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 |

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 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 |

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) |

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 |

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 |

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 |

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.

## 3 comments:

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.

Sadly,

JzB

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

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.

Cheers!

JzB

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