1. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-1 l Chapter 14 l Time Series: Understanding Changes over Time

2. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-2 Time Series Analysis  Goals Understand the past Forecast the future  Different from Cross-Sectional Data Time-series data are not independent of each other Not a random sample Does not satisfy the random-sample assumption for confidence intervals (in Chapter 9) or hypothesis testing (in Chapter 10) New methods are needed to take account of the interdependence

3. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-3 Cross-Sectional and Time-Series  Cross-Sectional Data Expect next observation to be about S away from  Time-Series Data Next will probably not be about S away from (not a random sample) S S S S

4. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-4 Forecasting  Use a model A system of equations that can produce data that “look like” your time series data Estimate the model Your forecast will be the expected (mean) value of the future behavior of the model The forecast limits are the confidence limits for your forecast (if your model can produce them) Computed from the appropriate standard error If model is correct, the future observation has a 95% chance of being within the forecast limits

5. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-5 Trend-Seasonal and Box-Jenkins  Trend-Seasonal Analysis Direct and intuitive, with four components: (1) Long-term Trend, (2) repeating Seasonal, (3) medium-term wandering Cyclic, and (4) random Irregular Forecast comes from extending the Trend and Seasonal  Box-Jenkins ARIMA Process Flexible, but complex, probability models for how current value of the series depends upon Past values, past randomness, and new randomness A better way to describe the Cyclic component Forecast is Expectation of random future behavior, given past data

6. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-6 Example: Computer Revenues  Steady growth not perfectly smooth Nonlinear (curved) Suggests constant growth rate  Logarithm of revenues Log plot looks linear if constant growth rate Can use regression to model relationship Points appear randomly distributed about the line, so serial correlation is not a problem Fig 14.1.4, 6

7. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-7 Example: Retail Sales  U.S. Retail Sales (Monthly) Growth Repeating seasonal variation High in December Low in January, February  Seasonally-Adjusted Sales Growth Seasonal pattern removed Shows how sales went up (or down) relative to what you expect for time of year Fig 14.1.7, 8

8. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-8 Example: Interest Rates  U.S. Treasury Bills, Yearly Generally rising Substantial variation Cyclic pattern Rising and falling Increasing magnitude Not perfectly repeating Not expected to continue rising indefinitely! Fig 14.1.9

9. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-9 Trend-Seasonal Analysis  Decompose a Time Series into Four Components Data = Trend  Seasonal  Cyclic  Irregular Trend Long-term behavior (often straight line or exponential growth) Seasonal Repeating effects of time-of-year Cyclic Gradual ups and downs, not repeating each year, not purely random Irregular Short-term, random, nonsystematic noise

10. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-10 Ratio-to-Moving-Average Method  Moving Average Represents Trend and Cyclic Eliminates Seasonal and Irregular by averaging a year  Divide Data by Moving Average Represents Seasonal and Irregular Group by season, then average, to obtain Seasonal  Seasonal Adjustment: Divide Data by Seasonal  Regress Seasonally-Adjusted Series vs. Time Represents Trend  Forecast by Seasonalizing the Trend Multiply (future predicted Trend) by (Seasonal index)

11. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-11 Example: Ford Motor Company  Time-series Plot Quarterly data with strong Seasonal pattern Revenues typically highest in second quarter Does not repeat perfectly (due to Cyclic and Irregular) Fig 14.2.1

12. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-12 Example: Moving Average  Averages one year of data 2 quarters before to 2 quarters after each data value Smooths the data, eliminating Seasonal and Irregular Shows you Trend and Cyclic Original data Moving average Fig 14.2.4

13. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-13 Example: Seasonal Index Fig 14.2.6  Average “Ratio-to-Moving-Average” by Quarter Seasonal index for each quarter, repeating each year Shows how much larger (or smaller) this quarter is compared to a typical period throughout the year

14. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-14 Example: Seasonal Adjustment  Divide Data by Seasonal Index To get Seasonally Adjusted Value Eliminates the expected seasonal component Shows changes that are not due to expected seasonal effects Original data Seasonally adjusted series Fig 14.2.7

15. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-15 Example: Trend Line  Regress Seasonally-Adjusted Data vs. time The resulting line can be extended into the future This gives a Seasonally-Adjusted Forecast Fig 14.2.8 Seasonally adjusted forecast Seasonally adjusted series Trend line

16. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-16 Example: Forecast  Seasonalize the Trend Multiply Trend by Seasonal Index Can be extended into the future Use future predicted Trend with quarterly Seasonal index Fig 14.2.9 Original data Seasonalized trend Forecast

17. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-17 Box-Jenkins ARIMA Processes  A Collection of Linear Statistical Models Can describe many different kinds of time-series Including medium-term “cyclic” behavior Compared to trend-seasonal analysis, Box-Jenkins Has a more solid statistical foundation Is more flexible Is somewhat less intuitive Outline of the steps involved Choose a type of model and estimate it using your data Forecast using average future random behavior of this model Find standard error (variability in this future behavior) Find forecast limits, to include 95% of future behavior

18. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-18 Random Noise Process  A Random Sample, with No Memory Data = (Mean value) + (Random Noise) Y t =  +  t The long-term mean of Y is  Mean

19. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-19 Autoregressive (AR) Process  Remembers the Past, Adds Random Noise Data =  +  (Previous value) + (Random Noise) Y t =  +  Y t–1 +  t The long-term mean value of Y is  –  Mean

20. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-20 Moving-Average (MA) Process  Remembers Previous Noise, Adds New Noise Data =  + (Random Noise) –  (Previous Noise) Y t =  +  t –  t–1 The long-term mean value of Y is  Mean

21. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-21 ARMA Process  Autoregressive Moving Average Process Remembers the Past, Previous Noise, Adds New Noise Data =  +  (Previous value) + (Noise) –  (Previous Noise) Y t =  +  Y t–1 +  t –  t–1 The long-term mean value of Y is  –  Mean

22. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-22 Example: Unemployment  Estimated ARMA Process for this Time Series Y t =  +  Y t–1 +  t +  t–1 where random noise has standard deviation 0.907

23. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-23 Example (continued)  Random Simulations from Estimated Process Look similar to actual unemployment rate history Because of estimation using actual data Looking at “what might have happened instead”

24. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-24 Example (continued)  Forecast and 95% Forecast Limits (10 years ahead) Using the average of random future possibilities And their lower and upper 95% limits Forecast

25. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-25 Example (continued)  Three Simulations of the Future With forecast and 95% Forecast Limits To see how forecast represents future possibilities

26. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-26 Pure Integrated (I) Process  A Random Walk from the Previous Value Data =  + (Previous value) + (Random Noise) Y t =  + Y t–1 +  t Over time, Y is not expected to stay close to any long-term mean value

27. Irwin/McGraw-Hill © Andrew F. Siegel, 1997 and 2000 14-27 ARIMA Process  Autoregressive Integrated Moving Average  Remembers its Changes The differences, Y t – Y t–1, follow an ARMA process Over time, Y is not expected to stay close to any long-term mean value