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Multiple Regression Forecasts
• Materials for this lecture• Demo
Lecture 8Multiple Regression.XLSX• Read Chapter 15 Pages 8-9 • Read all of Chapter 16’s Section 13
Black Swans (BSs) -- Taleb• 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• 2008 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
Fat Tails and Forecasting– Mandelbrot – founder of fractile science observed
that financial markets had fatter tails than an normal distribution implies.
– Taleb warns that people tend to underestimate risk, especially when armed with statistical models built on normal distributions.
– Posner offer this warning: We live in an information rich environment so we must seek out the right balance between human intuition and computer analysis.
– These cautions are offered now because we are expected to use a normal distribution to simulate residuals from a multiple regression model.
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
Structural Variation
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
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 – needed by ag input 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
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 σe) 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
Structural Forecast 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, 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 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 with OLS• 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’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
Multiple Regression Forecasts
• Probabilistic forecast uses ŶT+I and σ (Std Dev) and assume a normal distrib. for residualsỸT+i = ŶT+i + NORM(0, σ)
orỸT+i = NORM(ŶT+i , σ)
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)
Regression Model for Growth• 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)
Regression Model For Decay Functions
• Some data display a decay pattern• Forecast them with multiple regression • Add an exogenous variable to capture
the growth or decay of forecast variable
• Decay functionŶ = 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, you 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• 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• I like to follow these simple rules, in this
order– Correct parameter signs based on sound
economic theory for al variables • For supply beta on price must be positive, etc.
– Student t ratios greater than 2.0 and/or P values for betas less than 0.05
– F ratio larger than 20.0– R2 as large as you can get– MAPE (mean absolute percent error) less
than 0.1 (or 10%)• For large models OLS is preferred
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