Yanping Chong, Oscar Jorda and Alan M. Taylor have tackled the perennial challenge of measuring the long run relationship between the real exchange rate and per capita income levels. From the abstract to The Harrod-Balassa-Samuelson Hypothesis: Real Exchange Rates and their Long-Run Equilibrium:

Frictionless, perfectly competitive traded-goods markets justify thinking of purchasing power parity (PPP) as the main driver of exchange rates in the long-run. But differences in the traded/non-traded sectors of economies tend to be persistent and affect movements in local price levels in ways that upset
the PPP balance (the underpinning of the Harrod-Balassa-Samuelson hypothesis, HBS). This paper uses panel-data techniques on a broad collection of countries to investigate the long-run properties of the PPP/HBS equilibrium using novel local projection methods for cointegrated systems. …

…These semi-parametric methods isolate the long-run behavior of the data from contaminating factors such as frictions not explicitly modelled and thought to have effects only in the short-run. Absent the short-run effects, we find that the estimated speed of reversion to long-run equilibrium is much higher. In addition,
the HBS effects means that the real exchange rate is converging not to a steady mean, but to a slowly to a moving target. The common failure to properly model this effect also biases the estimated speed of reversion downwards. Thus, the so-called “PPP puzzle” is not as bad as we thought.

A lot of people have worked on this issue. In Chinn and Johnston (1997), we review some of the pre-1997 literature.

The challenge Louis Johnston and I tackled was combining cross-sectional and time series evidence on the long run relationship between the real exchange rate a number of variables, including per capita income. This is more difficult than it first sounds, since one had to deal with issues of integrated series (the real exchange rate and relative income levels appear to be I(1) series), but the testing procedures for estimating long run relationships in panel time series was not fully developed at the time. Now, of course, panel cointegration testing and estimation procedures are programmed into most statistical packages.

That doesn’t quite solve all the problems, however. And in this paper, Chong, Jorda and Taylor address this specific issue:

Traditionally, cointegration is examined through the prism of a vector error-correction model (VECM) but VECMs are difficult to extend to panels because they are parametrically intensive. In addition, parametrically tractable specifications and the structure of VECMs restrict the range of dynamics and half-lives that can be estimated from the data. …

The authors circumvent this problem by using a semi-parametric approach to characterizing the short run dynamics, which are not of intrinsic interest to them. Interested readers can see the details in Section 2 of the paper (which is entails some perseverence).

Using a panel of 21 OECD countries over the 1973-2008 period, the authors find a full panel long run “elasticity” of 0.57 to 0.78 (for the US as base country, and for the “rest-of-the-world” as base country, respectively).

My one comment about the interpretation of the paper is that the identified effect is not literally the Harrod-Balassa-Samuelson effect. As discussed in Chinn and Johnston, the HBS effect is literally the relationship between the intercountry relative productivity differentials between the traded and nontraded sectors. What has often been observed is that the relative productivity level of the traded sector often rises with per capita income. Hence, the short cut used in this paper, that higher income “stands in” for the relative productivity ratio.

Let me stress that this is a commonplace approach; however, it does complicate the interpretation of the long run elasticity that is estimated, since rising per capita income might induce higher demand for nontraded goods, and with imperfect factor mobility within the country, might also appreciate the exchange rate; for more, see Jose de Gregorio and Holger Wolf (1994) as well as Chinn (2009).

In addition to the new estimates of the long run elasticity, what the authors provide is alternative estimates of a half-life of a deviation from the long run real exchange rate. By allowing the long run real exchange rate to move with the relative per capita income term, and using the semi-parametric procedure to correctly purge the impulse response functions of the short run frictional effects, they find that the half lives are between 3 to 5 quarters shorter than what has been found elsewhere in the PPP literature. This last finding is similar in flavor to what Louis Johnston and I found, using relative productivity differentials as a proxy for the HBS effect.

Does this tell us anything about the economics of exchange rate deviations from the long run values? I think it hints at something. For a long time, many argued that the persistence in deviations from PPP was consistent with the importance of “real” shocks. The fact that including a relative income term shortens the estimated half-lives considerably suggests that what deviations from fundamentals occur could be due to monetary factors.

And of course, these results have implications for some current issues. Productivity (or more specifically per capita income) trends could be of importance in thinking about a certain type of “equilibrium” real exchange rate estimates pertaining to, for instance, the RMB. [0]

The productivity effect is discussed in these earlier posts: [1] [2] [3]; and this 2000 paper on East Asian exchange rates.