In a previous post, I pointed out that state and local governments in the US have large asset positions — 33 percent of GDP in total, down from nearly 40 percent before the recession. This is close to double state and local debt, which totals 17 percent of GDP. Among other things, this means that a discussion of public balance sheets that looks only at debt is missing at least half the picture.
On the other hand, a bit over half of those assets are in pension funds. Some people would argue that it’s misleading to attribute those holdings to the sponsoring governments, or that if you do you should also include the present value of future pension benefits as a liability. I’m not sure; I think there are interesting questions here.
But there are also interesting questions that don’t depend on how you treat the existing stocks of pension assets and liabilities. Here are a couple. First, how how do changes in state credit-market debt break down between the current fiscal balance and other factors, including pension fund contributions? And second, how much of state and local fiscal imbalances are financed by borrowing, and how much by changes in the asset position?
Most economists faced with questions like these would answer them by running a regression.  But as I mentioned in the previous post, I don’t think a regression is the right tool for this job. (If you don’t care about the methods and just want to hear the results, you can skip the next several paragraphs, all the way down to “So what do we find?”)
Think about it: what is a regression doing? Basically, we have a variable a that we think is influenced by some others: b, c, d … Our observations of whatever social process we’re interested in consist of sets of values for a, b, c, d… , all of them different each time. A regression, fundamentally, is an imaginary experiment where we adjusted the value of just one of b, c, d… and observed how a changed as a result. That’s the meaning of the coefficients that are the main outputs of a regression, along with some measure of our confidence in them.
But in the case of state budgets we already know the coefficients! If you increase state spending by one dollar, holding all other variables constant, well then, you increase state debt by one dollar. If you increase revenue by one dollar, again holding everything else constant, you reduce debt by one dollar. Budgets are governed by accounting identities, which means we know all the coefficients — they are one or negative one as the case may be. What we are interested in is not the coefficients in a hypothetical “data generating process” that produces changes in state debt (or whatever). What we’re interested in is how much of the observed historical variation in the variable of interest is explained by the variation in each of the other variables. I’m always puzzled when I see people regressing the change in debt on expenditure and reporting a coefficient — what did they think they were going to find?
For the question we’re interested in, I think the right tool is a covariance matrix. (Covariance is the basic measure of the variation that is shared between two variables.) Here we are taking advantage of the fact that covariance is linear: cov(x, y + z) = cov(x, y) + cov(x, z). Variance, meanwhile, is just a variable’s covariance with itself. So if we know that a = b + c + d, then we know that the variance of a is equal to the sum of its covariances with each of the others. In other words, if y = Σ xn then:
(1) var(y) = Σ cov(y, xn)
So for example: If the budget balance is defined as revenue – spending, then the variance of some observed budget balances must be equal to the covariance of the balance with revenue, minus the covariance of the balance with spending.
This makes a covariance matrix an obvious tool to use when we want to allocate the observed variation in a variable among various known causes. But for whatever reason, economists turn to variance decompositions only in few specific contexts. It’s common, for instance, to see a variance decomposition of this kind used to distinguish between-group from within-group inequality in a discussion of income distribution. But the same approach can be used any time we have a set of variables linked by accounting identities (or other known relationships) and we want to assess their relative importance in explaining some concrete variation.
In the case of state and local budgets, we can start with the identity that sources of funds = uses of funds. (Of course this is true of any economic unit.) Breaking things up a bit more, we can write:
revenues + borrowing = expenditure + net acquisition of financial assets (NAFA).
Since we are interested in borrowing, we rearrange this to:
(2) net borrowing = expenditure – revenue + NAFA = fiscal balance – NAFA
But we are not simply interested in borrowing,w e are interested in the change in the debt-GDP ratio (or debt-GSP ratio, in the case of individual states.) And this has a denominator as well as a numerator. So we write:
(3) change in debt ratio = net borrowing – nominal growth rate
This is also an accounting identity, but not an exact one; it’s a linear approximation of the true relationship, which is nonlinear. But with annual debt and income growth rates in the single digits, the approximation is very close.
So we have:
(4) change in debt ratio = expenditure – revenue + NAFA – nominal growth rate * current debt ratio
It follows from equation (1) that the variance of change in the debt ratio is equal to the sum of the covariances of the change with each of the right-side variables. In other words, if we are interested in understanding why debt-GDP ratios have risen in some years and fallen in others, it’s straightforward to decompose this variation into the contributions of variation in each of the other variables. There’s no reason to do a regression here. 
