# The Standard Deviation of NGDP Growth During the Great Inflation

This post is a side bar to the * Remarkably Stable GDP Growth* series

Once again I have to thank Mark Sadowski for goading me into digging deeper, staring longer, and thinking harder about this topic than I otherwise would have. In comments to Part 3, Mark informs us that:

In three year periods ending in 1954 to 1978, which overlaps with the Great Inflation, the 12 quarter standard deviations of the compounded annual rate of change in NGDP are significantly *negatively* correlated with the average rate of change in NGDP. In other words NGDP became *less volatile* as its average rate of change *increased*.

Let’s have a look. Graph 1 is a scattergram of 12 Qtr average NGDP growth from this FRED page, measured as Compounded Annual Rate of Change [CARC] vs Std Dev for the years 1954 through 1978. A linear trend line is included.

**Graph 1 – 12 Q Avg CARC vs Std Dev**

At first glance it appears that Mark is right. But there is something strange about that data distribution. Do you see it?

Let’s look back to one of my earlier graphs showing the change in Std Dev over time for a moving 13 quarter kernel. I see a broad sweep up in St Dev from the mid 60’s to the early 80’s. Can a 12 Q kernel be very different? No, it can’t, as Graph 2 indicates.

**Graph 2 – 12 Q Avg CARC and Std Dev**

Twelve Qtr average CARC is in yellow, St Dev in blue. The basic CARC data is in grey. What we observe are 5 different realms, with Average CARC and Std Dev moving broadly together: a sharp up and down from ’50 to the early 60’s; up from ’64 to ’81; down ’82 to 87; flatish ’88 (or ’90) to ’08, and then the Great Recession. How can we have Std Dev negatively correlated with average CARC when they exhibit similar movement? That’s at the gross level. The small magnitude undulations, however, are in contrary motion. This is easiest to see in the wiggles from 1954 to ’60, and again in the great recession, but actually occurs throughout. It happens mainly because recessions bring CARC values down while boosting the Std Dev. But — this is not the explanation.

To understand what’s going on, consider the big drop in Std Dev from 6.51 in 1960 to 2.47 in Q1 1964 shown in Graph 2. Remember that 1964 date, it’s important. Now, let’s have another look at the CARC data from 1954 to 1978, presented in Graph 3.

**Graph 3 CARC and STD Dev, 1954 to 1982**

The CARC data from FRED is in dark blue**.** It moves up over the period, but not in a regular manner. There are two flatish periods from Q2 ’61 to Q3 ’70, and from Q2 ’72 to Q1 ,78. Averages for these periods are indicated with yellow horizontal lines. The data packet spans for the two periods are outlined in red. Std Dev is in bright blue. I’ve included a trend channel in green, just because it amuses me. Data for the two periods is summarized in the table below.

A higher CARC range leads to a slightly wider data packet, and hence a higher Std Dev.

The 60’s were recession free, and in that decade we observe that after Std Dev hits bottom in 1964, it moves in near lock-step with average CARC for the rest of the decade [easiest to see in Graph 2.]. After the 1970 recession, CARC stepped up into a new range. There was a recession in 1974, yet the data envelope only widened slightly. This is because inflation at the time kept NGDP values high, even in the trough, as this FRED graph illustrates.

Now, lets have another look at the scattergram of average CARC vs Std Dev, this time with the data properly parsed around that significant 1964 date I mentioned earlier, shown in Graph 4.

**Graph 4 – 12 Q Avg CARC vs Std Dev**

The values from 1954 to Q4 ’63 are in red, and from Q1 ’64 on in yellow. The original trend line is shown in blue, trend lines for the two sub sets are color coded with their respective data points.

The conclusion is that the apparent negative correlation between CARC and St Dev over the period of 1954 to 1978 is specious, and wholly due to the high multiple-recession-driven Std Dev values of the 50’s. The Std Dev drop of 1960 to ’64 occurs when the last of these gyrating data points fall out of the moving 12 quarter kernel.

After that, Std Dev is positively correlated with CARC, as I claimed in the first place

There’s a lot more to dig into here, and I’ll do that in a follow-up post.

JB, To tell you the truth I had a hard time understanding the ‘object of your exercise’. In this post I show that NGDP growth was stable during the Great Moderation. I consider only the period after 1960:

http://thefaintofheart.wordpress.com/2013/03/22/genie-in-a-bottle/

The object was to determine if there was, indeed, a strong negative correlation between NGDP and Std Dev. At this point, it looks as though there is not.

More to come.

I’ll read your post.

JzB

“In other words NGDP became *less volatile* as its average rate of change *increased*”.

JB This counterintuitive result is due to the fact that in the 1960s and 70s NGDP growth was not stationary. In that case summary statistics like StDev are ‘meaningless’. You would have to difference the growth rate to get a proper measure of StDev.

Marcus –

“In other words NGDP became *less volatile* as its average rate of change *increased*”.Nope. That is what Mark said. I have it the other way around, with higher NGDP growth correlated with higher Std Dev..

Not sure what you mean by “difference the growth rate.” Please help me here. I think it’s important.

Cheers!

JzB

Jazz

I have to admit I have lost track of why it’s important. But I do think you have done something admirable: you looked at a claimed statistical correlation, and looked for a more natural causal factor, and found the statistics was in effect comparing cats to oranges, which seems to be pretty much the way people do statistics anymore.

Dale –

Thanx. This is all leading up to something. We’ll see now important it is.

I’ll bring out the cats and dogs in the follow up.

Cheers!

JzB

(I think this went to your e-mail)

JB

You can only calculate meaningful averages and standard deviations in a time series if the series is stationary. Picture a series with a rising trend (like NGDP growth from the late 1950s to 1980). What will the ‘average rate of growth’ mean? Not much. The series will cross the ‘mean value’ only once. In a stationary series (like NGDP growth during the Great Moderation from 1984 to 2007) the series will fluctuate around the mean (cross it from below and from above several times).

To calculate meaningful statistics you first have to make the series stationary. In the case of NGDP growth between 1960 -1980, we accomplish that either by detrending the series (i.e.subtracting the actual value from the trend value or by differencing it (i.e. if you are calculating growth as percent quarter(t) over quarter(t-4), you now calculate it as percent Quarter(t) over quarter(t-4) minus percent quarter(t-4)-quarter(t-8).

By the way. To avoid the (non)stationary problem is the reason i did charts that showed growth dispersion by graphing growth in period t-1 agains growth in period t. It´s clear that NGDP growth dispersion during the Great Inflation was much larger than during the Great Moderation.

Marcus –

People calculate moving averages and Std Devs on time series data all the time.

The definition of the Great Moderation is based on low Std Dev compared to earlier periods.

What we observe in the 60’s and 70’s are a couple of stationary periods, and a lot of NGDP increase outside of those periods. Is it not valid to note that the standard dev of a rising or falling period is greater than in a stationary period?

Cheers!

JzB