# Two-Way-Fixed-Effect-Model

#### 2020-08-04

library(xtreg2way)

The xtreg2way package is an algorithm to efficiently estimate a two-way fixed effects model. This algorithm is adapted from the Matlab code written for this paper.

To begin, we will generate sample data. It’s likely you already have data if you wish to use this package; Skip to the bottom to see how this is used. First, we decide how many groups and time effects we want. These are the two effects in the “two-way fixed effects” model.

numgroups <- 1000
T <- 200

Next, we will assign each observation to a group and time effect. These are stored in hhid and tid. The variable w is a weight for each observation.

observations <- numgroups * T
e <- 1:observations
## Create groups and weights
hhid <- floor((e - 1) / T + 1)
tid <-  e - (hhid - 1) * T
w <- pracma::rand(n = numgroups, m = 1)
w <- w[hhid]

Now each observation has a group, time factor, and weight. Next, we will generate two random x values as our independent variables, based on these effects.

#Randomly create effects for groups
heffect <- pracma::randn(n = numgroups, m = 1)
teffect <- pracma::randn(n = T, m = 1)
#Generate independent variables
x1 <- pracma::randn(n = observations, m = 1) +
0.5 * heffect[hhid] + 0.25 * teffect[tid]
x2 <- pracma::randn(n = observations, m = 1) -
0.25 * heffect[hhid] + 0.5 * teffect[tid]

Finally, we will generate a random error term, and then create our y dependent variable from the variables created.

#Generate Random Error
autoc <- pracma::rand(n = numgroups, m = 1)
initialv <- pracma::randn(n = numgroups, m = 1)
u <- pracma::randn(n = observations, m = 1)
for (o in 1:observations) {
if (tid[o] > 1){
u_1 <- u[o-1]
} else {
u_1 <- initialv[hhid[o]]
}
u[o] <- autoc[hhid[o]] * u_1 + u[o]
}
# Generate dependent variable
y <- 1 + x1 - x2 + heffect[hhid] + teffect[tid] + u

The function xtreg2way has two uses. The first time you use it, you must specify, y,X,hhid, and tid. The weights vector w is assumed to be all 1’s if omitted.

#XTREG2WAY
output <- xtreg2way(y, data.frame(x1,x2), hhid, tid, w, noise="1")
#>   coefficients          se     tstat pval
#> 1    0.9971871 0.004560598  218.6527    0
#> 2   -0.9956326 0.005447905 -182.7551    0

Using the noise="1" parameter, we can see the regression coefficients for x1 and x2 printed here, as well as standard errors, t-stat values, and p-values. Additionally, the output variable has several objects.

betaHat is a vector of the regression coefficients. aVarHat is the asymptotic variance of the estimator y and X are the input variables projected onto the two-way dummy matrices. struc contains several matrices necessary to perform this regression again on different independent variables, for the same set of observations.

The second way to use xtreg2way is to use the struc object from the first run, but now on a different set of columns. Below, we see the same process but only using x1 now.

#XTREG2WAY second time
output2 <- xtreg2way(y, x1, struc=output\$struc)

Providing struc saves computation time, making the algorithm much faster. The output variable output2 now only contains betaHat and aVarHat.