Abstract
THE PERIOD of generalized floating of exchange rates, which began in early 1973, has been associated with wider fluctuations of both nominal and, especially, real exchange rates than early advocates of exchange rate flexibility had anticipated (Shafer and Loopesko (1983)). There is considerable debate concerning the causes of these fluctuations, and the extent to which they have exerted unfavorable effects. Erratic fluctuations in relative prices—including real exchange rates—may be undesirable because they have unfavorable effects on economic activity.1 A different question that has arisen is whether exchange rate levels are appropriate; that is, whether they are consistent with some notion of underlying equilibrium.
THE PERIOD of generalized floating of exchange rates, which began in early 1973, has been associated with wider fluctuations of both nominal and, especially, real exchange rates than early advocates of exchange rate flexibility had anticipated (Shafer and Loopesko (1983)). There is considerable debate concerning the causes of these fluctuations, and the extent to which they have exerted unfavorable effects. Erratic fluctuations in relative prices—including real exchange rates—may be undesirable because they have unfavorable effects on economic activity.1 A different question that has arisen is whether exchange rate levels are appropriate; that is, whether they are consistent with some notion of underlying equilibrium.
Williamson (1983) has argued strongly that the real exchange rates of the largest industrial countries have been significantly and persistently out of line with what he calls “fundamental equilibrium,” and he advocates that policies be directed at narrowing these misalignments. Williamson defines the fundamental equilibrium exchange rate as “that which is expected to generate a current account surplus or deficit equal to the underlying capital flow over the cycle [in the absence of abnormal internal demand conditions or trade restrictions]” (Williamson (1983, p. 14)). There may of course be considerable uncertainty, both on the part of central banks and private investors, as to what is the appropriate value for the equilibrium exchange rate. For instance, during the period of fixed rates for the currencies of major industrial countries, it may well have been the case that incorrect assessments of underlying values led central banks to defend inappropriate exchange rates. This issue will not be discussed here, however; it will be assumed that the fundamental equilibrium rate is known by all, and hence that there is agreement about the magnitude of misalignments.
One of the sources of misalignments may be government policies themselves: for instance, a policy of monetary restraint often causes real exchange rate appreciation during the transition period while domestic prices adjust (Williamson (1983, p. 54)). Though Williamson favors subordinating monetary policy, to a greater or lesser extent, to exchange rate objectives, another instrument that has often been put forward as a possible means of limiting misalignments is sterilized intervention. However, the effectiveness of this policy is in general thought to be quite limited, as Williamson himself notes. A recent study by a working group set up at the Versailles summit also concludes that the effect of sterilized intervention on the exchange rate is likely, at best, to be very modest (Report (1983)). Nevertheless, if the authorities are unwilling to adjust monetary policy in order to achieve their exchange rate objectives, those who wish to make a case for greater exchange rate management must address the question of the effectiveness of the sterilized intervention and, more generally, of the extent to which the behavior of the economy differs depending on whether such intervention takes place.
The present paper considers the effect of sterilized intervention in a model where it is assumed that changes in monetary policy can produce misalignments. The model also assumes imperfect substitutability between domestic and foreign bonds, which implies that the changes in stocks of assets brought about by sterilized intervention have some effect on the equilibrium value of the exchange rate. In contrast to much of the theoretical work in this area, sterilized intervention is viewed in this paper not as a one-time exchange market operation, but rather as a rule that the authorities consistently follow in attempting to limit departures of the real exchange rate from its long-run equilibrium level, subject to a “reserves constraint.” This constraint requires that reserves lost by the central bank through sales of foreign exchange must eventually be replenished.
It should be made clear at the outset that several issues will not be considered in this paper. There is another rationale for intervention: to stabilize “disorderly markets.” This role is of a very short-term nature and involves the central bank giving “tone” and “breadth” to the market. On occasion there may be few active traders, leading to erratic swings in the exchange rate as the random arrival of new orders meets little demand on the other side of the market. Intervention may dampen these swings and decrease the magnitude of bid-ask spreads. We will not be concerned with this aspect of intervention; our focus will be longer term. Furthermore, it is assumed here that both the monetary authorities and private investors correctly assess the long-run equilibrium value for the exchange rate.2 In practice, of course, the problem of determining the sustainable level of a country’s real exchange rate is a formidable one; for a detailed discussion of this issue, see International Monetary Fund (1984b). Furthermore, the possible costs of either volatility or misalignment of exchange rates are not discussed; an intervention policy directed at limiting misalignments is assumed to be given, and its implications for the dynamic behavior of the economy are examined.
The now-familiar overshooting model (see Dornbusch (1976)) implies that financial shocks, such as a shift in asset preferences or a change in monetary policy, will produce a larger initial change in the exchange rate than is necessary for long-run equilibrium. Overshooting occurs in his model because prices in goods and labor markets are more sluggish than those in financial markets. A money supply shock makes the nominal exchange rate initially overshoot its long-run equilibrium level while the domestic price level changes very little. As a result, the real exchange rate diverges from its long-run equilibrium level. If the speed of adjustment of prices is very slow, this misalignment in the real exchange rate may persist for a considerable period.
The Dornbusch model cannot be applied in its original form to the questions addressed in this paper since it only provides a role for intervention if intervention changes the money supply; sterilized intervention has no effect. To examine the effects of sterilized intervention, Dornbusch’s model needs to be extended to include assets denominated in domestic and foreign currencies that are not perfect substitutes, so that their uncovered interest parity relationship is modified by a risk premium that depends on the relative supplies of the assets. Such a model is usually called the portfolio-balance model, and it has been applied a number of times to observed exchange rate movements.3 In the portfolio-balance model, sterilized intervention has a role because it alters the mix of domestic and foreign currency assets held by private investors. In effect, the central bank that carries out the sterilized intervention swaps a bond in domestic currency for one in foreign currency. As long as investors do not consider that the two bonds are perfect substitutes, the change in the relative supply of the two assets requires an exchange rate change for equilibrium to be re-established.
It is important to note at the outset that the existing empirical evidence is weak concerning the existence of significant effects of sterilized intervention on the exchange rate (see Tryon (1983)). It has not been conclusively proved that sterilized intervention is useless as a policy instrument, however, and the model for the Federal Republic of Germany reported below provides an element of support for the hypothesis that sterilized intervention can influence the value of the exchange rate.
