Federal and non-Federal debt should be roughly equal

Last time, I looked at the use of "control" variables in regression, and tried to express something that's been bothering me about them for a couple months now:

• "it seems pretty strange to me, this idea that if you make something a control variable, it suddenly has no effect on the outcome."
• "the idea that you can just "control" some variables to find out how your one "independent" variable affects economic growth is ... absurd"

My objection was that the outcome doesn't change. The economic growth we got over the past 70 years is what it is. It doesn't change, for example, when you control for everything except Federal debt, and then you change to controlling for everything except corporate debt. The outcome doesn't change, period. The growth we got is the growth we got.

Maybe that's a little silly, I dunno. There would be different measures of correlation, comparing economic growth to Federal debt versus comparing economic growth to corporate debt. Sure. And maybe you can turn that into a story about the effect of Federal versus corporate debt, on economic growth. Something like we find in The Real Effects of Debt. Maybe you can. But I let it sit in the back of my mind for a couple months, and I still couldn't see it.

Jerry responded in comments on my previous post. He said okay, lets say X is the Federal debt, and Y is the private debt -- corporate and household -- that I was looking at. And he set up two examples, a linear one and a non-linear one.

In the linear example, economic growth is Z, and Z = X + 2Y is how the economy works.

Jerry says we can hold Y constant, and change X, and calculate the effect of that change on Z.

He says we can hold X constant, and change Y, and calculate the effect of that change on Z.

He says we can take the two effects and add them together, to calculate the total effect on Z.

So in other words, there is nothing wrong with the method: holding private debt constant while evaluating the effect of Federal debt; and holding the Federal debt constant while evaluating the effect of private debt; and then using or comparing the effects; there is nothing wrong with this method.

But Jerry did say that the total effect we calculate may or may not be close to accurate, depending on whether debt actually has a "linear" effect on economic growth. This is probably a much stronger objection than the one I made.

I think Jerry is right. His explanation is overwhelmingly good.

Still, in Jerry's example and (I presume also) in the Real Effects of Debt PDF, you can calculate how economic growth would have changed if Federal debt was constant while private debt was growing. And you can calculate how it would have changed if private debt was constant while the Federal debt was growing.

But I suspect you cannot know if either of these alternate realities are correct. Because, as Jerry says, regression is only "a 'mostly valid' way to examine a nonlinear system", and that's only if "you're studying small movements in the system. As the movements get larger, the regression gets less valid".

And if there is one thing we know about debt, it is that its effect is not linear.

Now let me throw something else into the mix. You can have a straight line, or a line that isn't straight. Linear, or non-linear. But the line that isn't straight may be almost straight, or it may be extremely curvy. Maybe Jerry's "mostly valid" case holds true more for curves that are almost linear than for those that are extremely non-linear. I want to suggest that it is not only whether you're "studying small movements" that affects the validity of your results; but also the degree of non-linearity of your source data.

Jerry would probably say that for extremely small movements, the line is very close to linear no matter what. Okay. But he also says that determining whether a movement is large or small is "a little subtle". In other words, the accuracy of the regression is a difficult thing to determine with confidence.

I suspect that a small, "mostly valid" movement can be much larger for a curve that is almost linear than for an extremely non-linear curve. Or, in other words, a small movement of any given size will be less accurate for data with a high degree of non-linearity than for data with a low degree of non-linearity.

Just throwing that into the mix.

I also want to suggest that debt might have had linear effects, the very first time somebody borrowed money. When the first payment was made against that debt, however, the effects immediately became much more complicated. Since the very first time a loan was repaid over a period of time, the effects of debt would have been non-linear. And the greater the "financialization" of the economy, the greater the degree of non-linearity in the effect of debt.

Put that in your pipe and smoke it.


I want to go back to my concern with seeing the results change when debt changes. I don't want to focus on calculating differences that may not even be valid. I want to see the differences in actual economic growth that are due to differences in debt growth. That, I think, would be useful.

Suppose we take debt other than Federal debt, and look at it relative to the Federal debt:

Graph #1: The Non-Federal Debt relative to the Federal Debt

Here we have a picture of the relation between debt other than Federal, and the Federal debt.

We do not attempt to calculate how it would look if Federal debt was constant while private debt was growing. We do not attempt to calculate how it would look if private debt was constant while the Federal debt was growing. We simply look at the actual relation between the private ("Non-Federal") and the Federal debt.

Note that the accuracy of this picture does not depend on anything being linear or non-linear. Nor does it depend on the degree of non-linearity of the source data. It is simply a picture of the historical record.

I want to compare this ratio to economic growth, and look for the effect of changing debt on growth. So I add to the graph a series showing the growth rate of real GDP:

Graph #2: Non-Federal to Federal Debt (blue), and RGDP Growth (red)

Next, to see the relation between the two, I convert the graph to a scatter plot:

Graph #3: The Non-Federal to Federal Debt Ratio (horizontal) and RGDP Growth (vertical)

And then bring the data into Excel and put a trend line on it:

Graph #4: Showing a Linear Trend Line

We get a downsloping trend: Economic growth is in the neighborhood of 4% annual when the Non-Federal debt is roughly equal to the Federal debt. Economic growth is half as much when Non-Federal debt is around five times greater than Federal.

The trend remains generally downsloping when we make the trend line non-linear:

Graph #5: Showing a Non-Linear Trend Line

Again, trend growth runs in the neighborhood of 4% when the debt ratio is low (near 1.0) and in the neighborhood of 2% when the debt ratio is high (near 5.0).


These graphs do not answer the question How much debt should we have. But they do offer a perspective on how debt is best distributed between Federal and non-Federal entities. The graphs suggest that the Federal and non-Federal debts should be roughly equal. Thus, if we are concerned that the Federal debt is too big, when the non-Federal debt is three to five times bigger, we would do well to concentrate on reducing the non-Federal debt first, and put off our concern with Federal debt until later.