Compute limited mobility bias corrected correlation between fixed effects
With a model like y = Xβ + Dθ + Fψ + ε, where D
and F are matrices with dummy encoded factors, one application of lfe is to
study the correlation cor(Dθ, Fψ). However, if we use
estimates for θ and ψ, the resulting correlation is biased.
The function bccorr computes a bias corrected correlation
as described in Gaure (2014).
bccorr( est, alpha = getfe(est), corrfactors = 1L:2L, nocovar = (length(est$X) == 0) && length(est$fe) == 2, tol = 0.01, maxsamples = Inf, lhs = NULL )
est |
an object of class '"felm"', the result of a call to
|
alpha |
a data frame, the result of a call to |
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. |
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 Dθ (vD) and
Fψ (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.
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 |
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.
Bias correction for IV-estimates are not supported as of now.
Note that if est is the result of a call to felm
with keepX=FALSE (the default), the correlation will be computed
as if the covariates X are independent of the two factors. This will be
faster (typically by a factor of approx. 4), and possibly wronger.
Note also that the computations performed by this function are non-trivial, they may take quite some time. It would be wise to start out with quite liberal tolerances, e.g. tol=0.1, to get an idea of the time requirements.
The algorithm used is not very well suited for small datasets with only a few thousand levels in the factors.
Gaure, S. (2014), Correlation bias correction in two-way fixed-effects linear regression, Stat 3(1):379:390, 2014.
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)
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