multiple regression forecasts materials for this lecture demo lecture 8multiple regression.xlsx read...
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![Page 1: 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](https://reader034.vdocuments.site/reader034/viewer/2022052401/56649e895503460f94b8eb6d/html5/thumbnails/1.jpg)
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
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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
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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.
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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
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Structural Variation
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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
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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
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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
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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
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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
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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’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
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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)
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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 for a probability density function (PDF)
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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
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Multiple Regression Forecasts
Single Log Form Log (Yt) = b0 + b1 T
Double Log FormLog (Yt) = b0 + b1 Log (T)
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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)
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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
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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
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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 =(∑ et
2)/T
R = 1 -
e
(Y - Y)
2t2
t=1
T
t2
t=1
T
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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 =(∑ et
2)/T
R = 1 -
e
(Y - Y)
2t2
t=1
T
t2
t=1
T
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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
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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.
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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 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