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chapter seven

chapter seven. The Two-Variable Model: Hypothesis Testing. The Classical Linear Regression Model. Assumptions The regression model is linear in the parameters The explanatory variables, X, are uncorrelated with the error term

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chapter seven

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  1. chapter seven The Two-Variable Model: Hypothesis Testing

  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 Xi, the expected value of the disturbance term is zero: E(u|Xi) = 0. See Fig. 7-1.

  3. Figure 7-1 Conditional distribution of disturbances ui.

  4. More Assumptions of the CLRM • The variance of each ui is constant (homoscedastic): var(ui) = σ2 • 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(ui, uj) = 0 or the ui are random. • The model is correctly specified (no specification error or specification bias).

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

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

  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

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

  9. Gauss-Markov Theorem • Given the assumptions of the CLRM, the OLS estimators are BLUE. • b1 and b2 are linear estimators. • E(b1) = B1 and E(b2) = B2 in repeated applications the means of the estimators converge to the true values (unbiased). • The estimator of σ2 is unbiased. • b1 and b2 are efficient estimators (minimum variance among linear unbiased estimators).

  10. Sampling Distributions of OLS Estimators • The OLS estimators are normally distributed under the assumption that the error term ui of the PRF is normally distributed • b1~ N(B1, σb12), b2~ N(B2, σb22) • ui~ N(0, σ2) • Follows from the Central Limit Theorem and the property that any linear function of a normally distributed variable is normally distributed

  11. Figure 7-4 (Normal) sampling distributions of b1 and b2.

  12. Hypothesis Testing • Suppose we want to test the hypothesis H0: B2 = 0 • As b2 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

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

  14. Coefficient of Determination or r2 • 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

  15. Coefficient of Determination or r2 • The Total Sum of Squares (TSS) is composed of the Explained Sum of Squares (ESS) and the Residual Sum of Squares (RSS) • The r2 indicates the proportion of the total variation in Y explained by the sample regression function (SRF)

  16. Figure 7-8 Breakdown of total variation Yi.

  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.

  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. 7-14 • 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.

  19. Figure 7-14 95% confidence band for the true Lotto expenditure function.

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