No Splashy

"Even my prefaces have prefaces."


I did not react well when Marcus Nunes reduced the "golden age" to a few years in the 1960s when spending was up and inflation hadn't yet kicked in.

I responded badly, I think, because my understanding of US economic performance over the past 60 years is an essential part of my story of what's wrong with our economy. Marcus's innocent remark felt like an assault on everything I hold true.

I can take that. I just can't take it being presented so casually. If you're gonna undermine everything I believe, you want to be clear and thorough and precise.

College Grades: An Example

Graph #1

First semester of college you get four Bs and one F. Your average is 2.4. Next semester, all Bs. Your cumulative average is 2.7. The next year all Bs again, both semesters. Cumulative average: 2.85.

It's hard to get that number up.

Two observations:

1. When only a few numbers are averaged together, an outlier will strongly affect the result.

2. When many numbers are averaged together, the result changes only slowly.

It's the same if you figure "moving" averages. The more years you consider, the smoother the resulting trend. (For an example, see this hover craft post.) It's the same principle as with your college grades: In a shorter period, fewer numbers are averaged together, so the oddball highs and lows show up pretty strongly in the result. But in the longer period, with more numbers in the mix, the average doesn't move as much.

And it's the same when you're figuring things other than averages. Growth rates, for example. Consider individual year-to-year changes and you're liable to see numbers all over the chart. You're *not* liable to see any clear trend.

On the other hand, compare the compound growth rate of the 30 years before 1980 to the 30 years after, and you're looking at two numbers which are probably not all that far apart. You can make claims about changes in economic performance, and yet you are looking at just two numbers. And maybe if you picked a different year the outcome would be different, too.

Even if you take a more thorough look at the numbers, as Stuart Staniford did, it sounds like an argument. It does not sound like an argument resolved.

That lets people say different things about growth. And that is counter-productive.

Growth Rate Difference

In a comment at that Staniford post, Stuart gave me the "compound annual growth rate" formula. And he gave me the phrase "compound annual growth rate", so I could find out more.

Eventually, I put the formula into an Excel function called StepRate. Using the function I could figure the growth rate for any two consecutive years, or for a series of years. So I could use it with FRED data, for example.

Recently I started using it to look at Real GDP growth rates: compound annual growth rates for increasingly longer periods: 1947-48, and then 1947-49, and then 1947-50 and like that, right up through 1947-2011. For the first few years this gives me splashy data. When the average is composed of relatively few numbers, outliers make a big difference, and the average varies much.

But as noted above, when enough years are figured in, the variability of the numbers fades, and a more representative "trend" value emerges. In Graph #2 below, for example, the first 20 years of the plot show a lot more variability than the next 30:

Graph #2: Growth Rate Difference (Early less Late)

This graph shows, for any given year, the growth rate of the prior years less the growth rate of the following years. In the early years, the "prior" rate is splashy, or variable. In the later years, the "following" rate is splashy. Subtract the one from the other, and splashiness shows up at both ends of the plot.

The middle section is tame by comparison. From the mid-1960s to the mid-1990s it shows limited variability. The plot is consistently close to the 1.0 level for 30 years -- suggesting that early growth was consistently about one percentage point better than later growth. The line briefly rises above 3.0 near the end -- meaning that prior growth was much better -- because of the severity of the recent decline and because of the fewness of numbers in the "following" calculation, toward the end.

Since and Until

In the back of my mind something was saying that my method -- subtraction of one growth rate from another -- might be somehow exaggerating differences. So I decided to compare early and late growth visually, rather than by subtraction.

Graph #3: Growth Rate from 1947 to Plotted Year (blue)
and Growth Rate from Plotted Year to 2011 (red)

The higher line here, the blue line, shows compound annual growth rates of Real GDP, the compound rate being figured from 1947 to each year plotted. The first three points on the blue line of Graph #3 show the compound annual growth rates for the periods 1947-48, and 1947-49, and 1947-50. Each subsequent point shows the compound growth rate of a longer period. The general trend shows decline.

At the right, the last blue point shows the rate for the full 1947-2011 period.

Notice that the blue line is splashy at first, for maybe 20 years, because of the fewness of numbers in the calculation. It becomes quiet as the count increases. This is exactly the behavior we would expect to see, given the prefacing remarks above.

The other line, the red line, is splashy at the right end and quiet at the left. You can probably guess the reason. The first point at left considers the full 1947-2011 period. The second point considers 1948-2011. The third considers 1949-2011. As we move right the red line considers fewer and fewer points. Eventually the quiet transforms into splashiness and, by the end, we consider only the growth rate for 2010-2011. In the last 20 years or so, the fewness of numbers pushes the line away from its long-term path. The fewness of the numbers, and the values of those numbers.

Again, the general trend shows decline.

The red line begins and the blue line ends at exactly the same value. That is because at those two points, the calculated rate considers the full 1947-2011 period. But as we  move to the right on the red line, the older RGDP values drop out of the calculation, and the red line trends down hill. In other words, economic performance has deteriorated with time.

