# fevcov: Compute limited mobility bias corrected covariance matrix... In sgaure/lfe: Linear Group Fixed Effects

## Description

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 variances var(Dθ), var(Fψ) and covariances cov(Dθ, Fψ). However, if we use estimates for θ and ψ, the resulting variances are biased. The function `fevcov` computes a bias corrected covariance matrix as described in Gaure (2014).

## Usage

 ```1 2 3 4 5 6 7 8``` ```fevcov( est, alpha = getfe(est), tol = 0.01, robust = !is.null(est\$clustervar), maxsamples = Inf, lhs = NULL ) ```

## 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`. `tol` numeric. The absolute tolerance for the bias-corrected correlation. `robust` logical. Should robust (heteroskedastic or cluster) residuals be used, rather than i.i.d. `maxsamples` integer. Maximum number of samples for expectation estimates. `lhs` character. Name of left hand side if multiple left hand sides.

## Details

The `tol` argument specifies the tolerance. The tolerance is relative for the variances, i.e. the diagonal of the output. For the covariances, the tolerance is relative to the square root of the product of the variances, i.e. an absolute tolerance for the correlation. If a numeric of length 1, `tol` specifies the same tolerance for all variances/covariances. If it is of length 2, `tol` specifies the variance tolerance, and `tol` the covariance tolerance. `tol` can also be a square matrix of size `length(est\$fe)`, in which case the tolerance for each variance and covariance is specified individually.

The function performs no checks for estimability. If the fixed effects are not estimable, the result of a call to `fevcov` is not useable. Moreover, there should be just a single connected component among the fixed effects.

`alpha` must contain a full set of coefficients, and contain columns `'fe'` and `'effect'` like the default estimable functions from `efactory`.

In the case that the `felm`-estimation has weights, it is the weighted variances and covariance which are bias corrected.

## Value

`fevcov` returns a square matrix with the bias corrected covariances. An attribute `'bias'` contains the biases. The bias corrections have been subtracted from the bias estimates. I.e. vc = vc' - b, where vc' is the biased variance and b is the bias.

## Note

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 biases will be computed as if the covariates X are independent of the 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.

If there are only two fixed effects, `fevcov` returns the same information as `bccorr`, though in a slightly different format.

## References

Gaure, S. (2014), Correlation bias correction in two-way fixed-effects linear regression, Stat 3(1):379-390, 2014. http://dx.doi.org/10.1002/sta4.68

`varvars` `bccorr`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30``` ```x <- rnorm(5000) 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 <- runif(nlevels(firm)) foo.eff <- rchisq(nlevels(foo),df=1) ## left hand side id.m <- id.eff[id] firm.m <- firm.eff[firm] foo.m <- foo.eff[foo] # normalize them id.m <- id.m/sd(id.m) firm.m <- firm.m/sd(firm.m) foo.m <- foo.m/sd(foo.m) y <- x + 0.25*x2 + id.m + firm.m + foo.m + rnorm(length(x),sd=2) z <- x + 0.5*x2 + 0.7*id.m + 0.5*firm.m + 0.3*foo.m + rnorm(length(x),sd=2) # make a data frame fr <- data.frame(y,z,x,x2,id,firm,foo) ## estimate and print result est <- felm(y|z ~ x+x2|id+firm+foo, data=fr, keepX=TRUE) # find bias corrections, there's little bias in this example print(yv <- fevcov(est, lhs='y')) ## Here's how to compute the unbiased correlation matrix: cm <- cov2cor(yv) structure(cm,bias=NULL) ```