So what do we find?
Here’s the covariance matrix for combined state and local debt for 1955 to 2013. “Growth contrib.” refers to the last term in Equation (4). To make reading the table easier, I’ve reversed the sign of the growth contribution, fiscal balance and revenue; that means that positive values in the table all refer to factors that increase the variance of debt-ratio growth and negative values are factors that reduce it. 
|Debt Ratio Growth||Growth Contrib.||Fiscal Balance||Revenue||Expenditure||NAFA & Trusts|
|Debt Ratio Growth||0.18|
|Growth Contrib. (-)||0.10||0.11|
|Fiscal Balance (-)||0.03||0.04||0.13|
|NAFA & Trusts||0.06||-0.05||0.13||-0.01||-0.14||0.23|
How do we read this? First of all, note the bolded terms along the main diagonal — those are the covariance of each variable with itself, that is, its variance. It is a measure of how much individual observations of this variable differ from each other. The off-diagonal terms, then, show how much of this variation is shared between two variables. Again, we know that if one variable is the sum of several others, then its variance will be the sum of its covariances with each of the others.
So for example, total variance of debt ratio growth is 0.18. (That means that the debt ratio growth in a given year is, on average, about 0.4 percentage points above or below the average growth rate for the full period.) The covariance of debt-ratio growth and (negative) growth contribution is 0.10. So a bit over half the debt-ratio variance is attributable to nominal GDP growth. In other words, if we are looking at why the debt-GDP ratio rises more in some years than in others, more of the variation is going to be explained by the denominator of the ratio than the numerator. Next, we see that the covariance of debt growth with the (negative) fiscal balance is 0.03. In other words, about one-sixth of the variation in annual debt ratio growth is explained by fiscal deficits or surpluses.
This is important, because most discussions of state and local debt implicitly assume that all change in the debt ratio is explained this way. But in fact, while the fiscal balance does play some role in changes in the debt ratio — the covariance is greater than zero — it’s a distinctly secondary role. Finally, the last variable, “NAFA & Trusts,” explains about a third of variation in debt ratio growth. In other words, years when state and local government debt is rising more rapidly relative to GDP, are also years in which those governments are adding more rapidly to their holdings of financial assets. And this source of variation explains about twice as much of the historical pattern of debt ratio changes, as the fiscal balance does.
Since this is probably still a bit confusing, the next table presents the same information in a hopefully clearer way. Here see only the covariances with debt ratio growth — the first column of the previous table — and they are normalized by the variance of debt ratio growth. Again, I’ve flipped the sign of variables that reduce debt-ratio growth. So each value of the table shows the share of variation in the growth of state-local growth ratios that is explained by that component. There is also a second column, showing the same thing for state governments only.
|Component||State + Local||State Only|
|Nominal Growth (-)||0.52||0.30|
|Fiscal Balance (-)||0.17||0.31|
|… of which: Interest||0.06||0.03|
|Trust Contrib. and NAFA||0.33||0.37|
|… of which: Pensions||0.01||0.02|
I’ve added a couple variables here — interest payments under expenditure and pension contributions under NAFA and Trusts. Note in particular the small value of the latter. Pension contributions are quite stable from year to year. (The standard deviation of state/local pension contributions as a percent of GDP is just 0.07, versus around 0.5 for nontrust NAFA.) This says that even though most state and local assets are in pension funds, pension contributions contribute only a little to the variation in asset acquisition. Most of the year to year variability is in governments’ acquisition of assets on their own behalf. This is helpful: It means that if we are interested in understanding variation in the growth of debt over time, or the role of assets vs. liabilities in accommodating fiscal imbalances, we don’t need to worry too much about how to think about pension funds. (If we want to focus on the total increase in state debt, as opposed to the variation over time, then pensions are still very important.)
If we compare the overall state-local sector with state governments only, the picture is broadly similar, but there are some interesting differences. First of all, nominal growth rates are somewhat less important, and the fiscal balance more important, for state government debt ratio. This isn’t surprising. State governments have more flexibility than local ones to independently adjust their spending and revenue; and state debt ratios are lower, so the effect on the ratio from a given change in growth rates is proportionately smaller. For the same reason, the effect of interest rate changes on the debt ratio, while small in both cases, is even smaller for the lower-debt state governments. 
