Henghsiu Tsai

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH( p , q ) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH( p , q ) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH( p , q ) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

A Note on Non-Negative Arma Processes

Recently, there has been much research on developing models suitable for analysing the volatility of a discrete-time process. Since the volatility process, like many others, is necessarily non-negative, there is a need to construct models for stationary processes which are non-negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non-negative kernel by non-negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non-negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non-negative almost surely.

Constrained Factor Models

This article considers estimation and applications of constrained and partially constrained factor models when the dimension of explanatory variables is high. Both the classical and approximate factor models are investigated. For estimation, we employ both the maximum likelihood and least squares methods. We show that the least squares estimation is based on constrained principal component analysis and provides consistent estimates for the model under certain conditions. The normality condition is not used in the derivation. We then propose likelihood ratio statistics to test the adequacy of factor constraints. The test statistic is developed under the normality assumption, but simulation results show that it continues to perform well even if the underlying distribution is Student-t. The constraints are useful tools to incorporate prior information or substantive theory in applications of factor models. In addition, the constraints also serve as a statistical tool to obtain parsimonious econometric models for forecasting, to simplify the interpretations of common factors, and to reduce the dimension. We use simulation and real examples to investigate the performance of constrained estimation in finite samples and to highlight the importance of noise-to-signal ratio in factor analysis. Finally, we compare the constrained model with its unconstrained counterpart both in estimation and in forecasting. This article has supplementary material online.

On continuous-time autoregressive fractionally integrated moving average processes

In this paper, we consider a continuous-time autoregressive fractionally integrated moving average (CARFIMA) model, which is defined as the stationary solution of a stochastic differential equation driven by a standard fractional Brownian motion. Like the discrete-time ARFIMA model, the CARFIMA model is useful for studying time series with short memory, long memory and antipersistence. We investigate the stationarity of the model and derive its covariance structure. In addition, we derive the spectral density function of a stationary CARFIMA process.

A note on the non-negativity of continuous-time ARMA and GARCH processes

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.

Inference of Seasonal Long-memory Aggregate Time Series

Time-series data with regular and/or seasonal long-memory are often aggregated before analysis. Often, the aggregation scale is large enough to remove any short-memory components of the underlying process but too short to eliminate seasonal patterns of much longer periods. In this paper, we investigate the limiting correlation structure of aggregate time series within an intermediate asymptotic framework that attempts to capture the aforementioned sampling scheme. In particular, we study the autocorrelation structure and the spectral density function of aggregates from a discrete-time process. The underlying discrete-time process is assumed to be a stationary Seasonal AutoRegressive Fractionally Integrated Moving-Average (SARFIMA) process, after suitable number of differencing if necessary, and the seasonal periods of the underlying process are multiples of the aggregation size. We derive the limit of the normalized spectral density function of the aggregates, with increasing aggregation. The limiting aggregate (seasonal) long-memory model may then be useful for analyzing aggregate time-series data, which can be estimated by maximizing the Whittle likelihood. We prove that the maximum Whittle likelihood estimator (spectral maximum likelihood estimator) is consistent and asymptotically normal, and study its finite-sample properties through simulation. The efficacy of the proposed approach is illustrated by a real-life internet traffic example.

Asymptotic Behavior of Temporal Aggregates in the Frequency Domain

Abstract The classical aggregation result by Tiao (1972, Asymptotic Behavior of Temporal Aggregates of Time Series, Biometrika 59, 525–531) is generalized for a weak set of assumptions. The innovations driving the integrated processes are only required to be stationary with integrable spectral density. The derivation is settled in the frequency domain. In case of fractional integration, it is demonstrated that the order of integration is preserved with growing aggregation under the same set of assumptions.

COMPARISON OF TWO DISCRETIZATION METHODS FOR ESTIMATING CONTINUOUS-TIME AUTOREGRESSIVE MODELS

We have applied the trapezium method to approximate integrals in an implementation of the EM algorithm proposed by Tsai and Chan (1999b) for estimating continuous-time autoregressive models, whose original implementation was based on Euler’s method for approximating integrals. It is well known that the trapezium method generally provides a second order approximation to an integral of a well-behaved functional of Wiener process, whereas the Euler method is generally of first order. Simulation results confirm that with increasing discretization frequency, the EM estimators based on the trapezium method converge to the (conditional) ML estimator at a faster rate than the EM estimators based on Euler’s method. However, with an appropriate choice of discretization frequency, the EM estimator based on Euler’s method outperforms both the EM estimator based on the trapezium method and the ML estimator in terms of biases and standard deviations of the estimates. An invariance property of the EM estimator based on the trapezium method is briefly discussed. Some key words: Trapezium method, Girsanov formula, Maximum likelihood estimation, Stochastic differential equations, irregularly sampled time series, Kalman filter.

A Three-Parameter Speeded Item Response Model: Estimation and Application

When given time constraints, it is possible that examinees leave the harder items till later and are not able to finish answering every item in time. In this paper, this situation was modeled by incorporating a speeded-effect term into a three-parameter logistic item response model. Due to the complexity of the likelihood structure, a Bayesian estimation procedure with Markov chain Monte Carlo method was presented. The methodology is applied to physics examination data of the Department Required Test for college entrance in Taiwan for illustration.

Using Credible Intervals to Detect Differential Item Functioning in IRT Models

Differential item functioning (DIF) occurs when individuals from different groups with the same level of ability have different probabilities of answering an item correctly. In this paper, we develop a Bayesian approach to detect DIF based on the credible intervals within the framework of item response theory models. Our method performed well for both uniform and non-uniform DIF conditions in the two-parameter logistic model. The efficacy of the proposed approach is demonstrated through simulation studies and a real data application.