George C. Tiao

Intervention Analysis with Applications to Economic and Environmental Problems

Abstract This article discusses the effect of interventions on a given response variable in the presence of dependent noise structure. Difference equation models are employed to represent the possible dynamic characteristics of both the interventions and the noise. Some properties of the maximum likelihood estimators of parameters measuring level changes are discussed. Two applications, one dealing with the photochemical smog data in Los Angeles and the other with changes in the consumer price index, are presented.

Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance

Abstract This article studies the problem of multiple change points in the variance of a sequence of independent observations. We propose a procedure to detect variance changes based on an iterated cumulative sums of squares (ICSS) algorithm. We study the properties of the centered cumulative sum of squares function and give an intuitive basis for the ICSS algorithm. For series of moderate size (i.e., 200 observations and beyond), the ICSS algorithm offers results comparable to those obtained by a Bayesian approach or by likelihood ratio tests, without the heavy computational burden required by these approaches. Simulation results comparing the ICSS algorithm to other approaches are presented.

Modeling Multiple Time Series with Applications

Abstract An approach to the modeling and analysis of multiple time series is proposed. Properties of a class of vector autoregressive moving average models are discussed. Modeling procedures consisting of tentative specification, estimation, and diagnostic checking are outlined and illustrated by three real examples.

Factors affecting the detection of trends: Statistical considerations and applications to environmental data

Detection of long-term, linear trends is affected by a number of factors, including the size of trend to be detected, the time span of available data, and the magnitude of variability and autocorrelation of the noise in the data. The number of years of data necessary to detect a trend is strongly dependent on, and increases with, the magnitude of variance (σN2) and autocorrelation coefficient (ϕ) of the noise. For a typical range of values of σN2 and ϕ the number of years of data needed to detect a trend of 5%/decade can vary from ∼10 to >20 years, implying that in choosing sites to detect trends some locations are likely to be more efficient and cost-effective than others. Additionally, some environmental variables allow for an earlier detection of trends than other variables because of their low variability and autocorrelation. The detection of trends can be confounded when sudden changes occur in the data, such as when an instrument is changed or a volcano erupts. Sudden level shifts in data sets, whether due to artificial sources, such as changes in instrumentation or site location, or natural sources, such as volcanic eruptions or local changes to the environment, can strongly impact the number of years necessary to detect a given trend, increasing the number of years by as much as 50% or more. This paper provides formulae for estimating the number of years necessary to detect trends, along with the estimates of the impact of interventions on trend detection. The uncertainty associated with these estimates is also explored. The results presented are relevant for a variety of practical decisions in managing a monitoring station, such as whether to move an instrument, change monitoring protocols in the middle of a long-term monitoring program, or try to reduce uncertainty in the measurements by improved calibration techniques. The results are also useful for establishing reasonable expectations for trend detection and can be helpful in selecting sites and environmental variables for the detection of trends. An important implication of these results is that it will take several decades of high-quality data to detect the trends likely to occur in nature.

Estimation of time series parameters in the presence of outliers

Outliers in time series can be regarded as being generated by dynamic intervention models at unknown time points. Two special cases, innovational outlier (IO) and additive outlier (AO), are studied in this article. The likelihood ratio criteria for testing the existence of outliers of both types, and the criteria for distinguishing between them are derived. An iterative procedure is proposed for detecting IO and AO in practice and for estimating the time series parameters in autoregressive-integrated-moving-average models in the presence of outliers. The powers of the procedure in detecting outliers are investigated by simulation experiments. The performance of the proposed procedure for estimating the autoregressive coefficient of a simple AR(l) model compares favorably with robust estimation procedures proposed in the literature. Two real examples are presented.

An ARIMA-Model-Based Approach to Seasonal Adjustment

Abstract This article proposes a model-based procedure to decompose a time series uniquely into mutually independent additive seasonal, trend, and irregular noise components. The series is assumed to follow the Gaussian ARIMA model. Properties of the procedure are discussed and an actual example is given.

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.

Decomposition of Seasonal Time Series: A Model for the Census X-11 Program

Abstract This paper shows that the linear filter version of the Census X-11 program for time-series decomposition can be approximately justified in terms of an additive model with stochastic trend, seasonal and noise components. Optimal estimates of the trend and seasonal components are obtained from the model and found to be in close agreement with the corresponding estimates for the Census procedure. This approach makes it possible to assess the appropriateness of the Census method. Two examples are given, one showing that the use of the X-11 procedure is largely appropriate and the other much less so.

Likelihood Function of Stationary Multiple Autoregressive Moving Average Models

Abstract Procedures to estimate parameters in multivariate autoregressive moving average (ARMA) models are developed. Gaussian errors are assumed. Exact maximum likelihood estimation procedures are developed for pure moving average models. Approximate procedures are obtained to estimate stationary mixed ARMA models. Properties of the estimates and an example 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.