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8.4. VAR ANALYSIS OF MUNDELL—FLEMING

249

8.4VAR analysis of Mundell—Fleming

Even though it required tons of algebra to solve, the stochastic MundellFleming with one-period nominal rigidity is still too stylized to take seriously in formulating econometric speciÞcations. Modeling lag dynamics in price adjustment is problematic because we don’t have a good theory for how prices adjust or for why they are sticky. Tests of overidentifying restrictions implied by dynamic versions of the Mundell— Fleming model are frequently rejected, but the investigator does not know whether it is the Mundell-Fleming theory that is being rejected or one of the auxiliary assumptions associated with the parametric econometric representation of the theory.10

Sims [129] views the restrictions imposed by explicitly formulated macroeconometric models to be incredible and proposed the unrestricted VAR method to investigate macroeconomic theory without having to assume very much about the economy. In fact, just about the only thing that you need to assume are which variables to include in the analysis. Unrestricted VAR estimation and accounting methods are described in Chapter 2.1.

The Eichenbaum and Evans VAR

Eichenbaum and Evans [41] employ the Sims VAR method to the Þve dimensional vector-time-series consisting of i) US industrial production, ii) US CPI, iii) A US monetary policy variable iv) US—foreign nominal

interest rate di erential, and v) US real exchange rate. They consid- (143) ered two measures of monetary policy. The Þrst was the ratio of the logarithm of nonborrowed reserves to the logarithm of total reserves.

The second was the federal funds rate. They estimated separate VARs using exchange rates and interest rates for each of Þve countries: Japan, Germany, France, Italy, and the UK with monthly observations from 1974.1 through 1990.5.

Here, we will re-estimate the Eichenbaum—Evans VAR and do the associated VAR accounting using monthly observations for the US, UK, Germany, and Japan from 1973.1 to 1998.1. All variables except inter-

10See Papell [117].

250	CHAPTER 8. THE MUNDELL-FLEMING MODEL

est rates are in logarithms. Let yt be US industrial production, pt be the US consumer price index, nbrt be the log of non-borrowed bank reserves divided by the log of total bank reserves, it − it be the 3 month USforeign nominal interest rate di erential, qt be the real exchange rate, and st be the nominal exchange rate.11 For each US—foreign country pair, two separate VARs were run–one using the real exchange rate and one with the nominal exchange rate. In the Þrst system, the VAR is estimated for the 5-dimensional vector xt = (yt, pt, nbrt, it − it , qt)0. In the second system, we used xt = (yt, pt, nbrt, it − it , st)0.12

The Þrst row of plots in Figure 8.8 shows the impulse response of the log real exchange rate for the US-UK, US-Germany, and US-Japan, following a one-standard deviation shock to nbrt. An increase in nbrt corresponds to a positive monetary shock. The second row shows the responses of the log nominal exchange rate with the same countries to a one-standard deviation shock to nbrt.

Both the real and nominal exchange rates are found to depreciate upon impact but the maximal nominal depreciation occurs some months after the initial shock. The impulse response of both exchange rates is hump-shaped. There is evidently evidence of overshooting, but it is di erent from Dornbusch overshooting which is instantaneous. This unrestricted VAR response pattern has come to be known as delayed overshooting.

Long-horizon (36 months ahead) forecast-error variance decompositions of nominal exchange rates attributable to orthogonalized monetary shocks are 16 percent for the UK, 24 percent for Germany, and 10 percent for Japan. For real exchange rates, the percent of variance attributable to monetary shocks is 23 percent for the UK and Germany, and 9 percent for Japan. Evidently, nominal shocks are pretty important in driving the dynamics of the real exchange rate.

