## Wednesday, September 14, 2016

### Putting numbers on vigor (6: Projecting GDP)

We now have a projection for household debt ten years into the future. Nine years, whatever. My idea is to look at the "household debt to GDP" ratio and see if I can predict that graph ten years out also, and then divide the debt out of it and end up with a prediction for GDP.

My first thought was to make a prediction based on a particular part of the debt-to-GDP ratio. Omit the early years when debt was particularly low. Omit the "Great Inflation" years when GDP was particularly high. And omit the "housing bubble" years when debt was particularly high. I'm left with the the years from 1986 to 2000, where the ratio runs straight and somewhat uphill.

 Graph #1: Household Debt as a Percent of GDP
It's easy to imagine the future if it follows a straight line. And it seems right, because the line runs uphill.

Then I looked at the "GDP to household debt" ratio and it seemed a better choice. The general trend of the line is down rather than up. So the changes in recent years are smaller, rather than bigger. I felt I could make a better prediction because the recent changes are smaller. It's the same numbers, just B divided by A instead of A divided by B -- I know that, but I went with it anyway. The first (rightmost) four worksheets in today's Excel file develop this idea.

What I imagine looks something like this:

 Graph #2: GDP as a Multiple of Household Debt
It's easier to see on this graph than the previous one: In the years before 1960 the blue line was high (but coming down) because debt was so low (but growing quickly)... From the mid-1960s to the mid-1980s the blue line was high because inflation pushed the GDP numbers up... and after 2000 the blue line went low because there was so much "housing bubble" debt.

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I was not comfortable with the idea of using 15- to 30-year old data to predict future data values. But I went through the motions in a spreadsheet. It turned out that both the exponential and linear trend lines (based on the 1986-2000 period) ran quite near the most recent data.

I could explain that to myself (no Great Inflation, etc.) but I had no great confidence in the explanation. So I decided to set this work aside and take another crack, using a different approach.

I decided to see how the trends look if I use only the most recent data.

I looked at the numbers. The last three show decrease. Before that, the numbers were pretty reliably increasing since 2009. If I wanted to avoid using numbers from the housing boom correction, I was left with only the last three values. So I went with that.

I was even more unhappy about using only three values than I was about using only the 1986-2000 data. So I was extra cautious. I figured I'd apply all of Excel's different trend line types to those three data points, and see what turned up.

The next five graphs show the three most recent values in red (at the right end of the blue line) and the older data in blue. The trend line based on the three red values is shown as a black line. The five graphs are the same except for the black line and its identifier in the legend.

Unbelievably, four of the five trend lines turned out good. That is, four of them are surprisingly similar to the red line on Graph #2, which was my previous best guess. I show all five.

 Graph #3a: Exponential Trend

 Graph #3b: Linear Trend

 Graph #3c: Log Trend

 Graph #3d: Power Trend

 Graph #3e: Polynomial Trend
The polynomial trend shows its penchant for making "bowl" shapes. It is far from the blue line for most years on the graph. As for the others, I was astounded. All four black lines run close to the blue line. The first three follow the general downsloping path since around 1960; the fourth follows it from the start.

In every case except the polynomial, the black trend line runs particularly close to the blue line in the 1986-2000 period that I first considered.

The linear trend line, my personal favorite, not only touches the last three values and clings tightly to the blue line for 1986-2000, but also meets the blue line in 1964. This line, in other words, is very much like my red line on Graph #2, only better.

With these results, I was no longer concerned that using only three data points was a problem. Instead, it appears that with these last three points the GDP-to-household-debt ratio has pretty well returned to its general trend. It looks like we are done recovering from the housing bubble.

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To get data for my ten-year projection, I modified each of the five graphs #3 above as follows:

1. Remove the dates and date formatting from the x axis.
2. Reselect the data so that the first data used to figure the trend is the first data shown on the graph; for accuracy, increase the number of digits displayed in the trendline label.
3. Copy the equation from the trendline label, use it calculate data for the projection, and show the resulting data as a new line on the graph to verify that it matches Excel's trend line.

After modification, the linear trend graph looks like this:

 Graph #4: Showing the Trend Calculation Developed and Tested
The others are on sheet "FRED Graph (4)" of the Excel file.

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After I developed and tested the calculations for all five trend lines, I gathered 'em up and put all five projections together on one graph:

 Graph #5: GDP to Household Debt (blue) and Five Projections
The blue line on graph #5 shows all the way back to the start of the FRED data. Now that you've seen the projections in context, I'll trim off the blue before 2005:

 Graph #6: GDP to Household Debt since 2005, and Five Projections
The five data projections look like three. It is hard to distinguish one from another without looking at the Excel file. The projection that sweeps up and to the right is the polynomial calculation (making another bowl). The projection that runs downward is really two lines: Linear and Exponential. The gray projection that runs flat is also two lines: Log, and Power.

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Almost done.

// the Excel file