Question

3) Consider the following linear regression: y =a + Bx + Show that minimizing the sum of squared residuals ( - ) to obtain OLS estimators of the slope and the intercept results in the following algebraic properties a) b) Ex = 0 = 0 4) You run the following regression: TestScore = a + (Female) + where TestScore is measured on a scale from 400 to 1000, and female is an indicator for the gender of the student. You find that B = 45. a) Interpret the estimated coefficient using plain English b) Suggest a way to put the magnitude of the estimated cocfficient in perspective so we can judge whether the estimated effect is economically large of small. 5) Explain why in a bivariate regression, the formula for the SER divides the 6) You are trying to estimate the effect of labour market experience (measured in years) on hourly wage. To do this, you estimate the following regression model: HourlyWage = B. + .(experience) + u You find that i = 0.235 and SEB.) = 0.042 a) Interpret the effect of experience on hourly wage using plain English b) Does the estimated intercept have an economic interpretation in this case? Explain c) Calculate the t-statistic for the null hypothesis of B, = 0. d) Given the value of your t-statistic, explain whether your estimated effect of experience is statistically significant [if it is, indicate the significance level). The critical values for the I-distribution at 10%, 5%, and 1% significance level are 1.64, 1.96 and 2.57 respectively. 7) Prove that Cov(a,b) = yes, where y = a + Bx+€. You need to carefully show your work. At any point you use an assumption to set a term equal to zero, you need to specify the assumption used Hint 1: Cov(X,Y)=E(X - E(X)]LY-ECYD- Hint 2:a y-Bris 8) We proved in the lecture that 8.- B.-22where v = (x − )(). Show that if -N(0 $), then Br-N( ) 9) What determines whether we should do a one-sided or two-sided hypothesis testing? Briefly explain

Answer 1

Given the model, y; = a + Bx; +u; The estimates of a and B are obtained by the method of ordinary least squares, by solving the following normal equations. - (1) Σui-0 -- δα Σκι -0 -- οβ consider (1) ΟΣu, - (2) “Σ" Σν, -α+βα). = 0 = δα δα -2Σ(ν: -α+ βx) = 0 με Σν-α-βα) = 0 So Σu, =Σ(-y) - Σ(ν-α+ βα) = 0 consider (2) ΟΣΑ -02-Σ(ν-α- βακ, - 0 OB 1.e. Σν-α- βxx, = 0 So Σμα. - Σ(ν-νικ, = Σν-α-βα, x = 0 Hence the result