# Problems on the model with trend

Dear Professor Pfeifer,

I’ve encountered some problems when estimating the model with specified trend. Would you please give any advice? Thank you.

1. I see in Smets and Wouters(2007)'s code that they wrote the measurment equations in Dynare as:
dy=y-y(-1)+ctrend;
dc=c-c(-1)+ctrend;
In this case, the mean value of observed data dy or dc is not equal to the value of ctrend, which means the residuals of the static equations is not 0. How does Dynare run the code?

2. When dealing with the model with specified trend, do I need to add balance growth rate path into the utility function such as Iacoviello and Neri(2010) did in Housing Market Spillovers:Evidence from an Estimated DSGE Model? Because I find that many other papers did not do this.

3. In the model contains the trend, could I use one-sided HP filter instead of first-difference filter? e.g. I use measurment equations in Dynare as:
in which dy represents the observed data after one-sided HP filter. It works in the model without trend. I wonder if it is still available in the model with specified trend.

4. I was setting a model based on a fast growing country, which has a average quarterly growth rate like 2%(yearly 8%) lasting 20 years (up to now). In this case, do I need to calibrate the balance growth rate to 1.02? Or this high rate could not be the real balance growth rate because the growth will be slow down in the future? Could I use the data of these 20 years to estimate a proper balance growth rate?

Thanks again for your time on this.

1. No, in the model everything is consistent. This part is about imposing cointegration in the model. If the mean in the data is different across the two variables, then shocks need to account for that.
2. Your model needs to feature a BGP in the first place. In Iacoviello/Neri you need to adjust the utility function to get one.
3. You could do that, but may have an issue identifying parameters related to the trend.
4. Have a look at А model without balanced growth - #6 by jpfeifer

Dear Professor Pfeifer,

(1) I’m sorry but I do not quite understand the meaning of “imposing cointegration”. It seems that there is no shocks about the trend in Smets and Wouters(2007)'s paper. I’m still confused how the code runs successfully when the residuals of static equations is not 0.

(2) I read in Iacoviello and Neri(2010) that they add linear trend G_c in households’ utility function. While in Smets and Wouters(2007) or some other papers, the authors do not add this trend parameter into households’ utility function. What is the difference in between ? And which case would be more acceptable?

(4) I read that post and I think that “there will be balanced growth in the long run and the current behavior is just convergence to this point” is more reasonable. However, the problem still exists. Do I have to calibrate the balance growth rate to a high value like1.02? And could I use the data of these 20 years to estimate a proper balance growth rate?

And I have got a new question (5):
If I consider only one growth trend in the model, e.g. the growth trend of land supply which could be calibrated to a quarterly growth rate 1% according to the observed data. And this will lead to a low growth rate of output(like 0.15% quarterly, far less than 2% which data displays) because the growth of land supply is a relatively small factor comparing to the growth of TFP. Is it feasible of setting model like this? How should I set measurment equations to match the data in this case?

Thanks again for time on this.

1. I don’t get your point. The residuals of the static equation are 0. The big issue is that you impose the trend growth rates for the variables to be the same (conintegration). That may not be true in the data.
2. In the SW model the source of the trends in variables is not specified. They write down the stationarized model version, which features a BGP. In Iacoviello/Neri (2010) they have different sources of trends that do not allow for a BGP unless you alter the utility function.
3. No, if you think high growth is about convergence, then you need to set the long-run growth to approximate around this equilibrium. You cannot infer this growth rate from the data, because you don’t cleanly observe it.
4. You need to think more about your modeling approach. Do you care about fluctuations around a BGP or about converge behavior to the BGP?

Dear Professor Pfeifer,

1. Sorry that I’m a new learner on this. I understand the things you said about conintegration now. But I’m still confused why the residuals of the static equation is 0. Isn’t the residual of measurment equation equal to negative value of ctrend which is less than 0? Did I misunderstand anything?

2. In your paper Fiscal News and Macroeconomic Volatility and Land-Price Dynamics And Macroeconomic Fluctuations by Liu et.al.(2013), this balance growth rate parameter is also not contained in utility function. I’m confused in what case should I add this growth rate parameter into the utility function.

3. I’m not sure if I understand it correctly on the sentence “you need to set the long-run growth to approximate around this equilibrium”. Do you mean that I should set long-run growth to the mean value of data, i.e. 2%?

4. I mean to see the impact under different growth rate of land supply. How should I endeavour to make things right?

Sincerely thank you for your time on this.

1. No, with `dy=y-y(-1)+ctrend;` you have in steady state `dy=ctrend;`. Thus, `dy` has a non-zero mean (a mean estimated from the data), which is fine.
2. Again, it depends on the trends, the production function and preferences. With unit elasticites there is often not an issue. For more details on preferences that are unproblematic, see Modeling trend growth in DSGE, what utility function should be adopted? - #2 by jpfeifer
3. No, the mean value in the data is equal to the trend growth rate plus a term depending on the convergence to the BGP. It will overstate the trend growth.
4. The impact of what? What exactly is the research question?

Dear Professor Pfeifer,