# 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

## 1 Answer

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.

• Thank you for your comment, Durden. What I think difficult is that it is commonly believed that foreign aid is related to the level of democracy. I think if I follow your solution, what I will get is "the effect of aid on democracy". However, what I want to know is "the effect of aid on democracy through inequality", because my hypotheses say that foreign aid firstly increases the inequality level and inequality level raises the democracy level. That's why I did not use simple regression, putting all variables in one equation. – user51966 May 31 '15 at 5:26
• I downvoted this answer for the reason that i) it only partially addresses the question (e.g. no mention of the appropriate statistical technique in this case) and ii) the statement made about democracy and inequality is just wrong. In unequal societies elites may have more power and thus keep down the level of democracy, whereas an increase in the level of democracy may lead to more equal distribution of wealth etc., i.e. a typical reverse causality problem. Stating that these two are independent goes against decades of research in political economy and development economics. – Andy May 31 '15 at 9:03
• @Andy Please recognize that my comment addresses the model in user51966's post, not the (causal) link between democracy and inequality in general. Your comments on the general case may be right, but they do not apply to the model in this question. – Durden May 31 '15 at 10:10
• 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
• Sorry to interrupt here, but what I want to test is two hypotheses written on the very first part of my question. In two equations I wrote in the question, it seems there is no effect of democracy on inequality, but since it's a model, it greatly simplifies the reality and I want to make them rich. Would you mind reading the first five lines of my question? I hope it will make everything clearer. – user51966 May 31 '15 at 23:56