Ruey S. Tsay

Testing and Modeling Threshold Autoregressive Processes

Abstract The threshold autoregressive model is one of the nonlinear time series models available in the literature. It was first proposed by Tong (1978) and discussed in detail by Tong and Lim (1980) and Tong (1983). The major features of this class of models are limit cycles, amplitude dependent frequencies, and jump phenomena. Much of the original motivation of the model is concerned with limit cycles of a cyclical time series, and indeed the model is capable of producing asymmetric limit cycles. The threshold autoregressive model, however, has not received much attention in application. This is due to (a) the lack of a suitable modeling procedure and (b) the inability to identify the threshold variable and estimate the threshold values. The primary goal of this article, therefore, is to suggest a simple yet widely applicable model-building procedure for threshold autoregressive models. Based on some predictive residuals, a simple statistic is proposed to test for threshold nonlinearity and specify the ...

Testing and Modeling Multivariate Threshold Models

Abstract Threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years. In this article I use predictive residuals to construct a test statistic for detecting threshold nonlinearity in a vector time series and propose a procedure for building a multivariate threshold model. The thresholds and the model are selected jointly based on the Akaike information criterion. The finite-sample performance of the proposed test is studied by simulation. The modeling procedure is then used to study arbitrage in security markets and results in a threshold cointegration between logarithms of future contracts and spot prices of a security after adjusting for the cost of carrying the contracts. In this particular application, thresholds are determined in part by the transaction costs. I also apply the proposed procedure to U.S. monthly interest rates and two river flow series of Iceland.

Functional-Coefficient Autoregressive Models

In this article we propose a new class of models for nonlinear time series analysis, investigate properties of the proposed model, and suggest a modeling procedure for building such a model. The proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics. A consistency result is given to support the procedure. For illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi-step ahead forecasts over other models.

Time Series Model Specification in the Presence of Outliers

Abstract Outliers are commonplace in data analysis. Time series analysis is no exception. Noting that the effect of outliers on model identification statistics could be serious, this article is concerned with the problem of time series model specification in the presence of outliers. An iterative procedure is proposed to identify the outliers, to remove their effects, and to specify a tentative model for the underlying process. The procedure is essentially based on the iterative estimation procedure of Chang and Tiao (1983) and the extended sample autocorrelation function (ESACF) model identification method of Tsay and Tiao (1984). An example is given. Properties of the proposed procedure are discussed.

A nonlinear autoregressive conditional duration model with applications to financial transaction data

Abstract This paper presents a new model that improves upon several inadequacies of the original autoregressive conditional duration (ACD) model considered in Engle and Russell (Econometrica 66(5) (1998) 1127–1162). We propose a threshold autoregressive conditional duration (TACD) model to allow the expected duration to depend nonlinearly on past information variables. Conditions for the TACD process to be ergodic and existence of moments are established. Strong evidence is provided to suggest that fast transacting periods and slow transacting periods of NYSE stocks have quite different dynamics. Based on the improved model, we identify multiple structural breaks in the transaction duration data considered, and those break points match nicely with real economic events.

Consistent Estimates of Autoregressive Parameters and Extended Sample Autocorrelation Function for Stationary and Nonstationary ARMA Models

Abstract A unified approach for the tentative specification of the order of mixed stationary and nonstationary ARMA models is proposed. For the ARMA models, an iterative regression procedure is given to produce consistent estimates of the autoregressive parameters. An extended sample autocorrelation function based on these consistent estimates is then defined and used for order determination. One of the advantages of this new approach is that it eliminates the need to determine, usually rather arbitrarily, the order of differencing to produce stationarity in modeling time series. Comparisons with other existing identification methods are discussed, and several samples are given.

Forecasting the U.S. Unemployment Rate

Abstract This article presents a comparison of forecasting performance for a variety of linear and nonlinear time series models using the U.S. unemployment rate. Our main emphases are on measuring forecasting performance during economic expansions and contractions by exploiting the asymmetric cyclical behavior of unemployment numbers, on building vector models that incorporate initial jobless claims as a leading indicator, and on utilizing additional information provided by the monthly rate for forecasting the quarterly rate. Comparisons are also made with the consensus forecasts from the Survey of Professional Forecasters. In addition, the forecasts of nonlinear models are combined with the consensus forecasts. The results show that significant improvements in forecasting accuracy can be obtained over existing methods.

Conditional Heteroscedastic Time Series Models

Abstract Under the traditional linear time series or regression setting, the conditional variance of one-step-ahead prediction is time invariant. Experience in conjunction with data analysis, however, suggests that the variability of a process might well depend on the available information. This reality has motivated extensive research to relax the constant variance assumption imposed by the traditional linear time series model, and several classes of generalized parametric models designed specifically for handling nonhomogeneity of a process have been proposed recently. In particular, the random coefficient autoregressive (RCA) models were widely investigated by time series analysts and the autoregressive conditional heteroscedastic (ARCH) models were investigated by econometricians. The interesting fact is that the ARCH processes can be regarded as special cases of the RCA model. In this article, I first give the relationship between these two types of models and show that the special feature of these t...

Bayesian Inference and Prediction for Mean and Variance Shifts in Autoregressive Time Series

Abstract This article is concerned with statistical inference and prediction of mean and variance changes in an autoregressive time series. We first extend the analysis of random mean-shift models to random variance-shift models. We then consider a method for predicting when a shift is about to occur. This involves appending to the autoregressive model a probit model for the probability that a shift occurs given a chosen set of explanatory variables. The basic computational tool we use in the proposed analysis is the Gibbs sampler. For illustration, we apply the analysis to several examples.

Nonlinear Additive ARX Models

Abstract We consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models. The procedures proposed use two backfitting techniques (the ACE and BRUTO algorithms) to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model. Simulated and real examples are used to illustrate the analysis.