11Interest rates for the US and UK are the secondary market 3-month Treasury Bill rate. For Germany, I used the interbank deposit rate. For Japan, the interest rate is the Japanese lending rate from the beginning of the sample to 1981.8, and is the private bill rate from 1981.9 to 1998.1

12Using BIC (Chapter 2, equation 2.3) with the updated data indicated that the VARs required 3 lags. To conform with Eichenbaum and Evans, I included 6 lags and a linear trend.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING

251

0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1	9	17	25	33	41
0.012
0.01
0.008
0.006
0.004
0.002
0
2	10	18	26	34	42

0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002	1	9	17	25	33	41
-0.004

0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1	9	17	25	33	41

0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002	1	9	17	25	33	41
-0.004
-0.006

0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002	1	9	17	25	33	41
-0.004

Figure 8.8: Row 1: Impulse response of log real US-UK, US-German, US-Japan exchange rate to an orthogonalized one-standard deviation shock to nbrt. Row 2: Impulse responses of log nominal exchange rate.

Clarida-Gali Structural VAR

In Chapter 2.1, we discussed some potential pitfalls associated with the unrestricted VAR methodology. The main problem is that the unrestricted VAR analyzes a reduced form of a structural model so we do not necessarily learn anything about the e ect of policy interventions on the economy. For example, when we examine impulse responses from an innovation in yt, we do not know whether the underlying cause was due to a shock to aggregate demand or to aggregate supply or an expansion of domestic credit.

Blanchard and Quah [15] show how to use economic theory to place identifying restrictions on the VAR, resulting in so-called struc-

252	CHAPTER 8. THE MUNDELL-FLEMING MODEL

tural VARs.13 Clarida and Gali [28] employ Blanchard-Quah’ structural VAR method using restrictions implied by the stochastic MundellFleming model. To see how this works, consider the 3-dimensional vector, xt = (∆(yt − yt ), ∆(pt − pt ), ∆qt)0, where y is log industrial production, p is the log price level, and q is the log real exchange rate and starred variables are for the foreign country. Given the processes that govern the exogenous variables (8.21) and (8.22), the stochastic Mundell-Fleming model predicts that income and the real exchange rate are unit root processes, so the VAR should be speciÞed in terms of Þrst-di erenced observations. The triangular structure also informs us that the variables are not cointegrated, since each of the variables are driven by a di erent unit root process.14

As described in Chapter 2.1, Þrst Þt a p-th order VAR for xt and get the Wold moving average representation

∞
jX
xt = (CjLj)²t = C(L)²t,	(8.45)
=0

where E(²t²0t) = Σ, C0 = I, and C(L) = P∞j=0 CjLj is the one-sided matrix polynomial in the lag operator L. The theory predicts that in

the long run, xt is driven by the three dimensional vector of aggregate supply, aggregate demand, and monetary shocks, vt = (zt, δt, vt)0.

The economic structure embodied in the stochastic Mundell-Fleming model is represented by

∞
jX
xt = (FjLj)vt = F(L)vt.	(8.46)
=0

Because the underlying structural innovations are not observable, you are allowed to make one normalization. Take advantage of it by setting E(vtv0t = I). The orthogonality between the various structural shocks is an identifying assumption. To map the innovations ²t from the unrestricted VAR into structural innovations vt, compare (8.45) and (8.46). It follows that

²t = F0vt ²t−j = F0vt−j Cj²t−j = CjF0vt−j = Fjvt−j.

13They are only identifying restrictions, however, and cannot be tested. 14Cointegration is discussed in Chapter2.6.

8.4. VAR ANALYSIS OF MUNDELL—FLEMING	253
To summarize
Fj = CjF0 for all j F(1) = C(1)F0.	(8.47)

Given the Cj, which you get from unrestricted VAR accounting, (8.47) says you only need to determine F0 after which the remaining Fj follow.

