1. Chapter 15 Forecasting Copyright © 2011 Pearson Addison-Wesley. All rights reserved. Slides by Niels-Hugo Blunch Washington and Lee University

2. 15-1 © 2011 Pearson Addison-Wesley. All rights reserved. What Is Forecasting? In general, forecasting is the act of predicting the future In econometrics, forecasting is the estimation of the expected value of a dependent variable for observations that are not part of the same data set In most forecasts, the values being predicted are for time periods in the future, but cross-sectional predictions of values for countries or people not in the sample are also common To simplify terminology, the words prediction and forecast will be used interchangeably in this chapter –Some authors limit the use of the word forecast to out-of-sample prediction for a time series

3. 15-2 © 2011 Pearson Addison-Wesley. All rights reserved. What Is Forecasting? (cont.) Econometric forecasting generally uses a single linear equation to predict or forecast Our use of such an equation to make a forecast can be summarized into two steps: 1.Specify and estimate an equation that has as its dependent variable the item that we wish to forecast: (15.2)

4. 15-3 © 2011 Pearson Addison-Wesley. All rights reserved. What Is Forecasting? (cont.) 2.Obtain values for each of the independent variables for the observations for which we want a forecast and substitute them into our forecasting equation: (15.3) Figure 15.1 illustrates two examples

5. 15-4 © 2011 Pearson Addison-Wesley. All rights reserved. Figure 15.1a Forecasting Examples

6. 15-5 © 2011 Pearson Addison-Wesley. All rights reserved. Figure 15.1b Forecasting Examples

7. 15-6 © 2011 Pearson Addison-Wesley. All rights reserved. More Complex Forecasting Problems The forecasts generated in the previous section are quite simple, however, and most actual forecasting involves one or more additional questions—for example: 1. Unknown Xs: It is unrealistic to expect to know the values for the independent variables outside the sample What happens when we don’t know the values of the independent variables for the forecast period? 2. Serial Correlation: If there is serial correlation involved, the forecasting equation may be estimated with GLS How should predictions be adjusted when forecasting equations are estimated with GLS?

8. 15-7 © 2011 Pearson Addison-Wesley. All rights reserved. More Complex Forecasting Problems (cont.) 3. Confidence Intervals: All the previous forecasts were single values, but such single values are almost never exactly right, so maybe it would be more helpful if we forecasted a confidence interval instead How can we develop these confidence intervals? 4. Simultaneous Equations Models: As we saw in Chapter 14, many economic and business equations are part of simultaneous models How can we use an independent variable to forecast a dependent variable when we know that a change in value of the dependent variable will change, in turn, the value of the independent variable that we used to make the forecast?

9. 15-8 © 2011 Pearson Addison-Wesley. All rights reserved. Conditional Forecasting (Unknown X Values for the Forecast Period) Unconditional forecast: all values of the independent variables are known with certainty –This is rare in practice Conditional forecast: actual values of one or more of the independent variables are not known –This is the more common type of forecast

10. 15-9 © 2011 Pearson Addison-Wesley. All rights reserved. Conditional Forecasting (Unknown X Values for the Forecast Period) (cont.) The careful selection of independent variables can sometimes help avoid the need for conditional forecasting This opportunity can arise when the dependent variable can be expressed as a function of leading indicators: –A leading indicator is an independent variable the movements of which anticipate movements in the dependent variable –The best known leading indicator, the Index of Leading Economic Indicators, is produced each month

11. 15-10 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting with Serially Correlated Error Terms Recall from Chapter 9 that when serial correlation is severe, one remedy is to run Generalized Least Squares (GLS) as noted in Equation 9.18: (9.18) If Equation 9.18 is estimated, the dependent variable will be: (15.7) Thus, if a GLS equation is used for forecasting, it will produce predictions of Y* T + 1 rather than of Y T+1 Such predictions thus will be of the wrong variable!

