Location Multiplicative Error Model(1,1): Asymptotic Properties of a Modified QMLE

Abstract

This paper extends the Multiplicative Error Model (MEM) by adding the location parameter. The minimum of a sample is proved to be a consistent estimator for this parameter, and used to truncate the data set. If the truncated data set contains none or a trivial proportion of zeroes, the autoregressive coefficients in this model are estimated by a Gaussian Quasi Maximum Likelihood Estimator (QMLE). Consistency and asymptotic normality of this estimator based on the truncated process are demonstrated under weak assumptions about random errors. If a large proportion of zeroes exist in the truncated data set, we adopt a Zero-Augmented distribution for the random errors. Such distribution consists of a probability mass assigned at zero and a continuous density function at nonzero values. We also propose a different QMLE without specifying the true density function, to estimate the probability mass and autoregressive coefficients. Similar asymptotic properties for this modified QMLE are established under weak assumptions. We present simulation studies and empirical analysis on IBM High Frequency trading data, to illustrate the asymptotic results and model improvement for both cases.