Chapter 8 Appendix: Forecasting
Think of what you could accomplish if you could see the future. Of course, no one can truly predict future events. However, in many situations a forecast can provide a good idea of the possibilities. Of course, forecasting patterns for the near future is much easier than predicting what might happen several years out.
Forecasting is used in many areas of business. Marketing forecasts future sales, the effect of various sales strategies, and changes in buyer preferences. Finance forecasts future cash flows, interest rate changes, and market conditions. The HRM department builds forecasts of various job markets, the amount of absenteeism, and labor turnover. Strategic managers forecast technological changes, actions by rivals, and various market conditions. Sometimes these forecasts are built on intuition and rules of thumb. But, it is better to use statistical techniques whenever possible.
The science of forecasting is dominated by two major approaches: time-series forecasting that identifies trends over time, and structural modeling that identifies relationships among the underlying variables. Many forecasts require the use of both techniques.
As shown in Figure 8.1A, by focusing on the underlying model, structural modeling seeks to identify the cause of changes. For example, if consumer income increases, the demand for our product shifts out, which results in more sales. If we know the shapes of the supply and demand curves, it is straightforward to predict how sales will increase.
Figure 8.2A shows a time-series-approach to the same issue of sales forecasting. In some ways it is simpler. We know nothing about the underlying model and have simply collected sales data for the past few months. The data is plotted over time. By fitting a trend line to the data, it is clear that sales are increasing. Assuming that this trend continues, it is easy to forecast sales for the next period.
It is often tempting to say that structural models are “better” than time series forecasts. After all, the model provides an understanding of the causal relationships. So we know that the increase in sales is occurring because of increasing consumer income. While this knowledge is valuable, the structural model does not tell us how fast income will increase in the next period. Hence, we end up using time-series techniques to estimate the trends in consumer income. Consequently, we need both techniques. As much as possible, find a structural model to explain the problem. Then use time-series methods to estimate the underlying trends. Plug these forecasts into the structural model to determine the outcome of the desired variables.
Structural Modeling
Modeling an underlying structure provides the most information and knowledge about a problem. Consider a simple physics problem: If you throw a ball at a certain angle, with a certain force, how far will it travel? You could try several experiments, timing each event and measuring the outcome. You could then use this data to make a forecast of future attempts. However, if you know the underlying model of gravity (e.g., Newton’s equations), then it is easy to determine the outcome of any attempt.
Many economic models have been developed to determine relationships that apply to business decisions. For instance, cost models are used to determine supply relationships, and consumer preferences generate demand curves. Demand for a product can be expressed as a function of several variables: price, income, and prices of related products. These relationships can be estimated with common statistical techniques (e.g., multiple regression).
Figure 8.3A presents the basic steps in using a structural model to forecast sales demand. First you need a model—in this case, a basic economic model. Then you need to collect data for each of the variables in the model. It is best if the underlying variables change over time, and you will need observations from several points in time. It is best to have at least 40 observations, but you get better results with more data. Next you use regression analysis to estimate the values of the model parameters. Finally, you plug in estimates of the independent variables to obtain a forecast of the future sales.
Time-Series Forecasts
When you do not have a structural model, or when you need to forecast the value of an underlying variable, you can use time-series techniques to examine how variables change over time. The basic process is to collect data over time, identify any patterns that exist, and then extrapolate this pattern for the future. The approach assumes that the underlying pattern will remain the same. For example, if income has been gradually increasing over time, it assumes that this increase will continue.
Figure 8.4A shows a time series consists of a number of observations made over a period of time. Four types of patterns often arise in time-series data: (1) trends, (2) cycles, (3) seasonal variations, and (4) random changes. A trend is a gradual increase or decrease over time. A cycle consists of up and down movements relative to the trend. Seasonal variations arise in many disciplines. For instance, agricultural production increases in the summer and fall seasons, and many industries experience an increase in sales in November and December due to holiday sales. Random components are variations that we cannot explain through other means. In some cases, the random component dominates the others, and forecasting is virtually impossible. For example, many people believe this situation exists for stock market prices.
