Monday, April 16, 2012

Correlations Across Time: How Stable are the Curves?

What is the Philips curve, and how do we know it's there?   It was originally discovered by Irving Fisher in 1926 when he noted the negative correlation between inflation and unemployment.  Of course, he was not the first to realize this connection between prices and employment, as Hume commented on this exact issue almost 200 years before:

In my opinion, it is only in the interval or intermediate situation, between the acquisition of money and the rise in prices, that the increasing quantity of gold or silver is favourable to industry. . . . The farmer or gardener, finding that their commodities are taken off apply themselves with alacrity to the raising of more. . . . It is easy to trace the money in its progress through the whole commonwealth; where we shall find that it must first quicken the diligence of every individual, before it increases the price of labour
For this reason, Milton Friedman often said that modern macroeconomics has made it just one derivative past Hume.  Instead of just focusing on the first derivative and changes in the price level, we now look at the second derivative and changes in the inflation rate.

Robert Hall took this one step further in his 1986 exposition on efficient monetary policy and, instead of looking at one more derivative, looked at one more parameter.  Instead of just looking at the levels of unemployment and inflation, he theorized on the relationship between the volatility of the two variables.  He hypothesized the existence of an efficient policy frontier, a trade-off between price stability and unemployment stability that would prevent both variables from settling down in the face of periodic random shocks.

But have either of these correlations held throughout time?  The Philip's curve worked originally very well in the 1960's to 1980's, but then broke down as stagflation struck and expected inflation shifted the "stable" Philip's curve.  Thus, there seems to be a severe issue with measuring the Philip's curve; where should one start and end the observation window?  The analysis can easily become utterly meaningless, as:

To see how meaningless correlation can be outside of Mediocristan, take a historical series involving two variables that are patently from Ex­ tremistan, such as the bond and the stock markets, or two securities prices, or two variables like, say, changes in book sales of children's books in the United States, and fertilizer production in China; or real-estate prices in New York City and returns of the Mongolian stock market. Measure correlation between the pairs of variables in different subperiods, say, for 1994, 1995, 1996, etc. The correlation measure will be likely to ex­hibit severe instability; it will depend on the period for which it was com­puted. Yet people talk about correlation as if it were something real, making it tangible, investing it with a physical property, reifying it. The same illusion of concreteness affects what we call "standard" deviations. Take any series of historical prices or values. Break it up into subsegments and measure its "standard" deviation. Surprised? Every sample will yield a different "standard" deviation. Then why do people talk about standard deviations? Go figure. 
Note here that, as with the narrative fallacy, when you look at past data and compute one single correlation or standard deviation, you do not notice such instability (Taleb, The Black Swan, my emphasis).

So, in this post, I want to look at the time series data and see how the correlation evolves over time.  This is important for both the Philip's curve and the efficient policy frontier, as one can see if either of those relationships actually holds across all time periods.

Monthly CPI and unemployment data are obtained from the St. Louis Federal Reserve website, and variabilities for each variable are measured by the standard deviation of the past year's worth of observations.  Correlations were then calculated in five year windows, such that a correlation coefficient on month t is the correlation between the variables of interest in months t-59 to t.  As the concept of a standard deviation is a bit abstract and not well understood, I took the logarithms of the standard deviations, to allow an explanation in terms of percentage increases in one variable leading to percent increases in another.

Below is a tool to gain a qualitative understanding of the evolution of the correlations.  Red denotes high numbers (strong positive correlation), while green denotes low numbers (strong negative correlation).  The black lines mark every 10 years to give a sense of scale in the colorful "time series".

As expected, the correlation coefficients fluctuated throughout history. For the Philips curve, old Keynesian theory would predict a negative correlation.  However, if there's a supply shock, both inflation and unemployment move in the same direction.  This makes sense as the two major supply shocks in recent history were the negative aggregate supply oil shock in the mid 1980's, as well as the positive aggregate supply shock in the 1990's.

With this in mind, we see that the Philip's curve relationship was actually quite stable up until the 1990's.  Although the oil price shock did force the correlation positive for a short period, it quickly reverted to a negative value.  However, from about 1990 on, the correlation between unemployment and inflation became consistently, if only weakly, positive.  Since both inflation and unemployment rose in that time period, this is another piece of evidence that suggests much of the aggregate supply gains in the 1990's were steadily reversed in the 2000's.

However, the relationship between the two volatilities was not as clear cut.  A log-log regression of the unemployment volatility versus the inflation volatility over the entire 60 years yields a slope of 0.44, with a 95% confidence interval between 0.346 and 0.540, suggesting that 1% increase in inflation volatility resulted in about a 0.44% increase in unemployment volatility.  Yet this general correlation masks the variance.  Around the 1980's and 2010, the correlation was incredibly positive, while in the 1970's and 2000's the correlation is very negative.

From this, general conclusions can be made.  First, policy is not efficient.  Even if there were an efficient policy frontier, we're not on it.  The many zones of positive correlation indicate that there's much more monetary policy can do to limit volatility in the two variables.  Second, that there are interesting things going on with transmission mechanisms that would cause uncertain inflation to translate to uncertain output.  Third, if there are severe risks to inflation volatility, it may be in our interest to lower unemployment volatility as well.  Moderating the relationship between these two variables may become one of the biggest benefits of NGDP targeting, as uncertainty along the Philips curve may cause movement towards higher levels of volatility.

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