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1. Multiple Regression Forecasts Materials for this lecture Demo Lecture 2 Multiple Regression.XLS 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. 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
4. 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 Dramtaic increase in cotton price in 2010
5. 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 – inputs sales businesses need this Demand for a product – sales and production Price of corn or cattle – feedlots, grain mills, etc. Govt. payments – Congressional Budget Office Exports or trade flows – international ag. business
6. 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
7. Multiple Regression Model
8. Steps to Build Multiple Regression Models Plot the Y variable in search of: trend, seasonal, cyclical 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 model & theory Forecasting wheat production, 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 Make the deterministic forecast Make the forecast stochastic for a probabilistic forecast
9. 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
10. Multiple Regression Forecasts Specify alternative values for X 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
11. 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)
12. Multiple Regression Forecasts Present probabilistic forecast as a PDF with 95% Confidence Interval shown here as the bars about the mean in a probability density function (PDF)
13. Growth Forecasts 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
14. Multiple Regression Forecasts Single Log Form Log (Yt) = b0 + b1 T Double Log Form Log (Yt) = b0 + b1 Log (T)
15. Decay Function Forecasts 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)
16. 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)
17. 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
18. 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
19. 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
20. 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.
21. 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
22. 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)