where A and B are conformably partitioned matrices of coefficients for the state and forcing variables, respectively. The solution method proceeds as follows. The matrix A can be diagonalized:
where Λ is a diagonal matrix of eigenvalues of A. It is straightforward to show that
where λ1 < 0 and λ2 > 0 (appropriate discount factor), so that the number of stable and unstable roots is equal to the number of predetermined and nonpredetermined state variables, and the unstable root λ2 corresponds to the state variable.52 The matrix V is composed of linearly independent left eigenvectors of A and, along with the matrix of coefficients on forcing variables B, takes the general form
The minimal state (fundamental) solution to this linear rational expectations system in equation (A1) for the nonpredetermined variable q has the following general and specific forms:
where Z is the vector of forcing variables in (A1) and where
c ≡ V21B1 + V22B2, and
See Buiter (1989) for details.
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Hamid Faruqee is an Economist in the Southeast Asia and Pacific Department; work on this paper began while he was an Economist in the Research Department. He received his Ph. D. from Princeton University. The author would like to thank Peter Clark, Leonardo Bartolini, Tamim Bayoumi, Jonathan Ostry, Stephen Tokarick, and Steven Symansky for helpful comments and/or discussions. The views expressed and any errors remaining, however, are the author’s sole responsibility.
See Dornbusch (1987) for a survey on PPP. See Breuer (1994) for a recent survey on the empirical evidence. In keeping with this literature, the real exchange is defined here as the currency-adjusted ratio of national price levels.
In a normative context, this basic identification problem also arises. As a policy target, PPP exchange rate rules provide an anchor to minimize potential misalignments. However, efforts to stabilize an inappropriate target for the real exchange rate have sometimes lead to increased macroeconomic instability. See Aghevli, Khan, and Montiel (1991), Montiel and Ostry (1991), and Calvo, Rein-hart, and Végh (1994).
See Masson, Kremers, and Home (1994) for an empirical analysis on the long-run determination of net foreign assets for the G-3 industrial countries.
The well-known transfer problem provides a useful illustration. Consider a country that experiences a steady-state decline in its stock of net foreign assets. The expenditure-reducing impact of this redistribution of wealth on domestic spending predominantly affects demand for domestically produced goods. Hence, this international transfer of wealth must be accompanied by a real depreciation at home and expenditure-switching toward home roods to aliow adjustment at full employment (internal balance) and an improved trade position to offset lower interest income from abroad (external balance).
See, for example, Hsieh (1982), Marston (1987), De Gregorio, Giovannini, and Wolf (1994), and Asea and Mendoza (1994) and the references cited therein for empirical and theoretical background on the Balassa-Samuelson effect.
See also Ostry (1988). For industrial countries, several studies have used a simulations approach to calculate real exchange rate trajectories compatible with macroeconomic balance. See, for example. Williamson (1990) and Bayoumi and others (1994) and the references therein.
The model is a continuous-time version of Mussa (1984), The analysis assumes imperfect substitutability in goods but not assets. In the case of imperfect substi-tutability in assets, portfolio effects provide the channel linking asset stocks and the exchange rate. Changes in asset positions reflecting changes the relative supplies of domestic and foreign debt, require changes in either the relative yield (risk premium) or the relative valuation (real exchange rate) to restore portfolio balance.
See Ostry (1988), Edwards (1989), and Khan and Ostry (1991) for further discussion regarding the effects of tariff changes and terms-of-trade disturbances. Note that the theoretical analysis is restricted to factors determining the relative price of imports versus exports. However, the subsequent empirical analysis generalizes the measure of international relative prices to also take into account factors that affect the relative price of traded versus nontraded goods.
Equivalently, equation (1) can be interpreted from the absorption approach. As in the transfer example, countries favor their own goods in consumption. Hence, an increase in domestic real absorption—relative to (fixed) output—must be accompanied by an increase in the relative price of domestic goods to ensure goods market equilibrium.
Net foreign assets here refer to both private and official holdings. Note that the real analysis here applies to both fixed and floating nominal exchange rate regimes.
In general, the underlying balance approach defines the equilibrium exchange rate as that value or trajectory consistent with both internal and external macroeconomic balance. Transitory or short-run fluctuations in income are omitted here but they could be added—see Mussa (1984). In that case, the equilibrium real exchange rate would explicitly require both full-employment output and balance of payments equilibrium.
The postulated equation for desired holdings of net external assets serves as a short cut. Including this target level pins down the steady-state level of net foreign assets, avoiding the indeterminacy feature associated with standard infinite horizon models with representative agents, where any distribution of wealth is self-replicating (i.e., multiple stationary states). See Giavazzi and Wyplosz (1984).
