Identities, Parameters and Regressions

A reader comments:

Your final jab in this post regresses the state output gap on the fiscal gap. You then conclude that there is a positive relation between the two and that this somehow implies that a reduction in gov’t spending is a drag on the economy. I’ll just point out that …. that gov’t spending is a component of GSP. Of course they’re positively related.

I think this comment reflects a commonplace confusion between identities, functional relationships, and reduced form coefficients.

For instance, consider that at the industry level for a state, such as in data reported by BEA here, total private (agriculture, mining, through manufacturing through other services) and government has to sum to total gross state product (in current lexicon state GDP). So, one would think that regressing total output on government output has to yield a positive coefficient. Run the regression over the 2005Q1-15Q3 period (the entire available sample) for Wisconsin:

GDPWI,t = 18829.71 – 0.215GOVWI,t + ut

Adj-R2 = -0.02. Bold denotes significance at 10% msl using HAC robust standard errors.

Lest you think this proves higher government spending causes less output, well, consider the same regression for Kansas.

GDPKS,t = 262893.4 + 5.895GOVKS,t + ut

Adj-R2 = 0.34. Bold denotes significance at 10% msl using HAC robust standard errors.

This is a lesson I learned as an undergraduate — do not appeal to an accounting identity for information about a coefficient.

Let’s turn to an example that is more familiar. We all know the expenditure side definition of GDP from macro:

Y ≡ C + I + G + EX – IM

I stress in my undergraduate courses that one cannot use identities to explain how the world works, or more concretely, when G changes, how does Y. For that, one needs a model.

So, the hapless regression-runner might regress Y on G, asserting one must get a positive coefficient since G is by construction a component of Y. But in fact different theories yield different implied coefficients.

Suppose aggregate supply is given by Y = Yn = ΦF(K,N), and aggregate demand is given by Y = fn(G, T, M/P), where K, N are given. Then the correlation between Y and G is … zero!

Suppose P is predetermined within a period, and aggregate demand takes the same form as above. Then the correlation between Y and G is positive, but not necessarily one (it’ll depend on the marginal propensity to consume, interest sensitivity of investment, interest sensitivity of money demand, etc.).

Suppose P depends positively on the output gap (i.e., a Phillips Curve holds), such that P = Pe + θ(Y-Yn). Then the correlation is positive, but (holding all else constant) smaller than that in the previous example.

Suppose P depends positively on the output gap, and the fiscal authorities rely on a “fiscal rule” that achieves counter-cyclical stabilization, e.g., G = φ(Y-Yn), φ < 0. Then the implied correlation between Y and G is now negative.

One could in principle estimate the reduced form coefficient in the first three cases (after accounting for omitted variables, etc.); the parameters (e.g., θ) would require determining an appropriate instrumental variable. The correlation can always be calculated; whether it’s meaningful is another question.

Bottom Line: Identities do not tell you about behavior. Inferring causality is hard, but if one wants to tell a story, one has to try to deal with the data in an intelligent way.