Four simultaneous equations with first differences and lagged variable

[Eva Michalikova]

Dear statalists,

I am trying to estimate coefficients of four equations:

M'(t) = alpha1 * M(t) + alpha2 * A(t) + alpha3 * T(t)
A'(t) = alpha4 * M(t) + alpha 5 * R(t) + alpha6 * T(t)
R'(t) = alpha7 * M(t) + alpha8 * T(t)
T'(t) = alpha9 * T(t-1)

There are derivatives on the left side and I will aproximate them as first differences (denoted as "diff"):

diff M = alpha1 * M + alpha2 * A + alpha3 * T
diff A = alpha4 * M + alpha 5 * R + alpha6 * T
diff R = alpha7 * M + alpha8 * T
diff T = alpha9 * T(t-1)

Data represent time serie between 2011 - 2021 (no panel).

I would like to kindly ask you what technique to choose - simultaneous regression - SEM? However, I am little confused about the structure of model. Are M, A, T and R exogeneous or not? what assumptions should be verified?


Thank you very much.

[Jared Greathouse]

https://stats.oarc.ucla.edu/stata/ex...ations-models/

[Jared Greathouse]

https://stats.oarc.ucla.edu/stata/ex...ations-models/

[Eva Michalikova]

Thank you.

I Will try 2sls and 3sls.

My last problem is the specification of model.

Are M, A, T and R exogeneous or not?
What assumptions should be verified?

[Joro Kolev]

What is the meaning of primes? How is M' different from M, for example?

[Eva Michalikova]

M', A', R' and T' are derivatives of M, A, R and T.
It is a system of 4 differential equations. The client wants these equations to be estimated as regression models.
I approximate the derivative on the left M' as difference between current and previous value (it means, M' is aproximated by M(t)-M(t-1)=diffM).
I looked up information on Wooldridge and here it says that "if we add lagged variable to equation, we can treat it as exogeneous with assumption that error term is uncorrelated with current exogeneous variables (which is standart)." So I suppose if I have variable diffM = M(t) - M(t-1) == M' on the left side, then variable M on the right side can be also treated as exogeneous...

[Joro Kolev]

This does not look like a regression system, because you are regressing past values, say M{t} - M{t-1}, on future values, say M{t}. This thing in itself is making all the regressors on the right hand side endegenous.

[Eva Michalikova]

Ok, I can convert lagged values in the first equation on the left side and solve different form of system (rest equation can be same...):

original form:
diff M =M(t) - M(t-1) = alpha1 * M(t) + alpha2 * A(t) + alpha3 * T(t)
diff A = A(t) - A(t-1) = alpha4 * M(t) + alpha 5 * R(t) + alpha6 * T(t)
diff R = R(t) - R(t-1) = alpha7 * M(t) + alpha8 * T(t)
diff T = T(t) - T(t-1)= alpha9 * T(t-1)

New form:
M(t) = 1 / (1-alpha1) M(t-1) + alpha2/(1-alpha1) * A(t) + alpha3 / (1-alpha1) * T(t)
diff A = A(t) - A(t-1) = alpha4 * M(t) + alpha 5 * R(t) + alpha6 * T(t)
diff R = R(t) - R(t-1) = alpha7 * M(t) + alpha8 * T(t)
diff T = T(t) - T(t-1)= alpha9 * T(t-1)

Now I have no past value on left and future value on right....

Is there any possibility how to estimate parameters of these system by regression analysis?