SHAZAM Simple Exponential Smoothing

Simple Exponential Smoothing

With the choice of a smoothing constant, a smoothed series can be generated by exponential smoothing. The smoothed value calculated for the final period can then be used as the forecast of future values.
Newbold [1995, pp. 711-712] discusses that some judgement is required in the selection of a smoothing constant. A recommendation is to try several different values and select a value that minimizes the sum of squared forecast errors.
The SHAZAM commands (filename: ESMOOTH.SHA) below use the data set on annual sales (in thousands of dollars) of Lydia E. Pinkham from 1931 to 1960. This data set was used in a previous example. A SHAZAM procedure (the set of commands between the PROC and PROCEND commands) uses the SMOOTH command to calculate a smoothed series by exponential smoothing. The smoothed series is saved with the EMAVE= option. Forecast errors are obtained and the sum of squared forecast errors is then calculated.

SAMPLE 1 30 * Read the sales data READ SALES / BYVAR 1806 1644 1814 1770 1518 1103 1266 1473 1423 1767 2161 2336 2602 2518 2637 2177 1920 1910 1984 1787 1689 1866 1896 1684 1633 1657 1569 1390 1387 1289 PROC EXPSMTH SAMPLE 1 30 SMOOTH SALES / WEIGHT=W EMAVE=S SAMPLE 2 30 * Generate 1-step ahead predictions GENR PREDICT=LAG(S) * Generate forecast errors GENR E=SALES-PREDICT * Calculate the sum of squared forecast errors STAT E / CP=SSE PRINT W SSE PROCEND SET NOOUTPUT NODOECHO * Try several different smoothing constants. GEN1 W=0.8 EXEC EXPSMTH GEN1 W=0.6 EXEC EXPSMTH GEN1 W=0.4 EXEC EXPSMTH GEN1 W=0.2 EXEC EXPSMTH STOP

The EXEC command executes the SHAZAM procedure commands for values of the smoothing constant, w, of 0.8, 0.6, 0.4 and 0.2.
The SHAZAM output can be viewed. The results are summarized in the table below. Note that the value in the column 1-w is the Newbold definition for the smoothing constant.

w 1-w Sum of Squared
Forecast Errors

0.8 0.2 1,405,769

0.6 0.4 1,761,367

0.4 0.6 2,421,086

0.2 0.8 3,570,107

The results show that the sum of squared forecast errors is smallest for w=0.8.

Comparison with ARIMA modelling

Mills [1990, pp. 154-5] discusses that simple exponential smoothing has an interpretation as the ARIMA(0,1,1) model:

(X_t - X_t-1) = e_t - e_t-1

where e_t is a random error.

Forecasts are calculated as:

X_N+h = X_N - a ê_N for h = 1, 2, 3, ...

where a is the estimate of and ê is the observed residual.

The SHAZAM commands (filename: EARIMA.SHA) for the ARIMA estimation and forecasting are given below.

SAMPLE 1 30 * Read the sales data READ SALES / BYVAR 1806 1644 1814 1770 1518 1103 1266 1473 1423 1767 2161 2336 2602 2518 2637 2177 1920 1910 1984 1787 1689 1866 1896 1684 1633 1657 1569 1390 1387 1289 * Estimate an ARIMA(0,1,1) model ARIMA SALES / NDIFF=1 NMA=1 NOCONSTANT COEF=ALPHA GEN1 S=SQRT($SIG2) * Forecasting ARIMA SALES / NDIFF=1 NMA=1 NOCONSTANT COEF=ALPHA SIGMA=S FBEG=30 FEND=34 STOP

The SHAZAM output can be viewed. The ARIMA estimation results show that the estimate of the smoothing parameter is -0.3. This estimate is not in the [0,1] range that describes the exponential smoothing method.

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SHAZAM output - Simple Exponential Smoothing

|_SAMPLE 1 30 |_* Read the sales data |_READ SALES / BYVAR 1 VARIABLES AND 30 OBSERVATIONS STARTING AT OBS 1 |_PROC EXPSMTH |_ SAMPLE 1 30 |_ SMOOTH SALES / WEIGHT=W EMAVE=S |_ SAMPLE 2 30 |_* Generate 1-step ahead predictions |_ GENR PREDICT=LAG(S) |_* Generate forecast errors |_ GENR E=SALES-PREDICT |_* Calculate the sum of squared forecast errors |_ STAT E / CP=SSE |_ PRINT W SSE |_PROCEND |_SET NOOUTPUT NODOECHO |_* Try several different smoothing constants. |_GEN1 W=0.8 |_EXEC EXPSMTH W 0.8000000 SSE 1405769. |_GEN1 W=0.6 |_EXEC EXPSMTH W 0.6000000 SSE 1761367. |_GEN1 W=0.4 |_EXEC EXPSMTH W 0.4000000 SSE 2421086. |_GEN1 W=0.2 |_EXEC EXPSMTH W 0.2000000 SSE 3570107. |_STOP

