Testing for heteroskedasticity in the tobit and probit models

Abstract

Non-constant variance across observations (heteroskedasticity) results in the maximum likelihood estimators of tobit and probit model parameters being inconsistent. Some of the available tests for constant variance across observations (homoskedasticity) are discussed and examined in a small Monte Carlo experiment.

Keywords:

• tobit
• probit
• heteroskedasticity
• score test
• Monte Carlo

Notes

See, for example 1 12. Failure of the normality assumption also results in estimator inconsistency. See, for example 2 11.

ê ^(j), j=1, 2, …, indicates different varieties of generalised residual.

The which appears in the implied versions of (5) and (6) can be dropped so that the statistic is invariant to the choice of f.

Where ‘the difference is in how the rows of are constructed’ [9, p. 773]. But the that Greene subsequently outlines is just 2 σ⁴ times the previous , which corresponds to the component of the vector of page 769. It follows that the two statistics presented by Greene 9 are the same.

The rejection rate is 1.01% when a nominal significance level of 5% is used.

• As

  The expectation of this term is the of Equation (9) after the null is imposed.

The expression presented in 19 as a test statistic corresponds exactly to the regression implemented in the Monte Carlo experiments of section 4. Their text in fact refers to an asymptotically equivalent, but numerically different, statistic requiring an artificial multivariate regression, unless p=1. Quite what calculations are actually undertaken by Pagan and Vella is explored in 23.

One of these statistics is n times the R ² from the artificial regression that produces LM ₂. In fact this is the statistic outlined in [10, pp. 456–457].

The simulations were carried out using a programme written in Gauss 7. It is available from the author on request.

The tables for n=300 and 600 are not included to economise on space. They are available from the author on request. Recall that and .

The exception to this pattern is with and n=150.

An additional advantage of this approach is that the assumption of normality for the u _i is not required to establish consistency.