Advantage of estimating on non-differenced data?

[cfp]

Is there any advantage to estimating on non-differenced data? Certainly estimating on differenced data is easier, and you'd expect it to be more robust too. What's gained by using non-differenced data? You gain one extra data point, but you have as many extra parameters to infer (the initial levels of the non-stationary series). You learn something about e.g. productivity levels I guess, but you could always back this information out from the first data point and the estimated series for productivity differences.

Maybe I'm missing something?

[cfp]

Thinking about it some more, I guess it must be (something like) that when you estimate on non-stationary data, if you miss the actual trend in one period then your residuals are punished in all future periods, which is not the case if you estimate with differenced data. Still seems a bit odd that it should make any difference to the ML estimates (if indeed it does).

[jpfeifer]

Hi, the point is that you usually have to map non-stationary data to a stationary model. How you do this is up to you. Some people simply use an HP-filter to get rid of the trend and directly map these data observations to model variables.
A different approach is to use a first difference filter, i.e. take differenced data and use an observation equation to map differenced data to the model variables (as e.g. in Smets/Wouters). Of coursem this implies different kinds of trends one assumes to be present.

In principle, using differenced or non-differenced data does not make a difference. Given the discarded starting value and the trend specification, differenced data can easily be transformed back. Even the problem with missing the actual trend is the same. By differencing, you exactly specify the kind of trend that you think is present in the data and which you take out by differencing. If this is not the actual trend, by removing it you miss future data points the same way as with non-differenced data in case of a misspecified trend.

From a practical point of view the problem rather is that if you try to match non-differenced data to your model, you might have trouble in Dynare as the non-differenced data does not possess a well-defined steady state around which to approximate.

[cfp]

Umm, I'm not sure what you wrote is the whole story. You have some nonstationary model, which you've made stationary around a stochastic trend by dividing through by that trend where appropriate. You're then left with a set of stationary equations in variables relative to trend, and in the growth rate of the trend itself. You then have two choices, you either 1) estimate using differences of the observed non-stationary series, which will be stationary combinations of the variables already in your model, or 2) use Dynare's ability to estimate unit root processes and estimate in levels.
I don't think that by differencing "you're exactly specifying the kind of trend that you think is present in the data" here, other than "it has a unit root". What you wrote makes sense for the stationary around a non-stochastic trend case, but all of the models I work with have stochastic trends (which is surely the standard these days).

[jpfeifer]

Hi, I think it even applies to the stochastic trend case. If you difference, you implicitly assume that your data either has a deterministic trend or a unit root (potentially with drift) that you want to take out by first difference filtering. The nice thing in principle is that you don't have to worry at this step, which of the two is present. However, the mean of the differenced series corresponds to either the drift term in the random walk process or the increase in the deterministic trend - both of which you have to take care of when writing down your model.

I may be wrong, but I don't see how using levels of the data would change anything. If your data trend was deterministic but you matched it to a detrended model based on a random walk with drift you will have a problem irregardless of using differenced or level data. The same holds true if your data follows a random with drift, but you base your model on a deterministic trend.

I don't think that by differencing "you're exactly specifying the kind of trend that you think is present in the data" here, other than "it has a unit root".

Maybe this is just a matter of semantics, but by using a first difference filter, you implicitly assume that this filter takes out the trend in the data, with the remainder being the business cycle fluctuation around this stochastic trend you are trying to explain with your model. Hence, you specify your trend present in the data is this unit root. In contrast, by using an HP-filter you specify a different kind of trend.

To phrase my point differently: Given the trend structure embedded in your model and the corresponding data, you are able to transform the model estimated on differenced data into a model estimated on level data just by "inverting" the trend filtering performed (using for example the information embedded in the model that the driving process follows a random walk with drift).

[cfp]

OK. Thanks. That's certainly the answer I was looking for. Does beg the question of why support for estimating non-stationary models with unitrootvars was included.