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)

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Coal and the Conservatives

Interesting election results in the UK over the weekend, where the Conservatives romped to victory. This was despite a widespread consensus that neither the Conservative or Labour party would get a majority. This was a triumph for uncertainty and random error over the deterministic, as none of the statistical forecasts appeared to deem such a decisive victory probable. The UK election is a lot harder to model, for numerous reasons, when compared to the US.

This means that a lot of pollsters and political forecasters will have to go back to the drawing board and re-evaluate their methods. Obviously, the models used to forecast the 2015 election could not handle the dynamics of the British electorate. However, there is a high degree of persistence within electuary constituencies. Let’s explore this persistence by looking at the relationship between coal and % Conservative (Tory) votes.

Following a tweet by Vaughan Roderick and using the methodology of Fernihough and O’Rourke (2014), I matched each of the constituencies to Britain’s coalfields creating a “proximity to coal” measure. What the plot below shows is striking. Being located on or in close proximity to a coal field reduces the tory vote share by about 20%. When we control (linearly) for latitude and longitude coordinates, this association decreases in strength, but not by much. For me, this plot highlights a long-standing relationship between Britain’s industrial revolution, the urban working class, and labour/union movement. What I find interesting is that this relationship has persisted despite de-industrialization and the movement away from large-scale manufacturing industry.

torycoal

> summary(lm(tory~coal,city))

Call:
lm(formula = tory ~ coal, data = city)

Residuals:
    Min      1Q  Median      3Q     Max 
-42.507 -10.494   2.242  10.781  29.074 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  42.9492     0.7459   57.58   <2e-16 ***
coal        -24.9704     1.8887  -13.22   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 14.36 on 630 degrees of freedom
Multiple R-squared:  0.2172,	Adjusted R-squared:  0.216 
F-statistic: 174.8 on 1 and 630 DF,  p-value: < 2.2e-16

# robust to lat-long?
> summary(lm(tory~coal+longitude+latitude,city))

Call:
lm(formula = tory ~ coal + longitude + latitude, data = city)

Residuals:
    Min      1Q  Median      3Q     Max 
-44.495  -8.269   1.485   9.316  28.911 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 246.4355    18.9430  13.009  < 2e-16 ***
coal        -15.1616     1.8697  -8.109 2.68e-15 ***
longitude     1.4023     0.4015   3.493 0.000512 ***
latitude     -3.8621     0.3651 -10.578  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12.76 on 628 degrees of freedom
Multiple R-squared:  0.3838,	Adjusted R-squared:  0.3809 
F-statistic: 130.4 on 3 and 628 DF,  p-value: < 2.2e-16

Brazil’s Host Advantage

brazil

If history can tell us anything about the World Cup, it’s that the host nation has an advantage of all other teams. Evidence of this was presented last night as the referee in the Brazil-Croatia match unjustly ruled in Brazil’s favour on several occasions. But what it is the statistical evidence of a host advantage?

To look at this, I downloaded these data from the Guardian’s website. With these, I ran a very simple probit model that regressed the probability of winning the world cup on whether the country was the host and also if the county was not the host but located in the same continent (I merge North and South America for this exercise). Obviously, this is quite a basic analysis, so I hope to build on these data as the tournament progresses and maybe and the 2010 data, and look at more sophisticated models.

> probitmfx(formula=winners ~ continent + hosts, data=wc)
Call:
probitmfx(formula = winners ~ continent + hosts, data = wc)

Marginal Effects:
             dF/dx Std. Err.      z   P>|z|   
continent 0.064425  0.027018 2.3845 0.01710 * 
hosts     0.315378  0.121175 2.6027 0.00925 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

dF/dx is for discrete change for the following variables:

[1] "continent" "hosts"   

The results are as we would expect. I am using the excellent mfx package to interpret the probit coefficients. Being the host nation increases the probability of being victorious by nearly 32%. So, going by historical trends, Brazil have a huge advantage for this world cup. If we look at countries in the same continent (think Argentina for this world cup) we see that there is a small advantage here, just over 6%.

Whether these results are robust to additional control variables and in the inclusion of fixed effects alongside heterogeneous time-varying effects is something I hope to probe.

The ivlewbel Package. A new way to Tackle Endogenous Regressor Models.