In the Dornbusch model, as has been mentioned, overshooting occurs because goods prices adjust more slowly than asset prices. The portfolio-balance model provides an additional reason for overshooting, namely the slow accumulation of assets, in particular the claims on foreigners that are the counterpart of current account surpluses. The dynamics of exchange rate adjustment arising from this channel were developed by Kouri (1976) and Branson (1976), and a synthesis of the two types of overshooting models is attempted in Henderson (1981) and Frenkel and Rodriguez (1982). The Henderson article explicitly considers the effect of intervention in a general overshooting model; however, intervention is treated as a once-and-for-all asset swap—in particular, a purchase of foreign money with home money—rather than the ongoing dynamic process, described here, aimed at limiting departures of the real exchange rate away from its long-run equilibrium level.
We proceed by first attempting to analyze a theoretical model that is the simplest possible generalization of that in Dornbusch (1976); a risk premium variable and an intervention rule of the type discussed above are included. Subsequently, a more complicated version of that model (with an endogenous current account) is estimated for Germany. The estimation results suggest that data for the floating rate period are consistent with the hypothesis that the Deutsche Bundesbank intervenes in such a way as to resist movements of the real exchange rate away from some “normal” level of international competitiveness; furthermore, sterilized intervention seems to have a small but statistically significant effect on the value of the deutsche mark. We then proceed to simulate the model’s behavior in the face of a monetary shock in order to gauge its economic significance, and to examine its effect on the dynamic behavior of the economy.
I. The Dynamics of Sterilized Intervention
The literature on foreign exchange market intervention is now voluminous. A long series of articles has considered Friedman’s (1953) contention that intervention should also be profitable to central banks (see Taylor (1982) for a recent contribution). On the question of whether sterilized intervention has any effect at all, a task force of representatives of the seven major industrial countries (Report (1983), cited above, and various background studies) has provided new empirical evidence, as well as a discussion of the rationale for intervention. Two recent papers have also surveyed the empirical literature on the effectiveness of sterilized intervention (Genberg (1981) and Solomon (1983)); that is, whether such intervention constitutes an additional instrument, separate from overall monetary policy, that is capable of moving the exchange rate in the direction desired by the authorities. All of the studies found that existing empirical evidence indicates at most a very limited degree of effectiveness for sterilized intervention. Genberg also raises the issue of a possibly destabilizing role for sterilized intervention if it is combined with a target for holdings of international reserves (see Genberg (1981, p. 458)). This point is not amplified by Genberg, beyond a reference to similarities between this case and the case of nonsterilized intervention with a preannounced path for a monetary target, treated in an earlier paper (Genberg and Roth (1979)). The idea behind each case is that if there are other targets that force intervention to be reversed at some future date, then its effects on the exchange rate will also be reversed, at a possibly inconvenient time.
There is general agreement that nonsterilized intervention does have an effect on the exchange rate; the effect of sterilized intervention, however, is more controversial. (Henceforth in the paper, “intervention” is used as shorthand for “sterilized intervention.”) The effect of intervention is illustrated in a model that is as close as possible to that of Dornbusch (1976), in particular his model with sticky prices and output that is driven by aggregate demand. Using his notation, the model in the appendix to his paper can be written:
The variables R, y, m, p, and e are the domestic short-term interest rate, real output, money supply, prices, and the nominal exchange rate (defined as the domestic currency price of foreign currency), respectively; starred variables are foreign; dots over variables indicate rates of change, bars long-run equilibrium levels. Greek letters refer to positive parameters, and lowercase letters refer to natural logarithms of variables. The model assumes a world without trend inflation and with constant potential real output and money supply; consequently, expected inflation does not appear in equations (3) and (4) as would otherwise be appropriate. The foreign price level is normalized at unity, so it does not appear explicitly in the model. Expectations of the change in the exchange rate, x, are formed rationally; they are given by:
Rational expectations make Rational expectations make a known function of the other parameters in the model.
We proceed by first making minor extensions to this model to introduce expectations of domestic price inflation explicitly and to take into account official intervention in the foreign exchange market. Instead of equation (2), which states that uncovered interest parity holds, we introduce a risk premium, assumed to depend on the private stock of net claims on foreigners, K - F, where K is total net foreign assets (assumed to be denominated in foreign currency) and F is the net foreign claims of the central bank, or foreign exchange reserves:
Such a specification could result from a portfolio-balance relationship where domestic and foreign interest-bearing assets are not perfect substitutes in the portfolios of the private sector. In order to be induced to hold more of foreign assets, domestic residents require a higher expected return on them; alternatively, the foreign portfolio preferences could be reflected in the risk premium term. In either case, this term should be scaled by portfolio size, but for simplicity it is specified here as being linear in K and F. The theory underlying such an equation is taken as given.4 In addition, an equation explaining the endogenous intervention behavior of the authorities is introduced, where s is defined as the real exchange rate (s = e +p*—p):
Central banks are assumed to resist movements away from the long-run equilibrium real exchange rate, which we assume to be a constant,
As for the formation of expectations, for the moment Dorn-busch’s specification for the expected exchange rate is retained in terms of the expected real exchange rate:
Then, an equivalent specification written in terms of nominal exchange rates is:
For the time being, we assume that expectations of inflation are exogenous with respect to the endogenous variables of the model. For instance, expectations of inflation could be based on the rate of growth of the money supply, as in Buiter and Miller (1982).5
Expected inflation also appears in equations for both output and actual inflation:
The model of equations (1), (2a) through (5a), and (6) can be reduced to a pair of differential equations in p and F. Equations (1), (2a), and (3a) express R, y, and the real exchange rate s in terms of p, F, and exogenous variables. First note that equations (2a) and (5a) can be solved for s:
The real exchange rate may differ from its equilibrium level either because real interest rates differ at home and abroad or because private net claims on foreigners are nonzero. The LM and IS curves, equations (1) and (3a) respectively, can be solved jointly to express R and y as functions of other endogenous variables and of m andṗe (the demand shift variable u is ignored from now on):
where Δ = Φσ + λ(1 - γ)> 0.
Equations (7), (8), and (9) can be solved together to express s, R, and y as quasi-reduced-form functions of p and F and exogenous variables. Substituting the results into equations (4a) and (6) (omitting exogenous variables) yields
where Γ = Δθ + δΦ>0, with the relevant signs above each element.
The trace is negative, and it can also be shown that the determinant is positive. Therefore the Routh-Hurwicz conditions for stability are satisfied, and the model is stable whatever the value of the intervention parameter β. Furthermore, it can be shown that the two roots must be real because the discriminant is always positive, whatever the parameter values.
