Nominal and Real GDP: A Barrier to a Statistical Approach

Scott Sumner regularly talks about how almost all discussions of inflation become much clearer in terms of NGDP. This is because people have a hard time differentiating between inflation as a result of more aggregate demand (demand-push) and inflation as a result of less aggregate supply (cost-pull). The difference is summarized in the textbook aggregate demand/aggregate supply diagrams below:

Aggregate demand expansion = Inflation

Aggregate supply contraction = inflation

The first kind of inflation changes NGDP, while the second has minimal impact. This way, when we are in a recession and demand more inflation, what we really mean is that we need more of the first kind of inflation because we need more NGDP. If we were in the second situation, we wouldn't be demanding more or less NGDP because the supply shock would have had minimal impact.

Another example in which NGDP makes explanations easier is in discussions of whether deflation is bad in an economy. Often times, liberal economists will point to the recent recession and say deflation is bad, while libertarians might point to the late 19th century, early 20th century and say that deflation is good. The more correct answer is that stable NGDP is best. So because the first kind of deflation reduced NGDP, it was bad, while the second type of deflation kept NGDP steady, and therefore was good.

While NGDP is simple, it makes it hard to statistically show NGDP boosts RGDP. You can't look at a graph and point to any correlation; a skeptic could just say that it's the RGDP that's driving the movements in NGDP, and not the other way around. In the end, to explain the relationship between nominal and real output in AD shocks, I have to find specific channels, such as nominal debt. On the other hand, inflation and output make much more sense in terms of trying to find statistical relationships. These concepts are far enough in people's minds that a relationship doesn't seem like a tautology. However, when you start directly talking about NGDP and RGDP, it's too easy for people to think the observed relationship between NGDP and RGDP is just because the second is a component of the first.

Food (Price Shocks) For Thought

Global food prices and inflatable BRICs

While Shanghai recovers from the aftershocks of the Haikui typhoon, many other areas in the world are dealing with record droughts and rising food prices. The bad weather has hit U.S. farmers hard, with corn futures last week surging 59% from mid-June and soybeans jumping 21%. This may soon have international spillovers as U.S. crops count for more than half of the export market in corn and soybeans, both important inputs for the food industry, especially meat markets.

Although the recent food price spike has been significant, global food reserves and good harvests in other crops will likely prevent it from causing mass starvation. Nonetheless, food inflation is now putting the heat on global central banks as they consider whether they need to tighten monetary policy to maintain inflation credibility, or whether they should stick to maintaining short term growth instead. The BRIC countries stand on a dangerous precipice, as their real growth in recent months has slowed dramatically. The Brazilian Central bank is dealing with inflation on its doorsteps as its own employees are striking and demanding a 23% wage hike to compensate for higher cost of living. India's growth engine is also losing its spark and global food prices coupled with an already poor monsoon season could push inflation further beyond the central bank's comfort levels.  China is barely holding on, and monetary tightening at this juncture would have serious implications for both broad growth and the stability of the shadow banking sector. Russia is dealing with its own drought and its central bank is also under pressure from IMF officials calling for a monetary tightening. These food inflation problems are compounded by a rise in the value of the dollar, making purchases of U.S. corn, whose futures are 6% more expensive than their 2008 peak, even more costly.

No doubt, the current situation is quite severe, but what can history tell us about how food price affect inflation in the BRIC countries? Econometric evidence suggests that world food prices are a key driver, more so than oil, of global inflation, but can we generalize to the BRIC countries in the current situation? The first thing to note is that global food prices, as measured by the IMF food price index, have been on a secular rise since 2000, but that in June, the last measured month, food prices were still below where they were during the 2008 or 2011 food price crises.

Given that average food expenditure as a percentage of income for Brazil, Russia, India, and China all hover around 25 to 35%, we should expect that increases in food price growth should quickly show up in each country's inflation rates . However, by looking at the time series for each CPI and the IMF food price index, we see that the time series do not match up well and that the real story is a bit more complex. In each graph, CPI year over year growth rates are plotted on the left axis, while food index year over year growth rates are plotted on the right axis.

I split the countries in the above two groups for more than aesthetic reasons. If you look carefully at the time series, you can see that in the first group, China and Russia, food price growth and inflation rates seem to move together at all levels of food price growth. Over the entire period, China's inflation rate and food price growth had a correlation value of 0.7, which is enough at the 99% confidence level. Russia's correlation is more limited, as inflation only starts to move in sync with food prices after 2007. But in the period of time since 2007, the correlation value is 0.18, which is enough at about the 90% confidence level. I call this the unconditional inflation group, as the correlation between food prices and inflation is not conditional on the rate of food price growth.

On the other hand, if you look at the second graph, there's less of a discernible pattern for India or Brazil. Food prices spike in 2004 and 2008, but neither of the magnitudes of the two countries' change in inflation match the large swing in food price. However, the time series do start to line up in times of crisis, such as in 2009. This is especially evident for India, as from 2009 on, its inflation rate seemed to move in tandem with the food price growth rate. I call this group the conditional inflation group, as the correlation between food prices and inflation seems to be conditional on whether food price growth is sufficiently high.

