## Saturday, March 19, 2016

### Japan's Phillips curve is also flattening

John Handley brought up Japan's Phillips curve as evidence against Noah Smith's claim that Japan is where macro theories "go to die" (except mine!) and my own claim that the Phillips curve is useless (in comments here) because it isn't stable. I saw John's graph and thought it was pretty striking:

This is over data from 1986 to the present (roughly). Maybe Japan's Phillips curve isn't flattening? But I plotted the full set of data I had from 1970 to the present (roughly):

The blue is from 1970 to 1986, the red is 1986 to 2000 and the green is 2000 to the present (roughly). John's data would be the red and green. The full data set is decidedly flattening, consistent with the general behavior in the information equilibrium model (see e.g. here). As monetary policy becomes ineffective, the impact of monetary expansion on output and employment becomes less and less dramatic.

Also, this is related to the economics joke: Japan's Phillips curve looks like Japan ...

...

Update 20 March 2016

Per the discussion with John Handley below, here are the 10-year interval (for the prior 10 years) Phillips curve slopes for Japan with standard errors:

The slope falls over time as mentioned previously (and as it does for the US). Here are the p-values for these slope values:

The most recent points don't reach a p-value of 0.05, but the hypothesis being tested is that the slope is different from zero -- i.e. we can't reject the null hypothesis that the slope is zero.

Update 21 March 2016

And without removing the VAT:

Without removing VAT and prior 5 years (mostly, the significance just goes away):

1. Interestingly, it seems that the slope of the Phillips Curve (implied by a linear regression) is higher for 2000-present than for my original calculation (1986-present). For the whole sample, a linear regression yields CPI = 0.9818*E - 62.827 whereas 2000-present yields CPI = 1.0167*E - 65.027. This probably has to do with inflation expectations between 2000 and now being stuck at around 0%, so the Phillips Curve should collapse to a basic linear relationship between inflation and employment. The fit is worse if I exclude 1986-1999, though (R2 = 0.491 vs. R2 = 0.5129). Using Core CPI for the whole sample gives a slightly flatter Phillips Curve (m = 0.9528), but a slightly higher R2 value (0.5315). For 2000-present, Core CPI has the worst fit (R2 = 0.4476) and the lowest slope (m = 0.6807).

2. In addition, the slope for 1986-1999 is about the same as the one for 1999-2000 (1.0122 vs. 1.0167), but the fit is significantly worse (R2 = 0.3408 vs. R2 = 0.491). For Core CPI, the 1986-1999 slope is 0.555 and the R2 is 0.1657. In short, you may have spoken a little bit too soon. Not only is the fit for 2000-present better in every case than 1986-1999, the slope is higher.

1. R² is not a good measure of a nonlinear function, and has lots of issues. What likely is changing is the number of outliers.

https://onlinecourses.science.psu.edu/stat501/node/258

Especially with time series, e.g.:

http://stats.stackexchange.com/questions/101546/what-is-the-problem-with-using-r-squared-in-time-series-models

2. The slope is still an issue; 2000-present (which I accidentally wrote as 1999-2000 above) had a marginally higher slope than 1986-1999. Given that, I think it's hard to say that the curve is flattening.

3. Not to mention that I was doing a linear regression, so I don't see what the issue with non-linear functions is.

4. Here's the graph for 2000-2015: https://dl.dropboxusercontent.com/u/92766758/2000-2015%20CPI%20Prediction.png

And the same for 1986-1999: https://dl.dropboxusercontent.com/u/92766758/1986-1999%20CPI%20Prediction.png

5. The hypothesis is that the slope of the Phillips curve is changing, so the data could very well be determined by a nonlinear function. Don't exclude that possibility with your choice of goodness of fit test.

Also, from that stats course link:

A "statistically significant" r2 value does not imply that the slope β1 is meaningfully different from 0.

You could try some other goodness of fit tests and see if your claims hold up.

6. The data just looks linear, and all of the other regressions that I bothered trying (log, quadratic, cubic, exponential) change neither the slope nor the R2 very much.

As for other goodness of fit tests, what would you suggest?

7. You could try p-values for the fit coefficients. Usually when I do this (I do this kind of thing at work), the intercept is really good, but the slope is indeterminate. Most stats packages have Ordinary Least Squares (OLS) function that produces some standard stats tables.

I use this:

https://reference.wolfram.com/language/ref/LinearModelFit.html

It looks like you are using excel, but not sure -- here's a detailed way to go about it with excel [pdf]:

8. I use both octave (OSS matlab) and excel, but in this case I've been using excel (importing fred into octave is a pain).

So, the p-value for the slope is 2.67164073687122E-56 in the whole sample, 9.9476652913359E-17 for 1986-1999, and 3.38072063600558E-28 for 2000-2015. The corresponding slopes are 0.98179, 1.012189, and 1.016664, respectively. So everything I said before seems to hold; 2000-2015 has a higher slope and a better fit. The t-stats tell the same story.

9. Those p-values are really small -- likely indicating spurious correlation. I extracted the standard errors and the p-values above for 10 year intervals.

The slope still appears to fall over time.

3. John & Jason, just eyeballing Jason's red+green vs his green points it's hard to see how the red+green could have a smaller slope than just green. Are you guys actually using the same data?

1. Jason's data is not different, he's just using the first differences whereas I used the actual values. I'm not sure how much of a difference this makes though.

