## T FoVt tj FoVtj Cj etj Cj FoVtj FjVtj

13They are only identifying restrictions, however, and cannot be tested.

14Cointegration is discussed in Chapter2.6.

To summarize

Given the Cj, which you get from unrestricted VAR accounting, (8.47) says you only need to determine Fo after which the remaining Fj follow.

In our 3-dimensional system, Fo is a 3 X 3 matrix with 9 unique elements. To identify Fo, you need 9 pieces of information. Start with, E = G'G = E(etet) = FoE(vtvt)Fo = FoFo where G is the unique upper triangular Choleski decomposition of the error covariance matrix E. To summarize

Let gjj be the ijth element of G and /j be the ijth element of Fo. Writing (8.48) out gives g2i = /i2i,o + /i22,o + /i23,o, (8.49)

gii gi2 = /ii,o/2i,o + /i2,o/22,o + /i3,o/23,o, (8.50)

gii gi3 = /ii,o/3i,o + /i2,o/32,o + /i3,o/33,o, (8.51)

gi2 gi3 + g22g23 = /2i,o/3i,o + /22,o/32,o + /23,o/33,o, (8.53)

G has 6 unique elements so this decomposition gives you 6 equations in 9 unknowns. You still need three additional pieces of information. Get them from the long-run predictions of the theory.

Stochastic Mundell-Fleming predicts that neither demand shocks nor monetary shocks have a long-run effect on output which we represent by setting /i2(1) = 0 and /i3 (1) = 0, where /j (1) is the ijth element of F(1) = Xj=o Fj. The model also predicts that money has no long-run effect on the real exchange rate /33(1) = 0. Since F(1) = C(1)Fo, impose these three restrictions by setting

/i3 (1) = 0 = Cii(1)/i3,o + Ci2(1)/23,o + Ci3(1)/33,o, (8.55) /i2 (1) = 0 = Cii(1)/i2,o + Ci2(1)/22,o + Ci3(1)/32,o, (8.56) /33 (1) = 1 = C3i(1)/i3,o + C32(1)/23,o + C33(1)/33,o. (8.57)

(8.49)-(8.57) form a system of 9 equations in 9 unknowns and implicitly define Fo. Once the Fj are obtained, you can do impulse response analyses and forecast error variance decompositions using the 'structural' response matrices Fj.

 1 month Supply Demand Money 36 months Supply Demand Money Britain Germany Japan 0.378 0.240 0.382 0.016 0.234 0.750 0.872 0.011 0.117 0.331 0.211 0.458 0.066 0.099 0.835 0.810 0.071 0.119

Clarida and Gali estimate a structural VAR using quarterly data from 1973.3 to 1992.4 for the US, Germany, Japan, and Canada Their impulse response analysis revealed that following a one-standard deviation nominal shock, the real exchange rate displayed a hump shape, initially depreciating then subsequently appreciating. Real exchange rate dynamics were found to display delayed overshooting.

We'll re-estimate the structural VAR using 4 lags and monthly data for the US, UK, Germany, and Japan from 1976.1 through 1997.4. The structural impulse response dynamics of the levels of the variables are displayed in Figure 8.9. As predicted by the theory, supply shocks lead to a permanent real deprecation and demand shocks lead to a permanent real appreciation. The US-UK real exchange rate does not exhibit delayed overshooting in response to monetary shocks. The real dollar-pound rate initially appreciates then subsequently depreciates following a positive monetary shock. The real dollar-deutschemark rate displays overshooting by first depreciating and then subsequently appreciating. The real dollar-yen displays Dornbusch-style overshooting. Money shocks are found to contribute a large fraction of the forecast error variance both the long run as well as at the short run for the real exchange rate. The decompositions at the 1-month and 36-month forecast horizons are reported in Table 8.1   Figure 8.9: Structural impulse response of log real exchange rate to supply, demand, and money shocks. Row 1: US-UK, row 2: US-Germany, row 3: US-Japan.

Mundell-Fleming Models Summary

1. The hallmark of Mundell-Fleming models is that they assume that goods prices are sticky. Many people think of MundellFleming models synonymously with sticky-price models. Because there exist nominal rigidities, these models invite an assessment of monetary (and fiscal) policy interventions under both fixed and flexible exchange rates. The models also provide predictions regarding the international transmission of domestic shocks and co-movements of macroeconomic variables at home and abroad.

2. The Dornbusch version of the model exploits the slow adjustment in the goods market combined with the instantaneous adjustment in the asset markets to explain why the exchange rate, which is the relative price of two monies (assets), may exhibit more volatility than the fundamentals in a deterministic and perfect foresight environment. Explaining the excess volatility of the exchange rate is a recurring theme in international macroeconomics.

