solutions chapter 4 - Universitetet i Oslo

1 60 chapter 4 Exercise solutions chapter 4, Exercise solutions , principles of Econometrics, 3e 61 EXERCISE (a) ()222 = = (b) To calculate 2R we need ()2iyy , () 20 Ny = = = Therefore, = (c) From 222 () 11ieNKRSSTSST = = we have, 22(1) (1 ) (20 2)SSTRNK === chapter 4, Exercise solutions , principles of Econometrics, 3e 62 EXERCISE (a) ( ) ( )yx =+ where 10xx = (b) ( ) ( )yx =+ where 10yy = (c) ( ) ( )yx =+ where and 1010yxyx == The values of 2R remain the same in all cases. chapter 4, Exercise solutions , principles of Econometrics, 3e 63 EXERCISE (a) 0120 115 6ybbx=+ =+ = (b) n22202()11(51) var( ) ()510ixxfNxx = + +=+ += se( ) (c) Using se( )ffrom part (b) and ( , 3) , 0 se( ) 6 ( , )cyt f = = (d) Using se( )ffrom part (b) and ( ,3) , 0 se( ) 6 ( , )cyt f = = (e) Using 01xx==, the prediction is 0 111 2y=+ =, and n22202()11(11) var( ) ()510ixxfNxx = + +=+ += se( ) 0 se( ) 2 ( , )cyt f = = Width in part (c) () = Width in part (e) () = The width in part (e) is smaller than the width in part (c), as expected.

2 Predictions are more precise when made for x values close to the mean. chapter 4, Exercise solutions , principles of Econometrics, 3e 64 EXERCISE (a) When estimating 0(),Ey we are estimating the average value of y for all observational units with an x-value of When predicting 0,y we are predicting the value of y for one observational unit with an x-value of The first exercise does not involve the random error 0;e the second does. (b) 120120 1 20()()()Ebbx Eb Ebxx+=+ = + ()21201 0201222222002222222200222222200022var()v ar( )var( ) 2 cov( , )2() () ()()(2)()()2()11()()iiiiiiiiibbxb xbxbbxxxxNxxxxxxxx NxxxxNxxxxxxxxxxNxxNxx+= + + =+ + =+ + = += + (c) It is not appropriate to say that 00 ()Eyy= because 0y is a random variable.

3 [][]0120 12000 ()Eyx xey= + + + = We need to include 0y in the expectation so that ()0000 120 1200 () () ()() yEyEyxx Ee = = + + + = chapter 4, Exercise solutions , principles of Econometrics, 3e 65 EXERCISE (a) If we multiply the x values in the simple linear regression model 12yxe= + + by 10, the new model becomes ()21**122 21010where10 and10yxexex x = + + = + + = = The estimated equation becomes ()21 1010bybx =+ Thus, 1 and 1b do not change and 2 and 2b becomes 10 times smaller than their original values. Since e does not change, the variance of the error term 2var( )e= is unaffected. (b) Multiplying all the y values by 10 in the simple linear regression model12yxe= + + gives the new model ()()()1210101010yxe = + + or **12yxe = + + where **112 210,10,10,10yyee = = = = The estimated equation becomes ()()12 101010yyb bx = = + Thus, both 1 and 2 are affected.

4 They are 10 times larger than their original values. Similarly, 1b and 2b are 10 times larger than their original values. The variance of the new error term is ()2var( )var10100 var( ) 100eee = = = Thus, the variance of the error term is 100 times larger than its original value. chapter 4, Exercise solutions , principles of Econometrics, 3e 66 EXERCISE (a) The least squares estimator for 1 is 12bybx= . Thus, 12ybbx=+, and hence (),yx lies on the fitted line. (b) Consider the fitted line 12 iiybxb=+. Averaging over N, we obtain y = ()()12121212 11iiiiyxbxbbNb x bbbbxNNNN=+ =+=+=+ From part (a), we also have 12ybbx=+. Thus, yy=. chapter 4, Exercise solutions , principles of Econometrics, 3e 67 EXERCISE (a) 020 ybx= (b) Using the solution from Exercise part (f) ()()()2222222 ( + + + + + = 2 22222 24677911352iy=+++++ = = (c) ()()()()222 2 2222 ( ) iiyyyyyyiiyyyyryyyy ==== The two alternative goodness of fit measures 2uR and 2 yyr are not equal.)

