From the data analyzed and presented below, there is virtually no correlation between quarterly real growth rates and quarterly inflation rates if the inflation rate is between -5%/yr and +10%/yr. However, there is a strong negative correlation between decade average real growth rates and the standard deviation of decade long inflation. In other words, what matters for growth is not the actual inflation rate (provide that it is low.) What matters is having a low standard deviation in the monthly inflation rate for an entire decade (or more.) The goal of this post is to show how these results were calculated.

__Consumer Price Index__

I've analyzed the monthly inflation in the consumer price index from 1913 to 2012 and graphed the inflation rate at each month. There are 1200 data points in the graph below. In addition to the actual data, I've plotted some black lines that represent the average inflation rate over the decade...the average monthly inflation rate in units of [per month]. The grey lines represent standard deviation about the average inflation rate. The brown lines represent the average yearly inflation rate over a given decade, such as 1913-1922, ..., 203-2012. Roughly, not exactly, the brown lines are 12 times larger in value than the black lines.

Here are some conclusions one can draw from the graph above:

(1) There has only been one decade (1923-1933) with an average price deflation less then 0%/yr.

(2) The standard deviation in the CPI was very large between 1913 and 1953, and was fairly small between 1953 and 2013, except for 1973-1982 and 2003-2012.

(3) The average inflation rate has been pretty stable between 1982 and 2012; however, the standard deviation has gone up significantly between 2003 and 2012. (Note that my guess in why the standard deviation has gone up is that the Federal Reserve is not correctly incorporating inflation in oil&natural gas prices in the CPI, which causes the Federal Reserve to occasionally mess up and lower interest rates when they should be increasing the interest rates. One of the goals of this blog has been to develop a new way of measuring inflation...i.e. measuring inflation in the price of useful work. But this will be the focus of one of the next posts I'm writing.)

In order compare the inflation rate with the growth rate, I've listed the decade average real growth rates from the same blocks listed above. For example, the real growth rate from 2003-2012 was 2.0%/yr, from 1993-2002 was 3.4%/yr, from 1983-1992 was 3.4%/yr, from 1973-1982 was 2.3%/yr, from 1963-1972 was 4.4%/yr, from 1953-1962 was 3.2%/yr, and from 1943-1952 was 2.2%/yr.

__The correlation between decade average standard deviation in inflation rate and decade average real growth rate is -0.70__. This means that there is a strong negative correlation between the standard deviation in the inflation rate and the real growth rate. On the other hand, the correlation between decade averaged inflation rate and decade average real growth rate is only -0.36. This means that there is only a weak negative correlation between the average inflation rate and the real growth rate. (As we'll see later, the correlation between the standard deviation of the inflation rate and the real growth rate is much more statistically significant than the the correlation between the quarterly inflation rate and the quarterly real growth rate. As I'll point out again in the conclusion section, this means that what's more important for growth is not the actual inflation rate...but knowing that the inflation rate is fixed in time.)

__FFT of CPI & Real Growth Rates__

In addition to graphing the CPI inflation data vs. time, I wanted to graph the CPI inflation data as a function of frequency. In other words, I wanted to figure out if one could find sinusoidal fluctuations in the data with a given frequency. So, I computed the Fast Fourier Transform (FFT) of the data presented above. The graph below is the FFT of the data between 1928 to 2012. There are 1024 months worth of data here. For those of you familiar with FFT analysis in Excel, the function requires a number of data points equal to the a power of 2...4, 8, 16, 32, 64, 128, 256, 512, 1024. So, I had to cut out some of the data in the graph above (1928-2012). The data set above was from 1912 to 2012.

In the graph below, the y-axis has units of [% per year] and the x-axis has units of [per year]. The larger the value on the y-axis, then the larger the amplitude of a oscillation of a given frequency. For example, in the graph below, one can easily distinguish a peak with a frequency of 1 per year. This means that there is a yearly oscillation in prices. On average, prices are inflate the most between April and June, and they inflate the least between October and December. It's no wonder that we have holidays during the winter...if it weren't for the holidays, then amplitude of the 1 year period oscillation in price inflation would likely be even greater.

If the inflation data were purely random, then one would expect a flat, straight line as a function of frequency...i.e. there is no preferred frequency in the data. However, this is clearly not the case. There are peaks of data with frequencies of 1 per year and 0.2 per year.

