Preference shock setting Cook Devereux 2016 Replicating

Cook 和 Devereux - 2016 - Exchange rate flexibility under the zero lower bou.pdf (1.5 MB)
1-s2.0-S0022199616300460-mmc1.pdf (143.6 KB)
Dear Professors and friends,
Hope you find the message well! I am trying to replicate the fig 8 in Cook and Devereux (2016) JIE paper with dynare. Here I’ve solved the optimal conditions for single currency area and use the MMC method to deal with the zero lower bound. In the paper we derived the result that preference shock epsilon_t enters natural rate of interest in eq. 12 and 15 (in difference. Here a markov process is assumed: (1-mu), but I would like to generally set the process of epsilon_t) and the law of motion of terms of trade in eq. 30 (in level).

Here is the question: if I do perfect_forecast with MMC method to deal the ZLB problem, aligned with your code in Gali 2015 ch5, I have to set the preference shock epsilon that pushes the home country into liquidity trap. In your code you let \bar{r}_t^n (directly) equals to -1 for 6 periods. But now with the 3-period preference shock I have to let eps = -1.5, -1.0, -0.5, 0, 0, 0,… such that \bar{r} = (coefficient) \times E_t(eps_{t+1} - eps_t) being negatively constant. I am confusing about it. Could you please help me the setting of preference shock?

Much appreciated!

To clarify, I am really confused about the example code in Gali_2015_chapter_5_commitment_ZLB.mod and Gali_2015_chapter_5_commitment.mod, where one of them sets a preference shock to eps_z and the other sets the natural rate of interest r_nat=-1 as a shock. I would like to focus on the ZLB case, but instead, I need to set the preference shock since it will enters the terms of trade.

The preference shock will immediately affect the natural rate. The example Gali provides is very stylized. Instead of specifying a shock to trigger the fall in the natural rate, he simply directy shocks the natural rate. In most models, that cannot be easily done.

Thank you professor, I managed to understand the difference settings in Gali’s work. Later I got nice replication of Cooks.

Thanks again.