Ordinary least squares

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Revision as of 11:18, 1 February 2008 (view source) NormanShiau (Talk | contribs) (New page: '''Ordinary least squares''' ('''OLS''') is is a well-known method for linear regression by minimizing squared squared residuals between the predicted and observed values. Assuming a line...) ← Older edit		Current revision (19:06, 19 February 2008) (view source) Amhamilt (Talk | contribs) (Redirecting to Least squares method)
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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:	+
-	:<math> \mathbf{Y}=\mathbf{X}\theta + \varepsilon </math>	+
-	The equation can rearranged to place error component on the left hand side and then take squares on both sides:	+
-	:<math> \varepsilon^2 = \|\mathbf{X}\theta-\mathbf{Y}\|^2</math>	+
-	To minimize the residuals, if there exist least-squares estimator <math>\widehat{\theta}</math>, the first-order condition of minimization should be satisfied. Thus,	+
-	:<math> \|\mathbf{X}\widehat{\theta}-\mathbf{Y}\|\mathbf{X}=\mathbf{0} </math>	+
-	:<math> \mathbf{X'X}\widehat{\theta}-\mathbf{X'Y}=\mathbf{0} </math>	+
-	The least-squares estimator is:	+
-	:<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=	+
-	*Greene, W.H., 2003, Econometric analysis, Prentice Hall, Upper Saddle River, N.J.	+

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1. REDIRECT least squares method