5 (d) , {}{} SSESST+= + = = The decomposition does not hold. chapter 4, Exercise solutions , principles of Econometrics, 3e 68 EXERCISE (a) Simple linear regression results: ()()2** (se) += Linear-log regression results: ()()()2** (se) += Quadratic regression results: ()( )22** (se) += (b) (i) (ii) Figure (a) Fitted line and residuals for the simple linear regression Figure (b) Fitted line and residuals for the linear-log regression chapter 4, Exercise solutions , principles of Econometrics, 3e 69 Exercise (b) continued (b) Figure (c) Fitted line and residuals for the quadratic regression (iii) Error normality tests Jarque-Bera: Simple linear.

6 JB = p-value = Linear log: JB = p-value = Quadratic: JB = p-value = (iv) Values of 2R are given in part (a) To choose the preferred equation we consider the following. 1. The signs of the response parameters 222, and : We expect them to be positive because we expect yield to increase over time as technology improves. The signs of the estimates of 222, and are as expected. 2. 2R: The value of 2 Rfor the third equation is the highest, namely 3. The plots of the fitted equations and their residuals: The upper parts of the figures display the fitted equation while the lower parts display the residuals.

7 Considering the plots for the fitted equations, the one obtained from the third equation seems to fit the observations best. In terms of the residuals, the first two equations have concentrations of positive residuals at each end of the sample. The third equation provides a more balanced distribution of positive and negative residuals throughout the sample. The third equation is preferable. chapter 4, Exercise solutions , principles of Econometrics, 3e 70 EXERCISE (a) Equation 1: 0 49 + = Equation 2: 0 (49) += Equation 3: 20 (49) + = (b) Equation 1: m1 = Equation 2: m1 == = Equation 3: m1 22 49 = = (c) Evaluating the elasticities at 49t= and the relevant value for 0 y gives the following results.

8 Equation 1: n1049 yy= = = Equation 2: n1 yy == = Equation 3: n2210 22 yy === (d) The slopes tdydt and the elasticities ttdytdt y give the marginal change in yield and the percentage change in yield, respectively, that can be expected from technological change in the next year. The results show that the predicted effect of technological change is very sensitive to the choice of functional form. chapter 4, Exercise solutions , principles of Econometrics, 3e 71 EXERCISE (a) For households with 1 child ln()(se) ( ) ( ) ( ) ( ) ( ) WFOODTOTEXPRt= = For households with 2 children: ln()(se) ( ) ( ) ( ) ( ) ( ) WFOODTOTEXPRt= = For 2 we would expect a negative value because as the total expenditure increases the food share should decrease with higher proportions of expenditure devoted to less essential items.

9 Both estimations give the expected sign. The standard errors for 12 and bb from both estimations are relatively small resulting in high values of t ratios and significant estimates. (b) For households with 1 child, the average total expenditure is and ()()[] ( ) 1 ln( )lnbb TOTEXPbbTOTEXP ++ + === + For households with 2 children, the average total expenditure is and ()()[] ( ) 1 ln( )lnbbTOTEXPbb TOTEXP ++ + === + Both of the elasticities are less than one; therefore, food is a necessity. chapter 4, Exercise solutions , principles of Econometrics, 3e 72 Exercise (continued) Figure (a) Figure (b) (c) Figures (a) and (b) display the fitted curve and the residual plot for households with 1 child.

10 The function linear in WFOOD and ln(TOTEXP) seems to be an appropriate one. However, the observations vary considerably around the fitted line, consistent with the low 2R value. Also, the absolute magnitude of the residuals appears to decline as ln(TOTEXP) increases. In chapter 8 we discover that such behavior suggests the existence of heteroskedasticity. Figures (c) and (d) are plots of the fitted equation and the residuals for households with 2 children. They lead to similar conclusions to those made for the one-child case. The values of JB for testing 0:H the errors are normally distributed are and for households with 1 child and 2 children, respectively. Since both values are greater than the critical value 2( ,2) =, we reject 0H.