KE-YOUNG CHU *
Many time series—whether they are weekly prices, monthly interest rates, or the annual transaction volumes of commodities—have certain regularities, partially obscured by noises. Decomposing a time series into its systematic part and the random part is useful for understanding the nature of the series as well as for predicting its future values. The time-series technique extensively discussed in Box and Jenkins (1970) deals with the identification and estimation of autoregressive integrated moving average (ARIMA) models, which can be used for predicting the systematic parts of time series.
ARIMA models are now used not only for predicting time series but also for analytical purposes. 1 In comparison with forecasting based on reduced forms derived from simultaneous equations models, ARIMA equations are very simple to use and require much less information for forecasting. Furthermore, ARIMA models are not entirely without an economic justification. It was shown by Tinbergen (1939), Zellner and Palm (1974), Palm (1977), and others that the “final equation” in an ARIMA form can be derived for each of the endogenous variables in a simultaneous equations model on a simple set of assumptions.
This paper reviews the relative merits of using ARIMA models for forecasting, compared with reduced forms derived from a simultaneous equations model. The paper also attempts to apply the method of ARIMA models to a short-run simulation of commodity prices. The results suggest that ARIMA models can yield quite satisfactory short-run forecasts for the monthly prices of some commodities and that they can be useful even for forecasting the yearly averages of monthly prices. 2
The paper is divided into three sections. Section I illustrates how the final equation for price in an ARIMA form can be derived from a structural system; it also reviews the relative merits of the two methods. Section II outlines the results obtained from applying the ARIMA scheme to short-run forecasting of the international prices of ten selected commodities (foods and raw materials). Section III summarizes the findings.
Brillembourg, Arturo, “Purchasing Power Parity Tests of Causality and Equilibrium” (unpublished, International Monetary Fund, December 27, 1976).
den Butter, F. A. G., “The Use of Monthly and Quarterly Data in an ARMA Model,” Journal of Econometrics, Vol. 4 (November 1976), pp. 311–24.
Dhrymes, Phoebus J., “Restricted and Unrestricted Reduced Forms: Asymptotic Distribution and Relative Efficiency,” Econometrica, Vol. 41 (January 1973), pp. 119–34.
Feige, Edgar L., and Douglas K. Pearce, “Economically Rational Expectations: Are Innovations in the Rate of Inflation Independent of Innovations in Measures of Monetary and Fiscal Policy?” Journal of Political Economy, Vol. 84 (June 1976), pp. 499–522.
Frenkel, Jacob A., “The Forward Exchange Rate, Expectations, and the Demand for Money: The German Hyperinflation,” American Economic Review, Vol. 67 (September 1977), pp. 653–70.
Naylor, Thomas H., Terry G. Seaks, and D. W. Wichern, “Box-Jenkins Methods: An Alternative to Econometric Models,” International Statistical Review, Vol. 40 (August 1972), pp. 123–37.
Nelson, Charles R. (1972), “The Prediction Performance of the FRB-MIT-PENN Model of the U. S. Economy,” American Economic Review, Vol. 62 (December 1972), pp. 902–17.
Nelson, Charles R. (1976), “The Interpretation of R2 in Autoregressive-Moving Average Time Series Models,” American Statistician, Vol. 30 (November 1976), pp. 175–80.
Palm, Franz, “On Univariate Time Series Methods and Simultaneous Equation Econometric Models,” Journal of Econometrics, Vol. 5 (May 1977), pp. 379–88.
Rose, David E., “Forecasting Aggregates of Independent ARIMA Processes,” Journal of Econometrics, Vol. 5 (May 1977), pp. 323–45.
Saboia, João Luiz Maurity, “Autoregressive Integrated Moving Average (ARIMA) Models for Birth Forecasting,” Journal of the American Statistical Association, Vol. 72 (June 1977), pp. 264–70.
