I have a question related to applying the measurement error consistently across separate observation equations. This should be clear in the context of an example. Suppose we are estimating a model given consumption, investment, output, and the Solow residual as observables. There is a closed economy without government purchases, so we need to add a measurement error to one among C, I, or Y. Suppose we add the measurement error to investment.
So, we have in a linearized system
I_obs = I + e_I_ME
Now, we could define Y_obs as either
Y_obs = C_Y*C + I_Y*I, or Y_obs = C_Y*C + I_Y*(I_obs)
where C_Y and I_Y are the steady-state shares. Version 2 takes into account that the construction of output using the expenditure approach should inherit the measurement error of investment.
The logic carries over to the measurement equation for the Solow residual. The Solow residual uses output minus inputs weighted by their shares, so the specification is slightly different depending on the choice of output.
Thus, what I mean by consistency is that, even if we only considered measurement error in investment, it should be embedded in the other series by virtue of their construction.
Of course, even if more sounds, perhaps this tends to be largely irrelevant in practical terms?
A related follow-up is that, even if one does this, it is still useful to use the output variable
Y_r = C_Y*C + I_Y*I
with the muted measurement error if we want to look at the forecast error variance decomposition of just structural shocks.
Thanks so much!