Estimating the Output Gap for the US and the Euro Area

Following after this post from Menzie Chinn, I decided to try to estimate the output gap for the US and the Euro Area.

This method has two basic steps. First you assume that the Phillips curve takes the form
$$\pi_t = \bar\pi + a (y_t - \bar y_t) + \epsilon_t$$
where $\pi_t$ is the current inflation rate, $\bar\pi$ is the inflation target and/or the average inflation rate over the sample period, $y_t$ is the natural log of real GDP, $\bar y_t$ is potential GDP, and $\epsilon_t$ is a random shock.

Using an HP filter to determine 'potential' GDP, you then run a regression to determine the slope of the Phillips curve, which allows you solve for potential GDP:
$$\bar y_t = y_t + \frac{\bar\pi-\pi_t}{a} + \frac{\epsilon_t}{a}$$
Since the shock term is still present, it is necessary to smooth the data again. I used an HP filter for both this as the previous smoothing (discerning potential GDP), but you can also use a lowess/LOESS filter, if you prefer, the results are not different enough to be worthwhile, although sometimes the lowess result seems more plausible when eyeballing.

I digress, this new filtered series should provide a plausible estimate of potential GDP somewhat like this:

One important note is that I used core CPI for the US and the GDP deflator for the EA. Using the GDP deflator for each works well, but core CPI provides an estimate closer to the CBO output gap for the US, so I stuck with it there. Also core CPI implies an implausibly small slope for the Phillips curve in the Euro Area, which results in large swings in potential GDP and implies a large and increasing output gap since 2008.

I also tried this same process for Japan and the UK, but both of them gave highly implausible estimates of potential GDP -- the slope of the Phillips curve for the UK was implausibly small (even using the GDP deflator), resulting in a path for potential output similar to that of the Euro Area using core CPI. Japan's Phillips curve appears of have a negative slope, making the recent small increase in inflation imply a hugely negative output gap ($\approx -40$%).

Also, here is a comparison of my estimate of the US output gap and that of the CBO, as well as a comparison of the same data using an HP filter and a lowess filter:

Note that this data starts Q1-1997 and ends Q1-2016, just like the previous data (the lowess filter in python has the unfortunate side effect of making the x-axis labels really annoying to deal with). If you would like to play around with the IPython notebook, here it is.