To test this hypothesis, we can generate 2-year backwards looking rolling correlations and see how they evolve through time. These price correlations are plotted below, with the value of the 2-year rolling price correlation plotted on the left axis and year over year change in the IMF food index on the right.

China and Russia:

India:

Brazil:

In these graphs, we see the difference between the groups in a different light. The value of the food correlation for China and Russia seem quite uncorrelated with food prices, whereas for India and Brazil the correlation between food prices and inflation is higher when food prices are higher. With further analysis, it can be shown that we can reject the null hypothesis that food prices don't affect the value of the food correlation for Brazil and India, but we fail to reject the same null hypothesis for China and Russia. This is the reason why Brazil and India are grouped together as conditional inflation countries. Food price changes affect their inflation rate only if food prices are growing quickly enough. On the other hand, Russia and China are unconditional inflation countries, as food prices strongly affect their inflation rates at all levels of food price growth. The scatter plots of correlation versus food price growth for India and Russia are particularly illustrative of this difference. First, India:

Second, Russia:

While India's food correlation values look to be affected by food price growth, Russia's food correlations seem to just cluster horizontally around values of y=-0.75 and y=0.5. With more detailed regression analysis of India's results, we obtain a 95% confidence interval of (-0.25, -0.05) for the intercept and a 95% confidence interval of (0.0053, 0.0156) for the slope. A similar regression for Brazil returns a 95% confidence interval of (-0.35, -0.18) for the intercept and a 95% confidence interval of (0.0022, 0.0141) for the slope. Both these numbers suggest that the effect is real: higher food price inflation is associated with a tighter positive relationship between food prices and inflation. Food prices are a convex predictor: little effect when prices are low, much stronger effect when they are high.

What implications does this have for food inflation and the BRIC countries? First, we should expect China's and Russia's inflation rates to be hit the hardest by any food price growth. They unconditionally inflate, which means that the historical relationships suggest that a rise in food prices will directly translate into higher inflation rates for those two countries. On the other hand, India and Brazil only conditionally inflate. Statistically significant relationships are unlikely to form at current food price growth levels, and we need to be looking at at least 10% year over year growth in food prices before we should expect each country's inflation to becomes statistically linked to global food prices. Therefore, their inflation rates will likely only rise after China and Russia's inflation rates rise. However, this analysis does not say anything about welfare costs to these countries. Given that India and Brazil have higher inflation rates than China or Russia, convex costs to inflation may end up leading to more damage in the conditional inflators than in the conditional inflators. Nonetheless, it shows that the relationship between food prices and broad inflation is not so clear cut, and that some statistical manipulation can be invaluable in teasing out the connection.

1970's Sumner-time Stagnation

Recently, Scott Sumner and Karl Smith have debated to what extent was the 1970's an era characterized by "stagnation" (AS shock), or just overexpansionary monetary policy. For Scott, much this debate seemed to be about retelling the stories that we learned in introductory economics. Scott reframes the issue in terms of above trend nominal GDP growth, and, along with other data about the labor markets, argues that the decade of inflation came about not because of stagflation or supply shocks, but rather because of overly loose monetary policy.

From the most basic AS/AD perspective, robust real growth in the 1970's is no indication that there was no stagnation. Given the accelerated NGDP growth, real GDP growth should have been higher, for reasons ranging from money illusion to sticky wages to sticky prices. In other words, the higher rates of nominal change meant that a short run equilibrium with higher real growth could be maintained. Had NGDP growth been slower, then we would not have been able to squeeze 3.2% RGDP growth out of the economy. In short, I disagree with the statement that:

I think Karl and I would both agree that (whether or not there was stagflation during the 1970s) under 5% NGDP targeting there definitely would not have been any stagflation."

Is there any reason for this? Scott's argument presupposes that you can decouple nominal growth from real growth. So when he says:

Growth was normal, and inflation was very high.  Rapid growth in AD explains roughly 100% of the inflation during the 1970s.  There was no stagflation, just inflation.

He presupposes the high real growth was not explained by the higher inflation. While, in the long run, higher inflation does not lead to a permanently higher level of growth, there is still a positive relationship between inflation and output in the short run. In introductory terms, this trade-off is the aggregate supply curve, and the correlation arises from increases in aggregate demand. If the high level of aggregate demand is the reason why inflation is high, the high level of AD is also the reason output is high. I don't see how they can be separated.

There are also many places one can look for data to support this hypothesis. The first is the classic shifting of the Philips curve. If the Philips curve moved outwards in the 1970's era, that meant you needed a higher level of inflation to reach any given unemployment rate. If the unemployment rate is a proxy for real growth, then this directly leads to the conclusion that higher levels of inflation allowed higher levels of real growth.

However, Scott prefers to focus on real GDP ("I had thought the word ‘stagflation’ meant high inflation plus slow output growth (due to slow growth in AS.)"), and on this issue the data still suggests that the higher NGDP growth could not be disentangled from the higher RGDP growth. The first graph is a scatterplot of 1960's data, with NGDP YoY growth on the x-axis, and RGDP YoY growth on the y-axis.


The following grpah is a scatterplot of 1970's data, with the same x and y axis data.

