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EX 6.3
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.IP a.
We have, @partial Y / partial X@ = @beta sub 2@ + @beta sub 3 (1/X)@ + 
@beta sub 5 (Z/X)@.  The elasticity follows as
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eta~~=~~Y over X~{partial Y} over {partial X}~~ =~~1 over X~
[ beta sub 2~+ ~beta sub 3 (1/X)]~ + ~beta sub 5 (Z/X)~]~ [~beta sub 1~ +
~~beta sub 2 X~ + ~beta sub 3 ln X~ + ~beta sub 4 Z~ + ~beta
sub 5 (Z~ln X)]
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.IP b.
Regress @Y sub t@ against a constant and @X sub t@ and obtain the residuals
@u hat sub t@ = @Y sub t@ @-~beta hat sub 1@ @-~beta hat sub 2 X sub t@.
Next regress @u hat sub t@ against a constant, @X sub t@, @ln X sub t@, @Z
sub t@, and @Z sub t~ln X sub t@.  Then
compute the unadjusted @R sup 2@.  The test statistic is LM = @n R sup 
2@, where @n@ is the number of observations.
.IP c.
Under the null hypothesis that @beta sub i@ = 0 for @i@ = 3 ... 5,
LM has the @chi sup 2@ distribution with 3 d.f.
.IP d.
Look up the @chi sup 2@ table for 3 d.f.  to obtain the critical value
LM\u*\d at the 5% level.  Reject the null if LM > LM\u*\d. Alternatively,
compute @p@-value = the area to the right of LM\u*\d in @chi sub 3 sup
2@ and reject the null if @p@-value is less than 0.05.
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EX 6.10
.br
.IP a.
@F sub c@  =  @{(ESSB~-~ESSA)/2} over 
{ESSA/(40~-~6)}~~=~~{(0.311974~-~0.309293)/2} over {0.309293/34}@  =  
0.147
.IP b.
Under the null hypothesis, this has the @F@-distribution with 2 d.f. for 
the numerator and 34 d.f. for the denominator.
.IP c.
@F sub 2,34 sup * (0.10)@  =  (2.44, 2.49).
.IP d.
Since @F sub c@ < @F sup *@, we cannot reject the null hypothesis.
.IP e.
Not rejecting the null implies that the coefficients for ln(UNEMP) and 
ln(POP) are jointly insignificant.
.IP f.
The @t@-statistic for ln(PRICE) is given by (1.557 @-@ 1)/0.230 = 2.42.  
For ln(INCOME) it is (4.807 @-@ 1)/0.708 = 5.38.  For ln(INTRATE) it is 
(0.208 @-@ 1)/0.058 = @-@13.66.
.IP g.
Under the null hypothesis, the @t@-statistics have the @t@-distribution 
with d.f. (for Model B) 40@-@4 = 36.
.IP h.
The critical value @t sup *@ for 36 d.f. and 5 percent level is (since 
the alternative is two-sided) in the range 2.021 to 2.042.
.IP i.
Since all the @t@-statistics are numerically above this we reject the
null hypothesis and conclude that the elasticities are significantly 
different from 1.  Since the elasticities for price and income are 
numerically greater than 1, they are elastic.  For interest rate it is 
inelastic.
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EX 6.17
.br
.IP a.
First regress PRICE against a constant, SQFT, and YARD and obtain @beta 
hat sub 1@, @beta hat sub 2@, and @beta hat sub 3@.  Then compute @u 
hat@ as PRICE @-~beta hat sub 1@ @-^beta hat sub 2@SQFT @-^beta hat sub 
3@YARD.
.IP b.
The test statistic is LM = @n ^R sup 2@ = 59 \(mu 0.115 = 6.785.  It is 
distributed as chi-square with 2 d.f.
.IP c.
From the chi-square table we have LM@nothing sup *@ = 5.99146.  Since LM 
LM@nothing sup *@, we reject the null hypothesis and conclude that 
either ln(SQFT), or ln(YARD), or both belong in the model.
.IP d.
The rule of thumb for inclusion is any new variable with @p@-value less 
than 0.50.  By this rule, ln(SQFT) should be included.  The new model is 
PRICE = @beta sub 1@ + @beta sub 2@SQFT + @beta sub 3@YARD + @beta sub 
4@ln(SQFT) + v.
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