Setting p = r* assures a stable level for the real exchange rate in steady state. Otherwise, there would exist a steady-state rate of real appreciation or depreciation equal to the long-run real interest rate gap in equation (4).
The forward-looking nature of the solution results from the fact that desired excess spending is affected by the expected rate of depreciation. An “asset price” interpretation for this condition treats the expectations term as the anticipated capital gains from holding the foreign asset. Alternatively, from an intertemporal viewpoint, the expected future real exchange rate (i.e., intertemporal relative prices) influences an agent’s optimal consumption allocations that maximize ifetime utility. See Mussa (1984).
The two variables q and
The analysis can also be revised to account for steady-state growth rather than a stationary level of income, in which case a nonzero current account can exist in steady state. In steady-state equilibrium (with a constant real exchange rate but income growth), the trade balance and current account as a share of GNP depend on the stable ratio of net foreign assets to income along the balanced growth path. Otherwise, the analysis is essentially the same as described in the text.
The model characterizes two basic types of fundamental shocks: those that affect the short-run trade balance (flow shocks) and those that affect the long-run net foreign asset position (stock disturbances), where only the latter type affect both q and nx in steady state. In practice, however, variables rarely fit neatly into either category and the empirics to follow do not require these types of shocks to be orthogonal. For the purposes of exposition, a fundamental variable is referred to as a determinant operating through the trade balance or net foreign assets depending on the primary channel through which that factor impacts on the real exchange rate.
Following Ostry (1988), this result suggests that the direct effect of a permanent x shock on the trade balance is fully offset by the indirect effect through the change in the real exchange rate in the model.
Moreover, the number of independent cointegrating vectors r must be such that 0 < r < < N. If there were exactly N such linearly independent combinations, then the set of variables must all be stationary (i.e., integrated of order zero or I(0)). If no combinations exist (r = 0), the series are independent difference-stationary (i.e., integrated of order one or I(1)) variables.
The basic long-run (cointegrating) relationship between the real exchange rate and its fundamental determinants based on the theoretical discussion is summarized by equation (7).
All variables measured as indices are expressed in log-levels using 1985 as the base year (data source for REER: International Financial Statistics).
Source: International Financial Statistics.
Source: International Financial Statistics. Canada was excluded because of lack of data. Note that the wholesale price index predominantly measures traded goods prices, while the consumer price index has a significant component of services, which are generally not traded. Hence, the ratio of the two indices compared with each country’s foreign counterpart serves as a proxy for the relative price structure in the United States and Japan compared with each country’s major trading partners. Specifically, if CPI = (Pt)α (Pn)1-α and WPI = (Pt)β (Pn)1-β where β > α, then the ratio will be an increasing function of the relative price of nontraded goods at home, (Pn/Pt)β-α.
Source: Bureau of Labor Statistics. Note that in levels, output per manhour is not directly comparable across countries (the index level is arbitrary); however, trend comparisons can be made.
To make explicit the relationship between REER and TNT, note that the latter by definition can be expressed: ln (CPI/ECPI*) - ln(WPI/EWPI*), where * indicates foreign variables and E is the nominal exchange rate. The first term in this expression is in fact REER (ignoring coverage issues). If long-run PPP were to hold in tradables (i.e., ln(WPI/EWPI*) ~ I(0)), while the Balassa-Samuelson effect was the main source of secular trends in the real exchange rate, then REER and TNT alone >would cointegrate with a coefficient of unity. However, if permanent shocks to tradables cause ln(WPI/EWPI*) to be nonstationary as well, then REER and TNT will cointegrate only when some other measure(s) is included, capturing permanent movements in the relative price of traded goods. The model in the previous section highlights potential sources for these latter long-run movements.
See De Gregorio, Giovannini, and Wolf (1994), and Micosi and Milesi-Ferreti (1994) for recent evidence on the relation between the relative price of nontraded goods and sector productivity differentials.
For the dollar real exchange rate, a case could be made for rejecting the null in favor of a trend-stationary alternative (p value near 0.10), given the low power of unit root tests. This result only highlights the near observational equivalence between trend-stationary and difference-stationary processes in finite samples. See Campbell and Perron (1991). However, the steady-state implications of a deterministic trend are quite unappealing.
Replacing TNT, cointegration estimates using with PROD (not reported) also support a finding of cointegration. However, the latter measure consistently overstates the comparative decline in relative productivity for the United States in the first half of the sample and consequently yields non-normal errors in the estimation. Hence, subsequent analysis is conducted with TNT as the proxy for productivity in the case of the United States.