Given its assumptions, this simple generalization of the Dornbusch model implies that the authorities can help to guide the exchange rate toward its long-run equilibrium value without inducing short-run fluctuations in that rate. Although the model makes the simplifying assumption that both the authorities and the private sector have a correct assessment of the long-run equilibrium, it also assumes that private investors either do not anticipate the intervention behavior of the authorities or, if they do, consider it has no effect. This unrealistic feature can be modified by considering a model with intervention and fully rational expectations regarding the exchange rate.
If expectations correctly take account of intervention, the order of the system increases from second to third order. Instead of equation (5a) we would have x = ė, and equation (2a) would constitute a differential equation describing the rate of change of the exchange rate:
Using equations (8) and (9) to substitute out for R and y, the model becomes a system of three first-order differential equations (ignoring the exogenous variables):
or
where
The signs of the coefficients of the characteristic equation depend on the sign of B, which in turn depends on a number of parameters in the IS, LM, and Phillips curves. Now, a3 is the product of the characteristic roots; since Δ>0, it is clear that a3 is positive. Hence there are either three roots with positive real parts or one positive and two negative ones; in the latter case the model has the saddle-point property. In any case, it is clear that one root must be real. Coefficients a1 and a2 have ambiguous signs. If B is positive, then a1> 0, but the sign of a1 is ambiguous; if B is negative, a1 is ambiguous, but a2 is negative. In either case, however, there is only one change of sign in the coefficients of the characteristic equation, which implies, by Descartes’s rule of signs, that there must be only one positive real root. It can also be shown that the other roots, if they are complex, must have negative real parts. Hence the model has the saddle-point property: there is one unstable root, associated with the rationally expected exchange rate; the other roots are stable.
We can isolate the effect of intervention—that is, nonzero values of β—using the root locus method (Krall (1970)). If we let
and K = – βψ (abandoning an earlier notation to be consistent with KralPs), then equation (13) can be rewritten as:
When there is no systematic intervention, β = 0, and the characteristic equation can be factored to give:
Since C is positive, it is clear that the roots to the quadratic must be real, since the discriminant B2 + 4C will be positive whatever the sign of B, and that one of them will be positive and the other negative, as was discussed above. The remaining root is equal to –μ.
We can differentiate equation (14) to evaluate the effect of a nonzero value of β on the characteristic roots. Evaluated at β = 0, the effect of increasing β on any one of the three roots Z will be given by:
Consider first the pair of roots to the quadratic factor in equation (14a); that is:
If Z is positive, then clearly dZ/dβ > 0, so increasing β will tend to increase the value of the positive (unstable) root. The root is in some sense a discount factor applied to future information (Sargent (1979)); intervention can be interpreted as making the distant future less important when forming expectations. The other root to the quadratic Z is negative, and the effect of intervention on it is ambiguous. On the one hand, if |Z| is either greater or less than both μ and Πσ/Δ, so that Z + μ and Z + Πσ/Δ have the same sign, then increasing β will make Z more negative—that is, will tend to speed up the adjustment to past shocks. This is necessarily the case when μ=Πσ/Δ On the other hand, if |Z| is included in an interval bounded by μ and Πσ/Δ so that Z + μ and Z + Πσ/Δ have opposite signs, then increasing β will have the opposite effect: it will tend to slow down adjustment speed. Turning to the third root to equation (14a), Z = –μ, the effect of making β nonzero is also ambiguous because both numerator and denominator of dZ/dβ can have either sign. However, when μ is either very large or close to zero, dZ/dβ < 0.
Turning now to the other limiting case, we can use the negative root locus to discover what happens when intervention resists exchange rate movements very strongly (Krall (1970, p. 66)). As β→∞ and hence K→–∞, arbitrarily good estimates of two of the roots will be given by:
The root described by equation (15) obviously becomes unbounded in a positive direction as β→∞ while the root given by equation (16) is increasingly negative as β→∞. Thus, in the limit, intervention does not lead to cyclical behavior in this model. The possibility exists, however, of an intermediate range for p where cyclical behavior may obtain.
This analysis suggests that in a Dornbusch model modified to include a risk premium, an intervention rule to stabilize the real exchange rate is likely to help lessen overshooting, if the latter occurs, and is unlikely to induce any subsequent cyclical patterns. These results seem to favor intervention. However, it should be borne in mind that the model is unrealistically simple, especially in two respects: expectations of inflation are exogenous, and, though intervention has been endogenized, the current balance has not, despite the two variables appearing symmetrically in the exchange rate equation. Now the dynamics of the current balance, and especially its perverse short-run response to exchange rate changes, are a key factor in the economy’s dynamic behavior when subject to shocks (Levin (1983)). To account properly for the J-curve, as well as for dynamics resulting from sticky prices, a fifth- or higher-order system is called for, and this takes us beyond the bounds of analytical tract ability. A more complex, empirical model is therefore used in what follows, and is analyzed numerically.
II. Intervention in a Modified Dornbusch Model for Germany
Recent data suggest that, while the Deutsche Bundesbank may not have attempted to stabilize the real exchange rate of the deutsche mark explicitly, ex post intervention was consistent with the reaction function analyzed in the preceding section. For example, after substantial real appreciation of the deutsche mark in 1978, the Bundesbank began to purchase U.S. dollars on the foreign exchange markets more actively. Conversely, in late 1980 and early 1981, sharp nominal and real depreciation of the deutsche mark led to persistent official sales of dollars. The evidence presented below also suggests that the risk premium variable seems to be important in the case of Germany, so that intervention may have been effective in influencing the exchange rate. The intervention function and portfolio-balance equation for the exchange rate are estimated simultaneously with a small set of equations representing a macroeconomic model of the German economy. The model is estimated subject to rational expectations for the exchange rate and for the price level.
The Estimated Model
For purposes of estimation, a discrete-time system analogous to the Dornbusch model was specified. As discussed in Wickens (1984), unless additional lags are arbitrarily introduced into the price equation, the model in this form may not produce overshooting, whereas the Dornbusch model, written in continuous time, would. The reason is that in discrete time the price level can in fact jump—for instance, at the point of a monetary shock—even if it does not immediately attain its equilibrium level. Whether or not overshooting occurs in Wickens’s model depends on the amount the price level jumps: if it increases sufficiently to cause real balances to fall in response to a positive monetary shock (and, as a result, the interest rate rises), then the exchange rate undershoots—admittedly not a very plausible case. In our model the price equation, discussed below, allows for jumps because it depends contemporaneously on the exchange rate as well as on the expected value of the price level in the following period, and these expectations are formed rationally. As we shall see below, whether the exchange rate overshoots or not depends on more than just the behavior of real balances in a model where output is endogenous and where there is a J-curve phenomenon in the determination of the current balance, as is the case in our model.
