Question

1.      Suppose that a researcher estimates a consumptions function and obtains the following results:

where C=Consumption,          Y_d=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 

Answer 1

 Given

C = 15 + 0.81Y_d                  

     (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 Y_d statistically using t ratios

set the null hypothesis H₀: b=0. (C and Y_d 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 [-t_0.025,n-2 , t_0.025,n-2] and should be rejected otherwise

from a t table we have

t_0.025,n-2 = t_0.025,17 =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 Y_d 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