where the shares of the traded and nontraded-goods are identical at home and abroad (θ = θ). The log real exchange rate can be decomposed as

q = (s + pT − pT ) + (1 − θ)(pN − pT ) − (1 − θ)(pN − pT ), (7.5)

where lower case letters denote variables in logarithms. We adopt the commodity arbitrage view of PPP (chapter 3.1) and assume that the law-of-one price holds for traded goods. It follows that the Þrst term on the right hand side of (7.5), which is the deviation from PPP for the traded good, is 0. The dynamics of the real exchange rate is then completely driven by the relative price of the tradable good in terms of the nontraded good.

The Balassa—Samuelson Model

Now, we need a theory to understand the behavior of the relative price of tradables in terms of nontradables. It turns out if, i) factor markets and Þnal goods markets are competitive, ii) production takes place under constant returns to scale, iii) capital is perfectly mobile internationally, iv) labor is internationally immobile but mobile between the tradable and nontradable sectors, then the relative price of nontradable goods in terms of tradable goods is determined entirely by the production technology. Demand (preferences) does not matter at all.

The theory is viewed as holding in the long run and therefore omit time subscripts. To Þx ideas, let there be only one traded good and one nontraded good. Capital and labor are supplied elastically. Let LT (LN ) and KT (KN ) be labor and capital employed in the production of the traded YT (nontraded YN ) good. AT (AN ) is the technology level in the traded (nontraded) sector. The two goods are produced according to Cobb-Douglas production functions

The balance of trade is assumed to be zero which must be true in the long run. Let the traded good be the numeraire. The small open

(130) Finally, plug (7.16) into (7.11) to get the solution for relative price of the nontraded good in terms of the traded good

where C is a positive constant. Now let a = ln(A), r = ln(R), and c = ln(C) and take logs of (7.17) to get the solution for the log relative price of nontraded goods in terms of traded goods

Over time, the evolution of the log relative price of nontradables depends only on the technology and the exogenous rental rate on capital. We see that there are at least two reasons why the relative price of non-tradables in terms of tradables should increase with a country’s income.

First, suppose that the economy experiences unbiased technological growth where aN and aT increase at the same rate. pN will rise over time if traded-goods production is relatively capital intensive (αN < αT ). A standard argument is that tradables are manufactured goods whose production is relatively capital intensive whereas nontraded goods are mainly services which are relatively labor intensive. Second, pN will increase over time if technological growth is biased towards the capital intensive sector. In this case, aT actually grows at a faster rate than aN . If either of these scenarios are correct, it follows that fast growing economies will experience a rising relative price of nontradables and by (7.5), a real appreciation over time.

The implications for the behavior of the real exchange rate are as follows. If the productivity factors grow deterministically, the deviation of the real exchange rate from a deterministic trend should be a stationary process. But if the productivity factors contain a stochastic trend (chapter 2.6) the log real exchange rate will inherit the random walk behavior and will be unit-root nonstationary. In either case, PPP will not hold in the long run.

When we take the Balassa—Samuelson model to the data, it is tempting to think of services as being nontraded. It is also tempting to think

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 217

that services are relatively labor intensive. While this may be true of some services, such as haircuts, it is not true that all services are nontraded or that they are labor intensive. Financial services are sold at home and abroad by international banks which make them traded, and transportation and housing services are evidently capital intensive.

7.4Long-Run Analyses of Real Exchange Rates

Empirical research into the long-run behavior of real exchange rates has employed econometric analyses of nonstationary time series and is aimed at testing the hypothesis that the real exchange rate has a unit root. This research can potentially provide evidence to distinguish between the Casselian and the Balassa—Samuelson views of the world.

Univariate Tests of PPP Over the Float

To test whether PPP holds in the long run, you can use the augmented Dickey-Fuller test (chapter 2.4) to test the hypothesis that the real exchange rate contains a unit root. Using quarterly observations of the CPI-deÞned real exchange rate from 1973.1 to 1997.4 for 19 high-income countries, Table 7.2 shows the results of univariate unit-root tests for US and German real exchange rates. Four lags of ∆qt and a constant were included in the test equation. The p-values are the proportion of the Dickey—Fuller distribution that lies to the left (below) τc. Including a trend in the test regressions yields qualitatively similar results and are not reported.

Statistical versus Economic SigniÞcance. Classical hypothesis testing is designed to establish statistical signiÞcance. Given a su ciently long time series, it may be possible to establish statistical signiÞcance of the studentized coe cients to reject the unit root, but if the true value of the dominant root is 0.98, the half-life of a shock is still over 34 years and this stationary process may not be signiÞcantly di erent from a true unit-root process in the economic sense.

If that is indeed the case, then in light of the statistical di culties surrounding unit-root tests, it can be argued that we should not even care whether the real exchange rate has a unit root but we should instead focus on measuring the economic implications of the real exchange rate’s behavior. What market participants care about is the degree of persistence in the real exchange rate and one measure of persistence is the half life.

The annualized half-lives reported in Table 7.2 are based on estimates adjusted for bias by Kendall’s formula [equation (2.81)].4 The average half-life is 3.7 years when the US is the numeraire country. That is, on average, it takes 3.7 years–quite a long time since the business cycle frequency ranges from 1.25 to 8 years–for half of a shock to the log real exchange rate to disappear. The average half-life is 2.6 years when Germany is the numeraire county.

