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Lecture Notes 7 Econ 410 – Introduction to Econometrics 1 The multiple regression model – Part II Hypothesis testing and confidence intervals for a single coefficient Recall that in the multiple regression model, the coefficient j β represents the expected change in i Y following a one-unit change in the regressor ji X , holding constant all the other regressors. We can test the hypothesis that this expected change takes on a specific value 0 , j β . If the alternative hypothesis is two sided, we have: 0 , 0 : j j H β β = vs. 0 , 1 : j j H β β To compute the t -statistic and the p -value, we need to know the standard error of j β ˆ . The OLS estimator j β ˆ is a random variable with a sampling distribution; the standard error ( ) j SE β ˆ is an estimator of the standard deviation of j β ˆ . As in the case with only one regressor, ( ) j SE β ˆ is usually computed by regression software. The t -statistic is defined as: ( ) j j j SE β β β ˆ ˆ 0 , And the p -value is: ( ) 0 , 0 , ˆ ˆ Pr 0 j act j j j H β β β β = ( ) act t Φ 2 As usual, the null hypothesis is rejected at the 5% significance level if 96 . 1 > act t or, equivalently, if the p -value is less than 0.05. Basically, we use the same procedure as in the model with one regressor. This is because, if the four least squares assumptions hold, in large sample the distribution of j β ˆ is approximately ( ) 2 ˆ , j j N β σ β . It follows that, in large samples, the t -statistic has a standard normal distribution. Confidence intervals can be constructed using the same procedure adopted in the case of one regressor, too. The 95% confidence interval for the coefficient j β is the set of values of j β that cannot be rejected by a two-sided 5% significance level hypothesis test. Equivalently, it is an interval that contains the true value of j β in 95% of all possible randomly drawn samples. In large samples, the distribution of j β ˆ is approximately normal, so the 95% confidence interval for j β is: ( ) j j SE β β ˆ * 96 . 1 ˆ ± Note: hypothesis testing and confidence intervals about one coefficient might give different results if the regressors in the model are changed.
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Lecture Notes 7 Econ 410 – Introduction to Econometrics 2 Test of joint hypothesis A joint hypothesis is a hypothesis that imposes two or more restrictions on the regression coefficients: , .... , ; : 0 , 0 , 0 m m j j H β β β β = = for a total of q restrictions vs. : 1 H at least one of the q restrictions under 0 H does not hold. Ex.: consider a model in which the regressors are 5. Then, the null and alternative hypotheses could be: 0 0 ; 0 : 5 3 1 0 = = = β β β and H vs. 0
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