simultaneous equations

[little_stats]

hi guys,
i need help regarding simultaneous equations.

for the following simultaneous equations:

R=b1+b2*Q+errorterm2
X=a1+a2*(Q+R)+errorterm1
Q=V-T+M

in order to get to reduced form, is it irrelevant which equation i substitute into other?
how to write IV estimator for b2?

[bryangoodrich]

You have 3 equations and a whole bunch of variables. If R, X, Q, V, T, and M are all unknown values, then 3 equations for 6 unknowns is a very underdetermined system (i.e., you cannot find a solution). If you have 2 unknowns it is overdetermined (multiple solutions). What are considered unknown and known here?

[SiBorg]

I haven't done this sort of thing in ages and nothing as complex as these, but my naive mind would suggest rewriting equation 2 by replacing R with b1+b2*Q+errorterm2. Rearrange this equation to give b2 = .... simplifying as you go along. Then substitute Q in your rearranged equation for V-T+M when you've finished.

Obviously, as Bryan says, it depends what you're aiming for as to what the best way to go about it is.

[little_stats]

so, i have data on R, X, Q, V, T, and M
need to estimate a1,a2, b1 and b2,
the third equation is an identity

or did i missunderstood the questions

[little_stats]

what is bothering me is that i don't know how to write IV estimator for a2, because Q+R , Q and R depend on other variables, help, any ideas?

[bryangoodrich]

+ Substitute (V-T+M) wherever you see Q in the first two equations.
+ Substitute the entirety of R into the second equation.
+ You estimate the first equation.
+ You use that to estimate the second equation.

[little_stats]

+ Substitute (V-T+M) wherever you see Q in the first two equations.
+ Substitute the entirety of R into the second equation.
+ You estimate the first equation.
+ You use that to estimate the second equation.

thank you, but i am also interested in writing the IV estimator first into the general formula: for example the ols estimator for b2 is (sum (Qi-Q)(Ri-R))/(sum(Qi-Q)^2)
where Q, R are means. this is what i want to write for a2, a general formula, but the (Q+R) is confusing me

[bryangoodrich]

Would you not just carry over the estimator for a1 and b1 when you substitute R? Seems pretty messy, but I've not had to do this application, so I'm not sure of an alternative perspective or any shortcuts.

[little_stats]

how about this

R=b1+b2*Q+errorterm2
X=a1+a2*(Q+R)+errorterm1
Q=X-T+M

how to write iv estimator for a2