In April 2012, I wrote this blog post demonstrating an approach proposed in Lewbel (2012) that identifies endogenous regressor coefficients in a linear triangular system. Now I am happy to announce the release of the ivlewbel package, which contains a function through which Lewbel’s method can be applied in R. This package is now available to download on the CRAN.

Please see the example from the previous blog post replicated in the below. Additionally, it would be very helpful if people could comment on bugs and additional features they would like to add to the package. My contact details are in the about section of the blog.

lewbel2

library(ivlewbel)

beta1 <- beta2 <- NULL
for(k in 1:500){
  #generate data (including intercept)
  x1 <- rnorm(1000,0,1)
  x2 <- rnorm(1000,0,1)
  u <- rnorm(1000,0,1)
  s1 <- rnorm(1000,0,1)
  s2 <- rnorm(1000,0,1)
  ov <- rnorm(1000,0,1)
  e1 <- u + exp(x1)*s1 + exp(x2)*s1
  e2 <- u + exp(-x1)*s2 + exp(-x2)*s2
  y1 <- 1 + x1 + x2 + ov + e2 
  y2 <- 1 + x1 + x2 + y1 + 2*ov + e1
  x3 <- rep(1,1000)
  dat <- data.frame(y1,y2,x3,x1,x2)
  
  #record ols estimate
  beta1 <- c(beta1,coef(lm(y2~x1+x2+y1))[4])
  #init values for iv-gmm
  beta2 <- c(beta2,lewbel(formula = y2 ~ y1 | x1 + x2 | x1 + x2, data = dat)$coef.est[1,1])
}

library(sm)
d <- data.frame(rbind(cbind(beta1,"OLS"),cbind(beta2,"IV-GMM")))
d$beta1 <- as.numeric(as.character(d$beta1))
sm.density.compare(d$beta1, d$V2,xlab=("Endogenous Coefficient"))
title("Lewbel and OLS Estimates")
legend("topright", levels(d$V2),lty=c(1,2,3),col=c(2,3,4),bty="n")
abline(v=1)

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"))

Detecting Weak Instruments in R

Weak Instruments

Weak Instruments

Any instrumental variables (IV) estimator relies on two key assumptions in order to identify causal effects:

  1. That the excluded instrument or instruments only effect the dependent variable through their effect on the endogenous explanatory variable or variables (the exclusion restriction),
  2. That the correlation between the excluded instruments and the endogenous explanatory variables is strong enough to permit identification.

The first assumption is difficult or impossible to test, and shear belief plays a big part in what can be perceived to be a good IV. An interesting paper was published last year in the Review of Economics and Statistics by Conley, Hansen, and Rossi (2012), wherein the authors provide a Bayesian framework that permits researchers to explore the consequences of relaxing exclusion restrictions in a linear IV estimator. It will be interesting to watch research on this topic expand in the coming years.

Fortunately, it is possible to quantitatively measure the strength of the relationship between the IVs and the endogenous variables. The so-called weak IV problem was underlined in paper by Bound, Jaeger, and Baker (1995). When the relationship between the IVs and the endogenous variable is not sufficiently strong, IV estimators do not correctly identify causal effects.

The Bound, Jaeger, and Baker paper represented a very important contribution to the econometrics literature. As a result of this paper, empirical studies that use IV almost always report some measure of the instrument strength. A secondary result of this paper was the establishment of a literature that evaluates different methods of testing for weak IVs. Staiger and Stock (1997) furthered this research agenda, formalizing the relevant asymptotic theory and recommending the now ubiquitous “rule-of-thumb” measure: a first-stage partial-F test of less than 10 indicates the presence of weak instruments.

In the code below, I have illustrated how one can perform these partial F-tests in R. The importance of clustered standard errors has been highlighted on this blog before, so I also show how the partial F-test can be performed in the presence of clustering (and heteroskedasticity too). To obtain the clustered variance-covariance matrix, I have adapted some code kindly provided by Ian Gow. For completeness, I have displayed the clustering function at the end of the blog post.

