I'm trying to run a series of simultaneous linear regressions, and I want to constrain the regression coefficients. For the standard ordinary least squares regression, the specification of the estimated coefficients in matrix form is
B = (X'.X)-1X'.Y
Where X are the independent variables, Y are the dependent variables, and B are the estimated coefficients for X on Y.
I also have seen in Amemiya's Advanced Econometrics (1985) that Constrained Linear Least Squares can be used to impose a constraint on regression coefficients for an independent regression, such that the constrained estimated coefficients are:
Bc = B - (X'.X)-1.Q.(Q'.(X'.X)-1.Q)-1.(Q'.B - C)
Where Bc is the constrained estimated coefficients, B are the standard OLS coefficients, Q is a matrix of known constants governing the linear constraints on the coefficients, and C are the right-hand-side terms for the constraints (such that Q'.Bc = C).
However, I would like to constrain coefficients over several regressions.
For example, I have the equations: Y1 = A0 + A1.X1 + A2.X2 Y2 = B0 + B1.X1 + B2.X2
The Amemiya approach would allow me to impose a constraint such as A0 + A1 + A2 = 1
However, I would like to impose a constraint like: A1 + B1 = 1
I would really appreciate it if someone could give me some insight in to how to do this?
Thanks in advance.