Problem on model with trend

Estimation
1

Dear Professor Pfeifer,

  1. In your paper A Guide to Specifying Observation Equations for the Estimation of DSGE Models, the equation (64) (on page 57)says the observation equation should be written as y_{t}^{obs}={{\tilde{y}}_{t}}-{{\tilde{y}}_{t-1}}+{{\tilde{\mu }}_{t}}, in which {\mu }_{t} is growth rate. However, in my model other variables like consumption and investment also have growth rate {\mu }_{t}, and their mean value of observed data are of course not equal to the mean value of output data. So I cannot get 0 residual if I write the code as c_{t}^{obs}={{\tilde{c}}_{t}}-{{\tilde{c}}_{t-1}}+{{\tilde{\mu }}_{t}} or i_{t}^{obs}={{\tilde{i}}_{t}}-{{\tilde{i}}_{t-1}}+{{\tilde{\mu }}_{t}}. How should I deal with this?

  2. In NERI_APPENDIX_E.pdf (164.2 KB), they get inflation equation (on page 11) written as \log {{\pi }_{t}}-{{\iota }_{\pi }}\log {{\pi }_{t-1}}=\beta \left( {{E}_{t}}\log {{\pi }_{t+1}}-{{\iota }_{\pi }}\log {{\pi }_{t}} \right)-\frac{\left( 1-{{\theta }_{\pi }} \right)\left( 1-\beta {{G}_{C}}{{\theta }_{\pi }} \right)}{{{\theta }_{\pi }}}\log \left( \frac{{{X}_{t}}}{X} \right).
    But in my calculation, I can only get \log {{\pi }_{t}}-{{\iota }_{\pi }}\log {{\pi }_{t-1}}=\beta \left( {{E}_{t}}\log {{\pi }_{t+1}}-{{\iota }_{\pi }}\log {{\pi }_{t}} \right)-\frac{\left( 1-{{\theta }_{\pi }} \right)\left( 1-\beta {{\theta }_{\pi }} \right)}{{{\theta }_{\pi }}}\log \left( \frac{{{X}_{t}}}{X} \right), in which the trend parameter {G}_{C} is missing. I check many times but still cannot find the mistake. Where did I go wrong?

Thank you for your time!

2
  1. Please consult the remark in Cointegration in the guide.
  2. So you got a different slope of the NK-PC?
3

Dear Professor Pfeifer,

  1. I’m not sure if that remark could solve the problem. Does it mean that I could add an adjustment parameter to make sure the data of consumption and investment has same trend with output, e.g. written as
    y_{t}^{obs}={{\tilde{y}}_{t}}-{{\tilde{y}}_{t-1}}+{{\tilde{\mu }}_{t}}
    c_{t}^{obs}={{\tilde{c}}_{t}}-{{\tilde{c}}_{t-1}}+{{\tilde{\mu }}_{t}}+{\log {\mu}_{c} }
    y_{t}^{obs}={{\tilde{y}}_{t}}-{{\tilde{y}}_{t-1}}+{{\tilde{\mu }}_{t}}+{\log {\mu}_{i} }
    OR I could just use observed data demeaned by their respective mean and write observation equation as
    y_{t}^{obs}={{\tilde{y}}_{t}}-{{\tilde{y}}_{t-1}}
    c_{t}^{obs}={{\tilde{c}}_{t}}-{{\tilde{c}}_{t-1}}
    i_{t}^{obs}={{\tilde{i}}_{t}}-{{\tilde{i}}_{t-1}}?

  2. In my calculation, the inflation equation in the model with trend is just the same as the model without trend, which means the trend does not impact this equation. But their equation has a growth rate in the slope. I was wondering where it cames out. What did I neglect ?

Thanks again for your time!

4
  1. If you don’t want to impose cointegration (and therefore that different shocks account for difference in means), you can simply used demeaned data. That is mostly equivalent (apart from efficiency considerations) to the first set of equations you posted. The big question is \mu_t. If there are growth rate/unit root shocks then you will need that term.
  2. But the discount factor in that model is \beta G_C, not just \beta