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

Yt = 0 + 1X1t + 2X2t + … + kX1k + ut

Yt : Dependent Variable Xit‘s : Explanitory Variables

i‘s: Parameters ut : 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 R2

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 1970-

1997, 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: Yt = + Xt + ut

• Y: Automobile Sales X: New Car Price

• Linear Demand Curve

Demand for New Cars

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60.0

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4000 5000 6000 7000 8000 9000 10000

Sales

Pric

e

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 OutputSUMMARY OUTPUT

Regression StatisticsMultiple R 0.59R Square 0.35Adjusted R Square 0.31Standard Error 1013.7142Observations 20

ANOVAdf SS MS

Regression 1 9794932.261 9794932Residual 18 18497098.29 1027617Total 19 28292030.55

Coefficients Standard Error t StatIntercept 10200.23 887.95 11.49X Variable 1 -30.2750 9.8062 -3.09

Interpreting Regression Results

• Yt = 10,200.23 - 30.275Xt (10.20)

– Parameter on X: -30.27– 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

• 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

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

Actual Sales and Ex Post Forecast

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991

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 1992 - 1994 and chart them

Ex Post and Ex Ante Forecasts

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993

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• t0.025 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

Ex Post and Ex Ante Forecasts

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993

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:Xt = f(Xt-1,Xt-2,Xt-3,...,1, 2,, 3...,ut)– ut : residual term, captures random components– Must specify form and lag length

• Linear form, lag length kXt = 0 + 1 Xt-1,+ 2Xt-2,+ …+ kXt-k + ut

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 Xt-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 pathLinear: Xt = + Tt

Exponential: X = e + Tt

Reciprocal: X = 1/[ + Tt]

Parabolic: X = 0 + 1 Tt,+ 2T2t

Dummy Variables: Seasonal

• These are “Intercept shifters” - they allow the intercept term 0 to vary systematically

• Single Equation Model with Quarterly Dummies:

Yt = 1Q1+2Q2+3Q3+4Q4+1X1t+…+kX1k+ut

• 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: Yt = + Xt + DR + 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

Empirical Results

Yt = - 31.66Xt - DR

(571.918) (6.233) (360.237)

Forecast with Recession Dummy

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1971 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991

Actual

Forecast

Forecast Comparison

No Dummy Dummy

1991 F 6406.77 4839.97

SEE 1013.71 643.83

SSR 18,497,098.29 7,046,970

R^2 0.35 0.75

Exercise: AR Models

• Data: U.S. Population 1948-1990

• 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?

Year Pop Lag1 Lag2 Lag3 Lag41948 147.201949 149.77 147.201950 152.27 147.201951 154.88 147.201952 157.55 147.20

• 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

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