18. In a study on determinants of children born in the Philippines,2 the following results were obtained:

The variables are:

ED = years of schooling of the woman LWH = natural logarithm of present

value of husbands earnings at marriage AMAR = age of the woman at marriage SURV = survival probability at age 5 in the province RURAL = residence in rural area (dummy variable); this variable is supposed to capture search and schooling costs Age = age of the woman

The explained variable is number of children born.

(a) Do the coefficients have the signs you would expect?

(b) Using the f-ratio for AMAR and the R for the two equations, can you tell how many observations were used in the estimation?

(c) Looking at the f-ratio of AMAR, can you predict the signs of the coefficients of the other variables if AMAR is deleted from the equation?

(d) Given that the dependent variable is number of children born, do you think the assumptions of the least squares model are satisfied?

19. In a study of investment plans and realizations in U.K. manufacturing industries since 1955, the following results were obtained:

R = 0.96 DW = 2.61

"B. L. Boulier an<3 M. R. Rosenzweig, "Schooling, Search and Spouse Selection: TesUng Economic Theories of Marriage and Household Behavior," Journal of Political Economy, August 1984, p. 729.

Appendix to Chapter 4

The Multiple Regression Model in Matrix Notation

Consider the multiple regression model with explanatory variables; y, = p,x,, + 2 2, + • • • + , + , / = 1, 2, . . . , n

A, = investment that firms anticipate they will complete in year t;

these plans are held at the end of year t - 1 /, = actual investment in year /

C, = measure of average level of under utilization of capacity

Figures in parentheses are standard errors.

(a) Interpret these results and assess whether or not knowledge of firms anticipated investment is helpful in explaining actual investment.

(b) How many observations have been used in the estimation?

(c) What is the partial correlation coefficient of /, with A, after allowing for the effect of C, - C, ,?

20. The demand for Ceylonese tea in the United States is given by the equation

log 0 = Po + Pi log Pc + 2 log P, + log Pb + p4 log r -I- m

where Q = imports of Ceylon tea in the United States Pc = price of Ceylon tea P, = price of Indian tea Pb = price of BraziUan coffee

Y = disposable income The following results were obtained from T = 22 observations, log Q = 2.837 - 1.481 log Pc + 1.181 log P, + 0.186 log Pb

(2 0) (0 987) (0 690) (0 134)

+ 0.257 log Y RSS = 0.4277

( 370)

log Q + log Pc= - 0.738 + 0.199 log Pg + 0.261 log Y RSS = 0.6788

(0 820) (0 155) (0 165)

Figures in parentheses are standard errors.

(a) Test the hypothesis P, = -1, pj = 0, and , 0 against p, for / = 1, 2, 3,4.

(b) Discuss the economic implications of these results.

or = Xp + u, where

= an X I vector of observations on the explained variable

X = an X matrix of observations on the explanatory variables

u = an X 1 vector of errors

P = a X 1 vector of parameters to be estimated We assume that:

1. The errors are llDlO.a), that is, independently and identically distributed with mean 0 and variance tr.

2. The jcs are nonstochastic and hence independent of the ms.

3. The jcs are linearly independent. Hence rank (XX) = rank \ = k. This implies that (XX) exists.

Under these assumptions the best (minimum variance) unbiased linear estimator (BLUE) of is obtained by minimizing the error sum of squares:

g = uu = (y - Xp)(y - Xp)

This is known as the Gauss-Markoff theorem.

We shall derive the formula for this estimator and show that it is a linear estimator, that it is unbiased, and that it has minimum variance among the class of linear unbiased estimators. That will complete the proof of the Gauss-Markoff theorem.

Derivation

We have Q = yy - 2pXy -I- pXXp. Using the formulas for vector differentiation derived in the Appendix to Chapter 2, we get

= 0 gives -2Xy + 2XXp = 0 or 0 = (XX)-Xy (4A.2)

Since (XX) ~ X is a matrix of constants, the elements of 0 are linear functions of the ys. Hence p is a linear estimator. Also, substituting (4A.1) into (4A.2), we get

0 = (XX)-X(Xp + u) = p + (XX)-Xu (4A.3)

Since £(u) = 0, we have ) = p. Thus p is an unbiased estimator. Also,

This can be written as