Question

1. Consider the model

Y_t = β₁ + β₂X_2t + β₃X_3t + u_t

and we suspect that the variance of the error term has the following form:

Var(u_t) = δ1 + δ₂Y_t-1 + δY_t-2

(a) Sketch the test proecedure for the presence of heteroskedasticity

in this form.

(b) Write out the GLS model.

Answer 1

Answer: a

The presence of heteroskedasticity affects the estimation and test of the hypothesis. The heteroskedasticity can enter into the data due to various reasons. The tests for heteroskedasticity assume a specific nature of heteroskedasticity. Various tests are available in the literature, e.g.,

1. Bartlett test

2. Breusch Pagan test

3. Goldfeld Quandt test

4. Glesjer test

5. Test based on Spearman’s rank correlation coefficient

6. White test

7. Ramsey test

8. Harvey Phillips test

9. Szroeter test

10. Peak test (nonparametric) test

so the first test procedure -

Detection of Heterosk.: Goldfeld-Quandt (GQ) test

It is carried out as follows:

Split the total sample of length T into two sub-samples of length T1 and T2. The regression model is estimated on each sub-sample and the two residual variances are calculated.

2. The null hypothesis is that the variances of the disturbances are equal, H0: σ₁² = σ²₂

3. The test statistic, denoted GQ, is simply the ratio of the two residual variances where the larger of the two variances must be placed in the numerator. GQ = s₁²/ s₂²

4. The test statistic is distributed as an F(T1 − k, T2 − k) under the null of homoscedasticity. • Big practical issue: where do you split the sample? It is often arbitrary, and it may crucially affect the outcome of the test

ANSWER 2: THIS IS SUTOREGRESSIVE DISTRIBUTED MODEL SO YOU CAN TAKE LAGGED VALUED OF Y

The GLS estimator is unbiased, consistent, efficient, and asymptotically normal with {\displaystyle \operatorname {E} [{\hat {\beta }}\mid \mathbf {X} ]=\beta } and {\displaystyle \operatorname {Cov} [{\hat {\beta }}\mid \mathbf {X} ]=(\mathbf {X} ^{\mathtt {T}}\Omega ^{-1}\mathbf {X} )^{-1}}. GLS is equivalent to applying ordinary least squares to a linearly transformed version of the data. To see this, factor {\displaystyle \mathbf {\Omega } =\mathbf {C} \mathbf {C} ^{\mathtt {T}}},