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Part 4: Prediction 4-1/22 Regression Models Professor William Greene Stern School of Business IOMS Department Department of Economics

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Part 4: Prediction 4-2/22 Regression and Forecasting Models Part 4 – Prediction

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Part 4: Prediction 4-3/22 Prediction Use of the model for prediction Use “x” to predict y based on y = β 0 + β 1 x + ε Sources of uncertainty Predicting ‘x’ first Using sample estimates of β 0 and β 1 (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.

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Part 4: Prediction 4-4/22 Base Case Prediction For a given value of x*: Use the equation. True y = β 0 + β 1 x* + ε Obvious estimate: y = b 0 + b 1 x (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.

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Part 4: Prediction 4-5/22 Prediction Interval for y|x* The usual 95% Due to ε Due to estimating β 0 and β 1 with b 0 and b 1 (Remember the empirical rule, 95% of the distribution within two standard deviations.)

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Part 4: Prediction 4-6/22 Prediction Interval for E[y|x*] The usual 95% Due to estimating β 0 and β 1 with b 0 and b 1 (Remember the empirical rule, 95% of the distribution within two standard deviations.)

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Part 4: Prediction 4-7/22 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.

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Part 4: Prediction 4-8/22 Simpler Formula for Prediction

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Part 4: Prediction 4-9/22 Uncertainty in Prediction 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.

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Part 4: Prediction 4-10/22 Prediction from Internet Buzz Regression

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Part 4: Prediction 4-11/22 Prediction Interval for Buzz =.8

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Part 4: Prediction 4-12/22 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.

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Part 4: Prediction 4-13/22 Interval Estimates for the Sample of Monet Paintings Regression Analysis: ln (US$) versus ln (SurfaceArea) The regression equation is ln (US$) = ln (SurfaceArea) Predictor Coef SE Coef T P Constant ln (SurfaceArea) S = R-Sq = 20.0% R-Sq(adj) = 19.8% Mean of ln (SurfaceArea) =

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Part 4: Prediction 4-14/22 Prediction for An Out of Sample Monet Claude Monet: Bridge Over a Pool of Water Lilies Original, 36.5”x29.”

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Part 4: Prediction 4-15/22 Predicting y when the Model Describes log y

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Part 4: Prediction 4-16/ x Prediction by our model = $17.903M Painting is in our data set. Sold for 16.81M on 5/6/04 Sold for 7.729M 2/5/01 Last sale in our data set was in May 2004 Record sale was 6/25/08. market peak, just before the crash.

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Part 4: Prediction 4-17/22

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Part 4: Prediction 4-18/ ” (2 feet 8 inches) 26.2” (2 feet 2.2”) 167” (13 feet 11 inches) 78.74” (6 Feet 7 inch) "Morning", Claude Monet , oil on canvas 200 x 425 cm, Musée de l Orangerie, Paris France. Left panel

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Part 4: Prediction 4-19/22 Predicted Price for a Huge Painting

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Part 4: Prediction 4-20/22 Prediction Interval for Price

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Part 4: Prediction 4-21/22 Use the Monet Model to Predict a Price for a Dali? 118” (9 feet 10 inches) 157” (13 Feet 1 inch) Hallucinogenic Toreador 26.2” (2 feet 2.2”) 32.1” (2 feet 8 inches) Average Sized Monet

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Part 4: Prediction 4-22/22 Forecasting Out of Sample Per Capita Gasoline Consumption vs. Per Capita Income, How to predict G for 2017? You would need first to predict Income for How should we do that? Regression Analysis: G versus Income The regression equation is G = Income Predictor Coef SE Coef T P Constant Income S = R-Sq = 88.0% R-Sq(adj) = 87.8%

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