Let the original regression be as follows:
Estimating (1) by OLS will yield estimates (coefficient values and the associated statistics like standard error, t values, etc.) for a1, c1 and c2. The numerical values of b1 and b2 can be easily derived from c1 and c2. But how to compute the standard errors of b1 and b2? Software packages can produce those statistics , but I am interested in knowing the procedure so that I can manually progam codes.
I see two possible ways:
1) fit Y=a1X1+(1-a1)(b1X2+b2X3) as a non-linear regression model what will give you the standard errors directly, or
2) use error propagation (https://en.wikipedia.org/wiki/Propagation_of_uncertainty)
Rantu -
The source for your regression,
Y=a1X1+(1-a1)(b1X2+b2X3)+(error term),
is of interest, because it looks very much like you have a weighted pair of regressions and an attempt to combine them into one regression. I mean, it resembles a James-Stein/shrinkage estimator, or other ways of combining estimates or predictions or forecasts. I looked back into my papers, and found a reference, Granger, C.W.J., and Newbold, P.(1986), Forecasting Economic Time Series, 2nd edition, Academic Press. In that book, I recall that they combined two forecasts, using a reference I just tracked down again. As I recall it was John M Bates, Clive WJ Granger, "The combination of forecasts," Journal of the Operational Research Society 20 (4), 451-468, 1969. For such cases you take two predictions, or two forecasts, and multiply one, say by w, and the other by 1-w, and the weight, w, is obtained by considering variances, and perhaps covariances, so that the prediction, or forecast - prediction in your case - which has the least variance (of the prediction error) gets the highest weight.
But if instead of what I think is called a shrinkage estimator, you try to put both predictions in the same regression equation, then I think you may have very large collinearity. Here, I think x1 is used for one prediction of y, and the predictors x2 and x3 are used for another prediction of y, if I'm guessing the origin of this problem correctly. You may be better off leaving these as separate regressions whose predictions you combine with the weights w and 1-w. Here you have "a1," which makes it look like one regression is y = x1 + e1, but perhaps a coefficient is missing or combined with something.
When I searched for that Bates and Granger reference, I noted a number of other similar sounding references. For predictions, as noted above, you could research shrinkage estimators or perhaps search on "James-Stein," I think.
Perhaps I am wrong that the regression equation here started out as something like y = cx1 + e1 and y = dx2 + gx3 + e2, and someone wanted to write one regression equation from them. But it looked extremely familiar.
Just a thought.
Also, I'd estimate the coefficient of heteroscedasticity in each case and use weighted least squares (WLS) regression, not OLS.
Cheers - Jim
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