Chapter 16: Time-Series Analysis

1. Chapter 16: Time-Series Analysis Sections 1-4,7

2. 16.1: The Importance of Business Forecasting • Time-Series Data: data obtained at regular periods of time. • Very often, we are trying to predict the future. The procedure is called “forecasting.” • The managerial topic is “strategic planning.”

3. Types of Forecasting • Qualitative: very subjective and judgment-oriented. Usually a panel of experts is polled and their opinions are _______. One process is called the “Delphi Method.” • Quantitative: uses historical data and mathematical techniques. • Time-series: base the future values of a variable entirely on past and present values of the variable. Us. • Causal: include other related variables in the model in addition to past values of the predicted variable.

4. 16.2: Component Factors of the Classical Multiplicative Time-Series Model • Assume that whatever is driving the data (some set of variables) will continue to behave “as usual.” • Nobody said anything about identifying the set of variables. • Figure out the past behavior and use it to predict future behavior.

5. Model and Components • The “most basic” model is the classical multiplicative model. • Yi = Ti * Ci * Si * Ii (Formula 16.2) • Yi is the dependent variable • Ti is the trend component • Ci is the cyclical component • Si is the seasonal component • Ii is the irregular component • Sometimes the subscript “i” is shown as a subscript “t”

6. Components • Table 16.1 (please learn) • Trend—long term behavior e.g. several years • Seasonal—regular behavior within a 12 month period • Cyclical—up and down behavior that repeats; intensity might not be constant • Irregular—similar to OLS residual; what’s left over after removing TSC.

7. What does the model mean? • The value of Y at any time is the product of Trend, Cyclical, Seasonal, and Irregular components at that time. • Annual data does not have Si. Quarterly or Monthly data does have Si.

8. 16.3: Smoothing the Annual Time Series • Remember: annual data has NO Seasonal component. • Plot the data. If there is no apparent trend component, then “Smoothing” is a good approach. • Example is Figure 16.2. There is no apparent trend. • The two techniques of interest are: Moving Averages and Exponential Smoothing.

9. Moving Averages • There are several ways to do this. We’ll use the text rules: • Select an odd number of observations to average. Call this odd number “L.” • Example: L = 3. • The first MA(3) = (Y1+Y2+Y3)/3 • The second MA(3) = (Y2+Y3+Y4)/3 • Etc. • Plot the value of the MA against the date, or period, of the middle value in the average.

10. More on Moving Averages • The first (L-1)/2 and the last (L-1)/2 observations will not have a smoothed value to plot against. • L should not be too large. What does your text recommend for maximum L? • Greater L means more smooth. • Moving Averages cannot be used to forecast.

11. Exponential Smoothing • ES can be used to forecast 1 period into the future. • All of the previously occurring data points are used to obtain each smoothed data point. • Newer observations are given more “weight.” • Formula 16.3 and 16.4.

12. 16.4: Least-Squares Trend Fitting and Forecasting • Y = T*C*S*I • If the data set shows no trend, try smoothing the data. • If the data set shows a trend, try fitting a trend model: • Least Squares (x = time, or some coded value) • Other, eg. Double Exponential

13. Least-Squares: Linear Trend • Typically code the X values as “0” for the first observation, “1” for the second, etc. • Linear—use everything you know about simple linear regression; there’s a nice interpretation on page 670. • Check the r2and p-values.

14. Quadratic Trend Model • Look at the scatter plot of the data. • Quadratic or 2nd degree polynomial—the model appears in Equation 16.6. • Check r2 and p-values of “F test.” • Interpretations are more difficult with this model.

15. Exponential Trend Model • “when a series increases at a rate such that the percentage difference from value to value is constant.” • The models are given on page 672. • Check r2 and p-values of “F test.”

16. Comparing Trend Models • pp 674-676 We will omit this material.

17. 16-7: Choosing an Appropriate Forecasting Model • Consider the linear trend, quadratic trend, and exponential trend models. • Plot the data and trend lines. • Exhibit 16.3: • Residual analysis. • SSE • MAD • Parsimony

18. Residual Analysis • Do First! • Residuals vs Fitted Values • Residuals vs Independent Variable (Time) • Figure 16.27. • If the trend model in question does a good job, the residuals represent “I” or the Irregular component from the multiplicative model.

19. SSE or Squared Differences • For comparing models that “pass” residual analysis. • Sum up the differences between actual and fitted y values. • Susceptible to influence by outliers. • Smaller SSE is better.

20. Mean Absolute Deviations • Not as susceptible to outliers as SSE. • Calculate absolute differences between actual and fitted y values. Sum them up and divide by the number of data points. • Smaller MAD is better.

21. Parsimony • Given two models that satisfy residual analysis and have comparable SSE and MAD scores, simpler is better. • KISS.

22. Text Example • Figure 16.28. • Small data set. • The authors like panels ________.

23. 16-8: Time Series Forecasting of Monthly or Quarterly Data • Consider Figure 16.24. • It shows typical data that requires a Seasonal component in the multiplicative model (Y=TCSI). • The data is quarterly. Thus the coded date is expressed in number of quarters. • Use dummy variables to tell the equation which quarter.

24. Equation 16.19 • What does it mean? • How do you use it? • This type of model captures both Trend and Seasonal. • How do you decide which model is best?