12. 15-11 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting with Serially Correlated Error Terms (cont.) If forecasts are to be made with a GLS equation, Equation 9.18 should first be solved for Y T before forecasting is attempted: (15.8) Next, substitute T+1 for t (to forecast time period T+1) and insert estimates for the coefficients, ρs and Xs into the equation to get: (15.9) Equation 15.9 thus should be used for forecasting when an equation has been estimated with GLS to correct for serial correlation

13. 15-12 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting Confidence Intervals The techniques we use to test hypotheses can also be adapted to create forecasting confidence intervals Given a point forecast, all we need to generate a confidence interval around that forecast are t c, the critical t-value (for the desired level of confidence), and S F, the estimated standard error of the forecast: (15.11) The critical t-value, t c, can be found in Statistical Table B-1 (for a two-tailed test with T-K-1 degrees of freedom)

14. 15-13 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting Confidence Intervals (cont.) Lastly, the standard error of the forecast, S F, for an equation with just one independent variable, equals the square root of the forecast error variance: (15.13) where: s 2 = the estimated variance of the error term T= the number of observations in the sample X T+1 = the forecasted value of the single independent variable = the arithmetic mean of the observed Xs in the sample Figure 15.2 illustrates an example of a forecast confidence interval

15. 15-14 © 2011 Pearson Addison-Wesley. All rights reserved. Figure 15.2 A Confidence Interval for

16. 15-15 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting with Simultaneous Equations Systems How should forecasting be done in the context of a simultaneous model? There are two approaches to answering this question, depending on whether there are lagged endogenous variables on the right-hand side of any of the equations in the system:

17. 15-16 © 2011 Pearson Addison-Wesley. All rights reserved. Forecasting with Simultaneous Equations Systems (cont.) 1.No lagged endogenous variables in the system: the reduced-form equation for the particular endogenous variable can be used for forecasting because it represents the simultaneous solution of the system for the endogenous variable being forecasted 2.Lagged endogenous variables in the system: then the approach must be altered to take into account the dynamic interaction caused by the lagged endogenous variables For simple models, this sometimes can be done by substituting for the lagged endogenous variables where they appear in the reduced-form equations If such a manipulation is difficult, however, then a technique called simulation analysis can be used

18. 15-17 © 2011 Pearson Addison-Wesley. All rights reserved. ARIMA Models ARIMA is a highly refined curve-fitting device that uses current and past values of the dependent variable to produce often accurate short-term forecasts of that variable –Examples of such forecasts are stock market price predictions created by brokerage analysts (called “chartists” or “technicians”) based entirely on past patterns of movement of the stock prices If ARIMA models thus essentially ignores economic theory (by ignoring “traditional” explanatory variables), why use them? The use of ARIMA is appropriate when: –little or nothing is known about the dependent variable being forecasted, –the independent variables known to be important cannot be forecasted effectively –all that is needed is a one or two-period forecast

19. 15-18 © 2011 Pearson Addison-Wesley. All rights reserved. ARIMA Models (cont.) The ARIMA approach combines two different specifications (called processes) into one equation: 1. An autoregressive process (AR): expresses a dependent variable as a function of past values of the dependent variable This is similar to the serial correlation error term function of Chapter 9 and to the dynamic model of Chapter 12 2.a moving average process (MA): expresses a dependent variable as a function of past values of the error term Such a function is a moving average of past error term observations that can be added to the mean of Y to obtain a moving average of past values of Y

20. 15-19 © 2011 Pearson Addison-Wesley. All rights reserved. ARIMA Models (cont.) To create an ARIMA model, we begin with an econometric equation with no independent variables: and then add to it both the autoregressive and moving-average processes: (15.17) where the θs and the φs are the coefficients of the autoregressive and moving-average processes, respectively, and p and q are the number of past values used of Y and ε, respectively

21. 15-20 © 2011 Pearson Addison-Wesley. All rights reserved. ARIMA Models (cont.) Before this equation can be applied to a time series, however, it must be ensured that the time series is stationary, as defined in Section 12.4 For example, a non-stationary series can often be converted into a stationary one by taking the first difference: (15.18) If the first differences do not produce a stationary series, then first differences of this first-differenced series can be taken—i.e. a second-difference transformation: (15.19)

22. 15-21 © 2011 Pearson Addison-Wesley. All rights reserved. ARIMA Models (cont.) If a forecast of Y* or Y** is made, then it must be converted back into Y terms For example, if d = 1 (where d is the number of differences taken to make Y stationary), then: (15.20) This conversion process is similar to integration in mathematics, so the “I” in ARIMA stands for “integrated” ARIMA thus stands for Auto-Regressive Integrated Moving Average –An ARIMA model with p, d, and q specified is usually denoted as ARIMA (p,d,q) with the specific integers chosen inserted for p, d, and q –If the original series is stationary and d therefore equals 0, this is sometimes shortened to ARMA

23. 15-22 © 2011 Pearson Addison-Wesley. All rights reserved. Key Terms from Chapter 15 Unconditional forecast Conditional forecast Leading indicator Confidence interval (of forecast) Autoregressive process Moving-average process ARIMA(p,d,q)