The presence of multiple cointegrating vectors suggests the presence of multiple long-run economic relationships between the set of variables or some subsets thereof. For example, if the fundamentals influence one another in a long-run sense, these variables may cointegrate separately from the real exchange rate. Hypothesis testing for exclusion restrictions is conducted to examine this issue further.
Interpreting the cointegrating vector as the empirical analog to equation (7), the point estimate on the long-run coefficient of NFA suggests a real interest rate of about 5 percent, based on an empirical estimate of γ obtained from regressing the trade balance (as a share of GNP) on lagged cyclical fluctuations in the real exchange rate. Meanwhile, the estimated coefficient on TNT in close to unity, as expected.
This second finding may have the following economic interpretation. Broadly speaking, permanent movements in TNT involve factors that affect the relative price of nontraded goods without necessarily affecting the relative price across traded goods. Meanwhile, equilibrium changes in NFA require movements in the relative price of imports versus exports without necessarily affecting the price of nontraded goods relative to traded goods (under the proviso that wealth effects are not biased in that regard).
The coefficients of cointegration are based on the long-run (I(1)) co-movements between the series over the sample period whereas the observed values of the variables comprise both long-run movements and short-run (I(0)) noise. Hence, to represent the common stochastic trend, filtered estimates of the fundamentals are subsequently used in the cointegrating relation rather than actual values. Specifically, the permanent component is smoothed using a centered three-year moving average of the fundamentals (the estimated coefficient of error correction for the dollar real exchange rate is about 0.8).
The empirical literature on convergence documents the significant narrowing in the initial dispersion of income per capita and productivity measures across industrial countries over the postwar period until 1973. See Dowrick and Nguyen (1989) and the references therein for a review.
Correspondingly, the evidence for catch-up and convergence has been much weaker since the productivity slowdown after 1973. See again Dowrick and Nguyen (1989). Perron (1990) estimates the slowdown in income growth using a nonlinear (breaking) trend for the G-7.
Masson, Kxemers, and Horne (1994) largely attribute the sustained decline in net foreign assets in the 1980s to the overall stance of U.S. fiscal policy and the associated increase in stock of public debt.
A caveat is warranted on the interpretation of the cyclical component and the notion of disequilibrium. To the extent that the stochastic trend captures changing steady states, deviations between the actual and trend values do not necessarily represent (flow) disequilibria, as shown in the model (stock versus flow shocks). If trend movements reflect sustainable adjustment, deviations from trend do reflect transitory (flow) disequilibria. The estimation of the trend cannot further distinguish between these sources of long-run variation, without a priori information regarding equilibrium in the fundamentals. Alternatively, cyclical fluctuations can certainly be interpreted in a longer-run sense in terms of stock disequilibrium.
Data from 1950 to 1972 confirm that U.S. payments deficits were largely accommodated by an increase in dollar reserve holdings abroad. The increase in dollar liabilities during that period was approximately $60 billion, while the decline in the gold stock was about $12 billion.
This result provides empirical support for the view summarized by Krugman (1990): Arguably it was the secular decline in the equilibrium real dollar that really broke up Bretton Woods: the overvaluation of the dollar in 1971 owed little to faster U.S. inflation since 1960, and much to a decline in the real dollar compatible with international equilibrium, (p. 168)
By construction, the cyclical component is stationary, and thus has a welldefined (time-invariant) second moment. Comparing variances before and after 1973 indicates that the transitory fluctuations (obtained from the fitted trend) in the dollar real exchange rate have been 16 times more variable under floating exchange rates.
Reiterating an earlier caveat: the estimated cyclical component for the period from 1980-85 need not reflect completely a divergence from a value compatible with flow equilibrium. To the extent that the desired U.S. net foreign asset position also declined around that time, real exchange rate overshooting would in part be a necessary element of the adjustment process; a stronger currency is needed initially to induce a sufficient current account deficit to ensure a convergent path in net external assets toward its new stock equilibrium value. See also the Appendix.
Stein (1993) provides estimates of the medium-term dollar real exchange rate, which tracks its actual path much more closely. Not surprisingly, the empirical analysis there places much greater emphasis on flow measures than on stock variables.
The estimated cointegrating vector excluding PROD retains a large coefficient on NFA and an opposite sign on the coefficient for TOT: REER, = 3.Q3NFAt - 0.56TOTt. The large coefficient on net foreign assets reflects the fact that its upward trend has clearly been outpaced by the rate of real appreciation in the yen.
The estimated cointegrating vector obtained by replacing PROD with TNT in the system is given by: REER,— 1.19TNTt. The long-run coefficient on TNT is near unity as expected and smaller than the coefficient on PROD (due to faster trend growth in the latter series). Note that the above vector happened to be rejected by exclusion tests (for lag length = 4). However, that result is not robust to alternate lag-length specifications.