Table 1 contains a list of the equations of our generalization of the Dornbusch model, written in discrete time. This model has been estimated using the full information maximum likelihood method on a sample of data for Germany for the third quarter of 1973 through the second quarter of 1982, and parameter estimates are reported in Table 2. In what follows, Δ is the first-difference operator, with Δx = x-1, and ŝ+1 indicates the expectation formed at t for the value of x in period t + 1. We use Δŝ+1 as a shorthand for ŝ+1–x.
Modified Dornbusch Model: Equations and Error Statistics, 1973-82
Critical value at the 1 percent level equals 164.7.
Modified Dornbusch Model: Equations and Error Statistics, 1973-82
| Equation | Root Mean-Square Error Statistic |
|---|---|
|
|
0.0179 |
|
|
0.0475 |
|
|
0.0111 |
|
|
0.0092 |
|
|
0.0197 |
|
|
0.0050 |
|
|
0.0243 |
|
|
0.1016 |
|
|
0.0334 |
|
|
0.0096 |
| Full model fit statistics Carter-Nagar R2 | 0.9993 |
| Log-likelihood test of overidentifying restrictions1 | 668.0 |
Critical value at the 1 percent level equals 164.7.
Modified Dornbusch Model: Equations and Error Statistics, 1973-82
| Equation | Root Mean-Square Error Statistic |
|---|---|
|
|
0.0179 |
|
|
0.0475 |
|
|
0.0111 |
|
|
0.0092 |
|
|
0.0197 |
|
|
0.0050 |
|
|
0.0243 |
|
|
0.1016 |
|
|
0.0334 |
|
|
0.0096 |
| Full model fit statistics Carter-Nagar R2 | 0.9993 |
| Log-likelihood test of overidentifying restrictions1 | 668.0 |
Critical value at the 1 percent level equals 164.7.
Modified Dornbusch Model: Parameter Estimates, 1973-82
See Note to Table 1.
Modified Dornbusch Model: Parameter Estimates, 1973-82
| Parameter Estimates (ratio to standard error in parentheses)1 | ||||||||
| ψ | 0.0521 | (2.40) | ξ | 0.0631 | (0.72) | Φ | 1.6555 | (32.04) |
| β1 | 0.2512 | (2.00) | σ | -0.3554 | (12.84) | λ | -0.2045 | (2.17) |
| μ | 0.2532 | (3.87) | δ | 0.0462 | (4.41) | β4 | 0.1853 | (4.29) |
| Π | 0.8056 | (16.93) | β3 | 0.3582 | (7.59) | β5 | 0.0832 | (1.01) |
| Imposed Coefficients | ||||||||
| a0 | 1.23 | c 1 | 0.00604 | β9 | 0.8660 | |||
| b0 | 2.23 | β6 | 0.1623 | β10 | 0.1430 | |||
| η | 0.30 | β7 | 0.3071 | β11 | 1.5960 | |||
| C o | 0.02052 | β8 | 0.4316 | β12 | 1.8780 | |||
See Note to Table 1.
Modified Dornbusch Model: Parameter Estimates, 1973-82
| Parameter Estimates (ratio to standard error in parentheses)1 | ||||||||
| ψ | 0.0521 | (2.40) | ξ | 0.0631 | (0.72) | Φ | 1.6555 | (32.04) |
| β1 | 0.2512 | (2.00) | σ | -0.3554 | (12.84) | λ | -0.2045 | (2.17) |
| μ | 0.2532 | (3.87) | δ | 0.0462 | (4.41) | β4 | 0.1853 | (4.29) |
| Π | 0.8056 | (16.93) | β3 | 0.3582 | (7.59) | β5 | 0.0832 | (1.01) |
| Imposed Coefficients | ||||||||
| a0 | 1.23 | c 1 | 0.00604 | β9 | 0.8660 | |||
| b0 | 2.23 | β6 | 0.1623 | β10 | 0.1430 | |||
| η | 0.30 | β7 | 0.3071 | β11 | 1.5960 | |||
| C o | 0.02052 | β8 | 0.4316 | β12 | 1.8780 | |||
See Note to Table 1.
Equation (C1) in Table 1 is the interest parity condition augmented by a risk premium; the expected (log) change in the exchange rate was multiplied by 4 to convert to annual rates for comparability with interest rates, and then both sides of the equation were divided by 4. The exchange rate is a trade-weighted effective rate; the foreign interest rate is taken to be the Eurodollar rate. The risk premium on foreign currency assets, ψ(a0k—b0f—c0t), is assumed to be proportional to net foreign assets (the current account, cumulated from a benchmark figure, minus foreign exchange reserves) minus a time trend that proxies the growth in wealth, where, as before, lowercase letters denote variables in natural logarithms. This term is a log-linear approximation to (K - F)/W, where K, F, and W are the levels of the cumulated current account, foreign exchange reserves, and wealth, respectively. The coefficients a0 and b0 are constants used in the log-linearization of net foreign assets through a Taylor series expansion about sample means. Coefficient c0 is also imposed; it was estimated as the trend growth in money over the sample period, roughly 8 percent at an annual rate. The constant term in the equation captures the difference in scale of money and wealth.
The intervention function (C2) is specified such that movements away from the equilibrium level of the real exchange rate are resisted, but that intervention is constrained by the level of foreign exchange reserves. Excessive declines or increases in reserves are symptomatic of exchange rate pressure and may lead to destabilizing expectations. Moreover, when reserves are zero, intervention would cease to be possible while, from the point of view of the economy as a whole, very high levels of reserves may be associated with large forgone earnings on alternative investments. It is assumed that there is some target reserve level
The price equation (C3) combines elements of price stickiness and forward-looking expectations. Variable p is a value-added deflator, the largest component of which is wages. Average wages are assumed to contain an element of inertia because of emulation and overlapping contracts, for instance; it is also assumed that they anticipate future developments of consumer prices. The value-added deflator is therefore modeled in the following way:
Since domestic residents consume both domestic and foreign goods, the consumption deflator pc can be written as follows:
on the assumption that there is a one-quarter lag between the production of foreign goods and their consumption domestically, and that the deflator is a fixed-weight geometric average of the domestic and foreign components. It is further assumed that expectations of consumer prices are formed rationally; combining equations (17) and (18) yields equation (C3). It is clear that the price level in this formulation is not predetermined (unless Π = 1) because it is affected by contemporaneous movements in the exchange rate as well as by anticipation of future movements in p itself. There is an element of sluggishness to the response of price movements that increases with increases in the value of Π.
