Why I use Panel/Multilevel Methods

I don’t understand why any researcher would choose not to use panel/multilevel methods on panel/hierarchical data. Let’s take the following linear regression as an example:

y_{it} = \beta_{0} + \beta_{1}x_{it} + a_{i} + \epsilon_{it},

where a_{i} is a random effect for the i-th group. A pooled OLS regression model for the above is unbiased and consistent. However, it will be inefficient, unless a_{i}=0 for all i.

Let’s have a look at the consequences of this inefficiency using a simulation. I will simulate the following model:

y_{it} = 1 + 5 x_{it} + a_{i} + \epsilon_{it},

with a_{i} \sim N(0, 3) and \epsilon_{it} \sim N(0, 1). I will do this simulation and compare the following 4 estimators: pooled OLS, random effects (RE) AKA a multilevel model with a mixed effect intercept, a correlated random effects (CRE) model (include group mean as regressor as in Mundlak (1978)), and finally the regular fixed effects (FE) model. I am doing this in R, so the first model I will use the simple lm() function, the second and third lmer() from the lme4 package, and finally the excellent felm() function from the lfe package. These models will be tested under two conditions. First, we will assume that the random effects assumption holds, the regressor is uncorrelated with the random effect. After looking at this, we will then allow the random effect to correlate with the regressor x_{it}.

The graph below shows the importance of using panel methods over pooled OLS. It shows boxplots of the 100 simulated estimates. Even when the random effects assumption is violated, the random effects estimator (RE) is far superior to simple pooled OLS. Both the CRE and FE estimators perform well. Both have lowest root mean square errors, even though the model satisfies the random effects assumption. Please see my R code below.

remc

# clear ws
rm(list=ls())

# load packages
library(lme4)
library(plyr)
library(lfe)
library(reshape)
library(ggplot2)
# from this:

### set number of individuals
n = 200
# time periods
t = 5

### model is: y=beta0_{i} +beta1*x_{it} + e_{it}
### average intercept and slope
beta0 = 1.0
beta1 = 5.0

### set loop reps
loop = 100
### results to be entered
results1 = matrix(NA, nrow=loop, ncol=4)
results2 = matrix(NA, nrow=loop, ncol=4)

for(i in 1:loop){
  # basic data structure
  data = data.frame(t = rep(1:t,n),
                    n = sort(rep(1:n,t)))
  # random effect/intercept to add to each 
  rand = data.frame(n = 1:n,
                    a = rnorm(n,0,3))
  data = join(data, rand, match="first")
  # random error
  data$u = rnorm(nrow(data), 0, 1)
  # regressor x
  data$x = runif(nrow(data), 0, 1)
  # outcome y
  data$y = beta0 + beta1*data$x + data$a + data$u  
  # make factor for i-units
  data$n = as.character(data$n)
  # group i mean's for correlated random effects
  data$xn = ave(data$x, data$n, FUN=mean)
  # pooled OLS
  a1 = lm(y ~ x, data)
  # random effects
  a2 = lmer(y ~ x + (1|n), data)
  # correlated random effects
  a3 = lmer(y ~ x + xn + (1|n), data)
  # fixed effects
  a4 = felm(y ~ x | n, data)
  
  # gather results
  results1[i,] = c(coef(a1)[2],
                  coef(a2)$n[1,2],
                  coef(a3)$n[1,2],
                  coef(a4)[1])
  ### now let random effects assumption be false
  ### ie E[xa]!=0
  data$x = runif(nrow(data), 0, 1) + 0.2*data$a
  # the below is like above
  data$y = beta0 + beta1*data$x + data$a + data$u  
  data$n = as.character(data$n)
  data$xn = ave(data$x, data$n, FUN=mean)
  a1 = lm(y ~ x, data)
  a2 = lmer(y ~ x + (1|n), data)
  a3 = lmer(y ~ x + xn + (1|n), data)
  a4 = felm(y ~ x | n, data)
  
  results2[i,] = c(coef(a1)[2],
                  coef(a2)$n[1,2],
                  coef(a3)$n[1,2],
                  coef(a4)[1])  
}
# calculate rmse
apply(results1, 2, function(x) sqrt(mean((x-5)^2)))
apply(results2, 2, function(x) sqrt(mean((x-5)^2)))

