# Forecast error variance decomposition under stochastic growth trend

Greetings. Suppose we have a model with an explicitly specified stochastic trend, as detailed in Section 5 of “A Guide to Specifying the Observation Equations for the Estimation of DSGE Models”. Further suppose that we demean the data, so that the observation equation (for, say, output) takes the form

y_t^{obs} = log y_t - log y_{t-1} + g_t - \overline{g}

where g_t is the rate of technological progress and g_bar is the mean. Here, y_t represents the detrended output y_t=Y_t/X_t, so that g_t=\log(X_t/X_{t-1}).

Now, suppose that we want to calculate the forecast error variance decomposition of output. Generally speaking, under the detrending procedure y_t moves in lockstep with X_t and therefore cannot be used.

Though y_t^{obs} adjusts for the detrending procedures, it is expressed as a growth rate, whereas we want a unit in levels. So, it seems that we really want the FEVD of the cumulative response of y_t^{obs}.

In the presence of a deterministic trend, like Smets and Wouters, the observation equation (demeaned) would simply be

y_t^{obs} = log y_t - log y_{t-1}

and we could simply calculate the FEVD of log y_t, as they do in the paper.

So, am I correct that for FEVD–and for impulse responses, for that matter–we should compute the FEVD of the cumulative response of y_t^{obs} and, if so, what is the easiest way to calculate that in a Dynare mod file?

Warm regards,
Mario

The literature typically decomposes the growth rates. As you wrote, the level has a unit root and therefore no valid decomposition can be performed. A linear trend does not alter that fact.

Greetings:

I certainly understand that you need a stationary series for the FEVD. However, there are alternatives to growth rates. Smets and Wouters (2007) in Figure 1 seem to report the conditonal FEVD for output detrended by labor-augmenting technological progress \gamma^t. So, I took their original code from the AER repository and simply added commands to obtain CFEVD at several horizons for both y and dy and compared:

VARIANCE DECOMPOSITION (in percent)
ea      eb      eg     eqs      em   epinf      ew
y      29.14    1.56    4.09    9.88    2.38    6.40   46.54
dy     14.70   20.18   26.86   21.67    6.10    4.42    6.07

CONDITIONAL VARIANCE DECOMPOSITION (in percent)
Period 1:
ea      eb      eg     eqs      em   epinf      ew
y      15.04   24.41   34.43   18.97    5.15    1.91    0.10
dy     15.04   24.41   34.43   18.97    5.15    1.91    0.10
Period 4:
ea      eb      eg     eqs      em   epinf      ew
y      22.04   11.06   16.21   32.08    8.75    6.32    3.55
dy     15.12   21.76   29.56   21.07    5.78    3.72    3.00
Period 40:
ea      eb      eg     eqs      em   epinf      ew
y      30.42    1.74    4.36   11.03    2.66    7.15   42.62
dy     14.66   20.24   26.94   21.74    6.12    4.43    5.87
Period 100:
ea      eb      eg     eqs      em   epinf      ew
y      29.21    1.57    4.10    9.92    2.39    6.43   46.38
dy     14.70   20.18   26.86   21.68    6.10    4.42    6.06


It is clear that the findings for y approximately reproduce those of Figure 1, and that these differ substantially from those of dy except for period 1, where they must coincide. In particular, the component for wage markup shocks increases dramatically with the horizon for y, whereas they do not for dy.

This is why I would like to know how to obtain the analogue under a stochastic growth trend. The key difference is that, with a deterministic growth trend, the transformation used for detrending is independent of technology shocks, whereas they are not under a stochastic trend.

But, given an observation equation in demeaned first differences of the form

Y_{t,obs} = \log Y_t - \log Y_{t-1} + g_t - \overline{g} ;

we should be able to cumulate Y_{t,obs} to get the desired outcome.

Now, perhaps I am erring somewhere, but I am simply am looking for the analogue of the object used by Smets and Wouters (2007), since they focus on output in (detrended) level rather than growth rate.

Many thanks and warm regards,
Mario

usmodel.mod (12.7 KB)

I think you need to approach this problem from an economic instead of a statistical perspective. In Smets and Wouters (2007), the “level” is actually a deviation from the (deterministic) trend. It’s variance decomposition is an object you may be interested in. In a model with a stochastic trend, you can of course also consider the “level” deviation from the stochastic trend. An alternative is growth rates as in e.g. Garcia-Cicco et al. (2010).

Dear Johannes Pfeifer,

Thank you very much. That response helped me rethink the issue in a new light. The transformed variables, i.e. y_t=Y_t/X_t, where X_t is the nonstationary (labor-augmenting) productivity process is a deviation from the stochastic trend and can therefore be compared to the deviation from the deterministic trend in Smets and Wouters (2007).

Smets and Wouters (2007) also consider stationary technology shocks. To make the comparison more complete, one can have both a stochastic trend and also shocks to the level of technology z_t. That way, deviations from the stochastic trend can respond on impact to technology even though, by construction, they net out much of an effect of a shock to the growth rate of technology.

So, for an analysis along the lines as SW, one can have a stochastic trend alongside shocks to the level of neutral productivity and focus on the deviation of stochastic trend. In this case, a conditional forecast error variance decomposition is very informative as the contribution of the shock matters greatly with the horizon.

And, of course, one can also examine the forecast error variance of growth rates, which have a different information content.

I note that the model in in your paper “Fiscal news and macroeconomic volatility” featured both stationary and nonstationary technology shocks. You choose to conduct the FEVD in terms of growth rates in Table 3. Though I now generally regard this issue as closed, I ask in closing if this choice mostly driven by comparability to the literature or also because of substantive features of the research question/model?

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In our paper, we wanted to have the variance decomposition for a concept that we can also observe empirically. Of course, you can compute the FEVD also for deviations from the stochastic trend. But the stochastic trend is not observed, so you need to rely on the model to extract the trend in the first place.

Greetings,

Thanks, that is very helpful! This gets into the very deep issues surrounding the choice of examining

1. a business cycle component obtained under a model-implied decomposition
2. a business cycle component obtained from a model-independent decomposition

There is merit in both of these–and perhaps in examining both. As the example in SW demonstrates, the variance decomposition can look quite different depending on which measure we focus on.

Thanks again! This discussion has been very insightful, and hopefully will be useful to other visitors in the forum. If we expand on these issues in the future, it should be in a new thread.

Warm regards,
Mario