The standard lagged adjustment formulation for the demand for real money balances is given in equation (C5), while the domestic short-term interest rate is explained by a policy reaction function in equation (C6). The latter assumes that interest rates are adjusted both to resist movements in the differential with rates prevailing abroad and to limit deviations from monetary targets, here proxied by a uniform trend over the sample period. Monetary settings may be adjusted to external factors so as to avoid potential pressures on the exchange rate brought on by large interest differentials vis-à-vis other countries; if so, monetary aggregate targets may not be hit exactly. It should be noted that in this form the model relaxes the assumption made by Dornbusch that interest rates move instantaneously to equate money demand with exogenous money supply. Here interest rates respond to other factors, including external ones, and some monetary accommodation on the part of the authorities will tend to limit the degree of overshooting, as has been discussed by Papell (1983). Such behavior seems to characterize the historical period more accurately than strict monetary targeting. The simulations performed below will, however, replace this reaction function by the Dornbusch assumption that interest rates are set solely to achieve monetary targets, based on the equilibrium demand for money function. This is discussed more fully below.
Equations (C7) and (C8) represent the determination of the cumulated current balance. The coefficients of these equations are imposed on the small macroeconomic model, being derived through partial simulations of a much larger model of the German economy.6 The variable Q is a synthetic competitiveness variable which is a function of current and lagged values of the real exchange rate; its second order lag structure incorporates the J-curve effect implicit in the larger model. The rest of the model is estimated subject to the prior restrictions imposed on the trade submodel. Given the highly aggregated nature of the cumulated current balance, the coefficients necessarily had to be derived (in the manner described) from the considerably more disaggregated model.
In treating next period’s expected exchange rate ŝ+1 and price level
As is evident from the above description, the model is quite parsimonious. The aim was to keep it as small as possible, and to keep it linear without sacrificing too much realism or explanatory power. Some experimentation was necessary concerning the lagging of variables—for instance, output is lagged in the price equation—and the form of reaction functions. Rather than including unrestricted time trends, we fitted trends to the money supply and to real output over the sample: the estimated slope coefficients, c0 and c1, respectively, were imposed in the estimation of the full model. It proved impossible to estimate sensible parameters for the price equation when all were unrestricted. We therefore imposed a value for η, 0.3. This parameter measures the purely domestic cost influences on consumer prices; while somewhat implausibly low, it gave a higher likelihood for the full model than values of η closer to unity.
Table 2 presents estimates of the model’s parameters obtained by the full information maximum likelihood method.7 The estimates have the expected signs, and most are significantly different from zero. The estimate of ψ in equation (C1) tends to support the hypothesis that there is a risk premium related to net foreign asset stocks. The intervention equation, (C2), seems to correspond to Bundesbank behavior over the sample period; a more general equation (not reported) also included the change in the nominal exchange rate as an explanatory variable, consistent with the hypothesis that the authorities attempt to smooth short-run fluctuations. Such behavior does not show up in our sample of quarterly data, however, because the parameter estimate was insignificantly different from zero.8 The estimated price equation, (C3), implies considerable inertia, since Π is closer to unity than to zero; it is, however, significantly different from both polar cases and does include a forward-looking element. The effect of excess demand is positive, as expected, but not statistically significant; such was also the case when current, rather than lagged, excess demand was included. The output equation, (C4), was originally estimated to allow lagged adjustment. However, the speed of adjustment parameter was insignificantly different from unity, and the equation was simplified to its current form. It implies a strong negative effect of the real interest rate on economic activity and a significant positive effect of the real exchange rate, both operating within the current quarter. Such a strong contemporaneous effect seems somewhat implausible. The money demand estimates are conventional, and resemble those reported for Germany in Atkinson and others (1984). Finally, the interest rate reaction function embodies the effects of foreign rates and of domestic money targets, but the parameter capturing the latter is not well determined.
Clearly, of primary interest here is the possible effect of intervention on the exchange rate. The risk premium parameter, Ψ, is of the correct sign and is significantly different from zero at the 1 percent level. It suggests that a 1 percent change in the cumulated current account will lead to a 0.05 percent change in the spot exchange rate, other factors given. Evaluated at the mean level of reserves for 1983, this would imply that a once-and-for-all DM 1 billion purchase of foreign currency assets by the Bundesbank would lead to an immediate depreciation of roughly 0.08 percent in the deutsche mark’s effective exchange rate. As a standard of comparison, two previous studies imply that (sterilized) intervention of a similar size would cause a depreciation of 0.07 percent (Branson, Halttunen, and Masson (1977)) and 0.003 percent (Obstfeld (1983)).9 The small size of this coefficient seems, on the face of it, to be consistent with the negligible effect found by many previous studies (see also Tryon (1983)). That the risk premium parameter is very small does not necessarily imply that the impact of intervention will be small when the model is solved under rational expectations. It should be stressed that, in principle, agents solve for the entire future path of the economy under rational expectations, so knowledge that the authorities will intervene to limit real exchange rate movements in the future may bring about substantial changes in current variables relative to a situation where the authorities do not resort to such intervention. It is hard to judge on the basis of this one parameter estimate what will be the influence of an intervention rule, as opposed to the effect of a one-time intervention.
In order to examine its effect, the model was simulated with and without official intervention in the foreign exchange market on the assumption that market participants know the structure of the model and the future values of the exogenous variables. In the absence of uncertainty about parameters and about exogenous variables, our simulations therefore involve calculating perfect-foresight solutions to the model. We do so using the algorithm of Blanchard and Kahn (1980).10
In Dornbusch’s overshooting model the monetary authorities are assumed to target the money supply strictly, and interest rates adjust to equate money demand with the exogenous supply. The historical data imply that interest rates have not been determined in this way; in practice the authorities also attempt to achieve other objectives, and the money supply is to some extent endogenous, with interest rate fluctuations limited by central bank accommodation. It is for this reason that it proved necessary to estimate the interest rate as a policy reaction function with the money supply adjusting partially to demand.
