**5**

**NONSPHERICAL DISTURBANCES AND HETEROSCEDASTICITY**

**5.1 INTRODUCTION**

The discussion in the previous chapters was based on the assumption that the disturbance vector in the linear model **y = Xβ + ε** is such that the conditional distribution ** ε_{i}|X** is independently and identically distributed with zero mean and constant variance

**σ**. The implication of this assumption is that the variance of e does not change with changes in the conditional expectation,

^{2}**. A plot of**

*E*(y |X)**ε**versus

**should therefore exhibit a random scatter of data points. The random disturbances under this assumption are referred to as spherical disturbances. This chapter deals with alternative methods of analysis under violations of this assumption. The implication here is that**

*E*(y |X)*Var*(

**ε |X**) =

**=**

*σ*^{2}Ω**Σ**, where

**Ω**is a positive definite, symmetric matrix.The random disturbances under this assumption are referred to as nonspherical disturbances. Although the general method presented in this chapter can be extended to instrumental variables regression very easily, we will assume that the explanatory variables that are used in the model are exogenous. We will deal with two cases of nonspherical disturbances.

1. *Heteroscedasticity:* Here, the disturbances are assumed to have different variances. The variance of the disturbance may, for example, be dependent upon the conditional mean ** E(y |X)**. For example, this will happen in the case when the disturbances are assumed to follow the binomial distribution. Recall that if a random ...