Wen-Jen Tsay

The spurious regression of fractionally integrated processes

This paper extends the theoretical analysis of the spurious regression and spurious detrending from the usual I(1) processes to the long memory fractionally integrated processes. It is found that when we regress a long memory fractionally integrated process on another unrelated long memory fractionally integrated process, no matter whether these processes are stationary or not, as long as their orders of integration sum up to a value greater than 0.5, the t ratios become divergent and spurious effects occur. Our finding suggests that it is the long memory, instead of nonstationarity or lack of ergodicity, that causes such spurious effects. As a result, spurious effects might happen more often than we previously believed as they can arise even between stationary series while the usual first-differencing procedure may not completely eliminate spurious effects when data possess strong long memory.

Long memory story of the real interest rate

Abstract This paper reexamines the time series properties of the US ex post real interest rate. The estimation of the ARFIMA model using the Conditional Sum of Squares (CSS) method reveals that the ex post real interest rate can be well described using a fractionally integrated process.

A Generalized ARFIMA Process with Markov-Switching Fractional Differencing Parameter

We propose a general class of Markov-switching-ARFIMA (MS-ARFIMA) processes in order to combine strands of long memory and Markov-switching literature. Although the coverage of this class of models is broad, we show that these models can be easily estimated with the Durbin–Levinson–Viterbi algorithm proposed. This algorithm combines the Durbin–Levinson and Viterbi procedures. A Monte Carlo experiment reveals that the finite sample performance of the proposed algorithm for a simple mixture model of Markov-switching mean and ARFIMA(1, d, 1) process is satisfactory. We apply the MS-ARFIMA models to the US real interest rates and the Nile river level data, respectively. The results are all highly consistent with the conjectures made or empirical results found in the literature. Particularly, we confirm the conjecture in Beran and Terrin [J. Beran and N. Terrin, Testing for a change of the long-memory parameter. Biometrika 83 (1996), pp. 627–638.] that the observations 1 to about 100 of the Nile river data seem to be more independent than the subsequent observations, and the value of differencing parameter is lower for the first 100 observations than for the subsequent data.

Maximum likelihood estimation of stationary multivariate ARFIMA processes

This article considers the maximum likelihood estimation (MLE) of a class of stationary and invertible vector autoregressive fractionally integrated moving-average (VARFIMA) processes considered in Equation (26) of Luceño [A fast likelihood approximation for vector general linear processes with long series: Application to fractional differencing, Biometrika 83 (1996), pp. 603–614] or Model A of Lobato [Consistency of the averaged cross-periodogram in long memory series, J. Time Ser. Anal. 18 (1997), pp. 137–155] where each component y i, t is a fractionally integrated process of order d i , i=1, …, r. Under the conditions outlined in Assumption 1 of this article, the conditional likelihood function of this class of VARFIMA models can be efficiently and exactly calculated with a conditional likelihood Durbin–Levinson (CLDL) algorithm proposed herein. This CLDL algorithm is based on the multivariate Durbin–Levinson algorithm of Whittle [On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix, Biometrika 50 (1963), pp. 129–134] and the conditional likelihood principle of Box and Jenkins [Time Series Analysis, Forecasting, and Control, 2nd ed., Holden-Day, San Francisco, CA]. Furthermore, the conditions in the aforementioned Assumption 1 are general enough to include the model considered in Andersen et al. [Modeling and forecasting realized volatility, Econometrica 71 (2003), 579–625] for describing the behaviour of realized volatility and the model studied in Haslett and Raftery [Space–time modelling with long-memory dependence: Assessing Irelands wind power resource, Appl. Statist. 38 (1989), pp. 1–50] for spatial data as its special cases. As the computational cost of implementing the CLDL algorithm is much lower than that of using the algorithms proposed in Sowell [Maximum likelihood estimation of fractionally integrated time series models, Working paper, Carnegie-Mellon University], we are thus able to conduct a Monte Carlo experiment to investigate the finite sample performance of the CLDL algorithm for the 3-dimensional VARFIMA processes with the sample size of 400. The simulation results are very satisfactory and reveal the great potentials of using the CLDL method for empirical applications.

