# Instrumental variables and mixed/multilevel models

I want to estimate a growth model to model the growth trajectories of individuals $j$ over multiple time points $t$ by applying a standard mixed/mutilevel model (also known as random coefficient model):

\begin{align} Y_{tj} &= \beta_{0_j} + \beta_{1_j}A_{tj} + \beta_{2_j}X_{tj} + \beta_{3_j}Z_{tj} + e_{tj} \\ \beta_{0_j} &= \beta_0 + u_{0_j} \\ \beta_{1_j} &= \beta_1 + u_{1_j} \\ \beta_{2_j} &= \beta_2 + u_{2_j} \\ \beta_{3_j} &= \beta_3 + u_{3_j} \end{align}

$A_{tj}$ is a linear growth function (i.e., time point of observation: $1,2,3, ..., t$). $X_{tj}$ is an exogenous covariate. $Z_{tj}$ is an endogenous covariate. Let's further assume that I have reasons to believe that one of the independent variables on level 1, $Z_{ij}$, is endogenous.

I am wondering whether or not I can use an instrumental variable approach (using the lag of the endogenous variable as an instrument) to deal with the endogeneity of $Z_{ij}$. However, I have not found any references or examples. Is this generally possible, and how can I change the standard R code for mixed models to do this? Currently I'm using the function call lmer(Y ~ X + Z + (1 + X + Z | ID), data=data).

Gelman & Hill (2006), Chapter 23.4 (pdf) show how to do this by applying a Bayesian approach. I would be interested in references and R code implementing a frequentist approach to control for endogeneity by using instrumental variables (i.e., lags of endogenous variables as instruments) within a multilevel model.

• I have formatted your question with $L_AT^EX$. Please double-check that I haven't mistakenly introduced errors to your formulae. – Sycorax says Reinstate Monica Dec 3 '13 at 14:16
• I haven't worked with endogenous/exogenous variables. Having said that: Have you thought of using lme instead of lmer so that you can define an AR=1 correlation structure for your errors and see if that takes care of your endogeneity? – usεr11852 says Reinstate Monic Dec 7 '13 at 2:35
• Thanks for the comment. In fact I am doing this, but I did not want to complicate things. Unfortunately, this does not solve the endogeneity issue. Here is a paper illustrating why lagging variables does not help too much: business.otago.ac.nz/econ/seminars/Abstracts/2013/… – majom Dec 7 '13 at 9:05
• Are there any updates to this problem? A frequentist approach in R would be great. – sam May 18 '16 at 14:57
• @ sam: Me and my colleagues started working on a R package which implements some approaches. An early version can be found at cran.r-project.org/web/packages/REndo/index.html. – majom May 19 '16 at 8:41