Statistical Inference and Regression Analysis: GB.3302.30

1. Statistical Inference and Regression Analysis: GB.3302.30 Professor William Greene Stern School of Business IOMS Department Department of Economics

2. Statistics and Data Analysis Part 6 – Regression Model-1 Conditional Mean

3. U.S. Gasoline Price 6 Months 5 Years

4. Impact of Change in Gasoline Price on Consumer Demand? • Demand for gasoline • Long term vs. short term • Income • Elasticity concepts • Demand for food

5. Movie Success vs. Movie Online Buzz Before Release (2009)

7. Internet Buzz and Movie Success Box office sales vs. Can’t wait votes 3 weeks before release

8. Is There Really a Relationship? BoxOffice is obviously not equal to f(Buzz) for some function. But, they do appear to be “related,” perhaps statistically – that is, stochastically. There is a covariance. The linear regression summarizes it. A predictor would be Box Office = a + b Buzz. Is b really > 0? What would be implied by b > 0?

9. Covariation – Education and Life Expectancy Causality? Covariation? Does more education make people live longer? Is there a hidden driver of both? (Per capita GDP?)

10. Using Regression to Predict The equation would not predict Titanic. Predictor: Overseas box office = a + b Domestic box officeThe prediction will not be perfect. We construct a range of “uncertainty.”

11. Conditional Variation and Regression • Conditional distribution of a pair of random variables • f(y|x) or P(y|x) • Mean function, E[y|x] = Regression of y on x.

12. Expected Income Depends on Household Size X=4 X=3 X=2 X=1 y|x ~ Normal[ 20 + 3x, 42 ], x = 1,2,3,4; Poisson

13. Average Box Office by Internet Buzz Index= Average Box Office for Buzz in Interval

14. Linear Regression?Fuel Bills vs. Number of Rooms

15. Independent vs. Dependent Variables • Y in the model • Dependent variable • Response variable • X in the model • Independent variable: Meaning of ‘independent’ • Regressor • Covariate • Conditional vs. joint distribution

16. Linearity and Functional Form • y = g(x) • h(y) =  + f(x) • y =  + x • y = exp( + x); logy =  + x • y =  +  (1/x) =  + f(x) • y = e x, logy =  + log x. • Etc.

17. Inference and Regression Least Squares

18. Fitting a Line to a Set of Points Yi Gauss’s methodof least squares. Residuals Predictionsa + bxi Choose and tominimize the sum of squared residuals Xi

19. Least Squares Regression

21. Least Squares Algebra

22. Least Squares

23. Normal Equations

24. Computing the Least Squares Parameters a and b (We will use sy2 later.)

25. Least Absolute Deviations

26. Least Squares vs. LAD

27. Inference and Regression Regression Model

28. b Measures Covariation Predictor Box Office = a + b Buzz.

29. Interpreting the Function a = the life expectancy associated with 0 years of education. No country has 0 average years of education. The regression only applies in the range of experience. b = the increase in life expectancy associated with each additional year of average education. b a The range of experience (education)

30. Covariation and Causality Does more education make you live longer (on average)?

31. Causality? Correlation = 0.84 (!) 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

32. Inference and Regression Analysis of Variance

33. Regression Fits Regression of salary vs. years Regression of fuel bill vs. number of experience of rooms for a sample of homes

34. Regression Arithmetic

35. Variance Decomposition

36. Fit of the Equation to the Data

37. Regression vs. Residual SS

38. Analysis of Variance Table

39. Explained Variation • The proportion of variation “explained” by the regression is called R-squared (R2) • It is also called the Coefficient of Determination • (It is the square of something – to be shown later)

40. ANOVA Table

41. Movie Madness Fit

42. Regression Fits R2=0.360 R2=0.522 R2=0.424 R2=0.880

43. R Squared Benchmarks • Aggregate time series: expect .9+ • Cross sections, .5 is good. Sometimes we do much better. • Large survey data sets, .2 is not bad. R2 = 0.924 in this cross section.

44. Correlation Coefficient

45. Correlations rxy = 0.6 rxy = 0.723 rxy = +1.000 rxy = -.402

46. R-Squared is rxy2 • R-squared is the square of the correlation between yi and the predicted yi which is a + bxi. • The correlation between yi and (a+bxi) is the same as the correlation between yi and xi. • Therefore,…. • A regression with a high R2 predicts yi well.

47. Squared Correlations rxy2 = 0.36 rxy2 = 0.522 rxy2 = .924 rxy2 = .161

48. Movie Madness Estimated equation Estimated coefficients a and b S = se = estimated std. deviation of ε Square of the sample correlation between x and y N-2 = degrees of freedom Sum of squared residuals, Σiei2 S2 = se2

49. Software