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')

BMR: Bayesian Macroeconometrics in R

The recently released BMR package, short for Bayesian Macroeconometrics with R, provides a comprehensive set of powerful routines that estimate Bayesian Vector Autoregression (VAR) and Dynamic Stochastic General Equilibrium (DSGE) models in R.

The procedure of estimating both Bayesian VAR and DSGE models can represent a great computational burden. However, BMR removes a lot of this burden, performing the most computationally demanding procedures using C++, which is ported into R with the Rcpp package in a manner similar to that of the recently released STAN package.

Despite the complexity of these models, the package itself is very easy to use. Furthermore, the package’s author has provided an awesome vignette that explains both the theory underlining these models, and examples of their use.

Malthus in 21st Century Europe

Reverend Thomas Malthus is well known for his pessimistic views on population growth and economic welfare. The ubiquitous ‘Malthusian model’ is simple and lucid tool which offers an explanation as to why living standards showed no substantive improvement between the Neolithic revolution and the 19th century.

A consequence of the Malthusian model’s popularity is that many people have overlooked the motivation and core point of Malthus’ analysis. Malthus’ aim, outlined in various editions of An Essay on the Principle of Population (notably the second), was to encourage social reforms which promoted later marriage, particularly amongst the poorest in society.

His argument was simple. Sex outside marriage was socially unacceptable, so marriage marked the beginning of sexual relations. Malthus, a clergyman, did not consider fertility control within marriage as an option. Therefore, the earlier couples married, the higher their fertility would be. If early marriage was common across society, this would (exponentially) increase the next generation and thereby reduce income per person because population has increased more than income. Ultimately, Malthus promoted later/delayed marriage as a tool for economic growth and encouraged the foundations of social institutions capable of enforcing this.

While Malthus is one of the most studied and remarked upon character in the history of economic thought, the consensus is that he got it wrong. The purpose of this blog post is to ask: Can any of Malthus’ ideas help us understand economic and demographic trends in the modern Europe?

In the 200 plus years since Malthus wrote the first essay fertility, the institution of marriage and economic conditions have changed immensely in Europe. We might expect economic conditions and marriage to be unrelated, since marriage is no longer a prerequisite for childbirth. Similarly, the economic cost of marriage is (or at least can be) lower than in preindustrial societies where dowries are involved. In addition, modern welfare systems offer child support which helps the poorest in society, something which Malthus, as an opponent of the English Poor Laws, may have disagreed with (although his opposition to poor relief softened somewhat over time).

The above graphic is consistent with Malthus’ hopes. When economic conditions (measured by unemployment change) decline people delay or postpone marital unions. Assuming that there is a delay in the proposition of marriage, it is better to use the lagged unemployment rate. However, this does not matter as there appears to also be a strong relationship between contemporaneous unemployment rate and the marriage rate.

While Malthus might have found the first plot somewhat comforting, we also know that it is socially acceptable for couples to have births outside of wedlock. The figure above demonstrates that despite the social acceptance of out of wedlock births, crude birth and marriage rates are still highly correlated modern Europe. With this in mind, the link between unemployment and fertility is illustrated in the below. Once again the link is clear.

One of the most striking aspects of these graphics is the Crisis. Since 2008, most European nations have experienced a collapse in economic conditions, and inevitable rise in unemployment. Despite this, the demographic trends still obey what would seem to be ‘Malthusian’ logic.