[R] Simalteneous Equation Doubt in R

[Berend Hasselman]

On 22-03-2012, at 09:15, Priya.Saha at zycus.com wrote:

> Hi List
> 
> l am interested in developing price model. I have found a research paper 
> related to price model of corn in US market where it has taken demand & 
> supply forces into consideration. Following are the equation:
> Supply equation:
> St= a0+a1Pt-1+a2Rt-1+a3St-1+a5D1+a6D2+a7D3+U1                -(1)
> Where   D1,D2,D3=Quarterly Dummy Variables(Since quarterly data are 
> considered)
> Here, Supply equation has 1 endogenous (St) & 6 exogenous variables  (P
> t-1,Rt-1,St-1,D1,D2,D3) 
> Demand Side: 
> Demand of corn is divided into 3 equations:
> Feed equation:
> Ft=b0+b1Pt+b2P(sm)t+b3Bt+b4COFt+b5Ht+a6D1+a7D2+a8D3+U2           -(2)
> here there are 2 endogenous variable(Ft, Pt) & 7 exogenous variables 
> (P(sm)t,Bt,COFt,D1,D2,D3) 
> Export equation:
> EXt= c0+c1Pt+c2EXt-1+c3Wt+c4DXt+c5GDPt+c6D1+c7D2+c8D3+U3         -(3)
> here there are 2 endogenous variable(EXt, Pt) & 7 exogenous variables (EX
> t-1,Wt,DXt,D1,D2,D3) 
> Food, Alcohol, Industry (FAI) Demand Equation:
> FAIt= d0+d1Pt+d2Etht+d3Popt+d4Tt+d5D1+d6D2+d7D3+U4      -(4)
> here there are 2 endogenous variable(FAIt, Pt) & 6 exogenous variable(Eth
> t,Popt,Tt,D1,D2,D3) 
> Price Equation: price of corn is determined by supply and demand 
> simultaneously, following is the reduced form equation: 
> Pt=µ0+µ1St+µ2Ft+µ3EXt+µ4FAIt+µ5Pt-1+µ6D1+µ7D2+µ8D3+U5               -(5)
> here there are 5 endogenous variable(St, Ft,EXt, FAIt, Pt) & 4 exogenous 
> variable(Pt-1,D1,D2,D3)
> Now my question is :
> By applying 3SLS in the price equation, it will show the impact of 
> variables on Pt which are mentioned in equation (5).But if l want to find 
> impact of ETHt from equation (4) on Pt , l'll have to substitute equation 
> (1),(2),(3),(4) in price equation(5), which manually becomes very tedious, 
> is there any way this could be done directly in R?

Your system can be written compactly as

St  =                    + zS
Ft  = b1Pt               + zF
EXt = c1Pt               + zEX
FAIt= d1Pt               + zFAI
Pt  = µ2Ft+µ3EXt+µ4FAIt  + zP

St is exogenous so can be ignored.
The system is linear and can be written as where the zXXX are the exogenous terms of the equation for XXX.

( Ft   )      ( 0   0  0 b1 )  ( Ft   )         ( zF   ) 
( EXt  ) = ( 0   0  0 c1 )  ( EXt  )      +  ( zEX  ) 
( FAIt )      ( 0   0  0 d1 )  ( FAIt )         ( zFAI ) 
( Pt   )      ( µ2 µ3 µ4  0 )  ( Pt   )         ( zP   ) 

(Note: read the stacked ( and ) as a single large ( or ))
or

y = A %*% y + z

which can be written as

y = solve(diag(4)-A) %*% z

You only need to construct the matrix A.

Berend