On improving time series forecasting

502

Memoranda

On Improving Time Series Forecasting

INTRODUCTION THE A R I M A MODELING or Box-Jenkins (BJ) approach to forecasting univariate time series is appropriate whenever the observations are statistically dependent upon or related to each other [7]. These methods use the statistical dependency to produce forecasts that are likely to be more accurate (at least for short term projections) than forecasts that ignore the statistical dependency. BJ procedures require: (1) identification, (2) estimation, (3) diagnostic testing, including the Ljung-Box test [26], and (4) forecasting. Past research suggests the need to evaluate forecasting methods in general [1, 3, 4, 5, 6, 8, 13, 20, 23, 24, 31]. In addition, studies of the application of BJ procedures to business and economic time series indicated certain serious problems in the application of BJ methods [2, 6, 16, 18, 25, 34]. Last, another study [21] compared the accuracy of various models for forecasting business and economic time series. The comparison was for a large sample of corporate time series of income indicating the relative usefulness of BJ procedures over alternative ARIMA modeling procedures. However, the problem of identifying the appropriate underlying ARIMA model for each set of time series data persists with BJ methods. In particular, the question of the appropriate degree of differencing needed to achieve stationary time series for identification underlies some of the difficulties in BJ application. Differencing represents one of the many transformation methods used to achieve stationarity. It applies when the variance of the original time series about a changing mean is relatively constant. Business and economic time series generally exhibit nonstationarity that require the obtaining of first or second differences to identify the underlying ARIMA model. Stated simply, the differencing is necessary to find a stationary ARIMA process that gives rise to the time series. In this paper, we illustrate a method for finding the appropriate degree of differencing. For nonstationary processes, the autocorrelations will die down very slowly, but, how slow is

slow? There is unfortunately no precise answer to this question available from the sample autocorrelations alone. In practice, the nature of the time series helps to provide some guidance that may be examined further when the parameters of the ARIMA model are estimated. If nonstationarity in the data is suspect, then the sample autocorrelations of the first differenced model arc plotted and examined for information about the appropriate stationary model. On occasion, second differences will have to be obtained and their sample autocorrelations examined. A check on the choice of first or second differences is provided then the parameters of the identified model is estimated. Forecasters need only to identify the lowest level of differencing for which a stationary model is apparent. This is because further differencing of a stationary series results in series which is also stationary. Overdifferencing a series merely alters the pattern of autocorrclation present in a stationary series and serves only to complicate the identification process (Jarrctt [22], pp. 300-302 and O'Donovan [28], pp. 135-136). Despite the variety of transformation methods, the identification of an appropriate transformation method is not a minor task in the final determination of the appropriate ARIMA model. Although computer software provide statistics as well as plots of ordinary and partial autocorrelation functions, identification of the appropriate level of differencing is often the most complex problem for forecasters, especially new forecasters. Thus, identification is a major obstacle to well conceived and useful applications of BJ methods to time series. In this study, the goal is to determine the sufficient order of differencing by unit root tests. We recognize that over-differcncing usually results in inefficient parameter estimates leading to inaccurate and imprecise forecasts. Since the unit root test is usually carried out in autoregressive representations, we limit our analysis to a set of time series which are known to be autoregressive (AR) models. No more than a single root at each stage of the analysis is employed and this method is similar to

Omega, Vol. 19, No. 5

503

tests employed by Ghysels [19] and Perron [29] of similar AR time series. Furthermore, this illustration with AR representation can then easily extend to general ARIMA settings as shown by Solo [33], and Said and Dickey [32].

hypothesis such that it is associated with the model of a first difference, i.e. the model with unit root. In turn, one would not reject the null hypothesis that a first difference is needed unless the tests reveal compelling evidence for rejection.

