Assumptions of the simple linear regression model
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.
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.
Expected value of b 2 Estimator for b 2 can be written: Taking expectations:
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
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.