1. 1 Multiple Regression Analysis y =  0 +  1 x 1 +  2 x 2 +...  k x k + u 2. Hypothesis Testing

2. 2 Variance of the OLS Estimators Now we know that the sampling distribution of our estimated coefficients are centered around the true parameters Want to know how accurate/reliable our estimators are This is called hypothesis testing

3. 3 So far, we know that given the Gauss- Markov assumptions, OLS is BLUE In order to do hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x 1, x 2,…, x k and u is normally distributed with zero mean and variance  2 : u ~ Normal(0,  2 )

4. 4 Classical Linear Model Under these assumptions, OLS is not only BLUE (best linear unbiased estimator), but is the minimum variance unbiased estimator (meaning most accurate among all possible models that give unbiased estimators)

5. 5

6. 6 Population vs Sample Mean Sample mean Variance Sample Variance Standard Deviation Standard Error True Parameter Coefficient Estimate

7. 7 The t Test

8. 8 The t Test (cont) Start with a null hypothesis For example, H 0 :  j =0 This null says that x j has no incremental effect on y, beyond the effects from other x’s

9. 9 The t Test (cont) (Important)

10. 10 t Test Besides our null, H 0, we need an alternative hypothesis, H 1, and a significance level H 1 :  j  0 If we want to have only a 5% probability of rejecting H 0 if it is really true, then we say our significance level is 5%

11. 11 t Test If the sample is not too small (>30 observations), Reject the null if the magnitude(=absolute value) of our t statistic is greater than 2. If the magnitude of our t statistic is less than 2, then we fail to reject the null

12. 12 If we reject the null, we say “x j is statistically significant at the 5% significance  level”, or simply “x j is statistically significant” If we fail to reject the null, we say “x j is statistically insignificant at the 5% level”, or simply “x j is statistically insignificant”.

13. 13 p-values An alternative to look up what percentile the t statistic is in the appropriate t distribution – this is the p-value. Roughly speaking, p-value is the probability that we would observe this t statistic (or more extreme values) if the null were true (no significant coefficient)

14. 14 For example, if p-value=0.04, This means that, if the null were true (no significant coefficient), your chance of seeing the results that you have seen is just 4%. So the coefficient most likely is significant.

15. 15 p-values Most computer packages will compute the p-value for you. If p-value is <0.05, the coefficient is significant, reject the null.

16. 16 A story Three people saw a black sheep on the hillside in Berkeley: Astronomer: All sheep are black. Biologist: Sheep in Berkeley are black. Mathematician: There exists at least one sheep in Berkeley, at least one side of whom is black.