Heteroscedasticity and serial correlation

Abstract

There are many situations occurring in practice when the simple structure of random variation assumed in (1.1) does not hold; examples will be given below. Suppose that instead of (1.1) we have

$$\left. {{}_{V\left( Y \right)\, = \,V{\sigma ^2},}^{E\left( Y \right)\, = \,a\theta ,}} \right\}$$

((9.1))

where V is a known n × n positive definite matrix, then the appropriate method of estimation is generalized least squares (GLS), leading to

$$\hat \theta g = {\left( {a'{V^{ - 1}}a} \right)^{ - 1}}a'{V^{ - 1}}Y.$$

((9.2))

If V = 1, then this reduces to the ordinary least squares (OLS) estimator

$$\hat \theta = a{\left( {a'a} \right)^{ - 1}}a'Y.$$

((9.3))

If (9.3) is used when (9.2) is appropriate, then (9.3) is still unbiased, but it is not efficient, and the OLS estimator of σ ² will in general be biased. All tests will therefore be invalid, Clearly, further research needs to be carried out on how serious the effects are of using OLS when (9.2) ought to be used, but this argument indicates the desirability of testing for departures from the assumption of V = I.