Alternative Seasonality Detectors Using SAS /ETS Procedures Joseph Earley, Loyola Marymount University, Los Angeles

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1 Alternative Seasonality Detectors Using SAS /ETS Procedures Joseph Earley, Loyola Marymount University, Los Angeles ABSTRACT Estimating seasonal indices is an important aspect of time series analysis. This paper presents an illustration of the role of regression, arima and spectral analysis as seasonality detectors within the PROC X procedure of the SAS System. The time series used is atmospheric CO concentrations (ppmv) derived in monthly samples collected at the Mauna Loa Observatory, Hawaii (Keeling 00), from January, through December, 00. Figure illustrates the time series. Figure S catterplot of CO vs date 0 CO date 00 0 INTRODUCTION Numerous methods have been developed to measure the presence of seasonality in time series data, such as decomposition, arima modeling, spectral analysis, and X methodology. This paper uses the X procedure to detect the seasonality pattern in monthly atmospheric CO concentrations (ppmv). The X procedure was originally developed by the U.S. Bureau of the Census starting in the 0's. The method has been widely used for the seasonal adjustment of both government and other data. The X methodology has been significantly enhanced by the collaborative efforts of both Statistics Canada and the Banco de Espana. In its current incarnation, the X methodology combines regression with arima error modeling, model diagnostics, and seasonal adjustment with postadjustment diagnostics. There is also an automatic modeling component based on the TRAMO method developed by the Banco de Espana. For the purpose of this paper, seasonality is the cyclical pattern in a time series that more or less repeats itself. The basic idea behind X decomposition is that the series components can be filtered from the original time series using a series of symmetrical moving average filters. The seasonal component of the series is treated as noise, which may be filtered from the series in order to get at the trend-cycle component. Using monthly data, a general outline of the X procedure would be( Jaditz,pp.- ):. Use a -month centered moving average as an estimate of the trend-cycle component.. Difference the original series and the centered moving average for an estimate of the sum of the seasonal and irregular components.

2 . Apply a -term moving average separately to each month in order to extract seasonal factors.. Further smooth the seasonal factors and subtract them to yield an estimate of the irregular components.. Further smooth the initial estimate of the trend and re-estimate the seasonals and irregulars.. Finally, the seasonal adjusted series is obtained by subtracting the final estimate of the seasonal from the raw series. The X procedure provides numerous parametric and non-parametric tests for identifiable seasonality. The stable seasonality test is a one-way analysis of variance using the seasons -months- as the factor. Large F values indicate that a significant amount of variation in the SI ratios is due to months - i.e. seasonality exists. Technically, the null hypothesis is that there is no effect due to months. The X procedure allows the user to automatically select the best arima model specification, based on a number of pre-selected models. PROC X selects the best model based on traditional goodness of fit statistics such as the AIC and the BIC. Following are the best five arima models selected according to minimum BIC. Best Five ARIMA Models Chosen by Automatic Modeling for variable CO Rank Estimated Model BIC ( 0,, )(, 0, 0) 0.0 ( 0,, )( 0, 0, ) 0. (,, 0)(, 0, 0) 0.0 ( 0,, )( 0, 0, 0) 0.00 ( 0,, )(, 0, ) 0. Final Automatic Model Selection Source of Model Estimated Model Automatic Model Choice ( 0,, )(, 0, 0) Figure illustrates the autocorrelation and partial autocorrelation functions. The spike at lag for each of these plots indicates the presence of monthly seasonality. Lag 0 0 AutoCorr Figure Lag 0 0 Partial The regarima component of PROC X has numerous applications. In addition to using the regression with arima errors model for series extension, regarima may be used for determining trading day effects, holiday effects, additive outlier detection, level shifts, as well as changes in these patterns. (Findley p.). Following are the parameter estimates and summary statistics from a regarima, followed by the selected arima model and F tests for seasonality and moving seasonality. Regression Model Parameter Estimates Standard Type Parameter Estimate Error t Value Pr> t Seasonal JAN

3 FEB <.000 MAR <.000 APR <.000 Seasonal MAY <.000 JUN <.000 JUL <.000 AUG <.000 SEP <.000 OCT <.000 NOV <.000 DEC <.000 Constant <.000 Chi-squared Tests for Groups of Regressors Regression Effect DF Chi-Square Pr > ChiSq Seasonal. <.000 ARIMA Model: (0 )( 0 0) Nonseasonal differences: Standard Parameter Lag Estimate Error t Value Pr > t Seasonal AR Nonseasonal MA <.000 Estimation Summary Number of Residuals Number of Parameters Estimated Variance Estimate.0E-0 Standard Error Estimate.E-0 Log likelihood -. AIC. AICC (F-corrected-AIC). Hannan Quinn.00 BIC. F-tests for seasonality Test for the presence of seasonality assuming stability. Sum of Mean Squares DF Square F-Value Between Months...00 ** Residual Total. ** Seasonality present at the 0. per cent level. Nonparametric Test for the Presence of Seasonality Assuming Stability Kruskal- Wallis Probability Statistic DF Level..00% Seasonality present at the one percent level. Moving Seasonality Test Sum of Mean Squares DF Square F-Value Between Years Error Moving seasonality present at the five percent level. COMBINED TEST FOR THE PRESENCE OF IDENTIFIABLE SEASONALITY IDENTIFIABLE SEASONALITY PRESENT The final seasonal factors -Average (Expressed as Percent) for the CO series were then derived as follows: JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

