Recall that a measure of fit is the sum of squared residuals: where. The F-test statistic may be written as:

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1 1 Joint hypotheses The null and alternative hypotheses can usually be interpreted as a restricted model ( ) and an model ( ). In our example: Note that if the model fits significantly better than the restricted model, we should reject the null. The difference in fit between the model under the null and the model under the alternative leads us to an intuitive formulation of the F-test statistic, for testing joint hypotheses.

2 Recall that a measure of fit is the sum of squared residuals: where The F-test statistic may be written as:

3 3 where = number of restrictions. Notice that if the restrictions are true (if the null is true), will be small, and we ll fail to reject. Another statistic which uses is the : This gives us yet another formula for the F-test statistic.

4 4 where: F = ( R Rrestricted ) / q (1 R ) /( n k 1) R restricted= the R for the restricted regression R = the R for the regression q = the number of restrictions under the null k = the number of regressors in the regression. The bigger the difference between the restricted and R s the greater the improvement in fit by adding the variables in question the larger is the F statistic.

5 Note: the textbook differentiates between homoskedasticity only and heteroskedasticity robust F-tests. We will ignore heteroskedasticity for simplicity. Example: are the coefficients on strat and exppup zero? 5 Unrestricted population regression (under H A ): score i = strat i + exppup i + 3 eng i + u i Restricted population regression (that is, under H 0 ): TestScore i = eng i + u i (why?) The number of restrictions under H 0 is q = (why?). The fit will be better (R will be higher) in the regression (why?) By how much must the R increase for the coefficients on strat and exppup to be judged statistically significant?

6 6 Restricted regression: score = eng, (1.0) (0.03) Unrestricted regression: R restricted = score = strat exppup 0.656eng (15.5) (0.48) (1.59) (0.03) R = , k = 3, q = so F = = ( R Rrestricted ) / q (1 R ) /( n k 1) ( ) / (1.4366) /(40 3 1) = 8.01

7 A commonly performed F-test is one which assesses whether the chosen model fits at all. 7 any one of the s not equal to zero Why shouldn t the intercept be restricted? This test will be performed by most regression software, and reported as F-test in the regression output usually along with a p-value.

8 8 R Code teachdata = read.csv(" attach(teachdata) restricted = lm(score ~ eng) = lm(score ~ strat + exppup + eng) Let s look at the full () model: summary()

9 Output: 9 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < e-16 *** strat exppup ** eng < e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 416 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 3 and 416 DF, p-value: <.e-16

10 Now, to perform the F-test of whether school spending matters or not: anova(, restricted) Output: Analysis of Variance Table 10 Model 1: score ~ strat + exppup + eng Model : score ~ eng Res.Df RSS Df Sum of Sq F Pr(>F) *** --- Signif. codes: 0 *** ** 0.01 *

11 11 The F q,n k 1 distribution: The F distribution is tabulated many places As n, the F q,n-k 1 distribution asymptotes to the distribution: The F q, and q /q q /q distributions are the same. For q not too big and n 100, the F q,n k 1 distribution and the distribution are essentially identical. q /q Many regression packages (including STATA) compute p-values of F-statistics using the F distribution You will encounter the F distribution in published empirical work.

12 1 Summary F = ( R Rrestricted ) / q (1 R ) /( n k 1) The homoskedasticity-only F-statistic rejects when adding the two variables increased the R by enough that is, when adding the two variables improves the fit of the regression by enough If the errors are homoskedastic, then the homoskedasticity-only F-statistic has a large-sample distribution that is /q. But if the errors are heteroskedastic, the large-sample distribution is a mess and is not /q q q

13 The one at a time approach of rejecting if either of the t- statistics exceeds 1.96 rejects more than 5% of the time under the null (the size exceeds the desired significance level) For n large, the F-statistic is distributed q /q (= F q, ) The homoskedasticity-only F-statistic is important historically (and thus in practice), and can help intuition, but isn t valid when there is heteroskedasticity 13