There is an intratemporal Debt in the model I build, growth rate of which follows AR(1) process. Denote at is the output trend over debt trend.
For simplicity, assume at Debt=Yt*, after detrending, **at is gone, then ** debt=y (small letter means stationary variable).
The growth rate of Debt never appears in modal since Debt is Not intertemporal.

However, I use growth rate Debt data to estimate , so Measurement equation will appear the Debt growth rate AR(1) shock. The estimation results seem all good.

Do you think there is any obvious problem with that(Estimated AR1 shock not in model but in measurement equation) ?

I am not fully getting your point. Please provide a full, precise algebraic description of your problem.
This is a measure of correctly specifying the observation equation. Note that

is irrelevant. The growth rate of hours typically also does not appear in the model but is still used for estimation. You can still define the growth rate of debt by comparing the stock of intertemporal debt in neighboring periods. If this is what you observe in the data, you are fine.

Many thanks for your reply. Your suggestion would very important to me, since It seems that no one has done it before. I don’t know if it is right.

Please find attached precise algebraic description of my problem. Basically, there is an AR(1) process only appearing in measurement equation to be estimated.

This opens up a lot of questions, mostly about identification. If phi_t is marginal costs determined in the rest of the model and D_t does not appear anywhere else in the model, it should not determine/affect the dynamics of anything else in the model and is therefore redundant. You will be estimating parameters that are not needed for the model. Maybe you should try with a calibrated model whether anything changes when you change these parameters that now only show up in the measurement equation.

A second question is whether Y_t and D_t really have different trends in the long-run? If yes, this would most probably kill the balanced growth path property. For that reason I am not completely convinced this setup is correct.

[quote=“jpfeifer”]This opens up a lot of questions, mostly about identification. If phi_t is marginal costs determined in the rest of the model and D_t does not appear anywhere else in the model, it should not determine/affect the dynamics of anything else in the model and is therefore redundant. You will be estimating parameters that are not needed for the model. Maybe you should try with a calibrated model whether anything changes when you change these parameters that now only show up in the measurement equation.

A second question is whether Y_t and D_t really have different trends in the long-run? If yes, this would most probably kill the balanced growth path property. For that reason I am not completely convinced this setup is correct.[/quote]

Dear Johannes,

I really appreciate you taking your time to answer my question in so detail. Many thanks.

Your description perfectly matches my model. Yes, the debt equation is **redundant **, and is only used for introducing debt data in measurement equation when estimate the model.

In the long run, Debt/Output data ratio is growing over time so they actually have different trend. Even though in Econometrica(2013) paper by Zheng Liu,etc, they assume Debt has same trend with output, in my model I do not want to assume that , since the estimation results would otherwise have been rather bad.

Could you please have a look attached modified measurement equation ? Now if I assume model Dt has same trend with output, but data debt matches the product of (a trend ratio) and ( Dt) in measurement equation, would this be a problem? Still need to estimate the AR1 …it is just like a measurement error but with persistence.

Back to the first, fundamental question: why do you want to use debt as an observable in the first place? If it does not play a role in your model, but rather only enters via a redundant equation appended to the model, you can simply leave it out. The only purpose it seems to serve is to estimate that particular AR1 process that would not be there otherwise.

Yes, you are right. I can simply leave it out. If doing so, I can use the observed debt data to match the** right hand side of left out equation**-----the product of marginal cost and output , which is assumed all from borrowing.
Could you please have a look attached pdf showing this change ? I put a trend adjustment in the measurement equation.

OK, when you use the debt relationship to inform your model about the movement of marginal costs, then it makes sense to include debt.
I am still not entirely convinced by your detrending. If debt and output grow at different rates, there should be a constant term in your observation equation that is currently missing. It might be easier to demean debt growth in the data and work with a mean 0 observation equation.