Stage 1:

Y = B*X + m

where B is the vector of coefficients, X is the matrix of explanatory variables and m is the residual.

Stage 2:

y = a*x + b*Yhat + e

where a is the vector of auxiliary coefficients, x is the matrix of auxiliary explanatory variables, Yhat is an explanatory variable defined as the predicted values from Stage 1, b is its coefficient and e is the residual.

A transformation, based on each observation's contribution to the likelihood function, is then used to adjust the standard errors in Stage 2 to account for the variability in Yhat.

I was wondering if anyone knows of a method that does a similar error correction when there are multiple estimated parameters included in the second stage. For example:

Stage 1:

Y = B*X + m

where notation is defined as above.

Stage 2:

y = a*x + b*Yhat + c*m + e

In other words, can we correct the standard errors when including the first stage's predicted values and the first stage's residual values as parameters in the second stage?

Similarly, can we correct the standard errors when including predicted values from multiple first-stages? As follows:

Stage 1a:

Y1 = B1*X1 + m1

Stage 1b:

Y2 = B2*X2 + m2

Stage 2:

y = a*x + b*Y1hat + c*Y2hat + e

Any references that might shed light on this issue would be greatly appreciated.

Thanks,

-TC