The Simple Linear Regression Model Specification and Estimation

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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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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.

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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 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.

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Expected value of b2
Estimator for b2 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 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
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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