Estimation of simultaneous equations model

My model is as follows: \begin{aligned} y&=a_0 + a_1x_1 + a_2x_2 \\ x_2 &= b_0 + b_1x_1 + b_2z \end{aligned}

I'm only interested in the effect of $x_2$ on $y$. More precisely, I want to see the combined effect of:

1. $x_2$ directly, and
2. $x_2$ implicitly (through $x_1$)

After estimating the coefficients, I have the following procedure:

1. solve equation 2 for $x_1$
2. substitute the resulting term in equation 1.
3. look at the "new" $x_2$ coefficient: $(a_1/b_1 + a_2)$

The model comes from theory and there's no possibility to include other variables as instruments.

I tried multiple equations OLS with HAC standard errors and got some good results, but I'm not at all convinced that this is the right way.

• Welcome to the site, @user26594. Can you clarify your question? We need something more specific than you're looking for some advice or comments. What is it you find unsatisfactory about your current results? Commented Jun 6, 2013 at 22:23
• Perhaps I'm being dense, but can't you just collapse your model into $y=a_0 + a_1x_1 + a_2z$ and go from there? Commented Jun 6, 2013 at 23:04
• The results were good and in line with Theory, but I have concerns that my estimates are biased (because of the endogeneity of x. @Matt Krause: no, this would just give me the direct effect of $x_1$ Commented Jun 7, 2013 at 7:49

The way you have your model set up (which are usually intended to reflect the causality in these setups), it looks like it's $x_1$ that has an implicit and explicit effect on $z$, not $x_2$ ($x_1$ affects $x_2$ which affects $z$, and $x_1$ also has a direct effect on $z$).
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• Thx for your response. The variable $z$ is exogenous and not so important for the model, but of course it also effects my estimated coefficients. My concerns are the unbiasedness of my estimates for $a_1$, $b_1$ and $a_2$ . The mediation mehthods (never heard of it before) look interesting, especially for checking whether this the causal effects are as expected. Commented Jun 7, 2013 at 8:16
• Why does $x_1$ has an effect on $z$? Could you not use something related to two-staged-least-squares with $x_1$ and $z$ being the IVs for $x_2$? If you dont want to use any instruments there is a approach called Rank-Order Instrument Variables: Rummery, Vella, Verbeek - Estimating the Returns to Education for Australian Youth via Rank-Order Instrumental Variables Commented Jun 7, 2013 at 8:34