Part 4: Prediction4-1/22

Regression ModelsProfessor William Greene

Stern School of Business

IOMS Department

Department of Economics

Part 4: Prediction4-2/22

Regression and Forecasting Models

Part 4 – Prediction

Part 4: Prediction4-3/22

Prediction

Use of the model for predictionUse “x” to predict y based on y = β0 + β1x + ε

Sources of uncertainty Predicting ‘x’ first Using sample estimates of β0 and β1 (and,

possibly, σ instead of the ‘true’ values) Can’t predict noise, ε Predicting outside the range of experience –

uncertainty about the reach of the regression model.

Part 4: Prediction4-4/22

Base Case Prediction For a given value of x*: Use the equation.

True y = β0 + β1x* + ε

Obvious estimate: y = b0 + b1x (Note, no prediction for ε)

Minimal sources of prediction error Can never predict ε at all The farther from the center of experience,

the greater is the uncertainty.

Part 4: Prediction4-5/22

Prediction Interval for y|x*

0 1

22

0 1 e N 2i 1 i

Prediction includes a range of uncertainty

ˆPoint estimate: y b b x*

The range of uncertainty around the prediction:

1 (x * x)b b x* 1.96 s 1+

N (x x)

The usual 95% Due to ε Due to estimating β0 and β1 with b0 and b1

(Remember the empirical rule, 95% of the distribution within two standard deviations.)

Part 4: Prediction4-6/22

Prediction Interval for E[y|x*]

0 1

22

0 1 e N 2i 1 i

Prediction includes a range of uncertainty

ˆPoint estimate: y b b x*.

The range of uncertainty around the prediction:

1 (x * x)b b x* 1.96 S

N (x x)

Not predicting ε.

The usual 95% Due to estimating β0 and β1 with b0 and b1

(Remember the empirical rule, 95% of the distribution within two standard deviations.)

Part 4: Prediction4-7/22

Predicting y|x vs. Predicting E[y|x]

Predicting y itself, allowing for in the prediction interval.

Predicting E[y], no provision for in the prediction interval.

Part 4: Prediction4-8/22

Simpler Formula for Prediction

0 1

22 20 1 e

Prediction includes a range of uncertainty

ˆPoint estimate: y b b x*

The range of uncertainty around the prediction:

1b b x* 1.96 s 1+ (x * x) SE(b)

N

Part 4: Prediction4-9/22

Uncertainty in Prediction

2 2 2e

1 1.96 s 1+ (x* x) (SE(b))

N

The interval is narrowest at x* = , the center of our experience. The interval widens as we move away from the center of our experience to reflect the greater uncertainty.(1) Uncertainty about the prediction of x(2) Uncertainty that the linear relationship will continue to exist as we move farther from the center.

x

Part 4: Prediction4-10/22

Prediction from Internet Buzz Regression

Part 4: Prediction4-11/22

Prediction Interval for Buzz = .8

0 1

2 2 2e

2 2 2

Predict Box Office for Buzz = .8

b +b x = -14.36 + 72.72(.8) = 43.82

1 s 1 (.8 Buzz) SE(b)

N

113.3863 1 (.8 .48242) 10.94

62

13.93

Interval = 43.82 1.96(13.93)

= 16.52 to 71.12

Data obtained separately

Buzz = 0.48242

Max(Buzz)= 0.79

Part 4: Prediction4-12/22

Predicting Using a Loglinear Equation

Predict the log first Prediction of the log Prediction interval – (Lower to Upper)

Prediction = exp(lower) to exp(upper)

This produces very wide intervals.

Part 4: Prediction4-13/22

Interval Estimates for the Sample of Monet Paintings

ln (SurfaceArea)

ln (

US$)

7.67.47.27.06.86.66.46.26.0

18

17

16

15

14

13

12

11

10

S 1.00645R-Sq 20.0%R-Sq(adj) 19.8%

Regression95% PI

Fitted Line Plotln (US$) = 2.825 + 1.725 ln (SurfaceArea)Regression Analysis: ln (US$) versus

ln (SurfaceArea) The regression equation isln (US$) = 2.83 + 1.72 ln (SurfaceArea)Predictor Coef SE Coef T PConstant 2.825 1.285 2.20 0.029ln (SurfaceArea) 1.7246 0.1908 9.04 0.000S = 1.00645 R-Sq = 20.0% R-Sq(adj) = 19.8%

Mean of ln (SurfaceArea) = 6.72918

Part 4: Prediction4-14/22

Prediction for An Out of Sample Monet

Claude Monet: Bridge Over a Pool of Water Lilies. 1899. Original, 36.5”x29.”

