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Forecasting Techniques: MA and Regression Methods

Forecasting Techniques: MA and Regression Methods. Lab for Su, Chapter 10. Three Summary Statistics. Mean Absolute Error (MAE) MAE = S |FE t | / n = S |F t - A t | / n Mean Square error (MSE) MSE = S (FE t ) 2 / n = S (F t - A t ) 2 / n Root Mean Square Error (RMSE)

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Forecasting Techniques: MA and Regression Methods

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  1. Forecasting Techniques: MA and Regression Methods Lab for Su, Chapter 10

  2. Three Summary Statistics • Mean Absolute Error (MAE) MAE = S |FEt| / n = S |Ft - At| / n • Mean Square error (MSE) MSE = S (FEt)2 / n = S (Ft - At)2 / n • Root Mean Square Error (RMSE) RMSE = SQRT[S (FEt)2 / n = S (Ft - At)2 / n]

  3. Moving Average Methods • Provides more efficient mechanical projections of short-term movements • Has advantage of flexibility and presents a more realistic picture of long-run movements • Data are not forced into any particular patterns MA: X*t = (1/n)Sni=1Xt-i =(1/n)[Xt-1 +Xt-2 +Xt-3 + ...+Xt-n] • Note this is not a centered moving average • Must only decide on n • Can be applied to first differences or % changes

  4. Moving Average Example • Start with an MA(4) forecast • For ease of coding, copy the car sales values to the MA column, then the out of sample MA forecast can be easily written and copied • Compute the within sample, one period ahead MAE, MSE, RMSE

  5. Naïve Forecasts and MA(4)

  6. Changing the Order of an MA Forecast • Economists refer to MA forecasts by the number of periods they use, which is called the “order” of the moving average • MA(2): Two period moving average • MA(3): Three period moving average • etc. • The forecast depends on the MA order

  7. Effect of Changing Order

  8. Regression Models • Represent functional relationships between economic variables • Usually estimated by OLS techniques • General Form Yt = b0 + b1X1t + b2X2t + … + bkX1k + ut Yt : Dependent Variable Xit‘s : Explanitory Variables bi‘s: Parameters ut : Stochastic Term

  9. Example: Automobile Sales • Replicate the regression results in section 4 on page 348 • Use the regression data analysis tool • Model: Yt = a + bXt + ut • Y: Automobile Sales X: New Car Price • Linear Demand Curve

  10. Procedure • Step 1: Copy the sales and price data to a new worksheet • Step 2: Start the regression data analysis tool • Specify correct ranges

  11. Regression Output

  12. Interpreting Regression Results • Yt = 10,200.23 - 30.275Xt (10.20) • Parameter on X: -30.27 • t-statistic: 3.08

  13. Ex Post Point Forecasts • To make an ex post forecast for 1991, simply plug the actual value of the price index for 1991 into (10.20) - Put in D22 • Yt = 10,200.23 - 30.275(125.3) = 6,406.77 • Note that ex post forecasts can be done for any year in the period for which data are available

  14. Evaluation of Ex Post Forecasts • Can also evaluate forecasts within sample • Copy the formula from D22 into D21 • Where in the regression output can you find this number? • Fill in the rest of column D with the Ex Post Forecasts and plot the actual sales and the Ex Post forecasts

  15. Summary Statistics • Already know how to calculate, but in this case the regression function has already done some of the heavy lifting • We saw where the Ex Post forecasts could be found, what about the forecast errors?

  16. Residuals and Forecast Errors • In the terminology of econometrics, ex post forecast errors are called residuals • The OLS estimator is designed to minimize the sum of the residuals squared - OLS estimates minimize MSE and RMSE • To find value of MSE, look on the ANOVA table, for the row labeled Residual and under the column labeled SS

  17. Ex Ante Point Forecasts • To generate these, must forecast X, as these forecasts are conditional on unknown future values (must pretend that the present is 1991 in this case) • How should X be forecast?

