Question

Consider the model

                                    E_t = α + β I_t + u_t      where u_t has a zero mean

                  Where E_t is aggregate expenditure on a good and I_t is total income. Given the following information

OLS Estimates Using 51 Observations

Dependent Variable E_t

Variable	Coeff	STD error	T Stat
constant	0.26644	0.3294	.809
It	0.06754	.0035	19.288

                  SSR = 157.90709

                  Unadjusted R² = 0.884

                  Adjusted R² = .881

OLS Estimates Using 51 Observations

Dependent Variable u²_t

u_t = OLS residuals from earlier regression

Variable	Coeff	STD error	T Stat
constant	-1.3779	2.2407	0.5415
Population	1.3723	.6714	0.465
(Population)2	-0.04724	.03086	.1877

                  SSR = 3684.4776

                  Unadjusted R² = .119

                  Adjusted R² = .082

a)   Write down in symbolic terms the auxiliary equation for the error variance implicit in the above information.

b)   State the null hypothesis that there is no heteroscedasticity.

c)   Calculate the value of a test statistic to test this hypothesis.

d)   What is the distribution and degrees of freedom of this statistic?

e)   Carry out the test at a 5% level of significance on whether heteroscedasticity is present or not.

Answer 1

Answer:

Given that,

Consider the model Et = α + β It + ut ,where ut has a zero mean.

Where Et is aggregate expenditure on a good and It is total income. Given the following information.

OLS Estimates Using 51 Observations

Dependent Variable Et.

Variable	Coeff	STD error	T Stat
Constant	0.26644	0.3294	0.809
It	0.06754	0.0035	19.288

SSR = 157.90709

Unadjusted R2 = 0.884

Adjusted R2 = 0.881

OLS Estimates Using 51 Observations

Dependent Variable ut2

ut = OLS residuals from earlier regression

Variable	Coeff	STD error	T Stat
Constant	-1.3779	2.2407	0.5415
Population	1.3723	0.6714	0.465
(Population)2	-0.04724	0.03086	0.1877

SSR = 3684.4776

Unadjusted R2 = .119

Adjusted R2 = .082

(a).

The auxiliary equation for the error variance implicit in the above information:

Auxiliary equation for yhe error variance is,

(b).

The null hypothesis that there is no heteroscedasticity:

: Atleast one pair is not zero

[or]

: There is no heteroscedasticity.

:There is heteroscedasticity.

(c).

The value of a test statistic to test this hypothesis:

This is white test for testing heteroscedasticity and test statistic.

Where,

d.f=k*=Number of regression in auxiliary equation.

d.f=2, n=51 observations given.

R2=0.119 (in auxiliary)

T=nR2

=51 0.119

T=5.95

(d).

The distribution and degrees of freedom of this statistic:

(e).

Carry out the test at a 5% level of significance on whether heteroscedasticity is present or not:

Pvalue=0.051047 (From excel formuala CHISQ.DIST.RT(5.95,2))

Clearly Pvalue > 0.05, hence we reject H0 or there is heteroscedasticity present in data.