2 2 2

2 2

lnSurface ln(36.5 29) 6.96461

Prediction 2.83 1.72(6.96461) 14.809

1Uncertainty 1.96 1.00645 1 (6.96461 6.72918) (.1908)

328

1.96 1.012942(1.003049) (.23453) (.1908)

1.96(1.008984)

1.977608

Prediction Interval = 14.809 1.977608

= 12.83139 to 16.786608

Part 4: Prediction4-15/22

Predicting y when the Model Describes log y

Predicted Price: Mean = Exp(a + bx )

= Exp(14.809 ) = $2

The inter

,700,641.

val predicts log price. What abo

78

Upper Limit

ut the

= Exp(

price?

14.809+1.9776)

= $19,513,166.53

Lower Limit = Exp(14.809-1.9776)

= $ 373,771.53

Part 4: Prediction4-16/22

39.5 x 39.125. Prediction by our model = $17.903MPainting is in our data set. Sold for 16.81M on 5/6/04 Sold for 7.729M 2/5/01Last sale in our data set was in May 2004Record sale was 6/25/08. market peak, just before the crash.

Part 4: Prediction4-17/22

http://www.nytimes.com/2006/05/16/arts/design/16oran.html

Part 4: Prediction4-18/22

32.1” (2 feet 8 inches)

26.2” (2 feet 2.2”)

167” (13 feet 11 inches)

78.74” (6 Fe

et 7 inch

)

"Morning", Claude Monet 1920-1926, oil on canvas 200 x 425 cm, Musée de l

Orangerie, Paris France. Left panel

Part 4: Prediction4-19/22

Predicted Price for a Huge Painting

Regression Equation: ln $ = 2.825 + 1.725 ln Surface Area

Width = 167 Inches

Height = 78.74 Inches

Area = 13,149.58 Square inches, ln = 9.484

Predicted ln Price = 2.825 + 1.725 (9.484) = 19.185

Predicted Price = exp(19.185) = $214,785,473.40

Part 4: Prediction4-20/22

Prediction Interval for Price

22 2

e

Prediction Interval for ln Price is

1Predicted ln Price 1.96 S 1 ln Area* ln Area ( )

ln Area* = ln (167 78.74) = 9.484

ln Area = 6.72918 (computed from the data)

S = 1.00645 (from

e SE bN

22 2

regression results)

SE(b) = 0.1908

119.185 1.96 (1.00645) 1 9.484 6.72918 (.1908)

328

19.185 2.228 = [16.957 to 21.413]

Predicted Price = exp(16.957) to exp(21.413) =

$23,138,304 to $1,993,185,600

Part 4: Prediction4-21/22

Use the Monet Model to Predict a Price for a Dali?

118” (9 feet 10 inches)

157

” (1

3 F

eet

1 in

ch)

Hallucinogenic Toreador

26

.2”

(2 f

ee

t 2

.2”) 32.1” (2 feet 8 inches)

Average Sized Monet

Part 4: Prediction4-22/22

Forecasting Out of Sample

Income

G

2750025000225002000017500150001250010000

8

7

6

5

4

3

S 0.370241R-Sq 88.0%R-Sq(adj) 87.8%

Regression95% PI

Fitted Line PlotG = 1.928 + 0.000179 Income

Per Capita Gasoline Consumption vs. Per Capita Income, 1953-2004.

How to predict G for 2017? You would need first to predict Income for 2017.

How should we do that?

Regression Analysis: G versus Income The regression equation isG = 1.93 + 0.000179 IncomePredictor Coef SE Coef T PConstant 1.9280 0.1651 11.68 0.000Income 0.00017897 0.00000934 19.17 0.000S = 0.370241 R-Sq = 88.0% R-Sq(adj) = 87.8%