LinearRegression4

# xk in this case we need to estimate it it is common

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Unformatted text preview: OLM 61 What If We Don't Know the Form of Heteroscedasticity? • GLS is great if we know what Var (u | x ) = σ 2h(x ) looks like • But in most cases, we don't know the exact form of heteroscedasticity • Therefore, in practice we need something else That “something else” is called “Feasible GLS” (FGLS): • First, we estimate the heteroscedasticity function (the weights) • Second, we perform weighted least squares (WLS) with the estimated weights VER. 10/23/2012. © P. KOLM 62 Feasible GLS (1/3) • In practice, it is common that we don’t know the form of the heteroscedasticity, h(x1,..., xk ) • In this case, we need to estimate it • It is common to assume a fairly flexible model parametric, such as Var (u | x 1,..., x k ) = σ 2exp(δ0 + δ1x 1 +… + δk x k ) Idea: Since we don’t know the δi ’s, we estimate them! VER. 10/23/2012. © P. KOLM 63 Feasible GLS (2/3) Our assumption implies that u 2 = σ 2exp (δ0 + δ1x 1 +… + δk x k )v where E (v | x 1,..., x k ) = 1 If we assume that v is independent of x 1,..., x k , we can take the logarithm () log u 2 = α0 + δ1x 1 + … + δk xk + e where E (e ) = 0 and e is independent of the xi ’s • Using û as an estimate of u , we estimate the δi ’s by OLS ˆ ˆ • The estimates of h are h = exp(g ) , where i i i ˆ ˆ ˆ ˆ gi = α0 + δ1x i 1 + … + δk x ik VER. 10/23/2012. © P. KOLM 64 Feasible GLS (3/3) Summary of the Feasible GLS procedure: 1. Estimate the model y = β0 + β1x1 +…+ βk xk + u ˆ and save the residuals, u , ˆ 2. Calculate log (u 2 ) ˆ 3. Regress log (u 2 ) on all of the independent variables and get the fitted values, ˆ ˆ g1,..., gn 4. Estimate the model y = β0 + β1x1 +…+ βk xk + u 1 1 using WLS with the weights ≡ ˆ hi ˆ exp (gi ) (Alternatively, we can transform the model by multiplying each observation i ˆ by 1 / exp(gi ) and do OLS) VER. 10/23/2012. © P. KOLM 65 Example (C8.11 in Wooldridge) Use the data in 401KSUBS.RAW to answer this question, restricting attention to single-person households (fsize = 1). Estimate the equation (i) nettfa = β0 + β1inc + β2age + β3age 2 + β4male + β5e401k by OLS. Report the usual standard errors as well as the heteroscedasticityrobust standard errors. Comment on any notable differences. Solution: In the following equation, estimated by OLS, the usual standard errors are in (⋅) and the heteroscedasticity-robust standard errors are in [⋅]: nettfa = 1.50 + 0.774inc − 1.60age + 0.029age 2 + 2.47male + 6.98e401k (15.3) (0.062) (0.770) (0.0090) (2.05) (2.13) [19.1] [0.101] [1.08] [0.0140] [2.05] [2.19] n = 2,017, R2 = 0.128. The heteroscedasticity-robust standard errors are larger than the OLS standard errors (almost twice as large for some variables). Most notably, the t statistic for VER. 10/23/2012. © P. KOLM 66 variable age changed from −2.07 to −1.47 and the coefficient becomes insignificant at 5% level when using the heteroscedasticity-robust standard error. Ordinary Least-squares Estimates Dependent Variable = net tfa R-squared = 0.1280 Rbar-squared = 0.125...
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