# shape data and do ggplot
model.names = data.frame(X2 = c("1","2","3","4"),
                         estimator = c("OLS","RE","CRE","FE"))
res1 = melt(results1)
res1 = join(res1, model.names, match="first")
res2 = melt(results2)
res2 = join(res2, model.names, match="first")

res1$split = "RE Valid"
res2$split = "RE Invalid"
res1 = rbind(res1, res2)

res1$split = factor(res1$split, levels =  c("RE Valid", "RE Invalid"))
res1$estimator = factor(res1$estimator, levels =  c("OLS","RE","CRE","FE"))

number_ticks = function(n) {function(limits) pretty(limits, n)}

ggplot(res1, aes(estimator, value)) + 
  geom_boxplot(fill="lightblue") +
  #coord_flip() +
  facet_wrap( ~ split, nrow = 2, scales = "free_y") +
  geom_hline(yintercept = 5) +
  scale_x_discrete('') + 
  scale_y_continuous(bquote(beta==5), breaks=number_ticks(3)) + 
  theme_bw() + 
  theme(axis.text=element_text(size=16),
        axis.title=element_text(size=16),
        legend.title = element_blank(),
        legend.text = element_text(size=16),
        strip.text.x = element_text(size = 16),
        axis.text.x = element_text(angle = 45, hjust = 1))
ggsave("remc.pdf", width=8, height=6)

IV Estimates via GMM with Clustering in R

In econometrics, generalized method of moments (GMM) is one estimation methodology that can be used to calculate instrumental variable (IV) estimates. Performing this calculation in R, for a linear IV model, is trivial. One simply uses the gmm() function in the excellent gmm package like an lm() or ivreg() function. The gmm() function will estimate the regression and return model coefficients and their standard errors. An interesting feature of this function, and GMM estimators in general, is that they contain a test of over-identification, often dubbed Hansen’s J-test, as an inherent feature. Therefore, in cases where the researcher is lucky enough to have more instruments than endogenous regressors, they should examine this over-identification test post-estimation.

While the gmm() function in R is very flexible, it does not (yet) allow the user to estimate a GMM model that produces standard errors and an over-identification test that is corrected for clustering. Thankfully, the gmm() function is flexible enough to allow for a simple hack that works around this small shortcoming. For this, I have created a function called gmmcl(), and you can find the code below. This is a function for a basic linear IV model. This code uses the gmm() function to estimate both steps in a two-step feasible GMM procedure. The key to allowing for clustering is to adjust the weights matrix after the second step. Interested readers can find more technical details regarding this approach here. After defining the function, I show a simple application in the code below.

gmmcl = function(formula1, formula2, data, cluster){
  library(plyr) ; library(gmm)
  # create data.frame
  data$id1 = 1:dim(data)[1]
  formula3 = paste(as.character(formula1)[3],"id1", sep=" + ")
  formula4 = paste(as.character(formula1)[2], formula3, sep=" ~ ")
  formula4 = as.formula(formula4)
  formula5 = paste(as.character(formula2)[2],"id1", sep=" + ")
  formula6 = paste(" ~ ", formula5, sep=" ")
  formula6 = as.formula(formula6)
  frame1 = model.frame(formula4, data)
  frame2 = model.frame(formula6, data)
  dat1 = join(data, frame1, type="inner", match="first")
  dat2 = join(dat1, frame2, type="inner", match="first")
  
  # matrix of instruments
  Z1 = model.matrix(formula2, dat2)
  
  # step 1
  gmm1 = gmm(formula1, formula2, data = dat2, 
             vcov="TrueFixed", weightsMatrix = diag(dim(Z1)[2]))
  
  # clustering weight matrix
  cluster = factor(dat2[,cluster])
  u = residuals(gmm1)
  estfun = sweep(Z1, MARGIN=1, u,'*')
  u = apply(estfun, 2, function(x) tapply(x, cluster, sum))  
  S = 1/(length(residuals(gmm1)))*crossprod(u)
  
  # step 2
  gmm2 = gmm(formula1, formula2, data=dat2, 
             vcov="TrueFixed", weightsMatrix = solve(S))
  return(gmm2)
}

