# Part 1: Simple Linear Model 1-1/301-1 Regression Models Professor William Greene Stern School of Business IOMS Department Department of Economics.

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Part 1: Simple Linear Model 1-1/301-1 Regression Models Professor William Greene Stern School of Business IOMS Department Department of Economics

Part 1: Simple Linear Model 1-2/301-2 Regression and Forecasting Models Part 1 – Simple Linear Model

Part 1: Simple Linear Model 1-3/30 Theory  Demand Theory: Q = f(Price)  “The Law of Demand” Demand curves slope downward  What does “ceteris paribus” mean here?

Part 1: Simple Linear Model 1-4/30 Data on the U.S. Gasoline Market Quantity = G = Expenditure / Price

Part 1: Simple Linear Model 1-5/30 Shouldn’t Demand Curves Slope Downward?

Part 1: Simple Linear Model 1-6/30 Data on 62 Movies in 2010

Part 1: Simple Linear Model 1-7/30 Average Box Office Revenue is about \$20.7 Million

Part 1: Simple Linear Model 1-8/30 Is There a Theory for This? Scatter plot of box office revenues vs. number of “Can’t Wait To See It” votes on Fandango for 62 movies.

Part 1: Simple Linear Model 1-9/30 Average Box Office by Internet Buzz Index = Average Box Office for Buzz in Interval

Part 1: Simple Linear Model 1-10/30 Deterministic Relationship: Not a Theory Expected High Temperatures, August 11-20, 2013, ZIP 10012, NY

Part 1: Simple Linear Model 1-11/30 Probabilistic Relationship What Explains the Noise? Fuel Bill = Function of Rooms + Random Variation

Part 1: Simple Linear Model 1-12/30 Movie Buzz Data Probabilistic Relationship?

Part 1: Simple Linear Model 1-13/30 The Regression Model y =  0 +  1 x +  y = dependent variable x = independent variable The ‘regression’ is the deterministic part,  0 +  1 x The ‘disturbance’ (noise) is . The regression model is E[y|x] =  0 +  1 x

Part 1: Simple Linear Model 1-14/30  0 = y intercept  1 = slope E[y|x] =  0 +  1 x y x Linear Regression Model

Part 1: Simple Linear Model 1-15/30 The Model  Constructed to provide a framework for interpreting the observed data What is the meaning of the observed relationship (assuming there is one)  How it’s used Prediction: What reason is there to assume that we can use sample observations to predict outcomes? Testing relationships

Part 1: Simple Linear Model 1-16/30 The slope is the interesting quantity. Each additional year of education is associated with an increase of 3.611 in disability adjusted life expectancy.

Part 1: Simple Linear Model 1-17/30 A Cost Model Electricity.mpj Total cost in \$Million Output in Million KWH N = 123 American electric utilities Model: Cost =  0 +  1 KWH + ε

Part 1: Simple Linear Model 1-18/30 Cost Relationship

Part 1: Simple Linear Model 1-19/30 Sample Regression

Part 1: Simple Linear Model 1-20/30 Interpreting the Model  Cost = 2.44 + 0.00529 Output + e  Cost is \$Million, Output is Million KWH.  Fixed Cost = Cost when output = 0 Fixed Cost = \$2.44Million  Marginal cost = Change in cost/change in output =.00529 * \$Million/Million KWH =.00529 \$/KWH = 0.529 cents/KWH.

Part 1: Simple Linear Model 1-21/30 Covariation and Causality Does more education make you live longer (on average)?

Part 1: Simple Linear Model 1-22/30 Causality? Height (inches) and Income (\$/mo.) in first post-MBA Job (men). WSJ, 12/30/86. Ht. Inc. Ht. Inc. Ht. Inc. 70 2990 68 2910 75 3150 67 2870 66 2840 68 2860 69 2950 71 3180 69 2930 70 3140 68 3020 76 3210 65 2790 73 3220 71 3180 73 3230 73 3370 66 2670 64 2880 70 3180 69 3050 70 3140 71 3340 65 2750 69 3000 69 2970 67 2960 73 3170 73 3240 70 3050 Estimated Income = -451 + 50.2 Height

Part 1: Simple Linear Model 1-23/30 b0b0 b1b1 How to compute the y intercept, b 0, and the slope, b 1, in y = b 0 + b 1 x.

Part 1: Simple Linear Model 1-24/30 Least Squares Regression

Part 1: Simple Linear Model 1-25/30 Fitting a Line to a Set of Points Choose b 0 and b 1 to minimize the sum of squared residuals Gauss’s method of least squares. Residuals YiYi XiXi Predictions b 0 + b 1 x i

Part 1: Simple Linear Model 1-26/30 Computing the Least Squares Parameters b 0 and b 1

Part 1: Simple Linear Model 1-27/30 b 0 =-14.36 b 1 = 72.718

Part 1: Simple Linear Model 1-28/30 Least Squares Uses Calculus

Part 1: Simple Linear Model 1-29/30 Least squares minimizes the sum of squared deviations from the line.

Part 1: Simple Linear Model 1-30/30 Summary  Theory vs. practice  Linear Relationship Deterministic Random, stochastic, ‘probabilistic’ Mean is a function of x  Regression Relationship Causality vs. correlation Least squares

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