Univariate tests using data from the post Bretton-Woods ßoat typically cannot reject the hypothesis that the real exchange rate is driven by a unit-root process. Using the US as the home country, only two of the tests can reject the unit root at the 10 percent level of signiÞcance.

The results are somewhat sensitive to the choice of the home (numeraire) country.5 Part of the persistence exhibited in the real value of the dollar comes from the very large swings during the 1980s. The real appreciation in the early 1980s and the subsequent depreciation was largely a dollar phenomenon not shared by cross-rates. To illustrate, the evidence for purchasing-power parity is a little stronger when Germany is used as the home country since here, the unit root can be rejected at the 10 percent level of signiÞcance for German real exchange rates with several European countries.

Univariate Tests for PPP Over Long Time Spans

One reason that the evidence against a unit root in qt is weak may be that the power of the test is low with only 100 quarterly observations.6

4Christiano and Eichenbaum [27] put forth this argument in the context of the unit root in GNP.

5A point made by Papell and Theodoridis [119].

6The power of a test is the probability that the test correctly rejects the null hypothesis when it is false.

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 219

Table 7.2: Augmented Dickey-Fuller Tests for a Unit Root in Post-1973 Real Exchange Rates

Notes: Half-lives are adjusted for bias and are measured in years. SigniÞcance at the 10 percent level indicated in boldface.

Table 7.4: Variance ratios and asymptotic standard errors of real dollar—sterling exchange rates. Lothian—Taylor data using PPIs.

can be attributed to a permanent (random walk) component. The asymptotic standard errors tend to overstate the precision of the variance ratios in small samples. That being said, even at the 20 year horizon VR20 for the dollar—pound rate is (using the asymptotic standard error) signiÞcantly greater than 0 which implies the presence of a permanent component in the real exchange rate. This conclusion contradicts the results in Table 7.3 that rejected the unit-root hypothesis.

Summary of univariate unit-root tests. We get conßicting evidence about PPP from univariate unit-root tests. From post Bretton—Woods data, there is not much evidence that PPP holds in the long run when the US serves as the numeraire country. The evidence for PPP with Germany as the numeraire currency is stronger. Using long-time span data, the tests can reject the unit-root, but the results are dependent on the number of lags included in the test equation. On the other hand, the pattern of the variance ratio statistic is consistent with there being a unit root in the real exchange rate.

The time period covered by the historical data span across the Þxed exchange rate regimes of the gold standard and the Bretton Woods adjustable peg system as well as over ßexible exchange rate periods of the interwar years and after 1973. Thus, even if the results on the long-span data uniformly rejected the unit root, we still do not have direct evidence that PPP holds during a pure ßoating regime.

Panel Tests for a Unit Root in the Real Exchange Rate

Let’s return speciÞcally to the question of whether long-run PPP holds over the ßoat. Suppose we think that univariate tests have low power

Nonnparametric bootstrap p-values in parentheses. Parametric bootstrap p-values

in square brackets. Bold face indicates signiÞcance at the 10 percent level.

is included. The asymptotic evidence against the unit root is very weak. Next, we test the unit root when the common time e ect is omitted. Here, the evidence against the unit root is strong when the deutschemark is the numeraire currency, but not for the dollar. The bias-adjusted approximate half-life to convergence range from 1.7 to 5.3 years, which many people still consider to be a surprisingly long time. Table 7.6 shows panel tests of PPP using the Im, Pesaran, and Shin test and the Maddala—Wu test. Here, I did not remove the common time e ect. These tests are consistent with the Levin-Lin test results. When the dollar is the numeraire, we cannot reject that the deviation from trend is a unit root. When the deutschemark is the numeraire currency, the unit root is rejected whether or not a trend is included. The evidence against a unit root is generally stronger when

the deutschemark is used as the numeraire currency.

Canzoneri, Cumby, and Diba’s test of Balassa-Samuelson

Canzoneri, Cumby, and Diba [21] employ IPS to test implications of the Balassa—Samuelson model. They examine sectoral OECD data for the US, Canada, Japan, France, Italy, UK, Belgium, Denmark, Sweden, Finland, Austria, and Spain. They deÞne output by the “manufacturing” and “agricultural, hunting forestry and Þshing” sectors to be traded goods. Nontraded goods are produced by the “wholesale and

7.4. LONG-RUN ANALYSES OF REAL EXCHANGE RATES 225

Table 7.7: Canzoneri et. al.’s IPS tests of Balassa—Samuelson

Notes: Bold face indicates asymptotically signiÞcant at the 10 percent level.

retail trade,” “restaurants and hotels,” “transport, storage and communications,” “Þnance, insurance, real estate and business,” “community social and personal services,” and the “non-market services” sectors.

Their analysis begins with the Þrst-order conditions for proÞt max-

imizing Þrms. Equating (7.12) to (7.13), the relative price of nontrad- (133) ables in terms of tradables can be expressed as

where k = K/L is the capital labor ratio. By virtue of the CobbDouglas form of the production function, Akα = Y/L is the average product of labor. Let xT ≡ ln(YT /LT ) and xN ≡ ln(YN /LN ) denote the log average product of labor. We rewrite (7.19) in logarithmic form as

−

¯

Table 7.7 shows the standardized t calculated by Canzoneri, Cumby and Diba. All calculations control for common time e ects. Their results support the Balassa—Samuelson model. They Þnd evidence that there is a unit root in pN − pT and in xT − xN , and that they are cointegrated, and there is reasonably strong evidence that PPP holds for traded goods.