lfe (version 2.9-0)

bccorr: Compute limited mobility bias corrected correlation between fixed effects

Description

With a model like \(y = X\beta + D\theta + F\psi + \epsilon\), where \(D\) and \(F\) are matrices with dummy encoded factors, one application of lfe is to study the correlation \(cor(D\theta, F\psi)\). However, if we use estimates for \(\theta\) and \(\psi\), the resulting correlation is biased. The function bccorr computes a bias corrected correlation as described in Gaure (2014).

Usage

bccorr(
  est,
  alpha = getfe(est),
  corrfactors = 1L:2L,
  nocovar = (length(est$X) == 0) && length(est$fe) == 2,
  tol = 0.01,
  maxsamples = Inf,
  lhs = NULL
)

Value

bccorr returns a named integer vector with the following fields:

corr

the bias corrected correlation.

v1

the bias corrected variance for the first factor specified by corrfactors.

v2

the bias corrected variance for the second factor.

cov

the bias corrected covariance between the two factors.

d1

the bias correction for the first factor.

d2

the bias correction for the second factor.

d12

the bias correction for covariance.

The bias corrections have been subtracted from the bias estimates. E.g. v2 = v2' - d2, where v2' is the biased variance.

Arguments

est

an object of class '"felm"', the result of a call to [felm](keepX=TRUE).

alpha

a data frame, the result of a call to getfe().

corrfactors

integer or character vector of length 2. The factors to correlate. The default is fine if there are only two factors in the model.

nocovar

logical. Assume no other covariates than the two factors are present, or that they are uncorrelated with them.

tol

The absolute tolerance for the bias-corrected correlation.

maxsamples

Maximum number of samples for the trace sample means estimates

lhs

character. Name of left hand side if multiple left hand sides.

Details

The bias expressions from Andrews et al. are of the form \(tr(AB^{-1}C)\) where \(A\), \(B\), and \(C\) are matrices too large to be handled directly. bccorr estimates the trace by using the formula \(tr(M) = E(x^t M x)\) where x is a vector with coordinates drawn uniformly from the set \(\{-1,1\}\). More specifically, the expectation is estimated by sample means, i.e. in each sample a vector x is drawn, the equation \(Bv = Cx\) is solved by a conjugate gradient method, and the real number \(x^t Av\) is computed.

There are three bias corrections, for the variances of \(D\theta\) (vD) and \(F\psi\) (vF), and their covariance (vDF).The correlation is computed as rho <- vDF/sqrt(vD*vF). The variances are estimated to a relative tolerance specified by the argument tol. The covariance bias is estimated to an absolute tolerance in the correlation rho (conditional on the already bias corrected vD and vF) specified by tol. The CG algorithm does not need to be exceedingly precise, it is terminated when the solution reaches a precision which is sufficient for the chosen precision in vD, vF, vDF.

If est is the result of a weighted felm() estimation, the variances and correlations are weighted too.

References

Gaure, S. (2014), Correlation bias correction in two-way fixed-effects linear regression, Stat 3(1):379:390, 2014.

See Also

fevcov()

Examples

x <- rnorm(500)
x2 <- rnorm(length(x))

## create individual and firm
id <- factor(sample(40, length(x), replace = TRUE))
firm <- factor(sample(30, length(x), replace = TRUE, prob = c(2, rep(1, 29))))
foo <- factor(sample(20, length(x), replace = TRUE))
## effects
id.eff <- rnorm(nlevels(id))
firm.eff <- rnorm(nlevels(firm))
foo.eff <- rnorm(nlevels(foo))
## left hand side
y <- x + 0.25 * x2 + id.eff[id] + firm.eff[firm] + foo.eff[foo] + rnorm(length(x))

# make a data frame
fr <- data.frame(y, x, x2, id, firm, foo)
## estimate and print result
est <- felm(y ~ x + x2 | id + firm + foo, data = fr, keepX = TRUE)
# find bias corrections
bccorr(est)