For the purposes of subsequent policy analysis, however, the interest rate reaction function is suppressed, and the estimated money demand equation, in its equilibrium form, is inverted to determine the interest rate. In other words, lags are eliminated from the demand for money equation, and the long-run demand for money function is used. While this may seem to be an arbitrary procedure, the justification for it is that if money were strictly controlled, it is doubtful that interest rates would exhibit the dynamic behavior implied by the inverted money demand function. Laidler (1982) has argued that estimated lags in short-run “money demand equations” do not reflect lags in adjustment of demand so much as the stickiness of prices, captured elsewhere in our model; they may also result from the money supply process (Gordon (1984)).
The dynamic properties of the model can conveniently be examined by calculating its characteristic roots, since the model is linear. A conventional model in difference-equation form will be stable provided all roots have modulus less than unity; a perfect-foresight model will be stable provided there are only as many characteristic roots with modulus greater than unity as there are nonpredetermined variables—in this case two (the exchange rate and the price level), since transversality conditions are imposed on these variables to prevent them from exhibiting explosive behavior. Complex roots are evidence that some variables will respond in cyclical fashion when the model is subjected to shocks. The characteristic roots of the model with the modifications described above, both with and without the estimated reaction function for intervention, are presented in Table 3. As expected, there are two unstable roots corresponding to the rationally expected exchange rate and domestic price level. Intervention increases the size of one of these roots, as was the case for the theoretical model developed in the first section. The other roots have modulus less than unity, but there are complex roots, indicating cyclical behavior. We will see below that a monetary shock does in fact induce quite long and pronounced cycles. Interestingly enough, intervention adds a pair of complex roots, suggesting that it may itself be the source of cyclical fluctuations in response to shocks.
Characteristic Roots of the Estimated Model with the Long-Run Money Demand Function Renormalized on the Interest Rate
Characteristic Roots of the Estimated Model with the Long-Run Money Demand Function Renormalized on the Interest Rate
| Without Intervention | With Intervention | ||
|---|---|---|---|
| Root | Modulus | Root | Modulus |
| 14.018 | 14.018 | 14.018 | 14.018 |
| 1.194 | 1.194 | 1.211 | 1.211 |
| 0.984 ± 0.056i | 0.986 | 0.963 ± 0.059i | 0.965 |
| 0.838 | 0.838 | 0.910 | 0.910 |
| 0.747 | 0.747 | 0.689 ± 0.013i | 0.689 |
| 0.670 | 0.670 | 0.221 | 0.221 |
| 0.221 | 0.221 | ||
Characteristic Roots of the Estimated Model with the Long-Run Money Demand Function Renormalized on the Interest Rate
| Without Intervention | With Intervention | ||
|---|---|---|---|
| Root | Modulus | Root | Modulus |
| 14.018 | 14.018 | 14.018 | 14.018 |
| 1.194 | 1.194 | 1.211 | 1.211 |
| 0.984 ± 0.056i | 0.986 | 0.963 ± 0.059i | 0.965 |
| 0.838 | 0.838 | 0.910 | 0.910 |
| 0.747 | 0.747 | 0.689 ± 0.013i | 0.689 |
| 0.670 | 0.670 | 0.221 | 0.221 |
| 0.221 | 0.221 | ||
The results of simulating the model when it is subjected to a domestic money supply shock are presented in Table 4.11 The nominal exchange rate does not overshoot its equilibrium value, either with or without intervention. In contrast to Wickens (1984), this is not the result of real balances falling in response to a positive shock to nominal balances but is due to some rise in the domestic price level and a substantial rise in output, combined with a large income elasticity of money demand (see Table 2). As noted above, the size of the contemporaneous output effect is somewhat implausible; however, estimation results rejected the hypothesis of lagged adjustment of output to the real interest rate and real exchange rate. The subsequent behavior of output and prices is not that of monotonic adjustment toward their equilibrium levels: there are strong cycles, and the exchange rate appreciates for a time before depreciating once again. The price level moves fairly steadily upward, but it exceeds its equilibrium level after three years; in consequence, the real exchange rate, which had depreciated by over 8 percent initially, has appreciated relative to baseline by almost that amount after five years. The cumulated current account balance is strongly cyclical, as expected given the J-curve.
Simulated Effects of a Sustained 10 Percent Increase in the German Money Supply
(Percentage deviations from baseline)
Simulated Effects of a Sustained 10 Percent Increase in the German Money Supply
(Percentage deviations from baseline)
| Time Period (quarter) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Variable | 0 | 1 | 2 | 3 | 4 | 7 | 11 | 15 | 19 | ∞ |
| Without Intervention | ||||||||||
| e | 9.87 | 8.27 | 6.87 | 5.68 | 4.68 | 2.69 | 1.81 | 2.22 | 3.39 | 10.00 |
| f | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| p | 1.42 | 2.69 | 3.81 | 4.81 | 5.69 | 7.76 | 9.56 | 10.71 | 11.45 | 10.00 |
| e +p*-p | 8.45 | 5.58 | 3.06 | 0.87 | -1.01 | -5.07 | -7.75 | -8.49 | -8.06 | 0.00 |
| y | 4.41 | 3.76 | 3.18 | 2.66 | 2.21 | 1.14 | 0.23 | -0.34 | -0.70 | 0.00 |
| R | -6.25 | -5.34 | -4.55 | -3.85 | -3.23 | -1.73 | -0.30 | 0.72 | 1.46 | 0.00 |
| k | -2.68 | -3.80 | -3.48 | -2.33 | -0.83 | 3.27 | 5.15 | 3.52 | 0.06 | 0.00 |
| With Intervention | ||||||||||
| e | 9.82 | 8.28 | 6.99 | 5.92 | 5.05 | 3.46 | 3.05 | 3.84 | 5.25 | 10.00 |
| f | -2.11 | -2.98 | -3.02 | -2.52 | -1.70 | 1.44 | 4.92 | 6.68 | 6.99 | 0.00 |
| p | 1.41 | 2.68 | 3.83 | 4.86 | 5.79 | 8.07 | 10.14 | 11.41 | 12.09 | 10.00 |
| e +p*-p | 8.41 | 5.60 | 3.16 | 1.06 | -0.74 | -4.61 | -7.09 | -7.57 | -6.84 | 0.00 |
| y | 4.42 | 3.77 | 3.18 | 2.66 | 2.18 | 1.04 | 0.00 | -0.01 | -0.01 | 0.00 |
| R | -6.24 | -5.28 | -4.41 | -3.61 | -2.89 | 1.08 | 0.63 | 1.69 | 2.26 | 0.00 |
