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- | '''Ordinary least squares''' ('''OLS''') is is a well-known method for linear regression by minimizing squared squared residuals between the predicted and observed values.
| + | #REDIRECT [[least squares method]] |
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- | Assuming a linear regression model:
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- | :<math> \mathbf{Y}=\mathbf{X}\theta + \varepsilon </math>
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- | The equation can rearranged to place error component on the left hand side and then take squares on both sides:
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- | :<math> \varepsilon^2 = \|\mathbf{X}\theta-\mathbf{Y}\|^2</math>
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- | To minimize the residuals, if there exist least-squares estimator <math>\widehat{\theta}</math>, the first-order condition of minimization should be satisfied. Thus,
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- | :<math> \|\mathbf{X}\widehat{\theta}-\mathbf{Y}\|\mathbf{X}=\mathbf{0} </math>
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- | :<math> \mathbf{X'X}\widehat{\theta}-\mathbf{X'Y}=\mathbf{0} </math>
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- | The least-squares estimator is:
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- | :<math>\widehat{\theta}=(\mathbf{X}'\mathbf{X})^{-1} \mathbf{X'Y}</math>
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- | However, using OLS estimators to estimate structural parameters causes biased and inconsistent because included endogenous variables in each equation are correlated with the disturbances.
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- | =Reference=
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- | *Greene, W.H., 2003, Econometric analysis, Prentice Hall, Upper Saddle River, N.J.
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Current revision
- REDIRECT least squares method