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RE: Simultaneous equation model with interaction terms [ reply ]
By: Arne Henningsen on 2015-08-23 05:52
[forum:42489]
Dear Janka:

Question 1:
This questions seems to be not about systemfit but about the correct model specification. If X1 and X2 are exogenous ("valid instruments"), you should use them as instrumental variables, as they are certainly correlated with the interaction terms ("good instruments").

Question 2:
eq1 <- Y1 ~ I(Y2*X1) + I(Y2*X2) + X3
eq2 <- Y2 ~ Y1 + X1 + X2 +X3
inst <- ~ X1 + X2 + X3 + Z1 + Z2

Best regards,
Arne

Simultaneous equation model with interaction terms [ reply ]
By: Janka Vanschoenwinkel on 2015-08-21 08:51
[forum:42475]
Dear forum members,

I would like to estimate the following 2 models:

Model 1:
Y1 = b10 + b11 * (Y2*X1) + b12*(Y2*X2)+ b13 * X3 + e1

Y2 = b20 + b21 * (Y1*X1) + b22*(Y1*X2)+ b23 * X3 + e2,

where Y1 and Y2 are two endogenous dependent variables,
X1 and X2 are exogenous explanatory variables that are interacted with the endogenous variables,
X3 is an exogenous variable,
Z1 and Z2 are the instruments,
e1 and e2 are two potentially contemporaneously correlated error terms,
and b10, b11, b12, b13, b20, b21, b22 and b23 are parameters to be estimated.

The way to do this in R Systemfit should be:
eq1 <- Y1 ~ I(Y2*X1) + I(Y2*X2) + X3
eq2 <- Y2 ~ I(Y1*X1) + I(Y1*X2) +X3
inst <- ~ X3 + Z1 + Z2
system <- list( eq1 = eq1, eq2 = eq2 )
reg2SLS <- systemfit( system, "2SLS", inst = inst, data = mydata

But in that case, X1 and X2 are not included in the instruments and I wondered whether this was correct. X1 and X2 are climate variables (for instance temperature in spring, temperature in autumn). They are therefore exogenous. But they are interacted with for instance irrigation. Irrigation is endogenous. Does this mean that when there is an interaction between X1 and Y2, and X2 and Y2, that X1 and X2 are not allowed to be put in the instruments part?



Model 2:
Y1 = b10 + b11 * (Y2*X1) + b12*(Y2*X2)+ b13 * X3 + e1

Y2 = b20 + b21 * Y1 + b22 * X1 + b23 * X2+ b24 * X3 + e2,

where Y1 and Y2 are two endogenous dependent variables,
X1 and X2 are exogenous explanatory variables that are only interacted with the endogenous variable Y2 in the first equation but that are not interacted in the second equation,
X3 is an exogenous variable,
Z1 and Z2 are instruments
e1 and e2 are two potentially contemporaneously correlated error terms, and b10, b11, b12, b13, b20, b21, b22, b23 and b24 are parameters to be estimated.

How do I enter this model in R, given the fact that in the first equation there is an interaction, and in the second equation, there is no interaction?

Thank you very much in advance!

Janka

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