# generate data.frame
n = 100
z1 = rnorm(n)
z2 = rnorm(n)
x1 = z1 + z2 + rnorm(n)
y1 = x1 + rnorm(n)
id = 1:n

data = data.frame(z1 = c(z1, z1), z2 = c(z2, z2), x1 = c(x1, x1),
                  y1 = c(y1, y1), id = c(id, id))

summary(gmmcl(y1 ~ x1, ~ z1 + z2, data = data, cluster = "id"))

Endogenous Spatial Lags for the Linear Regression Model

spatialestimates
Over the past number of years, I have noted that spatial econometric methods have been gaining popularity. This is a welcome trend in my opinion, as the spatial structure of data is something that should be explicitly included in the empirical modelling procedure. Omitting spatial effects assumes that the location co-ordinates for observations are unrelated to the observable characteristics that the researcher is trying to model. Not a good assumption, particularly in empirical macroeconomics where the unit of observation is typically countries or regions.

Starting out with the prototypical linear regression model: y = X \beta + \epsilon, we can modify this equation in a number of ways to account for the spatial structure of the data. In this blog post, I will concentrate on the spatial lag model. I intend to examine spatial error models in a future blog post.

The spatial lag model is of the form: y= \rho W y + X \beta + \epsilon, where the term \rho W y measures the potential spill-over effect that occurs in the outcome variable if this outcome is influenced by other unit’s outcomes, where the location or distance to other observations is a factor in for this spill-over. In other words, the neighbours for each observation have greater (or in some cases less) influence to what happens to that observation, independent of the other explanatory variables (X). The W matrix is a matrix of spatial weights, and the \rho parameter measures the degree of spatial correlation. The value of \rho is bounded between -1 and 1. When \rho is zero, the spatial lag model collapses to the prototypical linear regression model.

The spatial weights matrix should be specified by the researcher. For example, let us have a dataset that consists of 3 observations, spatially located on a 1-dimensional Euclidean space wherein the first observation is a neighbour of the second and the second is a neighbour of the third. The spatial weights matrix for this dataset should be a 3 \times 3 matrix, where the diagonal consists of 3 zeros (you are not a neighbour with yourself). Typically, this matrix will also be symmetric. It is also at the user’s discretion to choose the weights in W. Common schemes include nearest k neighbours (where k is again at the users discretion), inverse-distance, and other schemes based on spatial contiguities. Row-standardization is usually performed, such that all the row elements in W sum to one. In our simple example, the first row of a contiguity-based scheme would be: [0, 1, 0]. The second: [0.5, 0, 0.5]. And the third: [0, 1, 0].

While the spatial-lag model represents a modified version of the basic linear regression model, estimation via OLS is problematic because the spatially lagged variable (Wy) is endogenous. The endogeneity results from what Charles Manski calls the ‘reflection problem’: your neighbours influence you, but you also influence your neighbours. This feedback effect results in simultaneity which renders bias on the OLS estimate of the spatial lag term. A further problem presents itself when the independent variables (X) are themselves spatially correlated. In this case, completely omitting the spatial lag from the model specification will bias the \beta coefficient values due to omitted variable bias.

Fortunately, remedying these biases is relatively simple, as a number of estimators exist that will yield an unbiased estimate of the spatial lag, and consequently the coefficients for the other explanatory variables—assuming, of course, that these explanatory variables are themselves exogenous. Here, I will consider two: the Maximum Likelihood estimator (denoted ML) as described in Ord (1975), and a generalized two-stage least squares regression model (2SLS) wherein spatial lags, and spatial lags lags (i.e. W^{2} X) of the explanatory variables are used as instruments for Wy. Alongside these two models, I also estimate the misspecified OLS both without (OLS1) and with (OLS2) the spatially lagged dependent variable.

To examine the properties of these four estimators, I run a Monte Carlo experiment. First, let us assume that we have 225 observations equally spread over a 15 \times 15 lattice grid. Second, we assume that neighbours are defined by what is known as the Rook contiguity, so a neighbour exists if they are in the grid space either above or below or on either side. Once we create the spatial weight matrix we row-standardize.

