Wednesday, October 23, 2013

Real GDP and the Exponential Trend

Sometimes there are two lines on a graph that will fool you. They're not as close as they look. Numbers are much bigger these days than they once were, and as the numbers get bigger, on a graph the old numbers get squeezed down toward zero. The spaces between get squeezed down, too, so the lines look close together.

I'm thinking about Real GDP and the exponential trend.

Trend Determination

Table #1
I grabbed annual RGDP data from FRED the other day and started out looking at recessions and compound annual growth rates. Ended up looking at exponential trends. Wanted to look at the overall trend. And wanted to split the data into two parts, early and late, and compare the two trends.

Somewhat arbitrarily, perhaps, I settled on 1979 as the end point for the "early years" trend.

I made up a graph of RGDP values for the 1947-1979 period. Added an exponential trend line, displayed the trend formula, and formatted the formula with a larger font and increased decimal accuracy so I could get the numbers right.

Oddly, in order to get useful values in the Trend Line Formula, I had to remove the date labels from the graph. That left nothing for the x-axis labels but a series of integers beginning with 1.

Graph #1: Determining the Parameters for the 1947-1979 Trend
The blue line shows the RGDP values that I got from FRED. The black line is the exponential trend line calculated and drawn by Excel. The red line shows my use of the trend parameters returned by Excel, to duplicate the trend line.

The red line is well hidden by the black line. That's good; it means my calculation of the trend line closely mimics Excel's calculation. I did it right.

Getting the several versions of the trend parameters from Excel requires only that I revise the start-and end-dates of the plotted RGDP values. Excel regenerates everything else for me.

When Excel draws a trend line, it is based on all the years of the source data that are shown on the graph.That's why I showed RGDP for 1947-1979 only. If I showed 1947-2012, Excel would give me a different trend line.

My purpose in duplicating Excel's trend calculation is so that I can compare the 1947-1979 trend path to the full 1947-2012 display of the RGDP data. To do that, I have to have my own calculation for the trend line values.

So far, so good.

Extending the 1947-1979 Trend

Next, I want to display RGDP for all the years given in the FRED data. I want to put an exponential trend line on the graph also. But the trend will be based on the early years, the 1947-1979 period, as shown in Graph #1. This has the best growth performance of the periods shown in Table #1.

Graph #2: RGDP and the 1947-1979 Trend Extended to 2012
As the graph shows, RGDP (blue) started falling below trend (red) around 1980.

For this graph I can show the year values across the bottom, because we already know the trend parameter values.

A Trend Comparison

I had Excel add a new trend line, based on the blue line (RGDP) for the full 1947-2012 period.

As noted above, when Excel creates a trend line, the line is based on the displayed portion of the source data.This time the trend line (black) is based on the full period, and it hugs the blue line closely until after 2007 and the crisis. The new black trend line is lower than the red trend line because RGDP growth slowed around 1980.

Graph #3: RGDP 1947-2012 (blue) and Trend (black) and the 1947-1979 Trend (red)
The black line is a smoothed measure of the growth we actually had from 1947 to 2012. The red line is a smoothed estimate of the growth we would have had if not for the slowing of growth since 1980. Real GDP would have been over 20 trillion dollars, instead of the 15 or 16 trillion that we actually saw.

But we didn't actually see the 15 or 16 trillion, because growth slowed again after 2007. You can see the blue line falling below the black trend line between 2007 and 2012, just as it fell below the red line after 1980.

This slowing of real growth below trend after 2007 created what has been called an "output gap". I have identified the output gap on Graph #4 -- below the black line, and after 2007.

Graph #4: Trend thru 1979 (red), Full Period Trend (black), RGDP (blue), and the Output Gap

Evaluating the Difference

Note that the separation between RGDP (blue) and the early trend (red) increases as time goes by. (Perhaps this is most easily seen on Graph #2.) Could this growing separation be not really a growing sluggishness but rather a mirage -- a false impression created by time and distance and old numbers getting squeezed down toward zero? After all, the red and blue lines are not only less spread apart in the early years; they are also lower on the graph. The values are less, in the early years. If the values are less and the separation is less, maybe everything stays in proportion. How can we test this?

We can take the early years of the graph and multiply the values by some constant that moves the values up to where the actual (blue and red) values are in the later years. When we magnify the small values we also magnify the small discrepancies. The differences between RGDP and trend in the early years will appear larger. Will they then appear as large as the actual discrepancies of the later years? It's worth doing a graph to find out.