In our 3-dimensional system, F0 is a 3 × 3 matrix with 9 unique elements. To identify F0, you need 9 pieces of information. Start with, Σ = G0G = E(²t²0t) = F0E(vtv0t)F00 = F0F00 where G is the unique upper triangular Choleski decomposition of the error covariance matrix Σ. To summarize

Σ = G0G = F0F0 .	(8.48)
0

Let gij be the ijth element of G and fij,0 be the ijth element of F0. Writing (8.48) out gives

g2	= f2			+ f2			+ f2			,				(8.49)
11		11,0			12,0			13,0
g11 g12		= f11,0f21,0 + f12,0f22,0 + f13,0f23,0,												(8.50)
g11 g13		= f11,0f31,0 + f12,0f32,0 + f13,0f33,0,												(8.51)
g2	+g2		= f2			+ f2			+ f2			,		(8.52)
12		22		21,0			22,0			23,0
g12 g13 + g22g23 = f21,0f31,0 + f22,0f32,0 + f23,0f33,0,														(8.53)
g2	+g2		+ g2		= f2			+ f2			+ f2		.	(8.54)
13		23		33			31,0			32,0		33,0

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 e ect on output which we represent by setting f12(1) = 0 and f13(1) = 0, where fij(1) is the ijth element of F(1) = P∞j=0 Fj. The model also predicts that money has no long-run e ect on the real exchange rate f33(1) = 0. Since F(1) = C(1)F0, impose these three restrictions by setting

f13(1)	= 0 = c11(1)f13,0 + c12(1)f23,0 + c13(1)f33,0,		(8.55)
f12(1)	=	0 = c11(1)f12,0 + c12(1)f22,0 + c13(1)f32,0,	(8.56)
f33(1)	=	1 = c31(1)f13,0 + c32(1)f23,0 + c33(1)f33,0.	(8.57)

254	CHAPTER 8. THE MUNDELL-FLEMING MODEL

(8.49)—(8.57) form a system of 9 equations in 9 unknowns and implicitly deÞne F0. Once the Fj are obtained, you can do impulse response analyses and forecast error variance decompositions using the ‘structural’ response matrices Fj.

Table 8.1: Structural VAR forecast error variance decompositions for real exchange rate depreciation

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

8.4. VAR ANALYSIS OF MUNDELL—FLEMING

255

0.4		Supply, US-UK					0			Demand, US-UK					0.2			Money, US-UK
0.4							0								0.2
0.35								1	5	9	13	17	21	25	0.15
0.35							-0.05								0.15
0.3							-0.05								0.1
0.3

0.25							-0.1								0.05
															0.05
0.2
0.2															0
0.15															0
0.15							-0.15								-0.05	1	5	9	13	17	21	25
0.1							-0.15								-0.05	1	5	9	13	17	21	25
0.1															-0.1
0.05							-0.2								-0.1
0.05							-0.2
0															-0.15
0
1	5	9	13	17	21	25	-0.25								-0.2
0.2	Supply, US-Germany						0		Demand, US-Germany						0.3		Money, US-Germany
0.2							0								0.3
							-0.05	1	5	9	13	17	21	25
0.15							-0.05								0.2
															0.2

							-0.1								0.1
															0.1
0.1							-0.15
							-0.15
															0
0.05							-0.2								-0.1	1	5	9	13	17	21	25
							-0.25								-0.1
							-0.25
0							-0.3								-0.2
1	5	9	13	17	21	25									-0.2


-0.05							-0.35								-0.3
1.8	Supply, US-Japan						0		Demand, US-Japan						0.4		Money, US-Japan
1.8							0								0.4
1.6							-0.1	1	5	9	13	17	21	25	0.35
1.4							-0.2								0.3
1.2							-0.2								0.3
1.2							-0.3								0.25
1							-0.3								0.25
1							-0.4								0.2
0.8							-0.4

							-0.5
0.6							-0.5								0.15
															0.15
							-0.6
0.4							-0.6								0.1
															0.1

0.2							-0.7								0.05
0.2															0.05
0							-0.8								0
0															0
1	5	9	13	17	21	25	-0.9									1	5	9	13	17	21	25

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.

256	CHAPTER 8. THE MUNDELL-FLEMING MODEL

Mundell-Fleming Models Summary

1.The hallmark of Mundell-Fleming models is that they assume that goods prices are sticky. Many people think of Mundell— Fleming models synonymously with sticky-price models. Because there exist nominal rigidities, these models invite an assessment of monetary (and Þscal) policy interventions under both Þxed and ßexible 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.

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