Download presentation
Presentation is loading. Please wait.
Published byGabriel Armstrong Modified over 10 years ago
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
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.
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 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.
14
Expected value of b 2 Estimator for b 2 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 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
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
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.