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

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Expenditure by households of a given income on food

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

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Econometric model

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Combines the economic model with assumptions about the random nature of the data. Dispersion. Independence of y i and y j. x i is non-random.

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

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Properties of the error term

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Assumptions of the simple linear regression model

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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.

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Fitting a line

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

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The least squares estimators

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Least Squares Estimates When data are used with the estimators, we obtain estimates. Estimates are a function of the y t 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.

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Expected value of b 2 Estimator for b 2 can be written: Taking expectations:

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Variances and covariances

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Comparing the least squares estimators with other estimators Gauss-Markov Theorem: Under the assumptions SR1-SR5 of the linear regression model the estimators b 1 and b 2 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

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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.

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Estimating the variance of the error term

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Estimating the variances and covariances of the LS estimators

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