Statistical Inference and Regression Analysis: GB.3302.30 Professor William Greene Stern School of Business IOMS Department Department of Economics
Statistics and Data Analysis Part 6 – Regression Model-1 Conditional Mean
U.S. Gasoline Price 6 Months 5 Years
Impact of Change in Gasoline Price on Consumer Demand? • Demand for gasoline • Long term vs. short term • Income • Elasticity concepts • Demand for food
Internet Buzz and Movie Success Box office sales vs. Can’t wait votes 3 weeks before release
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?
Covariation – Education and Life Expectancy Causality? Covariation? Does more education make people live longer? Is there a hidden driver of both? (Per capita GDP?)
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.”
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.
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
Average Box Office by Internet Buzz Index= Average Box Office for Buzz in Interval
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
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.
Inference and Regression Least Squares
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
Computing the Least Squares Parameters a and b (We will use sy2 later.)
Inference and Regression Regression Model
b Measures Covariation Predictor Box Office = a + b Buzz.
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)
Covariation and Causality Does more education make you live longer (on average)?
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
Inference and Regression Analysis of Variance
Regression Fits Regression of salary vs. years Regression of fuel bill vs. number of experience of rooms for a sample of homes
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)
Regression Fits R2=0.360 R2=0.522 R2=0.424 R2=0.880
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.
Correlations rxy = 0.6 rxy = 0.723 rxy = +1.000 rxy = -.402
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.
Squared Correlations rxy2 = 0.36 rxy2 = 0.522 rxy2 = .924 rxy2 = .161
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