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Faster Variance-Covariance Matrices and Standard Errors

If you’ve ever bootstrapped a model to get standard errors, you’ve had to compute standard errors from re-sampled models thousands of times.

In such situations, any wasted overhead can cost you time unnecessarily. Note, then, the standard method for extracting a variance-covariance matrix from a standard linear model, stats:::vcov.lm:

vcov.lm = function(obj, ...) {
  so <- summary.lm(object)
  so$sigma^2 * so$cov.unscaled
}

That is, stats:::vcov.lm first summarizes your model, then extracts the covariance matrix from this object.

Unfortunately, stats:::summary.lm wastes precious time computing other summary statistics about your model that you may not care about.

Enter vcov, which cuts out the middle man, and simply gives you back the covariance matrix directly. Here’s a timing comparison:

library(microbenchmark)
set.seed(1320840)
x = rnorm(1e6)
y = 3 + 4*x
reg = lm(y ~ x)

microbenchmark(times = 100,
               vcov = vcov:::Vcov.lm(reg),
               stats = stats:::vcov.lm(reg))
# Unit: milliseconds
#   expr      min       lq     mean   median       uq       max neval
#   vcov 12.45546 14.16308 18.80733 14.72963 15.17740  50.64684   100
#  stats 37.43096 44.62640 52.31549 45.59744 46.99589 251.90297   100

That’s three times as fast, or about 30 milliseconds saved (on an admittedly dinky machine). That means about 30 seconds saved in a 1000-resample bootstrap – this example alone spent 3 more seconds using the stats method, i.e., 75% of the run time was dedicated to stats.

Bonus: Accuracy

In returning a covariance matrix, by using the indirect approach taken in stats, numerical error is introduced unnecessarily. The formula for covariance of vanilla OLS is of course:

\[ \mathbb{V}[\hat{\beta}] = \sigma^2 \left( X^T X \right) ^ {-1} \]

stats, unfortunately, computes this as essentially

covmat = sqrt(sigma2)^2 * XtXinv

The extra square root and exponentiation introduce some minor numerical error; we obviate this by simply computing sigma2 and multiplying it with XtXinv. The difference is infinitesimal, but easily avoided.

Let’s consider a situation where we can get an analytic form of the variance. Consider \(y_i = i\), \(i = 1, \ldots, n\) regressed with OLS against a constant, \(\beta\).

The OLS solution is \(\hat{\beta} = \frac{n+1}2\). The implied error variance is \(\sigma^2 = \frac{n}{n-1} \frac{n^2 - 1}{12}\), so the implied covariance “matrix” (singleton) is \(\mathbb{V}[\hat{\beta}] = \frac{n^2 - 1}{12(n - 1)}\), since $ ( X^T X ) ^ {-1} = $.

N = 1e5
y = 1:N 

reg = lm(y ~ 1)
true_variance = (N^2-1)/(12*(N - 1))

stat_err = abs(true_variance - stats:::vcov.lm(reg))
vcov_err = abs(true_variance - vcov:::Vcov.lm(reg))
#absolute error with vcov
#  (i.e., there's still some numerical issues introduced
#   by the numerics behind the other components)
vcov_err
#              (Intercept)
# (Intercept) 1.818989e-12

#relative error of stats compared to vcov
#  (sometimes the error is 0 for both methods)
stat_err/vcov_err
#             (Intercept)
# (Intercept)           2

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They may not be fully stable and should be used with caution. We make no claims about them.
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