Exponential Smoothing
Random variations make it difficult to see the underlying trend, seasonal, and cyclical components. One solution is to remove these variations with exponential smoothing. Exponential smoothing computes a new data point based on the previously computed value and the newly observed data value. The weight given to each component is called the smoothing factor. The higher the smoothing factor, the more weight that is given to the new data point. Typical values range from 0.2 to 0.30, although it is possible to use factors up to 1.0 (which would consider only the new value and ignore the old ones). Lower values (down to 0.01) put more weight on previous computations and result in a smoother estimate.
Current spreadsheets (e.g., Microsoft Excel) have procedures that will quickly estimate the moving average for a range of data. You simply mark the range of data, highlight the output range, and supply the smoothing factor. As shown in Figure 8.5A, it is then easy to graph the original and the smoothed data.
How do you choose the smoothing factor? The best method is to apply several smoothing factors (start with 0.10, 0.20, and 0.30), and then examine the accuracy of the result. The accuracy is typically measured as the sum-of-squared errors. For each smoothed column of data, compute (actual – smoothed)*(actual – smoothed) to get the squared-error on each row. Add these values to get the total. Now compare these totals for each of the smoothing factors. The smoothing factor with the smallest error is the one to use.
In practice, most data will have a trend component. In these cases, you need to use double exponential smoothing. Perform the first smoothing as usual and find the best smoothing factor. Then perform a second smoothing on the new (smoothed) column of data using the same smoothing factor. Figure 8.6A shows that the result follows the basic trend line but still incorporates the cyclical variations in the data.
The smoothed data can be used to forecast the dependent variable for future periods. It is wise to stick with forecasts only one or two periods ahead. Longer range forecasts are less likely to be accurate. The basic formula is given in Figure 8.7A. You need the smoothing factor (a) and the number of time periods ahead to forecast (t). Then you take the smoothed values at the last data point (single and double) and plug them into the formula. The result is a forecast that incorporates a linear trend along with the basic cyclical variations in the data.
If you want a forecast that utilizes only the trend (and none of the cyclical variations), you can use a simple regression technique. Use standard regression techniques with the observed data as the dependent (Y) variable and time (t) as the independent variable. Then use the computed parameters to estimate the predicted value at any future point in time. Again remember that linear trends may not continue for extended time periods, so keep your predictions down to a few periods. Figure 8.8A illustrates the process, using Excel to obtain the regression coefficients. The result is the prediction along the trend line, which ignores cyclical variations.
Once you compute the trend factor, you can subtract it from the original series to identify the cyclical, seasonal, and random components. Use the spreadsheet to plug in values for each time period and compute the trend line. Then subtract these values from the original series. If you plot the new series, you will see the data without the trend. It should be easier to see seasonal and cyclical patterns on this new chart.
Seasonal and Cyclical Components
Two powerful methods exist to decompose time series data into its trend, seasonal, and cyclical components. They are Box-Jenkins and Fourier analysis. You can purchase tools that will perform the complex calculations for these methods. Unfortunately, it would take many pages to describe either one of these techniques, so they are beyond the scope of this appendix. Just remember that it is possible to perform much more detailed analyses (and forecasts) of time series data. If you need them, hire an expert, or take a separate class in time series forecasting.
Exercises
1. Obtain a three-year set of monthly data from the Bureau of Labor Statistics Web site (http://stats.bls.gov) that is not seasonally adjusted (e.g., Producer Price Index). Transfer the data to a spreadsheet. Plot the data and include a trend line.
Sales data for the remaining exercises
Jan / Feb / Mar / Apr / May / Jun / Jul / Aug / Sep / Oct / Nov / Dec1998 / 414 / 382 / 396 / 530 / 551 / 396 / 365 / 415 / 424 / 485 / 684 / 802
1999 / 457 / 432 / 465 / 598 / 632 / 424 / 392 / 476 / 489 / 555 / 768 / 883
2000 / 505 / 477 / 534 / 636 / 696 / 466 / 442 / 506 / 531 / 610 / 825 / 973
2. Plot the sales data from the table. Draw one graph with a trend line and a second chart with three-period exponential smoothing.
3. Using the regression functions in the spreadsheet, estimate the trend line and produce a forecast for four periods ahead.
4. Use double exponential smoothing (damping of 0.3) and plot the new data.
5. Use the formula in Figure 8.7A to forecast sales for the next four periods using double exponential smoothing.