| k | -2.67 | -3.80 | -3.50 | -2.35 | -0.83 | 3.49 | 6.02 | 5.20 | 2.51 | 0.00 |
Simulated Effects of a Sustained 10 Percent Increase in the German Money Supply
(Percentage deviations from baseline)
| Time Period (quarter) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Variable | 0 | 1 | 2 | 3 | 4 | 7 | 11 | 15 | 19 | ∞ |
| Without Intervention | ||||||||||
| e | 9.87 | 8.27 | 6.87 | 5.68 | 4.68 | 2.69 | 1.81 | 2.22 | 3.39 | 10.00 |
| f | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| p | 1.42 | 2.69 | 3.81 | 4.81 | 5.69 | 7.76 | 9.56 | 10.71 | 11.45 | 10.00 |
| e +p*-p | 8.45 | 5.58 | 3.06 | 0.87 | -1.01 | -5.07 | -7.75 | -8.49 | -8.06 | 0.00 |
| y | 4.41 | 3.76 | 3.18 | 2.66 | 2.21 | 1.14 | 0.23 | -0.34 | -0.70 | 0.00 |
| R | -6.25 | -5.34 | -4.55 | -3.85 | -3.23 | -1.73 | -0.30 | 0.72 | 1.46 | 0.00 |
| k | -2.68 | -3.80 | -3.48 | -2.33 | -0.83 | 3.27 | 5.15 | 3.52 | 0.06 | 0.00 |
| With Intervention | ||||||||||
| e | 9.82 | 8.28 | 6.99 | 5.92 | 5.05 | 3.46 | 3.05 | 3.84 | 5.25 | 10.00 |
| f | -2.11 | -2.98 | -3.02 | -2.52 | -1.70 | 1.44 | 4.92 | 6.68 | 6.99 | 0.00 |
| p | 1.41 | 2.68 | 3.83 | 4.86 | 5.79 | 8.07 | 10.14 | 11.41 | 12.09 | 10.00 |
| e +p*-p | 8.41 | 5.60 | 3.16 | 1.06 | -0.74 | -4.61 | -7.09 | -7.57 | -6.84 | 0.00 |
| y | 4.42 | 3.77 | 3.18 | 2.66 | 2.18 | 1.04 | 0.00 | -0.01 | -0.01 | 0.00 |
| R | -6.24 | -5.28 | -4.41 | -3.61 | -2.89 | 1.08 | 0.63 | 1.69 | 2.26 | 0.00 |
| k | -2.67 | -3.80 | -3.50 | -2.35 | -0.83 | 3.49 | 6.02 | 5.20 | 2.51 | 0.00 |
The effect of intervention is small, especially over the initial few periods. Even though the Bundesbank intervenes by an amount exceeding 2 percent of its reserves in the initial quarter (over $1 billion), the effect on the exchange rate is only 0.05 percent. Differences widen as the simulation progresses, and after 20 quarters the exchange rate in both nominal and real terms is almost 2 percent higher (and hence closer to its equilibrium) than in the absence of intervention. Note that at this point reserves are some 7 percent higher than in the baseline, and that other variables—the price level, the nominal interest rate, and the cumulated current account—art farther from their equilibrium values than in the absence of intervention. It is also the case that the amplitude of the cyclical swings in the cumulated current account is larger for the simulation where the central bank intervenes.
The qualitative response of the model to the monetary shock will in fact depend on the values taken on by the parameters. Overshooting is more likely the less output and prices respond in the first instance, and the less sensitive is money demand to their movements. Conversely, the more interest elastic is money demand, the less interest rates must move to equate money demand and money supply, and hence the less period-to-period movement in the exchange rate there will be. The presence or absence of cycles will also depend on the configuration of parameter values.
Though our results are specific to a particular set of parameter values—in most cases estimated on the basis of historical data—they are suggestive of the following broader conclusions. First, the possibility of significantly limiting deviations of real exchange rates from their equilibrium levels—even assuming that these are known—seems extremely limited. Though our estimation results show a statistically significant effect for intervention on the exchange rate, the economic significance of the result seems small. Evidence presented in other papers cited also gives very little support for the effectiveness of intervention. Second, to the extent that intervention does have an effect, resisting movements in the real exchange rate may have perverse effects on other variables, slowing down their adjustment toward equilibrium (see also Frenkel (1983)). The simulation results of a positive monetary shock imply that the authorities, by intervening to resist real exchange rate misalignments, would have caused interest rates to be higher after 20 quarters than rates would otherwise have been, and the level of net foreign claims to be higher than its equilibrium value, implying subsequent current account deficits.
More generally, whatever the reason for the less than instantaneous adjustment in the economy—price stickiness, slow adjustment of trade volumes, gestation lags for investment—the dynamic effects of an intervention rule will be very complex, and may have unintended consequences for other variables. An evaluation of the consequences for economic welfare must go beyond just considering the costs of a misaligned real exchange rate. Even if (sterilized) intervention is an effective policy instrument independent of monetary policy itself, using it still incurs costs as well as benefits. Inhibiting movements in one variable has repercussions for the rest of the economy that must be evaluated from a general welfare perspective. Furthermore, the simulation results suggest that an intervention rule such as the one specified may be the source of cyclical fluctuations, since the characteristic roots in this case include an extra complex pair. If there are costs to such fluctuations, then they must be set against the costs of the misalignments that are being reduced by the intervention.
III. Conclusions
In this paper, the Dornbusch model (1976) was extended to include a role for asset supplies through a risk premium variable, and the dynamics of an intervention rule whereby the authorities attempt to resist movements in the real exchange rate were analyzed under assumptions of both regressive and rational exchange rate expectations. It was shown that the generalized model was stable under regressive expectations, whether or not the authorities intervened, and that the intervention rule did not itself generate cycles. Similarly, the model was shown to have the saddle-point property under rational expectations and, provided that intervention was sufficiently strong, intervention would not lead to cyclical fluctuations. These results tended to provide analytical support for the view that attempts to limit overshooting—through intervention rules to stabilize the real exchange rate—may be helpful.