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SHAZAM output - ARIMA estimation and forecasting

|_SAMPLE 1 30 |_* Read the sales data |_READ SALES / BYVAR 1 VARIABLES AND 30 OBSERVATIONS STARTING AT OBS 1 |_* Estimate an ARIMA(0,1,1) model |_ARIMA SALES / NDIFF=1 NMA=1 NOCONSTANT COEF=ALPHA ARIMA MODEL NUMBER OF OBSERVATIONS = 30 ...NOTE..SAMPLE RANGE SET TO: 1, 30 DEGREE OF DIFFERENCING = 1 NUMBER OF MA PARAMETERS = 1 NO CONSTANT TERM IS IN THE MODEL ESTIMATION PROCEDURE STARTING VALUES OF PARAMETERS ARE: 0.50000 MEAN OF SERIES = -17.83 VARIANCE OF SERIES = 0.4332E+05 STANDARD DEVIATION OF SERIES = 208.1 INITIAL SUM OF SQUARES = 2036201.0 ITERATION STOPS - RELATIVE CHANGE IN SUM OF SQUARESLESS THAN 0.1E-05 NET NUMBER OF OBS IS 29 DIFFERENCING: 1 CONSECUTIVE, 0 SEASONAL WITH SPAN 0 CONVERGENCE AFTER 14 ITERATIONS INITIAL SUM OF SQS= 2036201.0 FINAL SUM OF SQS= 1128563.6 R-SQUARE = 0.0697 R-SQUARE ADJUSTED = 0.0364 VARIANCE OF THE ESTIMATE-SIGMA**2 = 40157. STANDARD ERROR OF THE ESTIMATE-SIGMA = 200.39 AKAIKE INFORMATION CRITERIA -AIC(K) = 10.670 SCHWARZ CRITERIA- SC(K) = 10.717 PARAMETER ESTIMATES STD ERROR T-STAT MA( 1) -0.30132 0.1760 -1.712 RESIDUALS LAGS AUTOCORRELATIONS STD ERR 1 -12 -.02 -.01 0.05 0.15 -.25 -.37 0.01 0.06 -.25 0.12 0.12 0.19 0.19 13 -24 -.29 0.10 0.07 -.05 -.21 0.04 0.07 -.10 -.04 0.08 -.01 0.01 0.24 25 -28 -.01 0.04 -.03 0.01 0.27 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 2 0.01 1 .910 15 20.41 14 .118 3 0.10 2 .953 16 20.60 15 .150 4 0.96 3 .812 17 23.90 16 .092 5 3.38 4 .496 18 24.01 17 .119 6 8.74 5 .120 19 24.48 18 .140 7 8.75 6 .188 20 25.46 19 .146 8 8.90 7 .260 21 25.66 20 .177 9 11.64 8 .168 22 26.52 21 .187 10 12.33 9 .195 23 26.52 22 .230 11 13.09 10 .218 24 26.54 23 .276 12 14.92 11 .186 25 26.55 24 .326 13 19.49 12 .077 26 26.92 25 .360 14 20.08 13 .093 27 27.23 26 .398 28 27.34 27 .446 CROSS-CORRELATIONS BETWEEN RESIDUALS AND (DIFFERENCED) SERIES CROSS-CORRELATION AT ZERO LAG = 0.94 LAGS CROSS CORRELATIONS Y(T),E(T-K) 1-12 0.27 -.01 0.04 0.16 -.20 -.42 -.10 0.06 -.21 0.04 0.15 0.21 13-24 -.22 0.01 0.08 -.03 -.22 -.02 0.08 -.07 -.06 0.06 0.02 0.01 25-28 -.01 0.03 -.02 0.01 LEADS CROSS CORRELATIONS Y(T),E(T+K) 1-12 -.02 0.00 0.09 0.08 -.35 -.35 0.03 -.02 -.20 0.15 0.17 0.10 13-24 -.24 0.12 0.05 -.11 -.19 0.06 0.04 -.11 -.02 0.08 0.00 0.01 25-28 0.01 0.03 -.02 0.01 ANALYSIS OF RESIDUALS VALUES RANGE FROM -510.0205 TO 370.5252 SAMPLE MOMENTS OF RESIDUALS (USING THE DIVISOR 29) : MEAN = -14.06923 VARIANCE = 38574.70 SKEWNESS = -0.1833068 KURTOSIS = 2.893125 STUDENTIZED RANGE = 4.483329 |_GEN1 S=SQRT($SIG2) ..NOTE..CURRENT VALUE OF $SIG2= 40157. |_* Forecasting |_ARIMA SALES / NDIFF=1 NMA=1 NOCONSTANT COEF=ALPHA SIGMA=S FBEG=30 FEND=34 ARIMA MODEL NUMBER OF OBSERVATIONS = 30 ...NOTE..SAMPLE RANGE SET TO: 1, 30 DEGREE OF DIFFERENCING = 1 NUMBER OF MA PARAMETERS = 1 NO CONSTANT TERM IS IN THE MODEL ARIMA FORECAST PARAMETER VALUES ARE: MA( 1)= -0.30132 FROM ORIGIN DATE 30, FORECASTS ARE CALCULATED UP TO 4 STEPS AHEAD FUTURE DATE LOWER FORECAST UPPER ACTUAL ERROR 31 862.852 1255.62 1648.39 32 611.019 1255.62 1900.23 33 432.969 1255.62 2078.28 34 287.117 1255.62 2224.13 STEPS AHEAD STD ERROR PSI WT 1 200.4 1.0000 2 328.9 1.3013 3 419.7 1.3013 4 494.1 1.3013 VARIANCE OF ONE-STEP-AHEAD ERRORS-SIGMA**2 = 0.4016E+05 STD.DEV. OF ONE-STEP-AHEAD ERRORS-SIGMA = 200.4 |_STOP

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