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The Simple Linear Regression Model Specification and Estimation

The Simple Linear Regression Model Specification and Estimation. Hill et al Chs 3 and 4. Expenditure by households of a given income on food. Economic Model. Assume that the relationship between income and food expenditure is linear: But, expenditure is random:

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The Simple Linear Regression Model Specification and Estimation

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  1. The Simple Linear Regression Model Specification and Estimation Hill et al Chs 3 and 4

  2. Expenditure by households of a given income on food

  3. Economic Model • Assume that the relationship between income and food expenditure is linear: • But, expenditure is random: • Known as the regression function.

  4. Econometric model

  5. Econometric model • Combines the economic model with assumptions about the random nature of the data. • Dispersion. • Independence of yi and yj. • xi is non-random.

  6. Writing the model with an error term • An observation can be decomposed into a systematic part: • the mean; • and a random part:

  7. Properties of the error term

  8. Assumptions of the simple linear regression model

  9. The error term • Unobservable (we never know E(y)) • Captures the effects of factors other than income on food expenditure: • Unobservered factors. • Approximation error as a consequence of the linear function. • Random behaviour.

  10. Fitting a line

  11. The least squares principle • Fitted regression and predicted values: • Estimated residuals: • Sum of squared residuals:

  12. The least squares estimators

  13. Least Squares Estimates • When data are used with the estimators, we obtain estimates. • Estimates are a function of the yt which are random. • Estimates are also random, a different sample with give different estimates. • Two questions: • What are the means, variances and distributions of the estimates. • How does the least squares rule compare with other rules.

  14. Expected value of b2 Estimator for b2 can be written: Taking expectations:

  15. Variances and covariances

  16. Comparing the least squares estimators with other estimators Gauss-Markov Theorem: Under the assumptions SR1-SR5 of the linear regression model the estimators b1 and b2 have the smallest variance of all linear and unbiased estimators of1 and 2. They are the Best Linear Unbiased Estimators (BLUE) of 1 and 2

  17. The probability distribution of least squares estimators • Random errors are normally distributed: • estimators are a linear function of the errors, hence they a normal too. • Random errors not normal but sample is large: • asymptotic theory shows the estimates are approximately normal.

  18. Estimating the variance of the error term

  19. Estimating the variances and covariances of the LS estimators

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