(a) Estimate a linear regression equation for each sample separately and for the pooled sample.

(b) State the assumptions under which estimation of the pooled regression is valid.

(c) Explain how you will test the validity of these assumptions using the data provided.

8. A researcher tried two specifications of a regression equation.

= a + +

= a + x + yz + u

Explain under what circumstances the following will be true. (A "hat" over a parameter denotes its estimate.)

(a) p = p.

(b) If , and , are the estimated residuals from the two equations

(c) p is statistically significant (at the 5% level) but p is not.

(d) p is statistically significant (at the 5% level) but p is not.

9. The model

y, = Po + PlJK + 2X2, + < +

was estimated by ordinary least squares from 26 observations. The results were

y, = 2 + 3.5xi, - 0.72, + 2.0x3,

( 9) (2 2) (1 5)

/-ratios are in parentheses and = 0.982. The same model was estimated with the restriction p, = P2. Estimates were:

y, = 1.5 + 3(x„ + X2,) - 0.6x3, = 0.876

(2 7) (2 4)

(a) Test the significance of the restriction p, = P2. State the assumptions under which the test is valid.

(b) Suppose that X2, is dropped from the equation: would the R rise or fall?

(c) Would the R rise or fall if X2, is dropped?

10. Suppose that the least squares regression of Y on x,, Xj, . . . , x yields coefficient estimates bjU = I, 2, . . . , k) none of which exceed their respective standard errors. However, the F-ratio for the equation rejects, at the 0.05 level the hypothesis that fc, = 2 = • • = * = 0.

(a) Is this possible?

(b) What do you think is the reason for this?

(c) What further analysis would you perform?

11. What would be your answer to the following queries regarding multiple regression analysis?

(a) I am trying to find out why people go bankrupt. I have gathered data from a sample of people filing bankruptcy petitions. Will these data enable me to find answers to my question?

(b) I want to study the costs of auto accidents. I have collected data from a sample of poUce auto accident reports. Are these data adequate for my purpose?

(c) I am trying to estimate a consumption function and 1 suspect that the marginal propensity to consume varies inversely with the rate of interest. Do I run a multiple regression using income and interest rate as explanatory variables?

(d) I am fitting a demand for food function for a sample of 1000 famiUes. I obtain an F? of only 0.05 but the regression program indicates that the F-statistic for the equation is very significant and so are the t-statistics. How can this be? Is there a mistake in the program?

(e) In the regression of F on x and z, should I leave one of them out?

(f) 1 know that depends linearly on x but I am not sure whether or not it also depends on another variable z. A friend of mine suggests that I should regress on jc first, calculate the residuals, and then see whether they are correlated with z. Is she correct?

12. A student obtains the following results in several different regression problems. In which cases could you be certain that an error has been committed? Explain.

(a) i?2 = 0.89, R\nM = 0.86

(b) \ = 0.86, ?,,234 = 0.82

(c) \ 2 = 0.23, ri,, = 0.13, R\,23 = 0.70

(d) Same as part (c) but r\2 = Q

13. Given the following estimated regression equations

C, = const. + 0.92 F, C, = const, -t- 0.84C, , C, , = const, -t- 0.78F, Y, = const. + 0.55C,„,

calculate the regression estimates of p, and P2 for

C, = po + P,F, + p2C, , + u,

*14. Instead of estimating the coefficients p, and P2 from the model

= a + p,x, + P2X2 +

it is decided to use ordinary least squares on the following regression equation:

= a + , :; + 2 2 + V

where xl is the residual from a regression of jc, and jcj and v is the disturbance term.

(a) Show that the resulting estimator of p2 is identical to the regression coefficient of on jc2.

(b) Obtain an expression for the bias of this estimator.

(c) Prove that the estimators of p, obtained from each of the two equations are identical.

-D. K. Benjamin and L. A. Kochin, "Searching for an Explanation of Unemployment in Interwar Britain," Journal of Political Economy, June 1979, pp. 441-478.

"P. A. Ormerod and G. D. N. Worswick, "Unemployment in Interwar Britain," Journal of Political Economy, April 1982, pp. 400-409.

*15. Explain how you will estimate a linear regression equation which is piece-wise linear with a joint (or knot) at x = xq if

(a) Xf, is known.

(b) Xo is unknown. *16. In the model

y« = iXi, + 2X2, + , +

the coefficients are known to be related to a more basic economic parameter a according to the equations

. + P2 = « P. + = -«

Assuming that the xs are nonrandom and that u, ~ IN(0, a), find the best unbiased Unear estimator a of a and the variance of d. 17. A study on unemployment in the British interwar period produced the following regression equation (data are given in Table 4.11):

= 5.19 + 18.3(B/W) - 90.0(log Q - log Q*)

(2.0) (4 46) (-8.3)

= 0.8 SER = 1.9 where SER = W S Sample period 1920-1938 (« = 19).

= unemployment rate B/W = ratio of unemployment benefits to average wage Q = actual output Q* = trend predicted output log Q - log Q* = captures unexpected changes in aggregate demand

The authors conclude that the high benefit levels are partly responsible for the high rates of unemployment. Critics of this study argued that when the single observation for 1920 is dropped the results change dramatically." The equation now is

= 7.9 + 12.9(B/W) - 87.0(log Q - log Q*)

(3.0) (2 4) (8.3)

R = 0.82 SER =1.7 Sample period 1921-1938 (n = 18).

Test whether the results are significantly different from each other.