1. Suppose that a researcher estimates a consumptions function and obtains the following results:
where C=Consumption, Yd=disposable income, and numbers in the parenthesis are the ‘t-ratios’
a. Test the significant of Yd statistically using t-ratios
b. Determine the estimated standard deviations of the parameter estimates
Given
C = 15 + 0.81Yd
(3.1) (18.7)
n=19 (sample size)
R2=0.99 (This signifies that out of 100% variations in consumption, our regression estimate can explain 99% variations in consumption)
a)
we test the significance of Yd statistically using t ratios
set the null hypothesis H0: b=0. (C and Yd are unrelated)
Against alternative hypothesis H1: b≠0
The appropriate test statistic under H0: b=0 would be
{ estimated ‘b’/SE (estimated ‘b') } Which follows a t distribution with (n-2) degrees of freedom so we have
t(n-2) = t = { estimated ‘b’/SE (estimated ‘b') } = 18.7 (given), now at 5% level of significance H0: b=0 will be accepted if
our t belongs to [-t0.025,n-2 , t0.025,n-2] and should be rejected otherwise
from a t table we have
t0.025,n-2 = t0.025,17 =2.110 but our given {estimated ‘b’/SE (estimated ‘b') }= 18.7(t ratio) is so high that it doesn’t lie in the interval [-2.110,2.110]. hence our null hypothesis is rejected and alternative is accepted and the relation is statistically significant and Yd is statistically significant.
b)
since for estimated ‘a’ , t = { estimated ‘a’/SE (estimated ‘a') } =3.1 (given)
and estimated ‘a’=15 so
3.1=15 SE (estimated ‘a')
Or, SE (estimated ‘a') = (15/3.1)=4.8387
Similarly for estimated'b' t= { estimated ‘b’/SE (estimated ‘b') }= 18.7(given)
Or, 18.7=0.81/SE (estimated ‘b')
Or, SE (estimated ‘b') = (0.81/18.7)= 0.0433
THUS THE ESTIMATED STANDARD DEVIATIONS OF THE PARAMETER ESTIMATORS ARE
SE (estimated ‘a') = 4.8387 and SE (estimated ‘b') =0.0433
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