Multiple cointegrating vectors and structural economic models: an application to the French franc U.S. dollar exchange rate.

• Author: Dibooglu, Selahattin

1. Introduction

   Structural models of the exchange rate have performed very poorly for the industrialized nations during the post-Bretton Woods period. The time series behavior of exchange rates seems to conform to other asset prices in that volatility is large and short-term changes seem to respond primarily to the "news." Using the Engle and Granger [16] technique, studies by Baillie and Selover [4], Baillie and McMahon [5], and Kim and Enders [21] among others, provided evidence that there are no long-run relationships between bilateral nominal exchange rates and the so-called fundamentals. More recently, papers by Baillie and Pecchenino [6] and Adams and Chadha [1] used the maximum-likelihood Johansen [18] and Johansen and Juselius [19] method to provide evidence of cointegration between exchange rates and some of the fundamentals. However, none of these papers is able to validate any of the standard models of exchange rate determination.

   Our departure from the previous literature is that we develop a simple modelling strategy that is useful in the presence of multiple cointegrating vectors. In our view, a well specified economic model indicates the number of cointegrating vectors that exist among a set of variables. Moreover, the presence of multiple cointegrating vectors conveys valuable information that should not be wasted. We extend the suggestion of Bagliano, Favero, and Muscatelli [3] and Smith and Hagan [26] and interpret each cointegrating vector as a behavioral or as a reduced form equation from a structural model. The technique is illustrated using U. S./French exchange rate and money market data. In doing so, it is shown that it is not possible to reject a popular structural exchange rate determination model. We demonstrate that in the presence of multiple cointegrating vectors, the theory can guide us in "identifying" the behavioral equations. The exactly identified long-run relationships can be properly considered to be behavioral equations resulting from a structural model of exchange rate determination. Given that these equations represent long-run properties of the data, an appropriate dynamic structure can be obtained by imposing this long-run analysis on the short-run behavior of the system. To that end we use conventional innovation accounting techniques (variance decomposition and impulse response functions) based on the correctly specified error-correction representation of the system.

   In order to introduce the key variables relevant for exchange rate determination, section II reviews the two-country Dependent Economy Model as developed by Dornbusch [13; 14]. Our use of the model is designed to be illustrative in that it suggests a convenient set of forcing variables, has a simple formulation, and contains some straightforward identification restrictions. Moreover, the illustrative model is general enough to be consistent with the Dornbusch "Overshooting" Model [15] or with the Mussa Monetary Model [24]. Section III discusses our modelling strategy and shows how it can be employed in the presence of multiple cointegrating vectors. We employ the technique to estimate a structural model of the Dollar/Franc exchange rate in section IV. To preview our results, we find two distinct cointegrating vectors linking the exchange rate to money supplies, output levels, interest rates, price levels, and productivity levels. We find reasonable support for our attempt to restrict the cointegrating vectors so as to identify (1) the money market equilibrium relationship and (2) the modified Purchasing Power Parity (PPP) relationship that is consistent with the Dependent Economy Model. In section V, we estimate the appropriately specified error-correction representation (with the restricted cointegrating vectors) in order to characterize the short-run properties of the system. Using innovation accounting techniques, we show that a modest proportion of exchange rate variability is explained by the fundamentals. Conclusions, limitations of the methodology, and directions for further research are contained in section VI.

2. An Illustrative Model

   Most conventional exchange rate determination models assume money market equilibrium conditions of the form:

   [m.sub.t] = k + [p.sub.t] + [Eta][r.sub.t] - [Lambda][r.sub.t]; [Eta], [Lambda] [greater than] 0 (1)

   [Mathematical Expression Omitted],

   where m is the domestic money supply; p is the domestic price level; y is domestic income; r is the domestic interest rate; t is a time subscript; (*) denotes the foreign counterpart of domestic variables; all variables except interest rates are expressed in logarithms, and k, [Theta], and [Lambda] are constants.

   A constrained version of the money market equilibrium relationship is the conventional two-country relative version:

   [Mathematical Expression Omitted].

