Forecasting Techniques: Single Equation Regressions Su, Chapter 10, section III

Regression Models Represent functional relationships between economic variables Usually estimated by OLS techniques General Form Y t =  0 +  1 X 1t +  2 X 2t + … +  k X 1k + u t Y t : Dependent Variable X it ‘s : Explanitory Variables  i ‘s: Parameters u t : Stochastic Term

Regression: Forecasting Ability Depends on the structure of the regression equation, including –Degrees of Freedom: Should be > 30 –Statistical Significance and sign of parameters –High Goodness of Fit Low Standard Error of Estimate High R 2

Forecasting with Regression Models Depends on choice of X’s, which is generally guided by economic theory –Example: According to the IS/LM model, what variables would be useful for forecasting GDP? Generally speaking, more data should be preferred

Some Useful Concepts I Ex Post Forecast: Extrapolation goes beyond sample period but not into future –Example: Sample period for regression is , forecast through 2000 Ex Ante Forecast: Extrapolation extends into future –Example: Sample period is 1990:1-2001:1, forecast through 2002:1

Some Useful Concepts II Predictive power of a regression model depends on its lag structure Conditional Forecasts: Some contemporaneous explanatory variables appear on RHS –Must also predict values for these contemporaneous explanatory variables Unconditional Forecasts: Only lagged explanatory variables appear on RHS

Some Useful Concepts III Point Forecast: Predicts a single number –Example: The Dow will be 1100 on July 1 Interval Forecast: Shows a numerical interval in which the actual value can be expected to fall –Example: The Dow will be between 1000 and 2000 on July 1 with 99% probability

Example: Automobile Sales Want to replicate the regression results in section 4 Use the regression data analysis tool to replicate the results on page 348 Model: Y t =  +  X t + u t Y: Automobile Sales X: New Car Price Linear Demand Curve

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

Regression Output

Interpreting Regression Results Y t = 10, X t (10.20) –Parameter on X: –t-statistic: 3.08

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 Y t = 10, (125.3) = 6, Note that ex post forecasts can be done for any year in the period for which data are available

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

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?

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

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?

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 and chart them

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)

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 t is just a value from a table in a statistics book  e has already been calculated by the regression program Text has wrong numbers

Forecast Interval

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”

AR Models: Functional Forms General: X t = f(X t-1,X t-2,X t-3,...,  1,  2,,  3...,u t ) –u t : residual term, captures random components –Must specify form and lag length Linear form, lag length k X t =  0 +  1 X t-1,+  2 X t-2,+ …+  k X t-k + u t Note that both No Change and Same Change naïve forecasts are special cases of this

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 X t-j are trivial and insignificant Rule of thumb: don’t use a k >4 because of econometric problems

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

Dummy Variables: Trends Uses a time variable T (=1,2,3,…) and extrapolates X along its time path Linear: X t =  +  T t Exponential:X = e  +  Tt Reciprocal:X = 1/[  +  T t ] Parabolic:X =  0 +  1 T t,+  2 T 2 t

Dummy Variables: Seasonal These are “Intercept shifters” - they allow the intercept term  0 to vary systematically Single Equation Model with Quarterly Dummies: Y t =  1 Q 1 +  2 Q 2 +  3 Q 3 +  4 Q 4 +  1 X 1t +…+  k X 1k +u t Can also use monthly dummies if Y is monthly Get a different forecast for each quarter

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

Example: Recession Dummy Model: Y t =  +  X t +  D R + u t 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

Empirical Results Y t =  X t -  D R ( ) (6.233) ( )

Forecast with Recession Dummy

Forecast Comparison

Exercise: AR Models Data: U.S. Population Available in a text file on Web page (tab2-1.txt) Step 1: Read file into Excel

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?

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

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

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