4 Test for the presence of residual seasonality. No evidence of residual seasonality in the entire series at the per cent level. F = 0.. No evidence of residual seasonality in the last years at the per cent level. F =. No evidence of residual seasonality in the last years at the per cent level. Table F of the PROC X procedure summarizes the various parametric and non-parametric tests for seasonality. Summary Measures Probability Statistic Level F-test for stable seasonality from Table B.:..0% F-test for stable seasonality from Table D.:.0.0% Kruskal-Wallis Chi Squared test for stable..0% seasonality from Table D.: F-test for moving seasonality from Table D.:. % Spectral analysis is a mathematical technique for estimating periodic components of time series. Following is a table from PROC X which illustrates the spectral analysis component of the methodology. Table G 0: 0*LOG(SPECTRUM) of the differenced, transformed original data spectrum estimated: JAN to DEC00. Type Approx. Spectrum Freq. Estimator Percent of Range None * None ** None **** None ******** None ********** None *********** None *********** None ************** None ******************* None **************************** Seasonal **************************************** None.0 -. ***************************** None.0 -. ********************** None.0 -. ******************* None ******************* None ******************* None.0 -. ****************** None ****************** None ******************** None.0 -. ************************* Seasonal ************************************ None.0 -. ********************** None.0 -. **************** None.0 -. ************ None ********* None.0 -. ******** None.0 -. ******* None ******** None.0 -. ********** None.0 -. ************** Seasonal ********************* None **************** None.0 -. ************ None.0-0. *********** None ************* None.0 -. ************** Trading.0-0. *********** None.0 -. ********* None.0 -. ********** None ***************

5 Seasonal ********************* None.0 -. *************** Trading.0 -. ************ None.0 -. ************* None.0 -. ****************** None *************** None.0-0. *********** None.0-0. *********** None.0 -. ************* None *************** Seasonal ************ None.0 -. ********** Trading.0 -. ********** None.0 -. ************ None ************** None.0 -. ************ None.0 -. ********** None ********* None.0 -. ********** None.0-0. *********** Seasonal *********** Finally, Figure illustrates the seasonal indices, detrended data by seasonal period, percent variation by seasonal period and residuals by seasonal period. Figure.0 Seasonal Indices Seasonal Analysis for CO M ultiplica tive M odel.0 Detrended Data, by Seasonal Period Percent Variation, by Seasonal Period Residuals, by Seasonal Period CONCLUSIONS PROC X is a powerful and easy to use statistical methodology. It is useful in determining whether or not seasonality exists in a time series as well as for estimating the seasonal factors. The researcher can, with a minimal amount of code, estimate and forecast values of a time series using the procedure. For the CO time series, PROC X using the power of the statistical methodologies of regression, arima and spectral analysis, confirms the existence of a -month sinusoidal seasonal pattern, a pattern frequently seen in climatological time series. COPYRIGHT INFORMATION SAS is a registered trademark of SAS Institute, Inc. in the USA and other countries. Indicates USA registration. Other brand or product names are registered trademarks or trademarks of their respective companies. REFERENCES Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C., and Chen,B.C.(), "New Capabilities and Methods of the X--ARIMA Seasonal Adjustment Program," Journal of Business and Economic Statistics,,- (with

6 Discussion). Gomez, V. and A. Maravall(a), "Program TRAMO and SEATS:Instructions for the User, Beta Version," Banco de Espana. Gomez, V. and A. Maravall(b), "Guide for Using the Programs TRAMO and SEATS, Beta Version," Banco de Espana. Gujarati, Damodar (00), Basic Econometrics, Fourth Edition, New York: McGraw-Hill Irwin, Inc. Jaditz, Ted, "Seasonality: economic data and model estimation", Monthly Labor Review, December, -. C.D. Keeling, T.P. Whorf and the Carbon Dioxide Research Group Scripps Institution of Oceanography (00). Ladiray, D. and Quenneville, B (00), "Seasonal Adjustment with the X- Method," New York: Springer-Verlag. Nelson, Charles R. (), Applied Time Series Analysis. San Francisco: Holden-Day, NC. SAS Institute Inc. ( and ), SAS/ETS Software: Applications Guides and, Version, First Edition, Cary, N.C.: SAS Institute Inc. SAS OnlineDoc TM : Version Chapter, The SPECTRA Procedure. SAS OnlineDoc TM : Version Chapter, The X Procedure. SAS OnlineDoc TM : Version Chapter, The X Procedure. CONTACT INFORMATION Joseph Earley Loyola Marymount University Los Angeles, California 00- Work Phone: 0-- Fax: jearley@lmu.edu