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

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

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

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

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

Multiple Regression Forecasts Structural model Ŷ = a + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + e Where X i ’s are exogenous variables that explain the variation of Y over the historical period Estimate parameters (a, b i ’s, and SEP e ) 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

Multiple Regression Model

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 Ac t = f(E(Price t ), Plt Ac t-1, E(P th Crop t ), Trend, Yield t-1 ) Harvested Ac t = a + b Plt Ac t Yield t = a + b T t Prod t = Harvested Ac t * Yield t Estimate and re-estimate the model Make the deterministic forecast Make the forecast stochastic for a probabilistic forecast

US Planted Wheat Acreage Model Plt Ac t = f(E(Price t ), Yield t-1, CRP t, Years t ) 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 Yield t-1, Trend masks the effects of Yield

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

Multiple Regression Forecasts Probabilistic forecast uses Ŷ T+I and SEP or Std Dev and assume a normal distrib. for residuals Ỹ T+i = Ŷ T+i + NORM(0, SEP T ) or Ỹ T+i = NORM(Ŷ T+i, SEP T )

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)

Growth Forecasts Some data display a growth pattern Easy to forecast with multiple regression Add T 2 variable to capture the growth or decay of Y variable Growth function Ŷ = a + b 1 T+ b 2 T 2 Log(Ŷ) = a + b 1 Log(T) Double Log Log(Ŷ) = a + b 1 T Single Log See Decay Function worksheet for several examples for handling this problem

Multiple Regression Forecasts Single Log Form Log (Y t ) = b 0 + b 1 T Double Log Form Log (Y t ) = b 0 + b 1 Log (T)

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 + b 1 (1/T) + b 2 (1/T 2 )

Forecasting Growth or Decay Patterns Here is the regression result for estimating a decay function Ŷ t = a + b 1 (1/T t ) or Ŷ t = a + b 1 (1/T t ) + b 2 (1/T t 2 )

Multiple Regression Forecasts Examine a structural regression model that contains Trend and an X variable Ŷ = a + b 1 T + b 2 X t does not explain all of the variability, a seasonal or cyclical variability may be present, if so need to remove its effect

Goodness of Fit Measures Models with high R 2 may not forecast well –If add enough Xs can get high R 2 –R-Bar 2 is preferred as it is not affected by no. Xs Selecting based on highest R 2 same as using minimum Mean Squared Error MSE =(∑ e t 2 )/T

Goodness of Fit Measures R-Bar 2 takes into account the effect of adding Xs where s 2 is the unbiased estimator of the regression residuals and k represents the number of Xs in the model

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.

Goodness of Fit Measures Summary of goodness of fit measures –SIC, AIC, and S 2 are sensitive to both k and T –The S 2 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

Goodness of Fit Measures MSE works best to determine best model for “in sample” forecasting R 2 does not penalize for adding k’s R-Bar 2 is based on S 2 so it provides some penalty as k increases AIC is better then R 2 but SIC results in the most parsimonious models (fewest k’s)