  18. Ex Ante Point Forecasts: Example • Step 1: Extend the time column to 1994 • Step 2: Calculate the forecasted X’s using the same change naïve forecasting model in column C • Step 3: Using the formula from above, calculate the Ex Ante forecasts for 1992 - 1994 and chart them

  19. Interval Forecasts • Instead of a line, can also display the range in which the forecast values will probably fall • These are called interval forecasts and are based on the variance of the regression • Based on (10.18)

  20. Interval Forecasts: Example • Must calculate average of X and sum of X - average(X) = x • First term of (10.18) is just ex ante forecast • t0.025 is just a value from a table in a statistics book • se has already been calculated by the regression program • Text has wrong numbers

  21. Forecast Interval

  22. Autoregressive Models • Even though they use sophisticated statistical techniques, these models are extrapolations • The explanatory variables (X’s) are lagged values of the dependent variable • Assumes that the time path of a variable is self-generating • Also called the “Chain Principle”

  23. AR Models: Functional Forms • General: Xt = f(Xt-1,Xt-2,Xt-3,...,b1, b2,, b3...,ut) • ut : residual term, captures random components • Must specify form and lag length • Linear form, lag length k Xt = b0 + b1 Xt-1,+ b2Xt-2,+ …+ bkXt-k + ut Note that both No Change and Same Change naïve forecasts are special cases of this

  24. AR Models: Determining Lag Length • The general form has an infinite number of parameters, but we never have this much data - model must be restricted to be used • Assume that the impact of some distant Xt-j are trivial and insignificant • Rule of thumb: don’t use a k >4 because of econometric problems

  25. Replication Exercise • Replicate results on page 352 • Note different sample periods

  26. Exercise: AR Models • Data: U.S. Population 1948-1990 • Available in an Excel file on Web page (tab2-1.xls) • Step 1: Read file

  27. Exercise: Creating Lag Variables • Best way is with formulas, although could copy as well • Population data are in column 2 • Step 2: Label columns 3-6 “Lag1”, “Lag2”, “Lag3” and “Lag4” • What value goes in C3? D4? E5? F6?

  28. C3 is the Lag1 value for 1949, which is the actual population in 1948 - population lagged one year • D4 is the Lag2 value for 1950, which is the actual population in 1948 - population lagged two years • Step 3:Fill in rest of lags using formulas

  29. Exercise: AR Regressions • Step 4: Replicate the regression results on page 352. Note: Watch sample period • Step 5: Calculate Ex Post forecasts for the sample period and RMSE for each method • Which has the lowest RMSE? • Step 6: Calculate Ex Ante population forecasts through 2025 and compare to Table 10.4

  30. Dummy Variables • Requires no additional economic data • Was discussed in chapter 2 • Two Types: • Trend • Seasonal / annual

  31. Dummy Variables: Trends • Uses a time variable T (=1,2,3,…) and extrapolates X along its time path Linear: Xt = a + bTt Exponential: X = ea + bTt Reciprocal: X = 1/[a + bTt] Parabolic: X = b0 + b1 Tt,+ b2T2t

  32. Dummy Variables: Seasonal • These are “Intercept shifters” - they allow the intercept term b0 to vary systematically • Single Equation Model with Quarterly Dummies: Yt = g1Q1+g2Q2+g3Q3+g4Q4+b1X1t+…+bkX1k+ut • Can also use monthly dummies if Y is monthly • Get a different forecast for each quarter

  33. Other Dummy Variables • Dummy variables can be useful tools in forecasting • Recall from the earlier section that the single equation forecast for new car sales was high for 1991 because it was a recessionary year • Can use a dummy variable for recessions to improve this forecast

  34. Example: Recession Dummy • Model: Yt = a + bXt + gDR + ut • Y: Automobile Sales X: New Car Price • DR: Recession Dummy, = 1 in years with troughs • Add new sheet to spreadsheet, copy Year, New Car Sales, New Car Price • Look at Table 7.1, p. 236 to create dummy

  35. Empirical Results Yt = 10,699.87 - 31.66Xt - 1893.29DR (571.918) (6.233) (360.237)

  36. Forecast with Recession Dummy

  37. Forecast Comparison

  38. Exercise: Trend Forecasting • Step 1: Create trend and trend squared variables in the spreadsheet • Step 2: Replicate the three regression results shown on page 354 • Step 3: Calculate a 100 year ahead Ex Ante forecast of U.S. population using each, and chart the time paths • How accurate are these forecasts

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