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.

How Easy is it to Raise LRAS?

There's recently been quite a bit of talk about greatly increasing the growth rate of GDP in the US over the next decade or so. The only way big aggregate demand stimuli could achieve this goal is if 1) output is currently far below potential in the US or 2) demand side stimulus can raise the long run level of output.

The first option is evidently not the case, given the fact that the labor market is reasonably tight -- the employment rate for Americans between the age of 15 and 64 has recovered most of what it lost in 2008, so output is clearly close to where it would be at full employment. By basically any reasonable account, output is pretty close to potential, so not much in the way of aggregate demand stimulus would have any effect other than increasing inflation. There are no credible models that I have ever seen that suggest that aggregate demand has any effect over the long run level of output, so it's almost fair to simply reject the hysteresis hypothesis simply based on its lack of theoretical reasoning. If this isn't enough, simply taking it to its logical conclusion should reveal some of the absurdity. If central banks, by increasing aggregate demand, could increase potential GDP, then they essentially have the power to decide the equilibrium level of employment, irrespective of demographics. Alternatively, they could be fueling innovation by printing loads of money and causing TFP growth to increase. Both options strike me as completely absurd; personally I see LRAS as completely deterministic to the Federal Reserve.

I will not, however, completely discount the role of fiscal policy. Certainly the Federal Government has the capability to engage in policies that have real effects. For instance, the government could permanently increase government spending as a percentage of GDP and induce everyone to work more by making them feel poorer. Also, the government could invest in a bunch of infrastructure, which could be treated as 'government capital' in all our Cobb-Douglas production functions and would, as such, be stimulative. Basically, the government could very well be boosting potential output, just not with anything that should be labeled 'demand' stimulus. In fact, this is probably true about any fiscal expansion that is ever undertaken -- we should only ever talk about the real effects of fiscal stimulus because everything else is the product of monetary-fiscal interaction. For example, any fiscal expansion will raise the 'natural rate of interest' in a New Keynesian model and the only reason the 'fiscal multiplier' will be any higher than that of a similarly calibrated RBC model is that the central bank refuses to raise the nominal interest rate one for one with the increase in the natural rate. In this sense, fiscal stimulus can almost have demand-side effects at the zero lower bound, but what is really happening is that the central bank fails to tighten monetary policy in the event of stimulus.

Basically, the only way for the government to increase potential output is by taking advantage of the real effects of fiscal policy -- people should stop preoccupying themselves with aggregate demand.

Using Demographics to Estimate Potential Output in Japan

I don't stray into empirical matters very frequently because I'm really not all that adept at them, but I thought it would be interesting to feed Japan's working age population growth into a basic Solow growth model.

Skip the following if you already understand the Solow model:
In the Solow model, it is assumed that output is produced using three inputs: capital, labor, and productivity. The production function is Cobb-Douglas for capital and labor (with constant returns to scale), but is multiplied by what's called the Total Factor of Productivity, or TFP, which represents technological progress. Defining output as $Y_t$, capital as $K_t$, labor as $N_t$, and TFP as $A_t$, the production function can be written as
$$ (1)\: Y_t = A_t K_t^\alpha N_t^{1-\alpha}$$
where $\alpha$ is capital's share in production and $1-\alpha$ is labor's share in production. Workers devote a constant share $c$ of production to consumption ($C_t$), so $ C_t = c Y_t $. Non-consumed income is used to increase the capital stock, which exogenously depreciates at $\delta$. Defining $s$, the share of income put toward investment in each period as $1-c$ allows us to write the capital accumulation equation as such:
$$ (2)\: K_{t+1} = (1 - \delta) K_t + s Y_t $$
The labor force is assumed to grow at constant rate $n$, so next period's labor forced is defined as
$$ (3)\: L_{t+1} = L_t (1 + n) $$
TFP is assumed to grow at constant rate $g$, so TFP evolves according to
$$ (4)\: A_{t+1} = A_t (1 + g) $$
Given $L_0$, $A_0$, and $K_0$, the economy will eventually converge to a balanced growth path in which all variables grow at the same rate as productivity: $g$. If one of the parameters ($\alpha$, $\delta$, $s$, $n$, or $g$) changes, then the economy will take time to adjust to new equilibrium levels.

Figure 1

As you can see in figure one, Japan began to see a secular decline in it's working age population growth rate in about 1990. This decline coincides roughly with Japan's lost decade -- the period between the mid 90s and the early 2000s characterized by low growth and high unemployment. Given the low working age population growth, it may be possible to explain some of this lack of economic activity with the Solow model. Assuming constant technological growth of 1%, a capital depreciation rate of 2.5%, a capital share of 33%, and a savings rate of 10% (I have no clue how close to accurate this calibration is, if someone wanted to find the average values of each variable over the last 20 years or so in Japan, I'll update them, but right now I can't be bothered to find the information myself), I was able to come up with an estimate of 'potential' output in Japan -- i.e. what Japanese output would be absent any shocks to productivity, government spending, or monetary policy (or natural disasters, which explain the 2011 output contraction).

Here are a couple of graphs relating actual output to demographically-adjusted potential output:

Figure 2

 

 

Figure 3

Figure 2 plots my estimate of potential output against actual output, assuming potential output was 2% above actual output in 1995 and figure 1 plots the output gap, or the percentage gap between actual and potential output. An interesting note here is that potential output, absent any demographic or technological changes, is predicted converge to a decay rate of roughly 0.5% per year and potential output is currently growth at about zero percent per year, meaning that, not only is potential growth for the next couple of years zero, the economy should be expected to shrink without being in a recession in the future. That is, unless the workforce stops decaying so quickly.

Another interesting observation is that Japan's lost decade seems to closely resemble the experience that the United States has had since the Great Recession. This is entirely unsurprising given that both periods are characterized by monetary policy ineffectiveness (the zero lower bound), but the post 2007 experience in Japan could possibly be used to predict the outcome of another large recession in the US absent monetary policy normalization. Perhaps more on this later.