2. There was a whole blow up on this blog between me and Mark Sadowski about first differences.

Without first differencing the data, you get spurious correlation. The p-values and R^2 are garbage. It might be problematic to first difference data in some cases (e.g. it destroys the information in the highly nonlinear QE, which was my point when addressing Mark Sadowski).

See e.g. here

http://www.jstor.org/stable/2526363?seq=1#page_scan_tab_contents

p-values on the order of 10^-50 are a major red flag for spurious correlation.

3. Strange... John Cochrane always seems extremely averse of first differences (or at least 'over-differencing' anyway see http://faculty.chicagobooth.edu/john.cochrane/research/papers/overdifferencing.pdf)

4. It's not strange. There's really no ambiguity here. You're goodness of fit measures show spurious correlation.

Cochrane's paper is in reference to multiple differencing. However, Cochrane does briefly mention spurious correlation:

As a result of that autocorrelation, the 30.44 t statistic is surely vastly inflated. The remaining columns give two better measures, which correct the OLS regression standard errors for the correlation of the residuals, giving much more sensible results. (Neither correction is aggressive enough, a lesson for another day.)

Emphasis mine. Inflated t-stat is equivalent to extremely low p-values. I agree that these stats aren't the end-all be-all of goodness of fit, and that sometimes you may have to fight through the spurious correlation and look at e.g. residuals or other measures (here is an example from me).

But my advice would be to quit now and check out some stats. Here is a post from Dave Giles, econometrics guru on spurious correlation:

http://davegiles.blogspot.com/2015/10/illustrating-spurious-regressions.html

5. I'm still confused. Regardless, even if I do run first differences (change from 12 months ago), the slope for 2005-2015 is 1.979641465 while the p-value is 10^-12.22488748 which doesn't seem to square with your chart above at all, unless I am misunderstanding what you mean by "for the prior 10 years."

6. If I'm reading Jason's chart correctly, his corresponding results are about 0.2 for the slope and about 0.2 for the p-value. John, did you use Excel for your calculations? If you can store it on Microsoft's "OneDrive" you can easily embed a downloadable version of that on your blog, which you can do from the OneDrive file explorer-like interface where you see folders and files as icons. You just put a check mark on it, and select "embed", and it generates a link you can paste directly into the HTML editing page for a blogger post or page. You can actually do all that for free using their free version of Excel (or used to be able to anyway -- I've since purchased a copy, so I'm not sure that's still the case).

Or I can just shut up and bite the bullet and learn how to collect the data you're both using so I can make my own calculation. :D

7. My data on Fred: https://research.stlouisfed.org/fred2/graph/?g=3QzI

You can export it as an .xls from there

8. Also, if that correlation is spurious, I will eat my hat (which conveniently doesn't exist).

9. Thanks John.

10. I did my own here, but I don't match either of you. I'm NOT going to eat my hat (several of which do exist), because chances are I screwed something up. I did 12 month differences in employment, but no differencing in the CPI column (since that's already in a 12-month difference format). Right off the bat, the data in my scatter plot doesn't look like Jason's. My regression results are on the tab called "Regression."

11. "...my scatter plot doesn't look like Jason's" of course his goes from 2000 to present (roughly), while mine goes from 2005.02 to 2015.02, but still I have ΔCPI values down around -2.5% and his green dots don't look like they cross -2.0%.

12. Tom: I removed the effects of the VAT increases in 1997 and 2014.

John: Spurious correlation is a term for a e.g. a crazy low p-value that is incongruous with the data. It happens due to cointegration. The series could still be correlated. Spurious correlation doesn't mean "not correlated" ... it means R^2 is not a valid goodness of fit measure.

In fact, my "quantity theory of labor" model says that inflation and labor force growth are correlated.

And in the plots above, the measures are correlated! It's just that the slope of that correlation changing.

13. Does removing the 2014 VAT increase involve eliminating some of the monthly data, or do you accomplish that by some other means?

14. Thanks. My first naive attempt was just to slide my interval forward so it's now from 2003.12 to 2013.12 to avoid 2014 altogether. I got slope=0.77 & p-value=1.1e-5 doing that.

I'll read up and see if I can do what you did.

15. Wren-Lewis on why excluding the tax hike might be misguided: http://mainlymacro.blogspot.com/2012/04/more-on-tax-increases-versus-spending.html

Basically, the Phillips Curve should operate no matter what causes the inflation and the only effects of the tax increase would be supply-side. In this case, the 'flexible price' rate of employment would fall, but the difference between the actual employment rate and the flexible price rate would actually increase because of higher inflation, at least if you like qualitative NK models.

16. Tested it. There is very little effect on the outcome if you leave the VAT in. Updated above. (Well, replied to comment first, then updated.)

4. I've just done my own calculations with 5-year instead of 10-year intervals and I think the Koizumi booms destroys the correlation that reasserts itself after 2008 http://ramblingsofanamateureconomist.blogspot.com/2016/03/more-on-japans-phillips-curve.html

My theoretical argument for why the Phillips curve dies during the Koizumi boom is that is wasn't cyclical (i.e., wasn't actually a boom), so no Phillips curve should have been observed. I have no idea how accurate this is though; I'm just rationalizing as best as I can without looking anything up.

1. I added a 5-year interval version above. Mostly, the significance goes away ...

5. You might be interest in this blog post (or at least the data in it):

http://www.hussmanfunds.com/wmc/wmc110404.htm