So now we have shown more rigorously what we suggested in the previous post: While the fiscal balance plays some role in explaining why state and local debt ratios rise at some times and fall at others, it is not the main factor. Nominal growth rates and asset acquisition both play larger roles.
Let’s turn to the next question: How do state and local government balance sheets adjust to fiscal imbalances? Again, this is just a re-presentation of the data in the first table, this time focusing on the third column/row. Again, we’re also doing the decomposition for states in isolation, and adding a couple more items — in this case, the taxes and intergovernmental assistance components of revenue, and the pension contribution component of NAFA. The values are normalized here by the variance of the fiscal balance. The first four lines sum to 1, as do the last three. In effect, the first four rows of the table tells us where fiscal imbalances come from; the final three tell us where they go.
|Component||State + Local||State Only|
|Revenue, of which:||0.94||1.01|
|Trust Contrib. and NAFA, of which:||1.04||0.92|
So what do we see? Looking at the first set of lines, we see that state-local fiscal imbalances are entirely expenditure-driven. Surprisingly, revenues are no lower in deficit years than in surplus ones. Note that this is true of total revenues, but not of taxes. Deficit years are indeed associated with lower tax revenue and surplus years with higher taxes, as we would expect. (That’s what the positive values in the “taxes” row mean.) But this is fully offset by the opposite variation in payments from the federal government, which are lower in surplus years and higher in deficit years. During the most recent recession, for example, aggregate state and local taxes declined by about 0.4 percent of GDP. But federal assistance to state and local governments increased by 0.9 percent of GDP. This was unexpected to me: I had expected most of the variation in state budget balances to come from the revenue side. But evidently it doesn’t. The covariance matrix is confirming, and quantifying, what you can see in the figure below: Deficit years for the state-local sector are associated with peaks in spending, not troughs in revenue.
Turning to the question of how imbalances are accommodated, we find a similarly one-sided story. None of the changes in state-local budget balances result in changes in borrowing; all of them go to changes in fund contributions and direct asset purchases.  For the sector as a whole, in fact, asset purchases absorb more than all the variation in fiscal imbalances; borrowing is lower in deficit years than in surplus years. (For state governments, borrowing does absorb about ten percent of variation in the fiscal balance.) Note that very little of this is accounted for by pensions — less than none in the case of state governments, which see lower overall asset accumulation but higher pension fund contributions in deficit years. Again, even though pension funds account for most state-local assets, they account for very little of the year to year variation in asset purchases.
So the data tells a very clear story: Variation in state-local budget balances is driven entirely by the expenditure side; cyclical changes in their own revenue are entirely offset by changes in federal aid. And state budget imbalances are accommodated entirely by changes in the rate at which governments buy or sell assets. Over the postwar period, the state-local government sector has not used borrowing to smooth over imbalances between revenue and spending.
 The interesting historical meta-question, to which I have no idea of the answer, when and why regression analysis came to so completely dominate empirical work in economics. I suspect there are some deep reasons why economists are more attracted to methodologies that treat observed data as a sample or “draw” from a universal set of rules, rather than methodologies that focus on the observed data as the object of inquiry in itself.
 I confess I only realized recently that variance decompositions can be used this way. In retrospect, we should have done this in our papers in household debt.
 Revenue and expenditure here include everything except trust fund income and payments. In other words, unlike in the previous post, I am following the standard practice of treating state and local budgets separate from pension funds and other trust funds. The last line, “NAFA and Trusts”, includes both contributions to trust funds and acquisition of financial assets by the local government itself. But income generated by trust fund assets, and employee contributions to pension funds, are not included in revenue, and benefits paid are not included in expenditure. So the “fiscal balance” term here is basically the same as that reported by the NIPAs and other standard sources.
 This is different from households and the federal government, where higher debt and, in the case of households, more variable interest rates, mean that interest rates are of first-order importance in explaining the evolution of debt ratios over time.
 It might seem contradictory to say that a third of the variation in changes in the debt ratio is due to the fiscal balance, even though none of the variation in the fiscal balance is passed through to changes in borrowing. The reason this is possible is that those periods when there are both deficits and higher borrowing, also are periods of slower nominal income growth. This implies additional variance in debt growth, which is attributed to both growth and the fiscal balance. There’s some helpful discussion here.
(This post is based on a paper in process. I probably will not post any more material from this project for the next month or so, since I need to return to the question of potential output.)