Observation Equation for demeaned, first differenced data

Estimation ML/Bayesian estimation
1

Hi all,

I have consulted Johannes’ guide to specify observation equations and I would have to ask a quick question regarding this. If I feed into the model first differenced, demeaned data, what would be the underlying observation equation? Would it be:

Case 1:

data_{QQ_{growth}} = \frac{q_t}{q_{t-1}} - 1

which is mean zero, based on the theoretical moments:

data_QQ_growth 0.0000 (mean) 0.0198 (std. dev) 0.0004 (variance)

Or is it case 2:

data_{QQ_{growth}} = \frac{q_{demeaned,t}}{q_{{demeaned},t-1}} - 1

where q_{demeaned,t} = \frac{q_t}{q_{SS}} and q_{SS} is the steady state of q_t.

My question is which version is the correct one?

Many thanks for your help.

Robert

2

If q is a level variable, then the first one is \frac{q_t-q_{t-1}}{q_{t-1}}, which is a growth rate.
The second one does not make sense. In that case q is in logs. The growth rate would then approximately be q_t-q_{t-1} as \frac{q_t-q_{t-1}}{q_{t-1}}\approx \log q_t-\log q_{t-1}

3

Thanks Johannes. Yes q is a level variable. I actually forgot to ask you the most important question why I wrote down those two versions. That is: for the observation equation in case one, do we have to specifically account for the mean zero feature in the data? Since the theoretical moment shows a mean of zero I’d say no. Is that correct?

Many thanks.

Rob

4

In steady state dataQQgrowth=\frac{q}{q}−1=0, so the object is mean 0 and therefore corresponds to mean 0 data.

5

Thanks a lot Johannes!