# What is the model similar to 2SLS, but not for IV?

I am analyzing the effect of foreign aid on democracy and I would like to test following hypothesis (I simplified the original version):

1. The amount of aid the country received and the corruption level determine the inequality level of the country (inequality is measured by Gini coefficient).
2. The inequality level (affected by aid) is related to the democracy level.

Firstly, I simply did two OLS analyses. ($i$ is the index for countries, $t$ is for years)

1. ${\rm Inequality}_{i,t} = \beta_0 + \beta_1 {\rm Aid}_{i,t} + \beta_2 {\rm Corruption}_{i,t} + \varepsilon_{i,t}$
2. $\textrm{Democracy}_{i,t} = \beta_3 + \beta_4 \textrm{Democracy}_{i,t-1} + \beta_5 {\rm Inequality}_{i,t} + \varepsilon_{i,t}$

We can say "Aid is related to Inequality" and "Inequality is related to Democracy" (let me suppose all coefficients are positive and statistically significant), but cannot say "how much is aid related to democracy" or "how much does Aid affect the level of democracy".

What kind of technique should I use to model for my purpose?

It seems 2SLS is similar to what I want to do, but since variables in the first equation are not independent from $\textrm{Democracy}_{i,t}$, I think 2SLS is inappropriate here.

I am familiar with both R and Rstan, so either using R packages or building a model with Stan would be fine.

Thank you.

• Would a seemingly unrelated regressions (SUR) model fit your goal? It allows to estimate the correlation between the two error terms which accounts for unobserved factors that correlate with both inequality and democracy. The 'systemfit' package implements SUR estimation. – stijn Jun 1 '15 at 7:02

What makes you believe that the "variables in the first equation are not independent from $Democracy_{i,t}$"? Because unless you oversimplified your model above, $Inequality_{i,t}$ is obviously not a function of $Democracy_{i,t}$ at all, hence there is no simultaneity. To estimate this model, all you have to do is replace $Inequality_{i,t}$ in (2) with the definition in (1), and estimate the reduced form.
• I don't understand what you want to tell me with this. The OP appears to be looking for a causal effect so your suggestion of estimating the reduced form doesn't work given that and and corruption clearly are not good instruments. What the reduced form will estimate is simply a mix-up of $\beta_5\cdot \beta_1$ and $\beta_5 \cdot \beta_2$ which neither has a natural interpretation nor a causal one. – Andy May 31 '15 at 14:02