Multiple Regression Forecasts

1. Multiple Regression Forecasts • Materials for this lecture • Demo Lecture 2 Multiple Regression.XLSX • Read Chapter 15 Pages 8-9 • Read all of Chapter 16’s Section 13

2. Structural Variation • Variables you want to forecast are often dependent on other variables Qt. Demand = f( Own Price, Competing Price, Income, Population, Season, Tastes & Preferences, Trend, etc.) Y = a + b (Time) • Structural models will explain most structural variation in a data series • Even when we build structural models, the forecast is not perfect • A residual remains as the unexplained portion

3. Structural Variation

4. Irregular Variation • Erratic movements in time series that follow no recognizable regular pattern • Random, white noise, or stochastic movements • Risk is this non-systematic variability in the residuals • This risk leads to Monte Carlo simulation of the risk for our probabilistic forecasts • We recognize risks cannot be forecasted • Incorporate risks into probabilistic forecasts • Provide forecasts with confidence intervals

5. Black Swans (BSs) • BSs low probability events • An outlier “outside realm of reasonable expectations” • Carries an extreme impact • Human nature causes us to concoct explanations • Black swans are an example of uncertainty • Uncertainty is generated by unknown probability distributions • Risk is generated by “known” distributions • Recent recession was a BSs • A depression is a BSs • Dramatic increases of grain prices in 2006 and 2007 • Dramatic increase in cotton price in 2010

6. Multiple Regression Forecasts • Structural model of the forecast variable is used when suggested by: • Economic theory • Knowledge of the industry • Relationship to other variables • Economic model is being developed • Examples of forecasting: • Planted acres – need by ag input sales businesses • Demand for a product – sales and processors • Price of corn or cattle – feedlots, grain mills, etc. • Govt. payments – Congressional Budget Office • Exports or trade flows – international ag. business

7. Multiple Regression Forecasts • Structural model Ŷ = a + b1 X1 + b2 X2 + b3 X3 + b4 X4 + e Where Xi’s are exogenous variables that explain the variation of Y over the historical period • Estimate parameters (a, bi’s, and SEPe) using multiple regression (or OLS) • OLS is preferred because it minimizes the sum of squared residuals • This is the same as reducing the risk on Ŷ as much as possible, i.e., minimizing the risk for your forecast

8. Structural ForecastModel

9. Steps to Build Multiple Regression Models • Plot the Y variable in search of: trend, seasonal, cyclical, structural, and irregular variation • Plot Y vs. each X to see the structural relationship and how X may explain Y; calculate correlation coefficients to Y • Hypothesize the model equation(s) with all likely Xs to explain the Y, based on knowledge of industry & theory • Wheat production forecasting model is Plt Act = f(E(Pricet), Plt Act-1, E(PthCropt), Trend, Yieldt-1) Harvested Act = a + b Plt Act Yieldt = a + b Tt Prodt = Harvested Act * Yieldt • Estimate and re-estimate the model with OLS • Make the deterministic forecast • Make the forecast stochastic for a probabilistic forecast

10. US Planted Wheat Acreage Model Plt Act = f(E(Pricet), Yieldt-1, CRPt, Yearst) • Statistically significant betas for Trend (years variable) and Price • Leave CRP in model because of policy analysis and it has the correct sign • Use Trend (years) over Yieldt-1, Trend masks the effects of Yield

11. Multiple Regression Forecasts • Specify alternative values for X’s and forecast the Deterministic Component • Multiply Betas by their respective X’s • Forecast Acres for alternative Prices and CRP • Lagged Yield and Year are constant in scenarios

12. Multiple Regression Forecasts • Probabilistic forecast uses ŶT+I and SEP or Std Dev and assume a normal distrib. for residuals ỸT+i= ŶT+i + NORM(0, SEPT) or ỸT+i= NORM(ŶT+i , SEPT)

13. Multiple Regression Forecasts • Present probabilistic forecast as a PDF with 95% Confidence Interval shown here as the bars about the mean for a probability density function (PDF)

14. Regression Model for Growth • Some data display a growth pattern • Easy to forecast with multiple regression • Add T2variable to capture the growth or decay of Y variable • Growth function Ŷ = a + b1T+ b2T2 Log(Ŷ) = a + b1 Log(T) Double Log Log(Ŷ) = a + b1 T Single Log See Decay Function worksheet for several examples for handling this problem

15. Multiple Regression Forecasts Single Log Form Log (Yt) = b0 + b1 T Double Log Form Log (Yt) = b0 + b1 Log (T)

16. Regression Model For Decay Functions • Some data display a decay pattern • Forecast them with multiple regression • Add an X variable to capture the growth or decay of forecast variable • Decay function Ŷ = a + b1(1/T) + b2(1/T2)

17. Forecasting Growth or Decay Patterns • Here is the regression result for estimating a decay function Ŷt = a + b1 (1/Tt) or Ŷt = a + b1 (1/Tt) + b2 (1/Tt2)

18. Multiple Regression Forecasts • Examine a structural regression model that contains Trend and an X variable Ŷ = a + b1T + b2Xtdoes not explain all of the variability, a seasonal or cyclical variability may be present, if so need to remove its effect

19. Goodness of Fit Measures • Models with high R2 may not forecast well • If add enough Xs can get high R2 • R-Bar2 is preferred as it is not affected by no. Xs • Selecting based on highest R2 same as using minimum Mean Squared Error MSE =(∑ et2)/T

20. Goodness of Fit Measures • R-Bar2 takes into account the effect of adding Xs where s2 is the unbiased estimator of the regression residuals and k represents the number of Xs in the model

21. Goodness of Fit Measures • Akaike Information Criterion (AIC) • Schwarz Information Criterion (SIC) • For T = 100 and k goes from 1 to 25 • The SIC affords the greatest penalty for just adding Xs. • The AIC is second best and the R2 would be the poorest.

22. Goodness of Fit Measures • Summary of goodness of fit measures • SIC, AIC, and S2 are sensitive to both k and T • The S2 is small and rises slowly as k/T increases • AIC and SIC rise faster as k/T increases • SIC is most sensitive to k/T increases

23. Goodness of Fit Measures • MSE works best to determine best model for “in sample” forecasting • R2 does not penalize for adding k’s • R-Bar2 is based on S2 so it provides some penalty as k increases • AIC is better then R2 but SIC results in the most parsimonious models (fewest k’s)