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CHAPTER 3 ECONOMETRICS x x x x x Chapter 2: Estimating the parameters of a linear regression model. Y i = b 1 + b 2 X i + e i Using OLS Chapter 3: Testing hypotheses about the parameters we estimated. The most common null (default) hypothesis is that a parameter is equal to zero. We test to see if we can reject the null hypothesis.

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The Classical Linear Regression Model 1. The regression model is linear in the parameters. Assumptions Y i = B 1 + B 2 ln(X i 2 ) + u i Y i = B 1 + B 2 X i B 3 + u i

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2. The explanatory variables are uncorrelated with the disturbance term. In general, we consider the explanatory variables fixed. Only the dependent variable is subject to random disturbances (i.e. stochastic.) 3. The expected value of the disturbance term is zero for any X i. E( u | X i ) = 0 For any X i, the disturbances are just as likely to be positive as negative

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4. The variance of each u i is constant. The model is homoscedastic. var( u i ) = σ 2 5. There is no correlation between two error terms.

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6. The model is correctly specified. We have included the right variables in the model. Variances and Stnd Errors of OLS Estimators We estimate regression coefficients. These estimators are random variables because their values change from sample to sample. Q UPS = 511 − 26.2 P UPS + 6.5 P FEDX 1 st sample Q UPS = 432 − 21.7 P UPS – 1.7 P FEDX 2 nd sample

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From this we estimate the variance of the parameter estimates : We estimate the variance of the disturbance term in the population from the residuals in the sample. Y i = b 1 + b 2 X i + e i Estimate var(b 1 )Estimate var(b 2 ) Note: OLS provides parameter estimates that are unbiased and efficient (minimum variance.)

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7. The disturbance term is normally distributed. One Last Assumption u i ~ N(0, σ 2 ) If so, then the estimators are normally distributed. b 1 ~ N(B 1, σ b1 2 )b 2 ~ N(B 2, σ b2 2 ) b1b1 Note: The stnd deviation of an estimator, σ b1, is usually called the standard error.

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We don’t know the variance of the population disturbance term, σ 2. We only have an estimate, σ 2. But, if we standardize the estimator by σ 2, the result follows the t-distribution (similar to the normal). ~ t n-2 b 1 – B 1 se(b 1 ) b 1 ~ N(B 1, σ b1 2 ) So we can’t use this normal distribution to test our hypothesis. So we use this t-distribution to test our hypothesis.

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