But is that because of our recent economic troubles? No, it is not. The blue line ends at the same value that the red line starts. But with the blue line, as we look to the left the more recent values drop out of the calculation. Thus the decline of the blue line through 2007 shows economic performance through 2007, but not any later.

Like the red line, the blue shows a general decline in real economic growth.

On Graph #3, after the initial splashiness of the blue line, the red and blue lines travel together until the 1990s, with the blue line about one percentage point higher than the red. Subtract red from blue, and you will have a growth rate difference of about one percent -- as we saw on Graph #2 above.

On Graph #3, near the end, the red line drops nearly to zero while the blue line remains above 3%. Subtract the red from the blue and in these late years we will see the growth rate difference rise to over three percent -- again, as we saw on Graph #2.

Sins of Omission

There are two causes of the splashiness we see on these graphs: the fewness of numbers used in a calculation, and the values of those numbers. If annual growth occurred at a constant rate, fewness could not give rise to splashiness. In that case, every plot of growth rates would look just like the long-term trend. But annual growth rates vary, and fewness makes that obvious.

After looking at the above graphs, I was still concerned that the recent bad years might excessively distort the picture. I was concerned, too, about Marcus's notion -- that a few good years at the start would excessively distort that end of the picture.

So I made a new graph, like Graph #3 except with the early and late years omitted.

Graph #4: Splashy in the Middle

The blue line here is like the blue line on Graph #3 except I omitted all the data before 1966. The first plot point on the blue graph shows growth for 1966-67. The last shows growth for 1966-2011.

You can see there's no blue line from 1947 to 1966. Of course, the fewness then begins with 1966-67 and the middle years are very splashy on this graph -- just the opposite of the earlier graphs!

By the 1980s there are enough data points that the splashiness is gone from the blue line. The compound annual growth rate hovers around 3%, dropping a bit at the end.

The red line has a similar story. I omitted all the data after 1992 for this line. The first point at left shows the growth rate for the 1947-1992 period. The last point at right shows the growth rate for 1991-92 only.

The red line starts out smooth, in the neighborhood of 3½%, but ends up splashy.

I omitted the years that I thought might be skewing my results, from the start of the one line and from the end of the other. I ended up with high growth early, slower growth late, and a splashy mess in the middle. I eliminated the years, but not the splashiness.

And growth is still better early than late.

The Moving Growth Rate

How to eliminate the splashiness? That was the question. It finally occurred to me to figure it like a moving average. Only I wouldn't take the average growth rate for a chunk of years; I would take the compound annual growth rate. Probably not much difference, really, but I was on a roll with my StepRate function.

I read one time that Kondratieff used a 9-year moving average when looking at his long wave. It was explained that nine years was about the length of the business cycle, and would help avoid getting cyclical distortion in the graph.

Since I'm looking for long-term trends, I went with 21 years. Two business cycles, and then some. And an odd number, so that there is always a "middle" year where I can plot the result.

Use of the 21-year period also eliminates most of the splashiness arising from the fewness of data points, as we saw above.

The first year on Graph #5 below shows the compound annual growth rate for the period 1947-1967. The second year shows 1948-1968. The last year shows the period 1991-2011. Each data point plotted considers the 21-year period is is centered in. Each point is a measure of two decades of Real GDP growth.

Graph #5: A 21-Year Moving Compound Annual Growth Rate

The growth rate starts high, above 4 percent. It ends low, barely above 2.5%.

The linear trend line shows a decline from over 4% growth to less than 2.5%. The red data line follows the path of the black trend line consistently from 1957 to 2001. There has been a gradual, long-term decline of economic growth.

Remember, this is a 21-year moving compound annual growth rate, and any given point is a measure of the growth that occurred from ten years before to ten years after that point. The low in 1983, for example, measures U.S. economic growth from 1973 to 1993. The high point in 1959 measures growth from 1949 to 1969.

There can be no question that economic performance was better early than late. There can be no question that the trend of real economic growth was consistent decline.


Early growth is consistently better than later growth. The overall trend of growth is down. There has been a gradual, long-term decline. There is no doubt that economic performance was better early than late.

And to Marcus I must say: I cannot accept your casual assertion that only the decade of the 1960s had anything like "golden" growth. Nor can I accept your Friedmanesque assertion that what looks like good growth was really no more than the precursor to inflation. I will continue to look upon the golden age as beginning even before the 1950s, after the second World War, around 1947.

And I will continue to assert that there was a weakening of economic performance as early as 1966-67 and this, rather than 1973-74, defines the end of the golden years.

By the way, I just discovered that Minsky said the same thing about the end of the golden age. Steve Keen writes:

We commenced deleveraging from 303% of GDP. After 3 years it is still 10% higher than the peak reached during the Great Depression. On current trends it will take till 2027 to bring the level back to that which applied in the early 1970s, when America had already exited what Minsky described as the “robust financial society” that underpinned the Golden Age that ended in 1966.

I love it when that happens.

// The Excel file with the StepRate function in VBA.