3. The dynamic stochastic version of the model is amenable to empirical analysis. The model provides a useful guide for doing unrestricted and structural VAR analysis.

Appendix: Solving the Dornbusch Model

From (8.9) and (8.11), we see that the behavior of i(t) is completely determined by that of s(t). This means that we need only determine the differential equations governing the exchange rate and the price level to obtain a complete characterization of the system's dynamics.

Substitute (8.9) and (8.11) into (8.6). Make use of (8.13) and rearrange to obtain

To obtain the differential equation for the price level, begin by substituting (8.58) into (8.9), and then substituting the result into (8.8) to get ^(144)

P(t) = n[6(s(t) — p(t)) + (y — 1)y — ^z* — J (p(t) — p) + gi- (8.59)

However, in the long run

0 = n[5(s — p) + (y — 1)y — ar * + g], the price dynamics are more conveniently characterized by p(t) = n

which is obtained by subtracting (8.60) from (8.59). Now write (8.58) and (8.61) as the system where

(8.62) is a system of two linear homogeneous differential equations. We know that the solutions to these systems take the form s(t) = s + ae0i, (8.63)

We will next substitute (8.63) and (8.64) into (8.62) and solve for the unknown coefficients, a, 0, and 0. First, taking time derivatives of (8.63) and (8.64) yields

Substitution of (8.65) and (8.66) into (8.62) yields

In order for (8.67) to have a solution other than the trivial one (a, P) = (0, 0), requires that

(146)^ where Tr(A) = -n(<5 + ct/A) and |A| = -n£/A otherwise, (A - 0I2)-1 exists which means that the unique solution is the trivial one, which isn't very interesting. Imposing the restriction that (8.69) is true, we find that its roots are

The general solution is s(t) = s + a1e0li + a2e02i, (8.72)

This solution is explosive, however, because of the eventual dominance of the positive root. We can view an explosive solution as a bubble, in which the exchange rate and the price level diverges from values of the economic fundamentals. While there are no restrictions within the model to rule out explosive solutions, we will simply assume that the economy follows the stable solution by setting a2 = ^2 = 0, and study the solution with the stable root

Now, to find the stable solution, we solve (8.67) with the stable root 0 = (A - 0i/2)(

When this is multiplied out, you get

Because a is proportional to P, we need to impose a normalization. Let this normalization be P = po — p where po = p(0). Then a = (po — p>)/0i A = — [po — p]/0A, where 0 = —0i. Using these values of a and P in (8.63) and (8.64), yields

where (so — s) = — [po — p]/0A. This solution gives the time paths for the price level and the exchange rate.

To characterize the system and its response to monetary shocks, we will want to phase diagram the system. Going back to (8.58) and (8.61), we see that s(t) = 0 if and only if p(t) = p, while p(t) = 0 if and only if s(t) — s = (1 + a/A5)(p(t) — p). These points are plotted in Figure 8.10. The system displays a saddle path solution. Figure 8.10: Phase diagram for the Dornbusch model.

Problems

1. (Static Mundell-Fleming with imperfect capital mobility). Let the trade balance be given by a(s + p* — p) — "0y. A real depreciation raises exports and raises the trade balance whereas an increase in income leads to higher imports which lowers the trade balance. Let the capital account be given by 6(i — i*), where 0 < 6 < to indexes the degree of capital mobility. We replace (8.3) with the external balance condition a(s + p* — p) — + 6(i — i*) = 0, that the balance of payments is 0. (We are ignoring the service account.) When capital is completely immobile, 6 = 0 and the balance of payments reduces to the trade balance. Under perfect capital mobility, 6 = to implies i = i* which is (8.3).

(a) Call the external balance condition the FF curve. Draw the FF curve in r, y space along with the LM and IS curves.

(b) Repeat the comparative statics experiments covered in this chapter using the modified external balance condition. Are any of the results sensitive to the degree of capital mobility? In particular, how do the results depend on the slope of the FF curve in relation to the LM curve?

2. How would the Mundell-Fleming model with perfect capital mobility explain the international co-movements of macroeconomic variables in Chapter 5?

### 3. Consider the Dornbusch model.

(a) What is the instantaneous effect on the exchange rate of a shock to aggregate demand? Why does an aggregate demand shock not produce overshooting?

(b) Suppose output can change in the short run by replacing the IS curve (8.7) with y = S(s — p) + Yy — ai + g, replace the price adjustment rule (8.8) with p = n(y — y), where long-run output is given by y = — p) + Yy — ai* + g. Under what circumstances is the overshooting result (in response to a change in money) robust?