Thompson, Howard E., and George C. Tiao, “Analysis of Telephone Data: A Case Study of Forecasting Seasonal Time Series,” Bell Journal of Economics and Management Science, Vol. 2 (Autumn 1971), pp. 515–41.
Zellner, Arnold, and Franz Palm, “Time Series Analysis and Simultaneous Equation Econometric Models,” Journal of Econometrics, Vol. 2 (May 1974), pp. 17–54.
Mr. Chu, economist in the Commodities Division of the Research Department, is a graduate of Kyung Hee University (Seoul) and Columbia University. Before joining the Fund, he was an instructor at Columbia University.
There have been studies on applications of time-series models for forecasting macroeconomic variables (Naylor, Seaks, and Wichern, 1972, and Nelson, 1972), for predicting birth rates (Saboia, 1977), and for analyzing telephone data (Thompson and Tiao, 1971); time-series models have also been used for specifying an econometric model (Zellner and Palm, 1974) and for evaluating economic causality (Brillembourg, 1976, Feige and Pearce, 1976, and Frenkel, 1977).
All the equations identified and estimated in this study exhibit stationarity: all of them are in autoregressive integrated moving average (ARIMA; p, d, q) forms with d = 0, or autoregressive moving average (ARMA; p, q1) forms, where p, d, and q indicate the orders of the AR part, differencing, and the MA part, respectively.
The nonstochastic second-order AR equations in Tinbergen (1939, p. 131) are essentially ARIMA (2, 0, 0) or ARMA (2, 0) processes.
The equation in (6) may be called a restricted final equation if the parameter estimates are derived from estimated structural coefficients.
The series used in preparing this paper are similar to but not identical with those published in International Monetary Fund, International Financial Statistics.
The sample periods for the estimation of the monthly models are not uniform. Bananas: January 1957–September 1976; beef: January 1957–November 1976; copra: January 1957–December 1972; maize: January 1957–December 1972; cocoa: January 1957–December 1973; tea: January 1957–September 1976; hides: January 1957–September 1976; jute: January 1957–September 1976; rubber: January 1957–September 1976; copper: January 1957–November 1976.
Beef: 2.5 years; cocoa: 3 years; hides: 3 years; rubber: 2.5 years; copper: longer than 3.5 years.
The sample autocorrelation function
The third-order regular autoregressive lag polynomials for bananas and beef have two imaginary roots and a real root each, and the autoregressive lag polynomials for bananas, tea (both second-order), and copper (fourth-order) have all imaginary roots, whereas all the other roots are real. The moduli of the imaginary roots range from 1.705 (tea, regular) to 2.975 (bananas, 3-month seasonal), and those of the real roots of the second-order, or higher-order, lag polynomials range from 1.006 (hides, regular) to 4.306 (bananas, 12-month seasonal). The real roots of the second-order or higher-order lag polynomials for bananas and hides are negative.
See Nelson (1976) for an interpretation of the statistic. Since the statistic is a measure of the degree of static simulation and is comparable with R2 in standard multiple regressions, the results of dynamic simulation may be different.
No attempt was made to find an annual model consistent with the estimated monthly model for any commodity. The annual models therefore may not be consistent with the monthly models in the way in which equation (6) is consistent with equation (3) in den Butter (1976).
The simulations are carried out for the ten-year period 1965–74, during which most prices showed both regular and unusual movements. This period is a subperiod of the sample period used for estimating the equations for all commodity prices.
The random walk model gives 0.839 and 0.923 for copra and maize, but the coefficients are lower than those associated with the monthly models. (See the Ra column and the Rl2a column of Table 4.) The difference between Rl2a and Ra (0.536–0.000 for bananas, 0.903–0.839 for copra, and 0.985–0.923 for maize) indicates the increase in the goodness of fit made possible by the use of the monthly model. Random walk models based on yearly price series would generate the simulated prices no better than those generated by the estimated annual AR or MA models in Table 3, since the estimated models maximize the coefficients of determination between the actual and the predicted prices.