White Error Standard Errors

Standard errors in the form of White or heteroskedasticity consistent standard errors ( HC se - Heteroskedasticity consistent standard errors ) - an econometric estimate of the covariance matrix (in particular standard errors) of OLS estimates of the parameters of a linear regression model that is consistent with heteroskedasticity of random model errors, alternative to the standard (classical) assessment, which in this case is untenable.

Content

Essence and Formula

The true covariance matrix of the OLS estimates of the parameters of the linear model in the general case is equal to:

$V({\hat {b}}_{OLS})=(X^{T}X)^{-1}(X^{T}VX)(X^{T}X)^{-1}$ ${\ displaystyle V ({\ hat {b}} _ {OLS}) = (X ^ {T} X) ^ {- 1} (X ^ {T} VX) (X ^ {T} X) ^ {- one}}$

where V is the covariance matrix of random errors. In case there is no heteroskedasticity and autocorrelation (i.e., when $V=\sigma ^{2}I$ ${\ displaystyle V = \ sigma ^ {2} I}$ ) the formula is simplified

${\hat {V}}({\hat {b}}_{OLS})={\sigma }^{2}(X^{T}X)^{-1}$ ${\ displaystyle {\ hat {V}} ({\ hat {b}} _ {OLS}) = {\ sigma} ^ {2} (X ^ {T} X) ^ {- 1}}$

Therefore, to estimate the covariance matrix in the classical case, it suffices to use the estimate of a single parameter - the variance of random errors: $s^{2}=ESS/(n-k)$ ${\ displaystyle s ^ {2} = ESS / (nk)}$ , which, as can be proved, is an unbiased and consistent assessment.

In the general case, however, some estimate of the unknown covariance matrix is needed. In particular, if heteroskedasticity is assumed in the absence of autocorrelation, the covariance matrix of random errors is diagonal and all diagonal elements $\sigma _{t}^{2}$ ${\ displaystyle \ sigma _ {t} ^ {2}}$ unknown. In this case, the general expression for the covariance matrix of estimates can be written as:

$V({\hat {b}}_{OLS})=(X^{T}X)^{-1}(\sum _{t=1}^{n}\sigma _{t}^{2}x_{t}x_{t}^{T})(X^{T}X)^{-1}$ ${\ displaystyle V ({\ hat {b}} _ {OLS}) = (X ^ {T} X) ^ {- 1} (\ sum _ {t = 1} ^ {n} \ sigma _ {t} ^ {2} x_ {t} x_ {t} ^ {T}) (X ^ {T} X) ^ {- 1}}$

White (White, 1980) showed that if instead of unknown variances of errors, the squares of the regression residuals are used in this formula, a consistent estimate is obtained:

${\hat {V}}({\hat {b}}_{OLS})=(X^{T}X)^{-1}(\sum _{t=1}^{n}e_{t}^{2}x_{t}x_{t}^{T})(X^{T}X)^{-1}$ ${\ displaystyle {\ hat {V}} ({\ hat {b}} _ {OLS}) = (X ^ {T} X) ^ {- 1} (\ sum _ {t = 1} ^ {n} e_ {t} ^ {2} x_ {t} x_ {t} ^ {T}) (X ^ {T} X) ^ {- 1}}$

It should be noted that this estimate is consistent only in the absence of autocorrelation of random errors (that is, as was described in the case of a diagonal covariance matrix of random errors). If there is also autocorrelation, then standard errors in the form of Newey West can be used.

Note

Sometimes the given formula for estimating the covariance matrix is adjusted by a factor $n/(n-k)$ ${\ displaystyle n / (nk)}$ . Such a correction theoretically allows obtaining more accurate estimates on small samples. At the same time, in large samples (asymptotically), these estimates are equivalent.

Literature

Magnus J.R., Katyshev P.K., Peresetsky A.A. Econometrics. - M .: Case, 2004 .-- 576 p.
William H. Greene. Econometric analysis. - New York: Pearson Education, Inc., 2003 .-- 1026 p.

White Error Standard Errors

Content

Essence and Formula

Note

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