1,Multiple Regression Analysis,y = b0 + b1x1 + b2x2 + . . . bkxk + u 2. Inference,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 testing, 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),3,CLM Assumptions (cont),Under CLM, OLS is not only BLUE, but is the minimum 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,5,Normal Sampling Distributions,6,The t Test,7,The t Test (cont),Knowing the sampling distribution for 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,8,The t Test (cont),9,t Test: One-Sided Alternatives,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 significance level is 5%,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 the critical value If the t statistic is less than the critical value then we fail to reject the null,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,12,Examples 1,Hourly Wage Equation H0: bexper = 0 H1: bexper 0,13,One-sided vs Two-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,14,yi = b0 + b1Xi1 + + bkXik + ui H0: bj = 0 H1: bj 0,c,0,a/2,(1 - a),-c,a/2,Two-Sided Alternatives,reject,reject,fail to reject,15,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” If we fail to reject the null, we typically say “xj is statistically insignificant at the a % level”,16,Examples 2,Determinants of College GPA colGPAcollege GPA(great point average), hsGPAhigh school GPA skippedaverage numbers of letures missed per week.,17,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 statistic is,18,Examples 3,Campus Crime and Enrollment H0: benroll = 1 H1: benroll 1,19,Examples 4,Housing Prices and Air Pollution H0: blog(nox) = -1 H1: blog(nox) - 1,20,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,21,Computing p-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 we would observe the t statistic we did, if the null were true,22,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 Many software,such as Stata or Eviews provides the t statistic, p-value, and 95% confidence interval for H0: bj = 0 for you,23,Testing a Linear Combination,Suppose instead of testing whether b1 is equal to a constant, you want to test if it is equal to another parameter, that is H0 : b1 = b2 Use same basic procedure for forming a t statistic,24,Testing Linear Combo (cont),25,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 More generally, you can always restate the problem to get the test you want,26,Examples 5,Suppose you are interested in the effect of campaign expenditures on outcomes Model is voteA = b0 + b1log(expendA) + b2log(expendB) + b3prtystrA + u H0: b1 = - b2, or H0: q1 = b1 + b2 = 0 b1 = q1 b2, so substitute in and rearrange voteA = b0 + q1log(expendA) + b2log(expendB - expendA) + b3prtystrA + u,27,Example (cont):,This is the same model as originally, but now you get a standard error for b1 b2 = q1 directly from the basic regression Any linear combinati