Thoughts on the Dollar: PPP and Thresholds

Includes: DBV, UDN, UUP
by: James Hamilton

As the dollar hits a new low against the euro, here are some thoughts on what arguments make sense, given our knowledge of the statistical properties of real exchange rates.

From Deutsche Bank's Exchange Rate Perspectives (February 22).

We see several themes pointing towards the current USD down cycle being close to an end:

  • The USD is close to the bottom of its historical valuation bands
  • We expect US growth to rebound in 2H 2008 with fixed-income and equity inflows into the US recovering in line
  • Interest rate differentials should move in favor of the USD by 2H 2008, as we expect the ECB to start cutting rates during Q2
  • The US trade deficit is narrowing
  • On the view that oil and commodities prices are at, or close to, their peaks for this cycle, a slowing in emerging market reserve growth will slow the demand for euros

Given the arguments above, we believe that the USD is likely to bottom in 2008 (with the exception of USD/JPY). ... the fact that [the] trade-weighted USD index is undervalued in PPP terms, and the cycle is now relatively mature, suggests the USD is close to a bottom.

I'm going to focus on the first point, which seems eminently reasonable, and seems validated by this graph:

From Deutsche Bank Exchange Rate Perspective (February 22, 2008) Figure 4.

I believe the blue line is the line implied by purchasing power parity [PPP], namely:

e = p - p* + c

where e is the log value of a currency (the inverse of the exchange rate), p and p* are the log price levels in the home and foreign countries, respectively, and c is a constant. Since p and p* are price indices and not price levels, one has to estimate the constant. The +/- 20% bands seem to bound the movements of the dollar.

If PPP holds in the long run, then the real currency value (in this case, of the dollar) should revert to a constant average value, or allowing for some omitted factors, to a trend. One can estimate the trend using OLS, to obtain the following picture (here I assume the DB TWI and the Fed's index are similar):

Source: Federal Reserve Board and author's calculations. Figure 1: Log real value of the dollar (broad), and 1973M01-08M01 trend.

However, as I've pointed out on a number of occasions, one has to be careful about estimating "trends". One can always do it, but it might not always be a good idea. It turns out that there's no "right" answer to whether in this case we should estimate deterministic trends (and hence assume that the real value of the dollar is a "trend stationary" process), or we should model the real dollar as a difference stationary process (a random walk is a difference stationary process).

The standard Augmented Dickey-Fuller test (optimal lag length selected by SBC at 1 lag, allowing for trend) fails to reject (the p-value is 0.76 using Mackinnon critical values). The Elliott-Rothenberg-Stock (1996) DF-GLS unit root test (allowing for a trend) fails to reject the unit root null, as do other tests (Phillips-Perron, etc.). On the other hand, a test with a trend stationary null hypothesis, namely the Kwiatkowski, Phillips, Schmidt and Shin [KPSS] test also fails to reject that null.

There's good reason to expect PPP not to hold, certainly in the short run, perhaps even in the long run (see this paper [pdf], and this post). After all, consumption bundles are not identical; there are nontraded goods, and transactions costs can prevent arbitrage. But there is hope, in the sense of an intermediate view -- that arbitrage kicks in when the real rate deviates sufficiently far away from the value implied by PPP.

From Taylor and Taylor's survey of PPP:

In empirical work on mean reversion in the real exchange rate, nonlinearity can be examined through the estimation of models that allow the autoregressive parameter to vary. For example, transactions costs of arbitrage may lead to changes in the real exchange rate being purely random until a threshold equal to the transactions cost is breached, when arbitrage takes place and the real exchange rate mean reverts back towards the band through the influence of goods arbitrage (although the return is not instantaneous because of shipping time, increasing marginal costs, or other frictions). This kind of model is known as "threshold autoregressive."

While this concept of bands of inaction seems to accord well with the bands drawn in Figure 4 above, Taylor and Taylor also observe:

Using a threshold autoregressive model for real exchange rates as a whole, however, could pose some conceptual difficulties. Transactions costs are likely to differ across goods, and so the speed at which price differentials are arbitraged may differ across goods (Cheung, Chinn and Fujii, 2001). Now, the aggregate real exchange rate is usually constructed as the nominal exchange rate multiplied by the ratio of national aggregate price level indices, and so, instead of a single threshold barrier, a range of thresholds will be relevant, corresponding to the various transactions costs of the various goods whose prices are included in the indices. Some of these thresholds might be quite small, for example, because they are easy to ship, while others will be larger. As the real exchange rate moves further and further away from the level consistent with PPP, more and more of the transactions thresholds would be breached, and so the effect of arbitrage would be increasingly felt. How might we address this type of aggregation problem? One way is to employ a well-developed class of econometric models that embody a kind of smooth but nonlinear adjustment, such that the speed of adjustment increases as the real exchange rate moves further away from the level consistent with PPP. Using a smooth version of a threshold autoregressive model, and data on real dollar exchange rates among the G-5 countries (France, Germany, Japan, the United Kingdom and the United States), Taylor, Peel, and Sarno (2001) reject the hypothesis of a unit root in favor of the alternative hypothesis of nonlinearly mean-reverting real exchange rates -- and using data just for the post-Bretton Woods period, thus solving the first PPP puzzle. They also find that for modest real exchange shocks in the 1 to 5 percent range, the half-life of decay is under 3 years, while for larger shocks the half-life of adjustment is estimated much smaller -- thus going some way towards solving the second PPP puzzle.

So, perhaps there is some reason to believe that these "20%" bands have some formal empirical (read econometric) basis. If one believes that these threshold approaches, then another question arises.

The discussion is usually couched in terms of the view that the exchange rate, rather than prices, adjust. This view is consistent with the idea that asset prices move fast, in contrast to sticky prices. But as elaborated in papers by Cheung et al. (2004) , and Engel and Morley (2001) (in a one-regime context), prices often seem to do the adjustment.

Work by UW student Deokwoo Nam is informative in this respect; from the abstract to "Can the real exchange rate be stationary within the band of inaction?," [pdf]:

We decompose the real exchange rate into the nominal exchange rate and relative price by imposing the symmetry assumption, and then model their dynamics as a bivariate Threshold Vector Error Correction Model [TVECM]. As expected by the evidence on nonlinearity of the real exchange rate, our empirical results provide evidence on a threshold cointegration of the nominal exchange rate and relative price. The surprising finding is that the roles of the nominal exchange rate and relative price in correcting the deviation from PPP, are distinct between outside and inside the band of inaction, relative to those in the linear framework. That is, it is only outside the band that the nominal exchange rate makes its significant contribution to the correction of the deviation from PPP. On the other hand, the correction by the relative price is insignicant outside the band, but there is some evidence that the relative price makes its significant adjustment to the deviation from PPP inside the band. This finding implies that it is the relative price, not the nominal exchange rate, that corrects the deviation from PPP within the band, if the correction is indeed made within the band of inaction.

Unfortunately, these results pertain to bilateral U.S. real exchange rate; we don't know how the aggregate trade weighted exchange rate behaves.

The bottom line: Maybe those bounds exist, but it's not clear what -- exchange rates or price levels -- will do the adjusting to reassert PPP.

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