EMPIRICAL TESTING OF THE MODEL

THE SAMPLE DATA AND EMPIRICAL RESULTS

To determine the appropriate level of integer differencing, the test for the presence of unit roots is performed. It is based on the notion that the model that gives rise to a time series is stationary in the mean or in logarithms of the mean of the time series. Thus, the test for unit roots enables a forecaster to determine the proper generating model. Statistical properties for time series with unit root are developed by Fuller [17], Dickey and Fuller [10, !1], and Evans and Saving [14, 15], Phillips [30] for different AR time series and the random walk hypothesis. Two competing versions of autoregressive representations of a time series {X,}, characterized in the levels of logarithms, are utilized in this study to test a single unit root, one with a constant mean, the other with a linear trend. Since most business and macroeconomic data are published quarterly, a four-lag autoregressive regression_is assumed to capture the possible seasonal effect. The tests of hypotheses were based on Dickey et al. [9] and Dickey and Pantula [12]. Hypothesis specified in this study and the associated regressions related to the testing procedure are detailed in the Appendix. We specify the null

The time series data utilized were collected from the International Financial Statistics of the International Monetary Fund, and consist of quarterly observations of the seasonally adjusted money supply (M) ranging from 1970 to 1979 for four nations, Japan, Korea, Taiwan, and Thailand. The data are transformed to natural logarithms to capture the tendency of mean and dispersion in proportion to an absolute level, as observed by Nelson and PIosser [27]. The foregoing discussion on unit root test is performed accordingly. EMPIRICAL RESULTS Table 1 presents the empirical results from the unit root detection of money supply for four countries. Six panels in the table represent the regression results from different versions of models, one group for the test of unit root without time variable in panels 1, 3, and 5, and another group for the unit root with a linear time trend in panels 2, 4, and 6. When In M is used as the underlying variable, i.e. no differencing, the regression coefficients and t values of the constant term and In M,_ ~ are

Table I. Unit root detection for In M, AIn M, and AAIn M (1970-1979) Series Panel 1 [In M] In M Japan

Time

-0.046

Panel 3 [AIn M] R2

AIn M

Time

Panel 5 [AAIn M] R2

AAIn M

Time

R2

0.440

(-3.523)* Korea

-0.005

0.071

(-0.599) Taiwan Thailand

- 0.016 (- 1.390) 0.003 (0.305)

--0.958 (-- 2.820)*

0.584

--0.974 (-- 2.568)

0.549

0.074 0.060 Panel 2 [In M]

Japan Korea Taiwan Thailand

-0.093

0.001

( - 1.90)

(0.995)

-0.467 ( - 2.997) -0.501 (-3.312)* -3.349 ( - 2.397)

0.032 (2.965) 0.027 (3.231) 0.011 (2.426)

--0.709 (- 3.214)*

Panel 4 [&In M]

0.55

Panel 6 [AAIn M]

0.458 0.287

-0.995 (-2.908)

-0.0007 (-0.986)

0.597

-0.990 (-2.528)

0.0008 (0.234)

0.550

0.311 0.214

-0.712 (-3.145)

0.0004 (0.114)

0.55

Nole$ :

(1) Values in parentheses represent t values of coefficients. (2) Critical values to reject the null hypothesis of unit root when n 1 50 is -2.60 for models without time, and -3.18 for models with time at 10% level Oee Fuller [17], p. 373). (3) Variables in brackets represent dependent variable. *Significant to reject the null hypothesis of unit unit at 10% level.

504

Memoranda

shown in Panel 1 for the version without time trend. When In M is used the underlying variable, the regression coefficients and t values of the constant term, In M,_ 1, and the linear time trend variable are presented in Panel 2. When Aln M is used as the dependent variable, i.e. with one differencing for the version with the trend, the regression coefficients and t values of the constant term and Aln M , _ ~ are provided in Panel 3 for version without time trend. When Aln M is used as the dependent variable, the regression coefficient and t values of the constant term, Aln M,_ 1, and the linear time trend variable are displayed in Panel 4 for version with time trend. Finally, Panels 5 and 6 represent the result of second order difference without and with time trend respectively. Results of the unit root tests are as follows. For the case of testing In M, it is observed that Japan data have stable univariate autoregressive representation without linear trend from Panel 1. For Taiwan it is also appropriate to fit stable univariate AR representations together with a linear time trend from Panel 2. Therefore, the money supply levels for Japan and Taiwan do not need any integer differencing. For the case of testing Ain M (first differenced series), it is also appropriate to fit univariate AR representation without the linear time trend for Korea from Panel 4. One integer differencing is considered as appropriate for-Korea. However, Thailand is the only country that failed to reject the null hypothesis at 10% or less level on all four cases (In M without t, In M with t, Aln M without t, and Aln M with t), this suggests the need of a second integer differencing. The unit test result rejects the null hypothesis without time trend when we use AAin M (a second order differenced series) for Thailand from Panel 5.

