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
• The estimates of h are h = exp(g ) , where
i i i 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
and save the residuals, u ,
2. Calculate log (u 2 )
3. Regress log (u 2 ) on all of the independent variables and get the fitted values, ˆ
4. Estimate the model y = β0 + β1x1 +…+ βk xk + u
using WLS with the weights
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
[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 =
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- Fall '14
- Regression Analysis, P. KOLM, Petter Kolm