A 2SLS when the instrumented variable has two interactions in the model

Question

I am using ivreg and ivmodel in R to apply a 2SLS.

I would like to instrument one variable, namely $x_1$, present in two interaction terms. In this example $x_1$ is a factor variable. The regression is specified in this manner because the ratio between $a$ and $b$ is of importance.

$$y = ax_1 x_2 + bx_1x_3 + cx_4 + e$$

For this instrumented variable I have two instruments $z_1$ and $z_2$. For both the following causal diagram is applicable (Z only has an indirect effect on Y through X).

What is for this problem the correct way to instrument $x_1$?

In the data

Translated to some (fake) sample data the problem looks like:

$$happiness = a(factor:income) + b(factor:sales) + c(educ) + e$$ $$=$$ $$(y = ax_1 x_2 + bx_1x_3 + cx_4 + e)$$

Where the instrument $z_1$ is urban and $z_2$ is size. Here I however become to get confused about how to write the regression.

For the first stage:

What is my dependent variable here?

For the second stage, should I do:

$$happiness = a(urban:income) + b(urban:sales) + c(educ) + e$$ $$happiness = a(size:income) + b(size:sales) + c(educ) + e$$

Or should I just do:

$$happiness = urban*(a:income+b:sales) + c(educ) + e$$ $$happiness = size*(a:income+b:sales) + c(educ) + e$$

Nevertheless, how should I specify this in R ?

library(data.table)
library(ivmodel)
library(AER)
panelID = c(1:50)   
year= c(2001:2010)
country = c("NLD", "BEL", "GER")
urban = c("A", "B", "C")
indust = c("D", "E", "F")
sizes = c(1,2,3,4,5)
n <- 2
library(data.table)
set.seed(123)
DT <- data.table(panelID = rep(sample(panelID), each = n),
                    country = rep(sample(country, length(panelID), replace = T), each = n),
                    year = c(replicate(length(panelID), sample(year, n))),
                    some_NA = sample(0:5, 6),                                             
                    Factor = sample(0:5, 6), 
                    industry = rep(sample(indust, length(panelID), replace = T), each = n),
                    urbanisation = rep(sample(urban, length(panelID), replace = T), each = n),
                    size = rep(sample(sizes, length(panelID), replace = T), each = n),
                    income = round(runif(100)/10,2),
                    Y_Outcome= round(rnorm(10,100,10),2),
                    sales= round(rnorm(10,10,10),2),
                    happiness = sample(10,10),
                    Sex = round(rnorm(10,0.75,0.3),2),
                    Age = sample(100,100),
                    educ = round(rnorm(10,0.75,0.3),2))        
DT [, uniqueID := .I]                                                         # Creates a unique ID     
DT <- as.data.frame(DT)

To make it slightly easier for someone to help who is not familiar with the packages, I have added how the structure of the two packages I use looks.

The structure of the second stage of ivreg is as follows:

second_stage <- ivreg(Happiness ~ factor:income + factor:sales + educ | urban:income + urban:sales + educ, data=DT)

The structure for ivmodel is:

second_stage<- ivmodel(Y=DT$Happiness,D=DT$factor,Z=DT[,c("urban","size")],X=DT$educ, na.action = na.omit) 

Any help with figuring out how to do this properly would be greatly appreciated!

Answer 1

You have an endogenous variable $x_1$ which is a factor, let's say that it takes 6 values ($x_1 = ``a",...``f")$. In your model it enters with an interaction with variables $x_2$ and $x_3$. Therefore your model is equivalent to

$$y = a_11_{(x_1=``a")}x_2+...a_51_{(x_1=``f")}x_2+b_11_{(x_1=``a")}x_3+...b_51_{(x_1=``f")}x_3+cx_4+e$$

where $1_{(x_1=``a")} = 1$ if $x_1=``a"$ and zero otherwise.

So it has $2\times6$ endogenous variables (the 12 interactions) and you need at least 12 instruments.

In your example you have an endogenous variable $factor$ and two good instrumental variables $urban$ and $size$. If they are good instuments for $factor$ then their interaction with $income$ and $sales$ is good for the same interaction with the instrumented variable. But still you need at least 12 instruments.

The first step of the 2SLS will be a OLS regression where the dependent variables are the 12 endogenous interactions $1_{(x_1=``a")}x_2,...1_{(x_1=``f")}x_3$ and the independent variables are all the instruments and the rest of the exogenous variables.

The second step is another OLS regression where the dependent variable is the original one and the independent variables are the exogenous variables + the fitted values of the first step.

Now, following your code example, suppose your instrument is also a factor, let's say it has 6 levels, you'll have enough variables to estimate it. In your code the variable Factor is numeric, so let's make it a factor first.

DT$Factor = as.factor(DT$Factor)

Based on your code, without instruments:

lm(happiness ~ Factor:income + Factor:sales + educ,data=DT)

Coefficients:
(Intercept)            educ  Factor0:income  Factor1:income  Factor2:income  Factor3:income  Factor4:income  Factor5:income   Factor0:sales  
   4.46555         3.25149        18.18265       -19.68570       -16.71225       -38.04578       -19.68150         2.98939        -0.10307  
 Factor1:sales   Factor2:sales   Factor3:sales   Factor4:sales   Factor5:sales  
  -0.16310        -0.03131        -0.12445        -0.14503        -0.08158

Now, in a 2SLS model using ivreg with the two instruments you provided:

ivreg(happiness ~ Factor:income + Factor:sales + educ | urbanisation:income + urbanisation:sales + size:income + size:sales + educ, data=DT)

Coefficients:
(Intercept)            educ  Factor0:income  Factor1:income  Factor2:income  Factor3:income  Factor4:income  Factor5:income   Factor0:sales  
    -6.690           8.736        -296.010         292.081          36.372        -183.167        -231.191         712.253          -1.522  
 Factor1:sales   Factor2:sales   Factor3:sales   Factor4:sales   Factor5:sales  
     2.428              NA              NA              NA              NA  

Warning message:
In ivreg.fit(X, Y, Z, weights, offset, ...) :
more regressors than instruments

You see that it can only estimate 8 endogenous variables, this is because we provided 8 instruments: urbanisation has 3 levels (hence 6 interactions) + 2 interactions for size.

Now consider using the variable urbanisation2 with 6 levels:

DT$urbanisation2 = rep(sample(c("A", "B", "C","D","E","F"), length(panelID), replace = T), each = n)

ivreg(happiness ~ Factor:income + Factor:sales + educ | urbanisation2:income + urbanisation2:sales + size:income + size:sales + educ, data=DT)

Coefficients:
   (Intercept)            educ  Factor0:income  Factor1:income  Factor2:income  Factor3:income  Factor4:income  Factor5:income   Factor0:sales  
     4.513e+00       3.321e+00       6.329e+00      -1.802e+01       1.045e+01      -1.154e+02       6.208e+01      -3.196e+01      -8.905e-03  
 Factor1:sales   Factor2:sales   Factor3:sales   Factor4:sales   Factor5:sales  
    -1.706e-01      -1.730e-01       2.692e-02      -3.410e-01      -3.671e-02

Now it works because you have enough instruments.