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 variablesQt. 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 + 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

Multiple Regression ModelPltAc = f(Price , Plt , IdleAcre , X )

HarvAc = f(PltAc )

Yield = f(Price , Yield )

Prod = Yield * HarvAc

Supply = Prod + EndStock

Price = a + b Supply

Domestic D = f(Price , Income / pop , Z )

Export D = f(Price , Y )

End Stock = Supply - Domestic D - Export D

t t-1 t-1 t t

t t

t t t-1

t t t

t t t-1

t t

t t t t

t t t

t t t t

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 isPlt 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

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

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 ŶT+I and SEP or Std Dev and assume a normal distrib. for residualsỸT+i = ŶT+i + NORM(0, SEPT)

orỸT+i = NORM(ŶT+i , SEPT)

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 T2 variable to capture the growth

or decay of Y variable• Growth function

Ŷ = a + b1T+ b2T2

Log(Ŷ) = a + b1 Log(T) Double LogLog(Ŷ) = a + b1 T Single Log See Decay Function worksheet for several examples for handling this problem

Multiple Regression Forecasts

Single Log Form Log (Yt) = b0 + b1 T

Double Log FormLog (Yt) = b0 + b1 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 + b1(1/T) + b2(1/T2)

Forecasting Growth or Decay Patterns

• Here is the regression result for estimating a decay functionŶt = a + b1 (1/Tt)

or Ŷt = a + b1 (1/Tt) + b2 (1/Tt

2)

Observed and Predicted Values for KOV

-50

0

50

100

150

Predicted Observed

Lower 95% Predict. Interval Upper 95% Predict. Interval

Lower 95% Conf. Interval Upper 95% Conf. Interval

Multiple Regression Forecasts

• Examine a structural regression model that contains Trend and an X variableŶ = a + b1T + b2Xt 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 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 =(∑ et

2)/T

R = 1 -

e

(Y - Y)

2t2

t=1

T

t2

t=1

T

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

R = 1 - (s / [( (Y - Y) ) / (T -1)])2 2t

2

t=1

T

s = (T

T - k) * [( e ) / T]2

t2

t=1

T

Goodness of Fit Measures

AIC = exp (2kT

) ( e / T)t2

t=1

T

SIC = T (kT

) ( e / T)t2

t=1

T

3.5

3.0

2.5

2.0

1.5

1.0

0.5

.05 .10 .15 .20 .25

Pen

alty

Fac

tor

k/T

SIC

AIC

s2

3.5

3.0

2.5

2.0

1.5

1.0

0.5

.05 .10 .15 .20 .25

Pen

alty

Fac

tor

k/T

SIC

AIC

s2

• 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 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

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)R2