# Problem on model with trend

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?

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

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 ?