1. Time Series Analysis Introduction Averaging Trend Seasonality

2. Lecture Objectives You should be able to : 1.Discuss the advantages and limitations of time series forecasting. 2.Use averaging, trend, and seasonality models appropriately. 3.Interpret the Bias, MAD, MAPE and Standard Error to evaluate a forecast.

3. Basic Forecasting Process Look at the data (Graph) Forecast (choose one or more methods) Evaluate (examine errors)

4. Time Series Sales Data PeriodSales 160 267 350 458 562 660 755 862 971 1065 Consider the following sales data for 10 time periods (quarters) What is a good forecast for Sales for the next period?

5. Naive Forecast Naive PeriodSalesForecast 160N/A 26760 35067 45850 56258 66062 75560 86255 97162 106571 11 65 How good is this forecast?

6. Evaluating the Forecast XYNaive Abs PercentSquared PeriodSalesForecastError 160 267607710.45%49.0 35067-171734.00%289.0 458508813.79%64.0 56258446.45%16.0 66062-223.33%4.0 75560-559.09%25.0 862557711.29%49.0 971629912.68%81.0 106571-669.23%36.0 11 650.567.2212.26%68.1 BIASMADMAPEMSE Standard Error (Square Root of MSE) =8.3 Error = Bias = Avg (Errors) MAD = Avg (Abs Errors) MAPE = Avg (Percent Errors) MSE = Avg (Squared Errors)

7. Moving Averages Moving Avg. Abs. PercentSquared PeriodSalesForecastError 160N/A 267N/A 350N/A 45859.01.01.72%1.0 56258.33.7 5.91%13.4 66056.73.3 5.56%11.1 75560.0-5.05.09.09%25.0 86259.03.0 4.84%9.0 97159.012.0 16.90%144.0 106562.72.3 3.59%5.4 1166.02.624.336.80%29.86 BIASMADMAPEMSE Standard Error (Square Root of MSE) =5.5 How does this 3-period moving average forecast compare to the Naive forecast?

8. Simple Exponential Smoothing alpha=0.3 Exponential PercentSquared PeriodSalesSmoothingErrorAbs. ErrorError 1 60N/A 2 6760.07.0 10.45%49.0 3 5062.1-12.112.124.20%146.4 4 5858.5-0.50.50.81%0.2 5 6258.33.7 5.92%13.5 6 6059.40.6 0.95%0.3 7 5559.6-4.64.68.37%21.2 8 6258.23.8 6.10%14.3 9 7159.411.6 16.40%135.6 10 6562.82.2 3.31%4.6 63.5 1.295.118.50%42.79 BIASMADMAPEMSE Standard Error (Square Root of MSE) =6.5

9. Interpretation Bias – indicates the direction of the errors. On average, is the forecasting technique underestimating or overestimating? Bias can be corrected. MAD – The average magnitude of error. MAPE – The average percent error. Error as a percent of the actual values of y. MSE – Mean Squared Error. SE – Square root of MSE. This is the standard deviation of the error terms. Useful for constructing confidence intervals.

10. Questions 1. Can Bias be greater than MAD? 2. If we know the Bias, can we figure out the MAD value? 3. Will Bias is lower for one technique than another, will MAD also be lower? 4. Answer the above questions for MSE and MAD instead of Bias and MAD.

11. Data with a Trend PeriodSales 160 288 350 4111 5135 690 7150 8149 9200 10190

12. Fitting a Trendline

13. Regression Output SUMMARY OUTPUT Regression Statistics Multiple R0.901106096 R Square0.811992197 Adjusted R Square0.788491221 Standard Error23.60928811 Observations10 ANOVA dfSSMSFSignificance F Regression119258.9121 34.55140.0004 Residual84459.1879557.3985 Total923718.1 CoefficientsStandard Errort StatP-valueLower 95% Intercept38.266716.12822.37270.04511.0749 Period15.27882.59935.87800.00049.2848

14. Seasonality QuarterSales (Y) 1 25 228 335 450 539 644 755 870 952 1060 1177 12100 1385 14100 15111 16140 Is there a trend? Is there seasonality?

15. Deseasonalizing

16. Trend and Seasonally Adjusted Forecasts

17. Questions How many seasons can there be in data? How many seasonal cycles are needed to determine if seasonality exists? What does a seasonal index of 1.2 mean?