As explained in the introduction to this chapter, we will confine our discus- sion of statistical forecasting methods to time-series models. A time-series is simply a time-ordered sequence of observations on a variable. In general,
Now try Technical Problems 4–5.
a time-series model uses only the time-series history of the variable of inter- est to predict future values. Time-series models describe the process by which these historical data were generated. Thus to forecast using time-series analy- sis, it is necessary to specify a mathematical model that represents the gener- ating process. We will first discuss a general forecasting model and then give examples of price and sales forecasts.
Linear Trend Forecasting
A linear trend is the simplest time-series forecasting method. Using this type of model, one could posit that sales or price increase or decrease linearly over time.
For example, a firm’s sales for the period 2007–2016 are shown by the 10 data points in Figure 7.1. The straight line that best fits the data scatter, calculated us- ing simple regression analysis, is illustrated by the solid line in the figure. The fitted line indicates a positive trend in sales. Assuming that sales in the future will continue to follow the same trend, sales in any future period can be forecast by extending this line and picking the forecast values from this extrapolated dashed line for the desired future period. We have illustrated sales forecasts for 2017 and 2022 ( ˆ Q 2017 and ˆ Q 2022) in Figure 7.1.
Summarizing this procedure, we assumed a linear relation between sales and time:
ˆ Q t 5 a 1 bt
Using the 10 observations for 2007–2016, we regressed time (t 5 2007, 2008, . . . , 2016), the independent variable expressed in years, on sales, the dependent variable expressed in dollars, to obtain the estimated trend line:
ˆ Q t 5 a 1 ˆ ˆb t
This line best fits the historical data. It is important to test whether there is a statistically significant positive or negative trend in sales. As shown in Chapter 4,
time-series model A statistical model that shows how a time- ordered sequence of observations on a variable is generated.
Sales
2022
Qˆ2017
2007
Qˆ Q
t Estimated trend line
2008 2009 2010 Time2011 2012 2013 2014 2015 2016 2017 2022
F I G U R E 7.1 A Linear Trend Forecast
it is easy to determine whether ˆb is significantly different from zero either by using a t-test for statistical significance or by examining the p-value for b . If ˆ b ˆ is positive and statistically significant, sales are trending upward over time. If ˆb is negative and statistically significant, sales are trending downward over time. However, if b ˆ is not statistically significant, one would assume that b 5 0, and sales are constant over time. That is, there is no relation between sales and time, and any variation in sales is due to random fluctuations.
If the estimation indicates a statistically significant trend, you can then use the estimated trend line to obtain forecasts of future sales. For example, if a manager wanted a forecast for sales in 2017, the manager would simply insert 2017 into the estimated trend line:
ˆ Q 2017 5 a ˆ 1 b ˆ 3 (2017) A Sales Forecast for Terminator Pest Control
In January 2016, Arnold Schwartz started Terminator Pest Control, a small pest- control company in Atlanta. Terminator Pest Control serves mainly residential customers citywide. At the end of March 2017, after 15 months of operation, Arnold decides to apply for a business loan from his bank to buy another pest- control truck. The bank is somewhat reluctant to make the loan, citing concern that sales at Terminator Pest Control did not grow significantly over its first 15 months of business. In addition, the bank asks Arnold to provide a forecast of sales for the next three months (April, May, and June).
Arnold decides to do the forecast himself using a time-series model based on past sales figures. He collects the data on sales for the last 15 months—sales are measured as the number of homes serviced during a given month. Because data are collected monthly, Arnold creates a continuous time variable by numbering the months consecutively as January 2016 5 1, February 2016 5 2, and so on. The data for Terminator and a scatter diagram are shown in Figure 7.2.
Arnold estimates the linear trend model Qt 5 a 1 bt and gets the following printout from the computer:
DEPENDENT VARIABLE: Q R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 15 0.9231 156.11 0.0001
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 46.57 3.29 14.13 0.0001 T 4.53 0.36 12.49 0.0001
The t-ratio for the time variable, 12.49, exceeds the critical t of 3.012 for 13 de- grees of freedom (5 15 2 2) at the 1 percent level of significance. The exact level of significance for the estimate 4.53 is less than 0.0001, as indicated by the p-value.
Thus the sales figures for Terminator suggest a statistically significant upward trend in sales. The sales forecasts for April, May, and June of 2017 are
April 2017: ˆ Q 16 5 46.57 1 (4.53 3 16) 5 119 May 2017: ˆ Q 17 5 46.57 1 (4.53 3 17) 5 123.6 June 2017: ˆ Q 18 5 46.57 1 (4.53 3 18) 5 128.1
The bank decided to make the loan to Terminator Pest Control in light of the sta- tistically significant upward trend in sales and the forecast of higher sales in the three upcoming months.
A Price Forecast for Georgia Lumber Products
Suppose you work for Georgia Lumber Products, a large lumber producer in south Georgia, and your manager wants you to forecast the price of lumber for the next two quarters. Information about the price of a ton of lumber is readily available. Using eight quarterly observations on lumber prices since 2014(III), you estimate a linear trend line for lumber prices through the 2016(II) time period.
Now try Technical Problem 6.
Qt Trend line: Qˆt = 46.57 + 4.53t
Qˆ18 = 46.57 + 4.53(18) = 128.1 Qˆ17 = 46.57 + 4.53(17) = 123.6 Qˆ16 = 46.57 + 4.53(16) = 119
Sales (homes serviced/month)
160 Qt
140 120 100 80 60 40
t
t 12 34 56 78 109 1112 1314 15
4656 7267 7766 6979 8891 10494 100113
3 5 12 16 120
2 4 6 7 8 9 1011 131415 1718
1
Month January 2016 February 2016 March 2016 April 2016 May 2016 June 2016 July 2016 August 2016 September 2016 October 2016 November 2016 December 2016 January 2017 February 2017 March 2017 20
0 F I G U R E 7. 2
Forecasting Sales for Terminator Pest Control
Your computer output for the linear time trend model on lumber price looks like the following printout:
DEPENDENT VARIABLE: P R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 8 0.7673 19.79 0.0043
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE INTERCEPT 2066.0 794.62 2.60 0.0407 T 25.00 5.62 4.45 0.0043
Both parameter estimates a and ˆ b ˆ are significant at the 5 percent significance level because both t-ratios exceed 2.447, the critical t for the 5 percent significance level. (Notice also that both p-values are less than 0.05.) Thus the real (inflation- adjusted) price for a ton of lumber exhibited a statistically significant trend upward since the third quarter of 2014. Lumber prices have risen, on average,
$25 per ton each quarter over the range of this sample period (2014 III through 2016 II).
To forecast the price of lumber for the next two quarters, you make the follow- ing computations:
Pˆ 2016 (III) 5 2066 1 (25 3 9) 5 $2,291 per ton Pˆ 2016 (IV) 5 2066 1 (25 3 10) 5 $2,316 per ton
As you can see by the last two hypothetical examples, the linear trend method of forecasting is a simple procedure for generating forecasts for either sales or price. Indeed, this method can be applied to forecast any economic variable for which a time-series of observations is available.