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

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.