ANSWERS TO EXERCISE 8.13

a.
Given sigma sub t sup 2  =  k YARD sub t.  Define w sub t = 1/sqrt
{(YARD sub t )}.  Next multiply each variable by w sub t and generate Y
sub t sup * = w sub t PRICE  sub t, X sub t1 sup * = wt, X sub t2
sup * = w sub t ln (SQFT sub t ), and X sub t3 sup * = w sub t ln
(YARD sub t ).  Then regress Y sub t sup * against X sub t1 sup *, X
sub t2 sup *, and X sub t3 sup *, with no constant term.

b.
Since the weights are known, WLS estimates are BLUE and hence most
efficient, that is, more efficient that OLS.

c.
For the Glesjer test, one possible assumption is that sigma sub t = alpha
sub 1 + alpha sub 2 SQFT sub t + alpha sub 3 YARD sub t.

d.
H sub 0 is alpha sub 2 = alpha sub 3 = 0.
e.
First regress PRICE  against a constant, ln(SQFT), and ln(YARD) and
compute the residuals u hat sub t = PRICE  sub t~- beta hat sub 1~-
beta hat sub 2  ln (SQFT sub t )~- beta hat sub 3 ln ( 
YARD sub t ).

Next estimate the auxiliary equation by
regressing | u hat sub t | against a constant, SQFT sub t, and
YARD sub t and compute LM = n R sup 2, where n is the number of
observations and R sup 2 is the unadjusted R-square in this regression.
Under H sub 0 this has a chi-square distribution with 2 d.f.

f.
Estimate sigma sub t by sigma hat sub t = alpha hat sub 1 + alpha
hat sub 2SQFT sub t + alpha hat sub 3 
YARD sub t.  Compute weight w
sub t = 1/sigma hat sub t.  Regress w sub t  PRICE  sub t against w
sub t, w sub t ln (SQFT sub t ), and w sub t ln ( YARD sub t ) with no
constant term.