•
12

Implications for Standard Errors
How then can we estimate the variance?
The trick, a clever one, is to estimate
with .
That is, the squared residual for each observation “i” is used to
estimate the population residual for observation “i”.
•
13

Implications for Standard Errors
The squared residual is not a very good estimator after all only one
observation is used to estimate each observation’s variance, but it turns
out to be good enough. The estimator is
which provides a consistent estimate of
in the presence of “arbitrary”
heteroskedasticity.
•
14

Implications for Standard Errors
(MLR case)
For comparison recall the earlier derivation leading to
Thus
This is the form we introduced earlier.
•
15

Implications for Standard Errors
(MLR case)
Intuition is the same and the formula is
residual for the i
th
observation from regressing x
j
on the other
independent variables
sum of squared residuals from the same regression
•
16

Using Robust Standard Errors
The robust standard error defined above have many names in the literature
•
White Std. Errrors
•
Eicker-White Std Errors
•
Huber SE
•
Robust SE
•
Heteroskedastic consistent standard errors,
•
etc. etc. etc.
Use the one’s in bold – it is easier to call them what they are.
17

Using Robust Standard Errors
Note:
•
Robust standard errors do not “correct” for heteroskedasticity, and
•
They are not adjusted versions of the original OLS standard errors.
These are a different variance estimator that happens to be consistent
in the presence of heteroskedasticity.
18

Using Robust Standard Errors
Finally,
Reporting robust standard errors does not affect coefficient estimates
in any way we just trade an inconsistent estimator of standard errors
for a consistent one.
Nothing has been done to account for types of heteroskedasticity like
known functions of X.
19

Using Robust Standard Errors
So why not just report robust standard errors all the time?
1.
In finite samples when the error term has a normal distribution, the
usual OLS t-stats have an exact t-distribution. In contrast, the robust
t-stats only have an asymptotic t-distribution thus
if the data are
really normally distributed
the OLS standard errors will be more
accurate and thus inference will be more accurate in finite samples.
20

Using Robust Standard Errors
So why not just report robust standard errors all the time?
2.
More generally, if the error terms is really homoscedastic there is an
efficiency gain from imposing this assumption. The efficiency gain
here comes from the variance of the estimates of the variance.
•
Remember that variance estimates themselves have a sampling
distribution and we would like this sampling distribution to have a
small variance, all else equal.
21

Using Robust Standard Errors, in
practice
Common to always report the robust standard error.

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- Fall '19
- Regression Analysis