Estimating Trending Variables In The Presence Of Fractionally Integrated Errors

This paper considers the problems of estimation and inference in the linear regression model with fractionally integrated errors. The ordinary least squares (OLS) and the first differenced (FD) estimators are studied. Relative to the OLS estimators, a substantial increase in the convergence rates of the coefficient estimator for the stochastic regressor can be achieved by the FD estimators when the error term is nonstationary. However, the preceding decisive results can not always sustain when the error term is stationary. We also find that the FD estimators can eliminate the spurious regression because the FD t-ratio for the coefficient estimators never diverges.

On the power of durbin-watson statistic against fractionally integrated processes

This paper provides the theoretical explanation and Monte Carlo experiments of using a modified version of Durbin-Watson ( D W ) statistic to test an 1 ( 1 ) process against I ( d ) alternatives, that is, integrated process of order d, where d is a fractional number. We provide the exact order of magnitude of the modified D W test when the data generating process is an I ( d ) process with d E (0. 1.5). Moreover, the consistency of the modified DW statistic as a unit root test against I ( d ) alternatives with d E ( 0 , l ) U ( 1 , 1.5) is proved in this paper. In addition to the theoretical analysis, Monte Carlo experiments show that the performance of the modified D W statistic reveals that it can be used as a unit root test against I ( d ) alternatives.

Coresidence With Husband’s Parents, Labor Supply, and Duration to First Birth

This article investigates the time to first birth, treating coresidence with husband’s parents and labor supply as endogenous and using representative data on Taiwanese married women born during 1933–1968. We use a full-information maximum likelihood estimator for a duration model with endogenous binary variables. Results controlling for endogeneity suggest that both coresidence and working are associated with a delay in childbearing, reversing the effect of coresidence on the timing of first birth but not that of working. Women in earlier cohorts tend to choose coresidency and not working, and an increasing number of women from later cohorts choose to do both or to work only.

SPURIOUS REGRESSION BETWEEN I(1) PROCESSES WITH INFINITE VARIANCE ERRORS

This paper considers spurious regression between integrated processes with stable errors. Our results show that the t-ratios diverge at the rate of âˆsT, which is identical to what Phillips (1986, Journal of Econometrics 33, 311–340) has obtained for the Gaussian case. Therefore, it is the long memory in the dependent variable and regressors, instead of the moment conditions of the error terms, that causes the spurious regression.

Forecasting Commodity Prices with Mixed-Frequency Data: An OLS-Based Generalized ADL Approach

This paper presents a generalized autoregressive distributed lag (GADL) model for conducting regression estimations that involve mixed-frequency data. As an example, we show that daily asset market information - currency and equity mar- ket movements - can produce forecasts of quarterly commodity price changes that are superior to those in the previous research. Following the traditional ADL lit- erature, our estimation strategy relies on a Vandermonde matrix to parameterize the weighting functions for higher-frequency observations. Accordingly, infer- ences can be obtained using ordinary least squares principles without Kalman fi ltering, non-linear optimizations, or additional restrictions on the parameters. Our fi ndings provide an easy-to-use method for conducting mixed data-sampling analysis as well as for forecasting world commodity price movements.

Maximum simulated likelihood estimation of the panel sample selection model

ABSTRACT Heckmans (1976, 1979) sample selection model has been employed in many studies of linear and nonlinear regression applications. It is well known that ignoring the sample selectivity may result in inconsistency of the estimator due to the correlation between the statistical errors in the selection and main equations. In this article, we reconsider the maximum likelihood estimator for the panel sample selection model in Keane et al. (1988). Since the panel data model contains individual effects, such as fixed or random effects, the likelihood function is more complicated than that of the classical Heckman model. As an alternative to the existing derivation of the likelihood function in the literature, we show that the conditional distribution of the main equation follows a closed skew-normal (CSN) distribution, of which the linear transformation is still a CSN. Although the evaluation of the likelihood function involves high-dimensional integration, we show that the integration can be further simplified into a one-dimensional problem and can be evaluated by the simulated likelihood method. Moreover, we also conduct a Monte Carlo experiment to investigate the finite sample performance of the proposed estimator and find that our estimator provides reliable and quite satisfactory results.