Taking our spatial weights matrix as defined, we want to simulate the following linear model: y = \rho Wy + \beta_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \epsilon, where we set \rho=0.4 , \beta_{1}=0.5, \beta_{2}=-0.5, \beta_{3}=1.75. The explanatory variables are themselves spatially autocorrelated, so our simulation procedure first simulates a random normal variable for both x_{2} and x_{3} from: N(0, 1), then assuming a autocorrelation parameter of \rho_{x}=0.25, generates both variables such that: x_{j} = (1-\rho_{x}W)^{-1} N(0, 1) for j \in \{ 1,2 \}. In the next step we simulate the error term \epsilon. We introduce endogeneity into the spatial lag by assuming that the error term \epsilon is a function of a random normal v so \epsilon = \alpha v + N(0, 1) where v = N(0, 1) and \alpha=0.2, and that the spatial lag term includes v. We modify the regression model to incorporate this: y = \rho (Wy + v) + \beta_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \epsilon. From this we can calculate the reduced form model: y = (1 - \rho W)^{-1} (\rho v + \beta_{1} + \beta_{2}x_{2} + \beta_{3}x_{3} + \epsilon), and simulate values for our dependent variable y.

Performing 1,000 repetitions of the above simulation permits us to examine the distributions of the coefficient estimates produced by the four models outlined in the above. The distributions of these coefficients are displayed in the graphic in the beginning of this post. The spatial autocorrelation parameter \rho is in the bottom-right quadrant. As we can see, the OLS model that includes the spatial effect but does not account for simultaneity (OLS2) over-estimates the importance of spatial spill-overs. Both the ML and 2SLS estimators correctly identify the \rho parameter. The remaining quadrants highlight what happens to the coefficients of the explanatory variables. Clearly, the OLS1 estimator provides the worst estimate of these coefficients. Thus, it appears preferable to use OLS2, with the biased autocorrelation parameter, than the simpler OLS1 estimator. However, the OLS2 estimator also yields biased parameter estimates for the \beta coefficients. Furthermore, since researchers may want to know the marginal effects in spatial equilibrium (i.e. taking into account the spatial spill-over effects) the overestimated \rho parameter creates an additional bias.

To perform these calculations I used the spdep package in R, with the graphic created via ggplot2. Please see the R code I used in the below.

library(spdep) ; library(ggplot2) ; library(reshape)

rm(list=ls())
n = 225
data = data.frame(n1=1:n)
# coords
data$lat = rep(1:sqrt(n), sqrt(n))
data$long = sort(rep(1:sqrt(n), sqrt(n)))
# create W matrix
wt1 = as.matrix(dist(cbind(data$long, data$lat), method = "euclidean", upper=TRUE))
wt1 = ifelse(wt1==1, 1, 0)
diag(wt1) = 0
# row standardize
rs = rowSums(wt1)
wt1 = apply(wt1, 2, function(x) x/rs)
lw1 = mat2listw(wt1, style="W")

rx = 0.25
rho = 0.4
b1 = 0.5
b2 = -0.5
b3 = 1.75
alp = 0.2
  
inv1 = invIrW(lw1, rho=rx, method="solve", feasible=NULL)
inv2 = invIrW(lw1, rho=rho, method="solve", feasible=NULL)

sims = 1000
beta1results = matrix(NA, ncol=4, nrow=sims)
beta2results = matrix(NA, ncol=4, nrow=sims)
beta3results = matrix(NA, ncol=4, nrow=sims)
rhoresults = matrix(NA, ncol=3, nrow=sims)

for(i in 1:sims){
  u1 = rnorm(n)
  x2 = inv1 %*% u1
  u2 = rnorm(n)
  x3 = inv1 %*% u2
  v1 = rnorm(n)
  e1 = alp*v1 + rnorm(n)  
  data1 = data.frame(cbind(x2, x3),lag.listw(lw1, cbind(x2, x3)))
  names(data1) = c("x2","x3","wx2","wx3")
  
  data1$y1 = inv2 %*% (b1 + b2*x2 + b3*x3 + rho*v1 + e1)
      
  data1$wy1 = lag.listw(lw1, data1$y1)
  data1$w2x2 = lag.listw(lw1, data1$wx2)
  data1$w2x3 = lag.listw(lw1, data1$wx3)
  data1$w3x2 = lag.listw(lw1, data1$w2x2)
  data1$w3x3 = lag.listw(lw1, data1$w2x3)

  m1 = coef(lm(y1 ~ x2 + x3, data1))
  m2 = coef(lm(y1 ~ wy1 + x2 + x3, data1))
  m3 = coef(lagsarlm(y1 ~ x2 + x3, data1, lw1))
  m4 = coef(stsls(y1 ~ x2 + x3, data1, lw1))
    