To examine these results empirically, a more complete macro-economic model was estimated for Germany under the assumption of rational expectations for both the exchange rate and for the gross domestic product (GDP) deflator. The risk premium parameter, through which intervention may have an impact on the exchange rate, was shown to be small but statistically significant by the usual criteria. The estimation results also suggest that, during the first decade of generalized floating, the intervention behavior of the Bundesbank, while possibly motivated by other concerns, has been consistent with resistance to real exchange rate movements. This model was then solved under the perfect-foresight assumption for a 10 percent permanent increase in the domestic money supply.
Instead of overshooting of the nominal exchange rate in response to a domestic monetary shock, the model produced a smaller nominal depreciation in the short run than in the long run, but the real exchange rate did depreciate substantially as prices took time to adjust. The intervention rule did tend to limit the deviation of the real exchange rate from its equilibrium level—which was unaffected by the monetary shock. However, the effects were small and were initially negligible. If the purpose of intervention is to limit nominal exchange rate overshooting, the model simulations provide little justification for its use. Furthermore, in the medium term the intervention rule produced adverse side effects, in particular slower adjustment of the current account balance and an increased tendency for cyclical fluctuations. The greater complexity of the empirical model, including a J-curve phenomenon on current balances, would seem to explain why there was a tendency toward cyclical behavior even though it did not exist in the simplest analytical examination of intervention.
It is important to underline the limitations of the analysis. A particularly simple strategy for intervention was examined—though one that has been advocated. If intervention does indeed have some identifiable effect, however small, other more complicated feedback rules—perhaps involving much larger intervention operations—might have clearly beneficial effects. Put somewhat differently, in a deterministic context, if there is an additional independent instrument, then its use will in general help to attain a higher value for the objective function that policymakers maximize. Optimal feedback rules were not derived, however, because such a deterministic setting is clearly not appropriate for that purpose. In particular, there was no attempt to model how behavior of individuals might change in response to changing uncertainty. In this paper the intervention rule is only allowed to affect the expectations of agents because the private sector is assumed to anticipate correctly the authorities’ actions. However, it may be a deliberate part of an intervention strategy to change the degree of uncertainty concerning exchange rate fluctuations: either by limiting transitory fluctuations and hence providing a more stable planning environment, or by adding an erratic element to exchange rate movements, to discourage speculation.
APPENDIX: Data Definitions and Sources
All data, unless otherwise stated, are for the Federal Republic of Germany and are taken directly from the magnetic tape corresponding to the series shown in the Monthly Report of the Deutsche Bundesbank.
Endogenous Variables
E Effective exchange rate against 23 trading partners
F Net external assets of the Deutsche Bundesbank, in billions of deutsche mark
P GDP deflator
Y Real GDP
M M3
R Three-month interbank interest rate
K Cumulated current account, in deutsche mark, calculated by summing current account flows from a benchmark figure
Q Synthetic competitiveness variable, calculated from data for the real exchange rate and parameters derived from the INTERLINK model of the Organization for Economic Cooperation and Development (OECD).
Exogenous Variables
R* Three-month Eurodollar deposit rate
P* Import price index in foreign currency terms
y* Real GDP of the seven major OECD economies excluding Germany (source: OECD, Main Economic Indicators)
t Time trend (equals 0 in third quarter of 1973, 1 in fourth quarter of 1973, and so on).
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Mr. Blundell-Wignall is employed at the Economic Planning Advisory Council in Canberra, Australia; he holds a Ph.D. from Cambridge University.
Mr. Masson, an economist in the Research Department, received his Ph.D. from the London School of Economics and Political Science. This work was begun when the authors were employed by the Organization for Economic Cooperation and Development (OECD), Paris.
However, a recent study by the International Monetary Fund (1984a) reviews exchange rate variability and international trade and concludes that there is little evidence of adverse effects.
Quite a different approach is used in Boughton (1984), where it is assumed that speculators do not have a firm notion of long-run equilibrium. Instead, they are assumed to readjust their assessments of the equilibrium exchange rate in line with observed exchange rate movements.
For an early survey, see Isard (1978). See also Branson and Halttunen (1979); Branson, Halttunen, and Masson (1977); Frankel (1982); Hooper and Morton (1982); Obstfeld (1983); and Blundell-Wignall (1984).
A mean-variance model with a similar property is described in Dornbusch (1983). The adjustments necessary to go from a model framed in terms of assets denominated in different currencies to a model using balance of payments data are discussed in the appendix to Hooper and others (1983).
In the empirical work reported in Section II, we generalize the model in a number of ways. In particular, it is assumed that both exchange rate and inflation expectations are formed rationally, conditional on the same information set.
The current account equation is derived by simulation of the OECD INTERLINK model. See OECD (1984). Variables for domestic income, foreign income, and the real exchange rate were successively shocked, holding other variables constant. The resulting values for the current account were compared with their baseline values, and the difference between them regressed on the shocks applied to the exogenous variables, with lags where appropriate; see Masson and Richardson (1985).
Using the RESIMUL program developed by C.R. Wymer (1977). Estimates of the parameters for expectations of the exchange rate and of the price level are not reported, nor are intercept coefficients, which are, however, included in all equations. Ratios of the parameter estimates to their standard errors, reported in Table 2, are asymptotically normally distributed.
Artus (1976) reports estimates of a similar equation over a shorter sample period of monthly data (April 1973-July 1975) in which both the deviation of the deutsche mark from its purchasing power parity level and the rate of change of the nominal exchange rate have significant coefficients.
These two studies model the bilateral dollar-deutsche-mark rate, and not the effective rate as is the case here. The effect quoted above for Obstfeld was obtained by scaling the number quoted in Tryon (1983) by the reserve money stock for 1983; it includes other model feedbacks and is not directly comparable, however, to our parameter ψ.
The exchange rate and the price level are the only truly “forward-looking” variables; however, the other state variables X are not predetermined in the sense that Buiter (1982) uses the concept, because they can jump in response to news available at t. We avoided this problem by creating a vector of new state variables X1 composed of the lagged values of the variables X. The elements of X1 are predetermined, and the model can then be written in the Blanchard-Kahn notation, with initial conditions imposed on X1 at time 0 and transversality conditions imposed on e and p such that they do not exhibit explosive behavior.
The effect of a foreign monetary shock is perhaps more interesting, because in this case the open market operation and the intervention operation are performed by different monetary authorities. Since our model does not contain equations for foreign income and money demand, this experiment could not be made, but its results should be qualitatively similar to the one performed but reversed in sign.