The peak in the data at a frequency of 1 per year is fairly easy to explain. Prices in the US oscillate over the year because the weather is not constant over the year. However, it's not as easy to explain the apparent peak in the data at 0.2 per year. Though, as you can see, it's a fairly broad peak, and is located close to where the amplitude is going up due to long-term trends in the inflation rate. But still, there's fairly convincing data above that inflation rates are not purely random: (1) Inflation rates oscillate over the year (2) Inflation rates oscillate with a period of roughly 4-6 years.

What's interesting is that last year when I did FFTs of yearly GDP growth rates and yearly growth rates in useful work for the US, I didn't see any peaks in the FFT data sets.

So, I decided to update the previous FFT of growth rates by including more data. In the FFT of the data set below, I've now included quarterly data on growth rates (from 1948 to 2012.)

I've also plotted the data above of the FFT of the CPI monthly data from 1928 to 1912. So, it should be noted that the time frames are slightly different and that there is a difference in the fact that one data set is monthly inflation vs. quarterly growth rates. The point in putting the data on the same graph is to point out that the FFT of the growth rate is different than the FFT of the inflation rate.

As the FFT data shows above, there are oscillation in both the inflation and the growth rates, but the oscillations occur at different frequencies. According to the data, there is no yearly oscillation in the real growth rate. So, even though there is a yearly oscillation in inflation rates (highest between April-June and lowest between Oct-Dec), there is no corresponding oscillation in the growth rate. This is interesting because it points out the fact that there can be yearly fluctuations in prices that don't effect growth rates.

__Correlation between Quarterly Inflation & Quarterly Real Growth Rates__

The next question is: what is the correlation between short-term inflation rate and the growth rate? In the graph below, I've plotted data points for the yearly inflation and growth rates (blue) as well as the quarterly inflation and quarterly growth rates when extrapolated to yearly rates (red.) The correlation between the quarterly inflation rate and the quarterly growth rate is -0.06. The correlation between the yearly inflation rate and the yearly growth rate is +0.24. In other words, there is effectively no correlation between real growth rates and the inflation rate, as can be seen graphically below. However, while there is no overall correlation between the data, there are some general trends. (1) Deflation greater than 5%/yr is likely to be bad. However, the question here is...was the deflation the cause or the effect of the negative growth? (2) Inflation rates above 10%/yr are likely to not be good for the economy. (3) Inflation rates between -5%/yr and 10%/yr are unlikely, by themselves, to be predict the growth rates...i.e. there's no correlation between quarterly growth and quarterly inflation in this range. (Even though, as mentioned earlier, there is a slight negative correlation between decade averaged inflation and decade averaged real growth.)

__Discussion & Conclusions__

It's important to separate the concept of growth from the concept of inflation. As shown in the above graphs, there is no correlation between the quarterly growth rate and the quarterly inflation rate when the inflation rate is between -5%/yr and 10%/yr. The question is: does this mean that inflation can take any value and have no effect on growth rates?

The answer to this question is clearly no. As shown earlier, large standard deviations in the inflation rate are correlated with low real growth rates. As well, large decade averaged inflation rates are weakly correlated with low real growth rates.

Therefore, we should limit the power of the Federal Reserve Board of Governors to inflate or to deflate the currency as they wish. I'm in favor of an inflation-target rule for the Federal Reserve. In other words, I think that the main goal of the Federal Reserve should be to maintain a constant inflation rate. While I personally would like to see a 0%/yr inflation rate on the price of useful work, I am not against the idea of maintaining a 2%/yr inflation rate in the CPI. While allowing for some exception in the case of war, I think that the Federal Reserve should be forced to maintain a fixed, low inflation rate. This way, financial markets can plan for the future. Not knowing the inflation rate 30 yrs from now means that it's really tough to make long-term planning decisions, such as building houses and building power plants. Think about it: would you be willing to loan money to a nuclear power plant at 7%/yr if you were afraid that the inflation rate in the future were 7%/yr...even though the inflation rate might only be 2-3%/yr right now?

__Based on the data presented above, my conclusion is that we should constrain the power of the Federal Reserve Board of Governors. By law, the Federal Reserve should be forced to maintain a constant, low inflation rate.__

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