   Although there is no a priori reason to expect symmetry in the behavioral coefficients, this widely used specification saves considerable degrees of freedom in a dynamic model. Moreover, the specification is compatible with the "missing money" literature if money demands shift in both countries [2]. Of course, it is possible to test the restrictions implied by equation (3).

   Recent work on the demand for money suggests that equation (3) holds only as long-run equilibrium relationship.(1) Thus, we depart somewhat from the original Dornbusch formulation in that our equation (3) is not intended to represent equilibrium at a point in time; rather, it represents the long-run/cointegrating relationship in the money market.

   Monetary Models of the exchange rate, such as Mussa [24] usually link national price levels (i.e., p and [p.sup.*]) by assuming the existence of PPP. However, the questionable performance of PPP during post-Bretton Woods period suggests the use of an alternative approach followed by the Dependent Economy Model of Dornbusch [13; 14]. Let the domestic price level be a weighted average of the prices of traded goods and non-traded goods:

   [Mathematical Expression Omitted]

   where

   [Mathematical Expression Omitted]

   and [p.sup.T] is logarithm of the price of traded goods; [p.sup.N] is logarithm of the price of non-traded goods; p is logarithm of the relative price of non-traded good; and [Theta] is a share parameter. A similar relationship exists for the prices in the foreign country.

   Assuming commodity arbitrage in the traded good only:

   [Mathematical Expression Omitted]

   where s is the logarithm of the domestic currency price of foreign exchange.

   Combining equations (4) and (5) and assuming that the share parameters for the two countries are equal (i.e., [Theta] = [[Theta].sup.*]), yields:

   [Mathematical Expression Omitted].

   Equation (6) is the modified Purchasing Power Parity (PPP) relationship; as long as there are no relative price changes, relative PPP will hold. However, relative price shocks (i.e., changes in [Rho] - [[Rho].sup.*]) can cause deviations from PPP.

   An important feature of the Dependent Economy Model is that the relative price of non-traded goods is determined by productivity differentials across sectors. To best understand the relationship between productivity and relative prices, consider Figure 1. The production possibility curve between tradables and non-tradables is given AB. Let the relative price of the non-traded good be given by the slope of line CD; hence, [[Rho].sup.t] = OC/OD. Note that line EF is parallel to CD. Assuming full employment, at this particular value of [[Rho].sub.t], production takes place at point H. As shown in the Figure, consumption takes place at point K (on the indifference curve labeled I). The market for the non-traded good clears since domestic production equals domestic demand. The excess demand for the tradable good can be satisfied by importing EC = KH units of tradables.

   The depiction of a nation with a trade deficit is necessarily a short-run phenomenon; given that a nation cannot forever borrow from abroad, long-run equilibrium requires that net imports equal zero. The dynamic adjustment of the system is such that the wealth reduction associated with the trade deficit leads to a reduction in expenditure levels. Let the Engle curve be given by line OK. Hence, any decline in expenditures must be accompanied by a relative price change so as to preserve equilibrium in the market for non-tradables. Thus, long-run equilibrium must occur on the production possibilities curve somewhere along line segment JH. The point is that there is a unique long-run relative price of non-tradables for any given set of preferences and technology. As can be inferred from Figure 1, given preferences, an increase in the relative productivity of tradables will be associated with a relatively high price of the non-traded good.

   Closing the model entails (i) linking domestic and foreign interest rates (usually through uncovered interest parity), (ii) the appropriate specification of the goods market clearing relationships (e.g., sticky prices versus full market clearing), and (iii) linking spot and forward exchange rates. We elect not to make any specific assumptions regarding these issues; our intent is to construct a model that is consistent with this framework. The error-correction representation can suggest the proper specifications regarding the dynamics of the model.

3. Cointegration and Structural Models: Methodological Issues

   Johansen [18] and Johansen and Juselius [19] provide a unified approach to the estimation and testing of cointegrating relationships. The procedure is particularly useful since: (i) it uses maximum likelihood techniques; (ii) it can detect and estimate multiple cointegrating vectors; and (iii) it allows us to test restrictions on the cointegrating vector(s).

   According to Engle and Granger [16], a single cointegrating vector has a straightforward interpretation as a long-run "equilibrium" relationship. Nevertheless, the...