A Monte-Carlo study of the properties of the Laplace estimator under non-ideal conditions with an application to foreign exchange forecasts

Abstract

Despite the wide usage of pre-testing in practice, the vicious risk properties of pre-test estimators are well-documented. Researchers have been attempting to search for alternatives that possess a better risk property than the pre-test estimator. These include the Stein-rule estimator and the Ridge Regression estimator. Among them, the Laplace estimator, interpreted as a Bayes estimator with double exponential prior density, has attractive risk properties over the pre-test estimator: it has the lowest risk among all estimators around the region where the risk of the Ordinary Least Square (OLS) estimator and the Restricted Least Square (RLS) estimator intersect. Moreover, instead of a binary choice algorithm of the pre-test estimator, the Laplace estimator continuously weights the RLS and the OLS according to their relative importance. This offers an advantage over some other shrinkage estimators such as the Stein rule estimator which, like the pre-test estimator, is a discontinuous function of the data. In parallel, a similar problem has puzzled data mining scientists for decades. To put data mining systems into real work, data miners usually impose their prior domain-driven knowledge to restrict the outcome of the system. However, data miners frequently encounter a conflict between the machine-generated result (data-driven knowledge) and their domain-driven knowledge learned from past experience. Given such conflicts, it is hard for data miners to determine which knowledge to accept. Although many have agreed that a yes/no selection renders the mining system unreliable, data miners have not found a formal treatment for the problem. The Laplace estimator can be a good treatment for both problems because it is theoretically justified and has a better risk property in some particular but important region of the parameter space where we do not know the source of knowledge to adopt. However, in the academic literature it was only shown that the Laplace estimator is better when the error term of the regression is correctly defined. Also, since the Laplace estimator is a shrinkage estimator, it is questionable whether the newly proposed shrinkage estimators with heavy computational work, such as the Least Angle (LA) estimator and the Generalized Ridge (GR) estimator, work better than the Laplace estimator. This motivates us to examine the robustness of the Laplace estimator under various non-ideal conditions, mostly with autocorrelated errors, and compare its performance with the traditional estimators as well as the LA and GR estimators. Using Monte Carlo simulations, we show that although the Laplace estimator is not the best estimator in terms of risk, it is the most stable estimator in different conditions in the sense that the risk property of the Laplace estimator do not change dramatically under different datasets. The Foreign Exchange (FX) Market is the world’s most traded market in terms of trading volume. There has been a long debate on whether past exchange rate data can predict future exchange rates. We select the exchange rate of the USD-JPY pair daily data and test for cointegration between the daily highs and lows and hence the significance of the error correction term. The out-of-sample performance with Modified Diebold-Mariano (MDM) test using different criteria results in different rankings among the estimators. It has been found that the Laplace estimator always ranks between the best and the worst. Hence, our out-of-sample empirical studies confirm that the Laplace estimator is a stable estimation procedure.

Date of Award	2 Oct 2008
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Tze-Kin Alan WAN (Supervisor)