Is There a Phillips Curve? If So, Which One?

One place that Paul Krugman and Chris House disagree is on the Phillips curve. Krugman (responding to a post by House) says:

New Keynesians do stuff like one-period-ahead price setting or Calvo pricing, in which prices are revised randomly. Practicing Keynesians have tended to rely on “accelerationist” Phillips curves in which unemployment determined the rate of change rather than the level of inflation.

So what has happened since 2008 is that both of these approaches have been found wanting: inflation has dropped, but stayed positive despite high unemployment. What the data actually look like is an old-fashioned non-expectations Phillips curve. And there are a couple of popular stories about why: downward wage rigidity even in the long run, anchored expectations.

House responds:

What the data actually look like is an old-fashioned non-expectations Phillips curve. 

OK, here is where we disagree. Certainly this is not true for the data overall. It seems like Paul is thinking that the system governing the relationship between inflation and output changes between something with essentially a vertical slope (a “Classical Phillips curve”) and a nearly flat slope (a “Keynesian Phillips Curve”). I doubt that this will fit the data particularly well and it would still seem to open the door to a large role for “supply shocks” – shocks that neither Paul nor I think play a big role in business cycles.

Simon Wren-Lewis also has something to say about this in his post from earlier today, Has the Great Recession killed the traditional Phillips Curve?:

Before the New Classical revolution there was the Friedman/Phelps Phillips Curve (FPPC), which said that current inflation depended on some measure of the output/unemployment gap and the expected value of current inflation (with a unit coefficient). Expectations of inflation were modelled as some function of past inflation (e.g. adaptive expectations) - at its simplest just one lag in inflation. Therefore in practice inflation depended on lagged inflation and the output gap.

After the New Classical revolution came the New Keynesian Phillips Curve (NKPC), which had current inflation depending on some measure of the output/unemployment gap and the expected value of inflation in the next period. If this was combined with adaptive expectations, it would amount to much the same thing as the FPPC, but instead it was normally combined with rational expectations, where agents made their best guess at what inflation would be next period using all relevant information. This would include past inflation, but it would include other things as well, like prospects for output and any official inflation target.

Which better describes the data? ...

[W]e can see why some ... studies (like this for the US) can claim that recent inflation experience is consistent with the NKPC. It seems much more difficult to square this experience with the traditional adaptive expectations Phillips curve. As I suggested at the beginning, this is really a test of whether rational expectations is a better description of reality than adaptive expectations. But I know the conclusion I draw from the data will upset some people, so I look forward to a more sophisticated empirical analysis showing why I’m wrong.

I don't have much to add, except to say that this is an empirical question that will be difficult to resolve empirically (because there are so many different ways to estimate a Phillips curve, and different specifications give different answers, e.g. which measure of prices to use, which measure of aggregate activity to use, what time period to use and how to handle structural and policy breaks during the period that is chosen, how should natural rates be extracted from the data, how to handle non-stationarities, if we measure aggregate activity with the unemployment rate, do we exclude the long-term unemployed as recent research suggests, how many lags should be included, etc., etc.?).