Lack-of-fit testing of the conditional mean function in a class of Markov multiplicative error models

Abstract

The family of multiplicative error models, introduced by Engle (2002, Journal of Applied Econometrics 17, 425–446), has attracted considerable attention in recent literature for modeling positive random variables, such as the duration between trades at a stock exchange, volume transactions, and squared log returns. Such models are also applicable to other positive variables such as waiting time in a queue, daily/hourly rainfall, and demand for electricity. This paper develops a new method for testing the lack-of-fit of a given parametric multiplicative error model having a Markov structure. The test statistic is of Kolmogorov–Smirnov type based on a particular martingale transformation of a marked empirical process. The test is asymptotically distribution free, is consistent against a large class of fixed alternatives, and has nontrivial asymptotic power against a class of nonparametric local alternatives converging to the null hypothesis at the rate of O (n –1/2). In a simulation study, the test performed better overall than the general purpose Ljung–Box Q-test, a Lagrange multiplier type test, and a generalized moment test. We illustrate the testing procedure by considering two data examples.