Question

w Add-ins Help FormatTell me what you want to do AaBbCcD AoBb .a-| = ::: . 으 田·LNormall1NoSpac . Heading 1 Heading 2 Title Subtitle Sut tie Subtitle Subtle i SUMMARY OUTPUT Regression Stotistics Multiple R R Square Adjusted R Square Standard Error Observations 0.989894408 0.97989094 0.946375839 0.093997207 ANOVA FSignificance F 5 1.291627011 0.258325402 29.23729686 0.009507436 Ss MS Regressiorn Residual Total 3 0026506425 0.008835475 8 1.318133436 p,value Lower 95% Upper 95% Coefficients Standard Error t Stot 2.000000 0.199040522 12.23566447 0.001175553 1.801957267 3.068828814 Intercept GPA Fin Major Gender NYC MajorXGender0.50000000.139726695 3.432145865 0.041472992 0.034889694 0.92 4235104 1000000 0.116964675 7.144529794 0.005645963 0.46342381 1.207891408 0.221410557 4.794091072 0.017265888 -1.766089586 0.356835166 0.115122597 -2.041317871 0.133876794 -0.601373298 0.131369669 0.130770483 -4.049701664 0.027116334 -0.945751481 -0.113411402 0.250000 -0,500000 14. (3 points) Consider the regression above. If we wanted to test the null hypothesis that gender has no effect on log earnings, how would write this joint null hypothesis. Write the restricted and the unrestricted equations. Using symbols, write the homoskedasticity-only formula for the test statistic. How many degrees of freedom (restrictions) are involved? What is the critical vale of this test at the 5% level? If our test statistic comes out to be less than the critical value, do we reject or fail to reject the joint null hypothesis?

Answer 1

Let be the coefficients of the regression model where $\beta_3$ and $\beta_5$ are the coefficients of the Gender and MajorXGender respectively. The joint null hypothesis is,

$H_0: \beta_3 = \beta_5 = 0$

The restricted equations is,

log(Earnings) = $\beta_0$ + $\beta_1$ GPA + $\beta_2$ Fin Major + $\beta_4$ NYC

The unrestricted equations is,

log(Earnings) = $\beta_0$ + $\beta_1$ GPA + $\beta_2$ Fin Major + $\beta_3$ Gender + $\beta_4$ NYC + $\beta_5$ MajorXGender

Test Statistic, F = [(RSS_R - RSS_UR)/q ] / [RSS_UR /(n-k-1)]

where RSS_R , RSS_UR are residual sum of squares for restricted and unrestricted equations, q is number of restricted coefficients, n is number of observations and k is number of predictors in the unrestricted model.

Number of restrictions = 2

n = 9 and k = 5

Numerator Degree of freedom = 2

Denominator Degree of freedom = 9 - 5 -1 = 3

Critical value of F at 5% level and df = 2,3 is 9.55

If our test statistic comes out to be less than the critical value, we fail to reject joint null hypothesis.