson,5,Normal Sampling Distributions,Economics 20 - Prof. Anderson,6,The t Test,Economics 20 - Prof. Anderson,7,The t Test (cont),Knowing the sampling distribution for
5、the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis For example, H0: bj=0 If accept null, then accept that xj has no effect on y, controlling for other xs,Economics 20 - Prof. Anderson,8,The t Test (cont),Economics 20 - Prof. Anderson,9,t Test: One-Sided A
6、lternatives,Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided H1: bj 0 and H1: bj 0 are one-sided H1: bj 0 is a two-sided alternative If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our s
7、ignificance level is 5%,Economics 20 - Prof. Anderson,10,One-Sided Alternatives (cont),Having picked a significance level, a, we look up the (1 a)th percentile in a t distribution with n k 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than t
8、he critical value If the t statistic is less than the critical value then we fail to reject the null,Economics 20 - Prof. Anderson,11,yi = b0 + b1xi1 + + bkxik + ui H0: bj = 0 H1: bj 0,c,0,a,(1 - a),One-Sided Alternatives (cont),Fail to reject,reject,Economics 20 - Prof. Anderson,12,One-sided vs Two
9、-sided,Because the t distribution is symmetric, testing H1: bj than c then we fail to reject the null For a two-sided test, we set the critical value based on a/2 and reject H1: bj 0 if the absolute value of the t statistic c,Economics 20 - Prof. Anderson,13,yi = b0 + b1Xi1 + + bkXik + ui H0: bj = 0
10、 H1: bj 0,c,0,a/2,(1 - a),-c,a/2,Two-Sided Alternatives,reject,reject,fail to reject,Economics 20 - Prof. Anderson,14,Summary for H0: bj = 0,Unless otherwise stated, the alternative is assumed to be two-sided If we reject the null, we typically say “xj is statistically significant at the a % level”
11、If we fail to reject the null, we typically say “xj is statistically insignificant at the a % level”,Economics 20 - Prof. Anderson,15,Testing other hypotheses,A more general form of the t statistic recognizes that we may want to test something like H0: bj = aj In this case, the appropriate t statist
12、ic is,Economics 20 - Prof. Anderson,16,Confidence Intervals,Another way to use classical statistical testing is to construct a confidence interval using the same critical value as was used for a two-sided test A (1 - a) % confidence interval is defined as,Economics 20 - Prof. Anderson,17,Computing p
13、-values for t tests,An alternative to the classical approach is to ask, “what is the smallest significance level at which the null would be rejected?” So, compute the t statistic, and then look up what percentile it is in the appropriate t distribution this is the p-value p-value is the probability
14、we would observe the t statistic we did, if the null were true,Economics 20 - Prof. Anderson,18,Stata and p-values, t tests, etc.,Most computer packages will compute the p-value for you, assuming a two-sided test If you really want a one-sided alternative, just divide the two-sided p-value by 2 Stat
15、a provides the t statistic, p-value, and 95% confidence interval for H0: bj = 0 for you, in columns labeled “t”, “P |t|” and “95% Conf. Interval”, respectively,Economics 20 - Prof. Anderson,19,Testing a Linear Combination,Suppose instead of testing whether b1 is equal to a constant, you want to test
16、 if it is equal to another parameter, that is H0 : b1 = b2 Use same basic procedure for forming a t statistic,Economics 20 - Prof. Anderson,20,Testing Linear Combo (cont),Economics 20 - Prof. Anderson,21,Testing a Linear Combo (cont),So, to use formula, need s12, which standard output does not have Many packages will have an option to get it, or will just perform the test for you In Stata, after reg y x1 x2 xk you would type test x1 = x2 to get a p-value for the test
