Standard Errors

Standard errors are the standard deviations of estimated coefficients.

Description

In the classical OLS model,, estimated coefficients are:

• univariate case:

• multivariate case:

Standard errors are the standard deviations of these coefficients.

Classical

Univariate

In the univariate case, standard errors are classically specified as:

Supposing the population Var(ε) is known and errors are homoskedastic, i.e. they are constant across all cases, this can be simplified.

Lastly, rewrite the denominator in terms of Var(X).

Var(ε) is unknown, so this term is estimated as:

,

1 degree of freedom is lost in assuming homoskedasticity of errors, i.e. ; and k degrees of freedom are lost in assuming independence of errors and k independent variables, which is necessarily 1 in the univariate case, i.e.:

This arrives at estimation as:

Multivariate

The classical multivariate specification is expressed in terms of (b-β), as:

That term is rewritten as (X^TX)^-1Xε.

E[εε^T|X] is not a practical matrix to work with, even if known. But if homoskedasticity and independence are assumed, i.e.: , then this simplifies to:

s² is unknown, so this term is estimated as:

This arrives at estimation as:

Robust

In the presence of heteroskedasticity of errors, the above simplifications cannot apply. In the univariate case, use the original estimator.

This is mostly interesting in the multivariate case, where E[εε^T|X] is still not practical. The assumptions made, when incorrect, lead to...

• OLS estimators are not BLUE
  • they are unbiased, but no longer most efficient in terms of MSE
• nonlinear GLMs, such as logit, can be biased
• even if the model's estimates are unbiased, statistics derived from those estimates (e.g., conditioned probability distributions) can be biased

Eicker-Huber-White heterskedasticity consistent errors (HCE) assume that errors are still independent but allowed to vary, i.e. Σ = diag(ε₁,...ε_n). Importantly, this is not a function of X, so the standard errors can be estimated as:

Note however that heterskedasticity consistent errors are not always appropriate. To reiterate, for OLS, classical estimators are not biased even given heteroskedasticity; if the model changes with introduction of robust standard errors, there must be a specification error. Furthermore, heterskedasticity consistent errors are asymptotically unbiased; they can be biased for small n.

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