Copyright © 2006 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin The Two-Variable Model: Hypothesis Testing chapter seven

7-2 The Classical Linear Regression Model Assumptions The regression model is linear in the parameters The explanatory variables, X, are uncorrelated with the error term Always true if X’s are nonstochastic (fixed numbers as in conditional regression analysis) Stochastic X’s require simultaneous equations models Given the value of X i, the expected value of the disturbance term is zero: E(u|X i ) = 0. See Fig. 7-1.

7-3 Figure 7-1 Conditional distribution of disturbances u i.

7-4 More Assumptions of the CLRM The variance of each u i is constant (homoscedastic): var(u i ) = σ 2 Individual Y values are spread around their mean with the same variance. See Fig. 7-2(a) Unequal variance is heteroscedasticity, Fig. 7-2(b) There is no correlation across the error terms. Or no autocorrelation. See Fig. 7-3 Cov(u i, u j ) = 0 or the u i are random. The model is correctly specified (no specification error or specification bias).

7-5 Figure 7-2 (a) Homoscedasticity (equal variance); (b) Heteroscedasticity (unequal variance).

7-6 Figure 7-3 Patterns of autocorrelation: (a) No autocorrelation; (b) positive autocorrelation; (c) negative autocorrelation.

7-7 Variances and Standard Errors The CLRM assumptions allow us to estimate the variances and standard errors of the OLS estimators. Note n - 2 is the degrees of freedom (or n – k) Standard error of the regression

7-8 Table 7-1 Computations for the lotto example.

7-9 Gauss-Markov Theorem Given the assumptions of the CLRM, the OLS estimators are BLUE. b 1 and b 2 are linear estimators. E(b 1 ) = B 1 and E(b 2 ) = B 2 in repeated applications the means of the estimators converge to the true values (unbiased). The estimator of σ 2 is unbiased. b 1 and b 2 are efficient estimators (minimum variance among linear unbiased estimators).

7-10 Sampling Distributions of OLS Estimators The OLS estimators are normally distributed under the assumption that the error term u i of the PRF is normally distributed b 1 ~ N(B 1, σ b1 2 ), b 2 ~ N(B 2, σ b2 2 ) u i ~ N(0, σ 2 ) Follows from the Central Limit Theorem and the property that any linear function of a normally distributed variable is normally distributed

7-11 Figure 7-4 (Normal) sampling distributions of b 1 and b 2.

7-12 Hypothesis Testing Suppose we want to test the hypothesis H 0 : B 2 = 0 As b 2 is normally distributed, we could use the standard normal distribution for hypotheses about its mean, except that the variance is unknown. Use the t distribution (estimator-value)/se

7-13 Figure 7-7 One-tailed t test: (a) Right-tailed; (b) left-tailed.

7-14 Coefficient of Determination or r 2 How good is the fitted regression line? Write the regression relationship in terms of deviations from mean values, then square it and sum over the sample The parts can be interpreted individually

7-15 Coefficient of Determination or r 2 The Total Sum of Squares (TSS) is composed of the Explained Sum of Squares (ESS) and the Residual Sum of Squares (RSS) The r 2 indicates the proportion of the total variation in Y explained by the sample regression function (SRF)

7-16 Figure 7-8 Breakdown of total variation Y i.

7-17 Reporting Results of Regression Analysis For simple regression in the Lotto example → For multiple equations and/or explanatory variables see Table II in schooltrans.doc.

7-18 Caution: Forecasting While we can calculate an estimate of Y for any given value of X using regression results As the X value chosen departs from the mean value of X, the variance of the Y estimate increases Consider the Lotto example, Fig Forecasts of Y for X’s far away from their mean and/or outside the range of the sample are unreliable and should be avoided.

7-19 Figure % confidence band for the true Lotto expenditure function.