Graph #5: Red and Blue Early Values Multiplied by 4 for Comparison with Late Values
I inserted some columns into the spreadsheet, to hold values that are four times RGDP of the early years, and four times the trend. These appear as the green and gold lines on the graph, above the early years of the red and blue. Note that green and gold run as high on the graph as red and blue, if not higher. The numbers are equally large. But the separation of RGDP from trend remains small in the early years.

The gold line hugs the green trend closely -- as you would expect, when the green line is based on the values shown in gold. The green line runs down the center of the variations shown by the gold. But also, there are no wide separations. In the whole of the green and gold, no separation of the lines is larger than the smallest separation of red and blue that appears in the 1980s.

So: Does everything stay in proportion? No. The separation of RGDP from trend is not dependent on the level of RGDP. The increase of separation in the late years is greater, out of proportion to the increase in the level of RGDP.

A Second Evaluation

I thought of another way to compare RGDP and trend: Look at RGDP relative to Trend. Look at the ratio. If the separation of RGDP from Trend increases in proportion as the trend increases, the ratio will run roughly flat. If the ratio is roughly flat, we will know that the separation is a mirage. But if the ratio shows that the separation is increasing, then we will know that RGDP is definitely falling behind its 1947-1979 trend.

To figure the ratio I will divide RGDP by Trend and show it as a percentage. When RGDP is precisely on trend, the graph will show 100% (meaning that RGDP is 100% of the Trend value). Where RGDP is above trend, its value will be more than 100% on the graph. And where RGDP is below trend, its value will be less than 100% on the graph.

For example, if the ratio shows a value of 102%, we know that RGDP is 2% above trend; and if the ratio shows a value of 97%, we know that RGDP is 3% below trend.

Graph #6: RGDP as a Percent of the 1947-1979 Trend
I evaluate this graph by imagining straight lines. I imagine a straight line at the 100% level from 1947 to 1975 or a little after. And I imagine a straight line starting around 1967 and downsloping so as to pass through the "center" of the wavy blue line from 1967 to 2012. The two trends I imagine don't meet at a point; they overlap from 1967 to 1975 or after. That's probably realistic: Changes in the economy are not usually instantaneous.

So we can say that RGDP clung to trend until the latter 1970s, or we can say it abandoned the trend by 1967. These are our choices. Either way, we must say that RGDP shows a significant fall from trend.

A Closer Look

There's a lot of white space on Graph #6. You could cut off everything below 60% of Trend, and everything above 110% of Trend, and still see all of the blue line. We should do that. It will allow us to fit more faint horizontal lines on the graph, to better gauge how much RGDP varies from Trend.

Graph #7: RGDP Within 5% of Trend from 1947 to 1979
In the early years, until just after 1979, RGDP is always within 5% of Trend. And the pattern is cyclical: It runs low, then high, then low, then high, then low again. The ratio hugs the 100% line. RGDP hugs the trend.

But after 1979, all bets are off. By the last years shown, RGDP falls to 70% of Trend and below. Income falls to 70% of trend, and below. If we use 1979 as the turning point, we see that the economy has been in decline for 34 years now. If we use 1967 as the turning point, the economy has been in decline for 46 years. By God, it's time to change that.


Jazzbumpa said...

There are a couple of easier ways to get at this. First you could have Excel fit curves on subsets of the data, frex up to whatever break point you chose, and then another beyond it.

Second, consider looking at the data on a log graph. That's one of the FRED transformations. Download that data set and use the Excel slope function to give slopes of any size subset you desire.

The slope is proportional to the growth rate.

You'll find a secular trend to lower growth over almost the entire data set, with a modest counter-trend up swing from the mid 80's to the early naughts. Exact dates depend on what subset size you select.

But the big picture is lower growth from a peak occurring between '67 [with a 5 year kernel] and '74 [with a 21 year kernel]

I used monthly data, and multiplied the slope values by 10,000 to get rid of leading zeros.

Aha - I feel a blog post coming on.

BTW, I've previously gone through more or less the same exercise as you have here. My conclusion is that the American economy is in the throes of a long and agonizing death.

On a cheerier note, the 5-year slope bottomed in '11 at close to zero [very small negative number] and has been moving steadily up since. But, alas, it is still below the level of most historical bottoms. The 8 year kernel has been flat for about 2 years, and will turn up some day.


The Arthurian said...

Thanks, Jazz. You've given me some things I never thought to try before in Excel. And I look forward to your new post!

Jazzbumpa said...

A couple more ways of thinking about this --

In the real world, exponential trends inevitably fail. The failure can be catastrophic, or something more gradual. Maybe what we are witnessing is a gradual trend failure as the slope continues to decline.

Or, from another point of view, we have been in recession for 21 of the past 46 years, or 46% of the time.