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FormulaSheetFinal

# FormulaSheetFinal - Formula sheet for final Classical...

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Formula sheet for final Classical Linear Regression (CLR) model The n observations iK i i X X Y , , , 1 K , n i , , 1 K = , on a dependent variable Y and K independent variables K X X , , 1 K satisfy (1) i iK K i i i u X X X Y + + + + = b b b L 2 2 1 1 for n i , , 1 K = . Assumption 1: n i u i , , 1 , K = are random variables with 0 ) ( = i u E Assumption 2: K k n i X ik , , 1 , , , 1 , K K = = are deterministic, i.e. non-random, constants. Assumption 3: (Homoskedasticity) All s u i ' have the same variance, i.e. for n i , , 1 K = (2) 2 2 ) ( ) ( s = = i i u E u Var Assumption 4 (No serial correlation) The random errors i u and j u are not correlated for all n j i , , 1 K = (3) 0 ) ( ) , ( = = j i j i u u E u u Cov For the CLR model with normal errors we make the additional assumption: Assumption 5. The random error terms n i u i , , 1 , K = are random variables with a normal distribution.

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Ordinary Least Squares (OLS) estimators in the simple CLR model and their sampling variance (4) = = - - - = n i i n i i i X X Y Y X X 1 2 1 ) ( ) )( ( ˆ b (5) X Y b a ˆ ˆ - = (6) 2 1 2 1 2 ) ( ) ˆ ( s a = = - = n i i n i i X X n X Var (7) = - = n i i X X Var 1 2 2 ) ( ) ˆ ( s b (8) 2 1 2 ) ( ) ˆ , ˆ ( s b a = - - = n i i X X X Cov Unbiased estimator of 2 s (9) = - = n i i e K n s 1 2 2 1 Substitution in the sampling variances gives the estimated sampling variances. The square root of these are the standard errors ) ˆ ( ), ˆ ( b a std std or in the multiple CLR model K k std k , , 1 ), ˆ ( K = b . Goodness-of-fit Predicted/computed value: (10) iK K i i i X X X Y b b b ˆ ˆ ˆ ˆ 2 2 1 1 + + + = L OLS residual: (11) iK K i i i i X X X Y e b b b ˆ ˆ ˆ 2 2 1 1 - - - - = L Properties of OLS residuals: (12) K k e X n i i ik , , 1 , 0 1 K = = = (with a constant term this implies sum of OLS residuals is 0) Decomposition of total variation: (13) = = = + - = - n i i n i i n i i e Y Y Y Y 1 2 1 2 1 2 ) ˆ ˆ ( ) ( Note: Y Y ˆ = Total Sum of Regression Error Sum of Squares (TSS) Sum of Squares (ESS) Squares (RSS) (14) TSS ESS TSS RSS R - = = 1 2
Confidence interval for k b in CLR model with normal errors

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FormulaSheetFinal - Formula sheet for final Classical...

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