# load packages
library(AER) ; library(foreign) ; library(mvtnorm)
# clear workspace and set seed
rm(list=ls())
set.seed(100)

# number of observations
n = 1000
# simple triangular model:
# y2 = b1 + b2x1 + b3y1 + e
# y1 = a1 + a2x1 + a3z1 + u
# error terms (u and e) correlate 
Sigma = matrix(c(1,0.5,0.5,1),2,2)
ue = rmvnorm(n, rep(0,2), Sigma)
# iv variable
z1 = rnorm(n)
x1 = rnorm(n)
y1 = 0.3 + 0.8*x1 - 0.5*z1 + ue[,1]
y2 = -0.9 + 0.2*x1 + 0.75*y1 +ue[,2]
# create data
dat = data.frame(z1, x1, y1, y2)

# biased OLS
lm(y2 ~ x1 + y1, data=dat)
# IV (2SLS)
ivreg(y2 ~ x1 + y1 | x1 + z1, data=dat)

# do regressions for partial F-tests
# first-stage:
fs = lm(y1 ~ x1 + z1, data = dat)
# null first-stage (i.e. exclude IVs):
fn = lm(y1 ~ x1, data = dat)

# simple F-test
waldtest(fs, fn)$F[2]
# F-test robust to heteroskedasticity
waldtest(fs, fn, vcov = vcovHC(fs, type="HC0"))$F[2]

####################################################
# now lets get some F-tests robust to clustering

# generate cluster variable
dat$cluster = 1:n
# repeat dataset 10 times to artificially reduce standard errors
dat = dat[rep(seq_len(nrow(dat)), 10), ]

# re-run first-stage regressions
fs = lm(y1 ~ x1 + z1, data = dat)
fn = lm(y1 ~ x1, data = dat)

# simple F-test
waldtest(fs, fn)$F[2]
# ~ 10 times higher!
# F-test robust to clustering
waldtest(fs, fn, vcov = clusterVCV(dat, fs, cluster1="cluster"))$F[2]
# ~ 10 times lower than above (good)

Further “rule-of-thumb” measures are provided in a paper by Stock and Yogo (2005) and it should be noted that whole battery of weak-IV tests exist (for example, see the Kleinberg-Paap rank Wald F-statistic and Anderson-Rubin Wald test) and one should perform these tests if the presence of weak instruments represents a serious concern. 

# R function adapted from Ian Gows' webpage:
# http://www.people.hbs.edu/igow/GOT/Code/cluster2.R.html.
clusterVCV <- function(data, fm, cluster1, cluster2=NULL) {
  
  require(sandwich)
  require(lmtest)
  
  # Calculation shared by covariance estimates
  est.fun <- estfun(fm)
  inc.obs <- complete.cases(data[,names(fm$model)])
  
  # Shared data for degrees-of-freedom corrections
  N  <- dim(fm$model)[1]
  NROW <- NROW(est.fun)
  K  <- fm$rank
  
  # Calculate the sandwich covariance estimate
  cov <- function(cluster) {
    cluster <- factor(cluster)
    
    # Calculate the "meat" of the sandwich estimators
    u <- apply(est.fun, 2, function(x) tapply(x, cluster, sum))
    meat <- crossprod(u)/N
    
    # Calculations for degrees-of-freedom corrections, followed 
    # by calculation of the variance-covariance estimate.
    # NOTE: NROW/N is a kluge to address the fact that sandwich uses the
    # wrong number of rows (includes rows omitted from the regression).
    M <- length(levels(cluster))
    dfc <- M/(M-1) * (N-1)/(N-K)
    dfc * NROW/N * sandwich(fm, meat=meat)
  }
  
  # Calculate the covariance matrix estimate for the first cluster.
  cluster1 <- data[inc.obs,cluster1]
  cov1  <- cov(cluster1)
  
  if(is.null(cluster2)) {
    # If only one cluster supplied, return single cluster
    # results
    return(cov1)
  } else {
    # Otherwise do the calculations for the second cluster
    # and the "intersection" cluster.
    cluster2 <- data[inc.obs,cluster2]
    cluster12 <- paste(cluster1,cluster2, sep="")
    
    # Calculate the covariance matrices for cluster2, the "intersection"
    # cluster, then then put all the pieces together.
    cov2   <- cov(cluster2)
    cov12  <- cov(cluster12)
    covMCL <- (cov1 + cov2 - cov12)
    
    # Return the output of coeftest using two-way cluster-robust
    # standard errors.
    return(covMCL)
  }
}

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')