  beta1results[i,] = c(m1[1], m2[1], m3[2], m4[2])
  beta2results[i,] = c(m1[2], m2[3], m3[3], m4[3])
  beta3results[i,] = c(m1[3], m2[4], m3[4], m4[4])
  rhoresults[i,] = c(m2[2],m3[1], m4[1])  
}

apply(rhoresults, 2, mean) ; apply(rhoresults, 2, sd)
apply(beta1results, 2, mean) ; apply(beta1results, 2, sd)
apply(beta2results, 2, mean) ; apply(beta2results, 2, sd)
apply(beta3results, 2, mean) ; apply(beta3results, 2, sd)

colnames(rhoresults) = c("OLS2","ML","2SLS")
colnames(beta1results) = c("OLS1","OLS2","ML","2SLS")
colnames(beta2results) = c("OLS1","OLS2","ML","2SLS")
colnames(beta3results) = c("OLS1","OLS2","ML","2SLS")

rhoresults = melt(rhoresults)
rhoresults$coef = "rho"
rhoresults$true = 0.4

beta1results = melt(beta1results)
beta1results$coef = "beta1"
beta1results$true = 0.5

beta2results = melt(beta2results)
beta2results$coef = "beta2"
beta2results$true = -0.5

beta3results = melt(beta3results)
beta3results$coef = "beta3"
beta3results$true = 1.75

data = rbind(rhoresults,beta1results,beta2results,beta3results)
data$Estimator = data$X2

ggplot(data, aes(x=value, colour=Estimator, fill=Estimator)) + 
  geom_density(alpha=.3) + 
  facet_wrap(~ coef, scales= "free") +
  geom_vline(aes(xintercept=true)) + 
  scale_y_continuous("Density") +
  scale_x_continuous("Effect Size") +
  opts(legend.position = 'bottom', legend.direction = 'horizontal')

Linear Models with Multiple Fixed Effects

Estimating a least squares linear regression model with fixed effects is a common task in applied econometrics, especially with panel data. For example, one might have a panel of countries and want to control for fixed country factors. In this case the researcher will effectively include this fixed identifier as a factor variable, and then proceed to estimate the model that includes as many dummy variables (minus one if an intercept is included in the modelling equation) as there are countries. Obviously, this approach is computationally problematic when there are many fixed factors. In our simple example, an extra country will add an extra column to the X matrix used in the least squares calculation.

Fortunately, there are a number of data transformations that can be used in this panel setting. These include demeaning each within unit observation, using first differences, or including the group means as additional explanatory variables (as suggested by (Mundlak 1978)). However, these approaches only work well when there is one factor that the researcher wants to include fixed effects to account for.

Simen Gaure offers a solution this problem that allows for multiple fixed effects without resorting to a computationally burdensome methodology. Essentially the solution involves an elaboration of the group demeaning transformation mentioned in the above. More technical details can be found here or by referring to Gaure’s forthcoming article in Computational Statistics & Data Analysis. Those interested in implementing this estimation strategy in R can use the lfe package available on CRAN.

In the below, I have included a simple example of how the package works. In this example, the model needs to be set up to calculate fixed effects for two factor variables. Obviously, adding 2,000 columns to the data frame is not a convenient way to estimate the model that includes fixed effects for both the x2 and x3 variables. However, the felm function tackles this problem with ease. Stata has a similar function to feml, areg, although the areg function only allows for absorbed fixed effects in one variable.

# clear workspace
rm(list=ls())
# load lfe package
library(lfe)

# create data frame
x1 <- rnorm(10000)
x2 <- rep(1:1000,10)
x3 <- rep(1:1000,10)
e1 <- sin(x2) + 0.02*x3^2 + rnorm(10000)
y <- 10 + 2.5*x1 + (e1-mean(e1))
dat <- data.frame(x1,x2,x3,y)

# simple lm
lm(y~x1)
# lm with fixed effects
felm(dat$y ~ dat$x1 + G(dat$x2) + G(dat$x3))

##############################################
# output
##############################################
# simple lm
> lm(y~x1)
Call: lm(formula = y ~ x1)
Coefficients:
  (Intercept)           x1  
        10.47       -36.95  
> # lm with fixed effects
> felm(dat$y ~ dat$x1 + G(dat$x2) + G(dat$x3))
dat$x1 
2.501