Multiple Regression Forecasts

1
Multiple Regression Forecasts

• Materials for this lecture
• Demo
• Lecture 3 Multiple Regression.XLS
• Read Chapter 15 Pages 8-9
• Read all of Chapter 16s Section 13

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Multiple Regression Forecasts

• Structural model of the forecast variable used
  when suggested by
• Economic theory
• Knowledge of the industry
• Relationship to other variables
• Economic model is being developed
• Examples
• Forecast planted acres for a crop
• Forecast demand for a product
• Forecast annual price of corn or cattle
• Forecast government payments for a crop
• Forecast exports or trade flows

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Multiple Regression Forecasts

• Structural model
• Y a b1 X1 b2 X2 b3 X3 b4 X4 e
• Where Xi are exogenous variables that explain the
  variation of Y over the historical period
• Estimate parameters (a, bis, and SEP) using
  multiple regression (OLS)
• OLS is preferred because it minimizes the sum of
  squared residuals
• This is the same as reducing the risk on Y as
  much as possible, i.e., minimizing the risk on
  the forecast

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Multiple Regression Model
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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
• 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

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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

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Multiple Regression Forecasts

• Specify alternative values for X and forecast the
  Deterministic Component
• Multiply Betas by their respective Xs
• Forecast Acres for alternative Prices and CRP
• Lagged Yield and Year are constant in scenarios

8
Multiple Regression Forecasts

• Probabilistic forecast uses YTI and SEP or Std
  Dev and assume a normal distrib. for residuals
• ?Ti YTi SEP NORM()
• or
• ?Ti NORM(YTi , SEP)

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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)

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Growth Forecasts

• Some data display a growth pattern
• Easy to forecast with multiple regression
• Add an X variable to capture the growth or decay
  of Y variable
• Growth function
• Y a b1Tb2T2
• Log(Y) a b1 Log(T) Double Log
• Log(Y) a b1 T Single Log
• See Decay Function worksheet for several
  examples for handleing this problem

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Multiple Regression Forecasts
Single Log Form Log (Yt) b0 b1 T
Double Log Form Log (Yt) b0 b1 Log (T)
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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
• Y a b1(1/T) b2(1/T2)

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Forecasting Growth or Decay Patterns

• Here is the regression result for estimating a
  decay function
• Yt a b1 (1/Xt)
• or
• Yt a b1 (1/Xt) b2 (1/Xt2)

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Multiple Regression Forecasts

• Examine a structural regression model that
  contains Trend and an X
• Y a b1T b2It does not explain all of the
  variability, a seasonal or cyclical variability
  may be present, if so need to remove its effect

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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

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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

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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.

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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

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Goodness of Fit Measures

• MSE works best to determine best model for in
  sample forecasting
• R2 does not penalize for adding ks
• 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 ks)