additional differencing was necessary but, also, indicated if a linear time trend should be included in the model. Thus, the notion that one A R I M A model could represent the time series for a particular class of time series is disputed. Proper model identification must be done for each time series for each nation studied. Research is underway to extend this analysis to generalized A R I M A models having longer lags by an augmented Dickey-Fuller [11] statistic. APPENDIX Two versions of hypotheses are specified, (1) and (2) for the case with a constant mean and (3) and (4) for the case with a linear trend. Version I t/o: ~(BXI - a)x, - ~o) = e,

(t)

H~: q(B×t - pB)(X, -/~o) = e,

(2)

and

Version 2

Ho: ¢p(B)(i -- B)(Xr X,:

~(aXl

-

(3)

~o - ~l t) = e,

pa)(x,-/]o

(4)

- P, t) = e,

where B is the backshift operator (BX, = X,_ 1), t = 1, 2 . . . . . n, e, is a series of i.i.d, random errors with mean zero and a constant variance, ~p(B)--- 1 - ~ o I B - ~ p 2 B 2 - c p 3 B 3 - q ~ 4 B ~, p < I, and rio, rl, ¢Pl, ~P2, ¢P3, ¢P4 are parameters. To perform the test for unit root, two alternative autoregressive specifications are made as follows: Version I

SUMMARY AND FINDINGS The purpose of this study was to indicate the importance of a test to determine the degree of differencing necessary to achieve a stationary time series for estimation and forecasting. Previous research, especially in the areas of forecasting time series data of corporate income and often in the areas of forecasting income, money supply, and production of an economy, indicated the need to facilitate the identification stage of the A R I M A modeling process. For the sample data collected and the time period covered, the unit root detection process enables forecasters to easily identify the underlying A R I M A model generating a particular time series. With ease of model identification, the A R I M A modeling process can now be applied more effectively on a wide variety of time series. The test enabled one to determine not only if

X,-X,_l=~o+(~o~+~2+~3+~,-

1)X,_ ~

(~2 + ~3 + ~ , X X , _ t - X,_ 2)

-(~03 + ~,XX,_2 - X,_ 3) -~,(X,_3- X,_,)

(5)

+ et

Version 2 X,-x,_l=#0+~lt

+(¢Pl + ~Pz+ ~P3+ ¢P4- I)X,_ - ( ~ + ~ + ~o4XX,_ i - X,_2) -(~3 + ~,Xx,_= - x,_,) - ~,(X,_ ~ X,_,) -

•4-¢t

(6)

In performing a hypothesis test to determine the need for differencing, we regress ( X , - X,_ j) on I, t (for Version 2 only), X,_l, ( X , _ , - X , _ 2 ) ,

Omega, Vol. 19, No. 5 (X,_2 - X,_3), and (X,_3 - X , _ , ) for t = 5, 6 . . . . . n. Testing of this unit root hypothesis is equivalent to testing that the coefficient of X,_ ~ in (5) or (6), ~p] + cp2 + (P3 + ¢P14 - 1, is zero. The large sample distribution of this estimator is shown in Fuller ([17], pp. 374). A t statistic can then be constructed to reject (or accept) the null hypothesis that oPt + ~P2+ ~P3+ c'P4- 1 = 0 .

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S Chen J Jarrett (March 1991) Department of Management Science University of Rhode Island Kingston, RI 02881.0802 USA