Heteroscedasticity-consistent standard errors
Encyclopedia
The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics
and econometrics
in the context of linear regression
and also time series analysis. The alternative names of Huber–White standard errors, Eicker–White or Eicker–Huber–White are also frequently used in relation to the same ideas.
In regression and time-series modelling, basic forms of models make use of the
assumption that the errors or disturbances ui have the same variance across all observation points. When this is not the case, the errors are said to be heteroscedastic, or to have heteroscedasticity, and this behaviour will be reflected in the residuals
estimated from a fitted model. Heteroscedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroscedastic residuals. The first such approach was proposed by White (1980), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
in the context of linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
and also time series analysis. The alternative names of Huber–White standard errors, Eicker–White or Eicker–Huber–White are also frequently used in relation to the same ideas.
In regression and time-series modelling, basic forms of models make use of the
assumption that the errors or disturbances ui have the same variance across all observation points. When this is not the case, the errors are said to be heteroscedastic, or to have heteroscedasticity, and this behaviour will be reflected in the residuals
estimated from a fitted model. Heteroscedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroscedastic residuals. The first such approach was proposed by White (1980), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation.Definition
Assume that we are regressing the linear regression model-
where X is the design matrix and β is a column vector of parameters to be estimated.
The ordinary least squaresOrdinary least squaresIn statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...
(OLS) estimator is
-
If the residualsErrors and residuals in statisticsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
all have the same variance σ2 and are uncorrelatedUncorrelatedIn probability theory and statistics, two real-valued random variables are said to be uncorrelated if their covariance is zero. Uncorrelatedness is by definition pairwise; i.e...
, then the least-squares estimates of β is called BLUEBlueBlue is a colour, the perception of which is evoked by light having a spectrum dominated by energy with a wavelength of roughly 440–490 nm. It is considered one of the additive primary colours. On the HSV Colour Wheel, the complement of blue is yellow; that is, a colour corresponding to an equal...
(best linear unbiased estimator).
However, all the necessary assumptions may not be valid. For example, suppose the residuals have variances σi2 and the OLS variance estimator is
where
then OLS estimator is not "best" in the sense of having minimum mean square error (but it is unbiased) and the OLS variance estimator does not provide a consistent estimate of the variance of the residuals.
There are many different kinds of heteroscedasticity, however, and one should use caution when constructing heteroscedastic robust standard errors. HC (heteroscedasticity-consistent) estimators are recommended to deal with this problem.
White's heteroscedasticity-consistent estimator
White's (1980) HC estimator, often referred to as HCE, has the estimator
-
The estimator can be derived in terms of the generalized method of momentsGeneralized method of momentsIn econometrics, generalized method of moments is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data...
(GMM).
See also
- Generalized estimating equations
- White testWhite testIn statistics, the White test is a statistical test that establishes whether the residual variance of a variable in a regression model is constant: that is for homoscedasticity....
— a test for whether heteroscedasticity is present.
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