## Saturday, July 23, 2011

### National Debt & Regression Models - Units of Measurement Matter!

In a recent post, titled "Debt and Delusion", Robert Shiller draws attention to a very important point amid the current bombardment of news about the debt crisis (crises?).

Referring to the situation in Greece, he comments:

"Here in the US, it might seem like an image of our future, as public debt comes perilously close to 100% of annual GDP and continues to rise. But maybe this image is just a bit too vivid in our imaginations. Could it be that people think that a country becomes insolvent when its debt exceeds 100% of GDP?

That would clearly be nonsense. After all, debt (which is measured in currency units) and GDP (which is measured in currency units per unit of time) yields a ratio in units of pure time. There is nothing special about using a year as that unit. A year is the time that it takes for the earth to orbit the sun, which, except for seasonal industries like agriculture, has no particular economic significance.

We should remember this from high school science: always pay attention to units of measurement. Get the units wrong and you are totally befuddled.

If economists did not habitually annualize quarterly GDP data and multiply quarterly GDP by four, Greece’s debt-to-GDP ratio would be four times higher than it is now. And if they habitually decadalized GDP, multiplying the quarterly GDP numbers by 40 instead of four, Greece’s debt burden would be 15%. From the standpoint of Greece’s ability to pay, such units would be more relevant, since it doesn’t have to pay off its debts fully in one year (unless the crisis makes it impossible to refinance current debt)." .........

Yes, units of measurement matter - always!

So, you earn 100K a year - is that 100K dollars or 100K Chilean pesos? (Check the current exchange rate, and weep!) You pay $1,000 for your rental accommodation - is that per month or per week? All obvious stuff, right? But let's not forget that Shillers' point also applies when we fit a regression model and interpret the estimated coefficients. Suppose we estimate a linear regression model of the form: y = b0 + b1X1 + b2X2 + u where y is measured in dollars, X1 is measured in Kg., and X2 is a (unitless) index. This is an equation. We can't add, or equate, quantities that have different units. So, every term in the equation must have the same units - dollars, in this example. This means that the coefficients must also have units of measurement. The intercept coefficient, b0, and the coefficient of X2 are both measured in dollars; but the coefficient of X1 has different units - dollars per Kg. here. You need to keep this in mind when interpreting the estimated values of these coefficients - which in the case of this linear model are also the marginal effects. Have a think about what this means when you have a model that's non-linear in the parameters. Of course, this is where a unitless measure of "impact" - such as an elasticity - comes in handy! Even if you're careful about this, it's still easy to slip up when you report a confidence interval for a regression coefficient. Take b1 from the model above, for example. The standard error that goes with the point estimate also has units of dollars per Kg. The critical value (tc) from the Student-t table is unitless. So, it makes sense, from a units viewpoint, to add (subtract) the product of tc and the standard error to (from) the estimate of b1. We then have our confidence interval. Suppose we compute a 95% confidence interval for b1 to be [1.34 , 1.98]. What we have to remember, and report, is that the confidence interval is actually [1.34 , 1.98]$/Kg.

Those units really do matter!