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Thursday, March 25, 2010
Cato has an interesting picture of real spending by HHS since 1962:
I used the same data they used and was able to replicate their picture. I also calculated real HHS spending as a percentage of real GDP:
The real spending is non-stationary as evidenced by a Dickey-Fuller test. In fact, real HHS spending is I(1), that is the first-difference is stationary:
This is not an indictment on its own because real GDP is also I(1) and therefore it is, on the surface, possible that real HHS spending can increase indefinitely as long as real GDP continues to. However, another source of potential concern is that real HHS spending as a percentage of real GDP is also non-stationary:
I am not sure what this means. This might be sustainable if the percentage of GDP that is transferred to the federal government is also non-stationary, but I am not sure that such a statistical finding makes sense because the percentage of real GDP that can be transferred to the federal government is limited to 100%.
Indeed, looking at the percentage of GDP transferred to the federal government in the form of receipts (both on and off budget) is stationary:
And yet the percentage of GDP that is spent by the Federal Government in the form of outlays (both on budget and off budget) is non-stationary:
The percentage of GDP that is outlays can be non-stationary if the government is able to borrow to finance its deficit. Thus, the deficit as a percentage of GDP is also non-stationary:
This is the real problem, I think (although I admit that I am not a macro-economist). It seems that the deficit as a percentage of GDP cannot be non-stationary indefinitely, any more than the percentage of real gdp spent by the HHS can be non-stationary indefinitely.
How long can the non-stationarity in the percent of GDP spent by HHS persist? That seems to me to be a very, very, very important question.
My guess is that you would find the same thing for private health care expenses, and I would bet you would find the same thing for private (and probably public) expenditures on education. It's a pretty classic case of cost disease. But yes, there is a limit to which that can continue.
I had a working paper on this about 15 years ago called "Does the Unemployment Rate have a Unit Root? No!" I was unable to get it published - I could document the effect but couldn't really get very specific with the distribution theory.
The problem is applying a unit root test to ratios. Ratios of normally distributed variables have Cauchy distributions, and therefore very fat tails and critical values that are far away from zero. The distribution theory for this has been known for decades. So, when you do a unit root test on the ratio of two stationary variables, your critical values are much further from zero than in the DF.
But ... if you have ratios of trending variables with additive normally distributed errors it gets much more complex. In a small sample, the errors dominate, and you have fat tails and critical values far from zero. As the sample size increases, the trend (doesn't matter if it's stochastic of deterministic) starts to dominate. The distribution is still fat-tailed, but it starts to compress from both ends. You end up getting something with a lot of kurtosis (some people describe this sort of distribution as broad-shouldered rather than fat-tailed).
I did Monte Carlo on this using the unemployment rate as an example. Turns out your critical values are highly dependent on the data generating mechanism, and the sample size. Basically, your DF critical values are useless.
And ... that's why your results here are difficult to interpret: the HHS percentage is a ratio of two integrated series.
Bang: I agree with your general assessment that the increase in spending by the government (whether for health or education) likely parallels similar increases in private sector spending.
My concern would be that private sector spending is voluntary and therefore likely capped by individual budget constraints.
On the other hand, public spending is not undertaken voluntarily and is, or has been, determined in a world without budget constraints. My point is that this type of spending pattern seems "unsustainble" to use a catch phrase of the day.
Tufte: Thanks for the input - I was unaware of the specific problems with using percentages in unit root tests, although I have always wondered about how a variable that has a natural lower and upper bound can be "non-stationary" in the long-run. I have seen other papers apply unit root tests to percentages, so the problems are either not recognized or are viewed as not being so important as to deny publication (except, as it turns out, in your case?).
If one takes the logs of I(1) series and test for cointegration would that not be the same as testing for a unit root in the ratio?
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