Looking at these two graphs, one can see that the relationship between NGDP and RDGP was very tight in both periods. While correlation does not prove causation, it's hard to think of any theoretical mechanism that could have decoupled NGDP from RGDP in either period.. Another interesting note is that the x-intercept of the 1970's data is much larger than the 1960's data. This suggests that you needed a higher level of nominal growth to hold RGDP steady in the 1970's than you needed in the 1960's. the 1970's was suffering from an overall supply shock relative to the 1960's.

As an interesting corollary to all of this, given the lack of a long-run relationship between higher nominal and real growth, you need accelerating nominal growth to prevent real growth from falling down. In a sense, in addition to the positive relationship between the first derivatives of NGDP and RGDP, there should also be a  positive relationship between the second derivatives of NGDP and RGDP.

Notably, these correlations are even stronger than the first two graphs. This lends more evidence to the argument that, had the Fed decided to decrease YoY NGDP growth, RGDP would have taken a hit. For all the power that lies within expectations, they can't take away all of the pain away from a NGDP disinflation.

What seems unexplained is why the 1970's slope is higher (with greater than 95 confidence) than the 1960's slope. If agents adapt, the same increase in NGDP growth should result in a lower increase in RGDP growth. By that theory, the 1970's slope should be less than the 1960's slope. This will be an interesting topic for future investigation.

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.

What is with inflation expectations? Analysis from Cleveland Fed Data

After working with TIPS data in a previous post, I became more interested in the relationship between inflation expectations and actual inflation over time.  Generally, as per a paper by Mankiw, Reis, and Wolfers, inflation expectations can be highly contentious, with uncertainty among Economists especially high in times of crisis.  However, even though the data suggests that inflation expectation are well correlated with past inflation, the hypothesis of rational expectations and inflation predictions is a bit more uncertain.  In the Mankiw et al. study, the short term predictions of the Michigan, Livingston, and Survey of Professional Forecasters seemed reasonably accurate; how does this accuracy carry over to longer term measures of inflation expectation?

However, since the TIPS data does not go back very far, I used the Cleveland Fed's inflation expectation data instead.  And as per some comparative analysis between the two measures, the Cleveland Fed data can be more descriptive in times of major change, which is when the stability of expectations is the most important.  From the CPI data, I calculated the actual inflation over the future timeframe of the expectation, and then associated this inflation rate with each month's inflation expectation data starting from January 1982.  Thus, for the 5-Year inflation expectation data for January 1982, the actual inflation was the average annual inflation from January 1982 to December 1986.  These two numbers formed a point.  I then took 60 of these points (five years), and used them to calculate a Pearson's r-value, a measure of the correlation between the two values.  For example, the set of data beginning with a point in January 1982 includes the inflation data from December 1992 to make the last calculation for actual inflation.  The movements of the different correlations are plotted below:

What is immediately apparent is that the stability of the relationship changes with time.  During the second half of the 80's, expectations matched reality quite well, with an r-value of over 0.9 for the 5 year expectation data.  This seems to match the rational expectations proposition that the expectation should be the reality.  However, beginning with the 90's, the correlation between inflation expectations and actual inflation became more negative.  What is striking is that the correlation kept on going down, reaching almost -0.8 with the 5 year expectation.  The correlation is consistent with the r value of about -0.73 with the TIPS data, lending credibility to the fact that Cleveland Fed predictions are theoretically robust.

What is more interesting than the correlations are the slopes; an increase in expected inflation predicts how much of an increase in actual inflation over the future period?  The data is graphed below.  A value of 1 means that for every percentage point increase in inflation expectations in a period, the actual inflation in the corresponding term is 1 percentage point higher.

Before looking at the actual numbers, what should we expect the slope to be?  In a world of perfect rational expectations, in which , the slope should be one.  Although there may be error, the expectation should, on average, match reality.  In the world of an inflation targeting central bank (with or without rational expectations), the slope should be 0, as actual inflation should always gravitate towards a constant.  This analysis of central banks is supported by comparisons between US (non-inflation targeting) and UK and Swedish (inflation targeting) central banks.  Below are both the time series of the slopes, as well as box and whisker plots showing the distributions of the slopes.

As can be seen from the time series, the slopes don't stay constant.  Rather, they jump around, with the 1-Year slope value substantially more volatile than the 5 or 10 year slopes.  In spite of this, the median of the slopes do lie around zero.  However, their distributions are skewed right, with the outliers mostly coming from the mid 1980's to the early 1990's time period, which was the time after the brunt of the Volcker disinflation.  The amazing convergence of measures at that time suggest it has something to do with people adjusting to a new monetary regime.  It is as if, in the transition to the fight against inflation, people were able to accurately predict the new stable regime, creating the high correlation as people lowered their expectations of inflation.  Later, as the regime became stable, the correlation became weaker as noise gained a proportionally larger effect.

But then what explains the negative slopes in more recent times?  One interpretation is that they're statistical anomalies: the 1 year data goes much farther and, while it does have a few blips into negative slope, they revert back to mildly positive relatively quickly.  Given this volatility, we will have to wait for more data to try to evaluate the impact of the regime on inflation and their expectations.