STAT 541: Sandwich Estimator

# Statistics 541: Sandwich Estimator

## Admistrivia

## Car data example

Example: Car data.
## Sandwich Estimator

#### The problem

- heteroskedasticity = fan shaped residuals
- usual estimator is "consistent." (Like who cares?)
- SEs are wrong! (Now this is important.)
- Hypothesis-tests are wrong, CIs are wrong
- Even Bonferonni doesn't work!

#### Example

- Suppose X mostly equals zero and sometimes equals 1
- Suppose Y = iid N(0,1)
- slope estimate = Y at (X =1)
- Suppose it is heteroskadastic, small variance at zero large at
one
- high probably of looking significant, having incorrect SEs,
making bad predictions.

#### First solution: weighted least squares

(Myers:7.1)
- suppose Y
_{i} = X_{i} beta + sigma_{i} Z_{i}
instead of Y_{i} = X_{i} beta + sigma Z_{i}
- Then Y
_{i}/sigma_{i} = X_{i}/sigma_{i} beta + Z_{i}
- But this is homoskadastic and we are done
- Where do the sigma
_{i}'s come from?

- theory hopefully
- estimation possibly. E.g. fit model to
Y
_{i}^{2}, or (Y_{i} -
Y-hat)^{2}. Then use predictions from this
model to weight regression.

#### Second solution: Sandwich estimator.

(White 1980, Long and Ervin 2000)
- Use usual LS estimators for Y
- beta-hat = (X'X)
^{-1}X'Y
- So var(beta-hat) = (X'X)
^{-1}X' var(Y_{i})
X(X'X)^{-1}
- Called sandwich estimator since the variance of Y is sandwiched
between the two inverses.
- Consistent for true variance of beta-hat

#### example revisited

(Foster Stine 2001)
- Suppose you fit then compute variance
- Oops! zero variance estimate at (X = 1)
- Better is compute variance, THEN fit

#### Third solution: Use both!

- First change by doing weighted least squares
- Then use sandwich on resulting Y's
- If your weights are wrong, you should still get good results.

Foster, D. P. and Stine, R. A. (2001) "Variable selection in data
mining: Building a predictive model for bankruptcy."
preprint.
Long, J. S. and Ervin, L. H. (2000), "Using heteroscedastic consistent
standard errors in the linear regression model," American
statistician, 54, 795 - 806.

White (1980) "A heteroscedastic-consistent covariance matrix estimator
and a direct test of heteroskedasticity," Econometrica, 48, 817 - 838.

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Last modified: Tue Feb 20 08:44:47 2001

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