LinearRegression4

1 x i1 xi 2 xik yi i 12 n is a random sample

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Unformatted text preview: d then use OLS o Estimating the transformed equation by OLS is an example of generalized least squares (GLS) VER. 10/23/2012. © P. KOLM 55 Recall: Classical Linear Regression Assumptions (Multivariable Case) • Population model is linear in parameters: y = β0 + β1x1 + β2x 2 +…+ βk xk + u [MLR.1] • {(x i1, xi 2,…, xik , yi ) : i = 1,2,…, n} is a random sample from the population model, so that yi =β0 + β1xi1 + β2xi 2 +…+ βk xik + ui [MLR.2] • E(u | x1, x 2,…xk ) = 0 , implying that all of the explanatory variables are [MLR.3] uncorrelated with the error • None of the x ’s is constant, and there are no exact linear relationships among them4 [MLR.4] • Homoscedasticity: Assume Var (u | x 1, x 2,..., x k ) = σ 2 [MLR.5] • Normality: u ∼ N (0, σ 2 ) [MLR.6] (needed for hypothesis testing, etc.) → MLR.1-MLR.5 are known as the Gauss-Markov assumptions → MLR.1-MLR.6 are called the classical linear model assumptions (CLM) VER. 10/23/2012. © P. KOLM 56 The Standard Case: Heteroscedasticity Is Known up to a Multiplicative Constant Assume MLR.1-MLR.4 are valid, but Var (ui | x i 1,..., x ik ) = σ 2h(x i 1,..., x ik ) → MLR.5 is violated → The trick is to figure out what h(xi1,..., xik ) ≡ hi (the “heteroscedasticity function”) looks like ∗ If we know hi , we define ui = ui hi and see that Var (ui∗ | x i 1,..., x ik ) = σ 2 because Var (ui | x i 1,..., x ik ) = σ 2hi • So, if we divided our whole model by hi we would have a model where the error is homoscedastic! VER. 10/23/2012. © P. KOLM 57 Original model: yi = β0 + β1xi1 +…+ βk xik + ui with Var (ui ) = σ 2hi Transformed model: yi hi = β0 1 hi + β1 xi1 hi +…+βk xik hi + ui hi or ∗ ∗ yi∗ = β0x 0 + β1xi∗1 +…+ βk xik + ui∗ ∗ 0 where x = 1 hi ∗ ij ,x = xij hi etc., and Var (ui∗ ) = σ 2 The transformed model satisfies the Gauss-Markov (MLR.1-MLR.5)5 VER. 10/23/2012. © P. KOLM 58 Generalized Least Squares • GLS will be BLUE since the transformed model satisfies MLR.1-MLR.5 • GLS is a weighted least squares (WLS) procedure where each squared residual is weighted by the inverse of Var (ui | x i ) VER. 10/23/2012. © P. KOLM 59 GLS and WLS We can use OLS on the transformed model ∗ ∗ yi∗ = β0x 0 + β1xi∗1 +…+ βk xik + ui∗ where Var (ui∗ ) = σ 2 . Minimizing the squared residuals, we obtain n min ˆˆ ˆ β0 , β1 ,..., βk ∑ (u ) i =1 = ˆ min ˆ ˆ β0 , β1 ,..., βk 2 ∗ i n = ∑( i =1 ˆ∗ ˆ ˆ∗ yi∗ − β0x 0 − β1x i∗1 −… − βk x ik ) 2 = 2 ⎞ n⎛ xi1 x ik ⎟ ⎜ yi 1 ⎟ ˆ ˆ ˆ ⎟ = ˆ min ˆ ∑ ⎜ − β0 − β1 −…−βk ⎜ ⎟ ˆ ,..., β ⎜h β0 , β1 ⎟ k i =1 ⎜ hi hi hi ⎠ ⎟ ⎝i = ˆ min ˆ ˆ β0 , β1 ,..., βk VER. 10/23/2012. © P. KOLM n ˆ ˆ ˆ ∑ 1 / hi (yi − β0 − β1x i1 −…−βk x ik ) 2 i =1 60 Remarks • The last expression is just the usual OLS problem except each observation is weighted by 1 / hi o This is called weighted least squares (WLS) VER. 10/23/2012. © P. K...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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