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

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.”

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.

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.

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”

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.

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.

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.

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.

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.

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.

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

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.

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.

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.”

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

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

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.

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.

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.

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

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

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.

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?