- •Preface
- •Contents
- •Chapter 1
- •1.1 International Financial Markets
- •Foreign Exchange
- •Covered Interest Parity
- •Uncovered Interest Parity
- •Futures Contracts
- •1.2 National Accounting Relations
- •National Income Accounting
- •The Balance of Payments
- •1.3 The Central Bank’s Balance Sheet
- •Chapter 2
- •2.1 Unrestricted Vector Autoregressions
- •Lag-Length Determination
- •Granger Causality, Econometric Exogeniety and Causal
- •Priority
- •The Vector Moving-Average Representation
- •Impulse Response Analysis
- •Forecast-Error Variance Decomposition
- •Potential Pitfalls of Unrestricted VARs
- •2.2 Generalized Method of Moments
- •2.3 Simulated Method of Moments
- •2.4 Unit Roots
- •The Levin—Lin Test
- •The Im, Pesaran and Shin Test
- •The Maddala and Wu Test
- •Potential Pitfalls of Panel Unit-Root Tests
- •2.6 Cointegration
- •The Vector Error-Correction Representation
- •2.7 Filtering
- •The Spectral Representation of a Time Series
- •Linear Filters
- •The Hodrick—Prescott Filter
- •Chapter 3
- •The Monetary Model
- •Cassel’s Approach
- •The Commodity-Arbitrage Approach
- •3.5 Testing Monetary Model Predictions
- •MacDonald and Taylor’s Test
- •Problems
- •Chapter 4
- •The Lucas Model
- •4.1 The Barter Economy
- •4.2 The One-Money Monetary Economy
- •4.4 Introduction to the Calibration Method
- •4.5 Calibrating the Lucas Model
- •Appendix—Markov Chains
- •Problems
- •Chapter 5
- •Measurement
- •5.2 Calibrating a Two-Country Model
- •Measurement
- •The Two-Country Model
- •Simulating the Two-Country Model
- •Chapter 6
- •6.1 Deviations From UIP
- •Hansen and Hodrick’s Tests of UIP
- •Fama Decomposition Regressions
- •Estimating pt
- •6.2 Rational Risk Premia
- •6.3 Testing Euler Equations
- •Volatility Bounds
- •6.4 Apparent Violations of Rationality
- •6.5 The ‘Peso Problem’
- •Lewis’s ‘Peso-Problem’ with Bayesian Learning
- •6.6 Noise-Traders
- •Problems
- •Chapter 7
- •The Real Exchange Rate
- •7.1 Some Preliminary Issues
- •7.2 Deviations from the Law-Of-One Price
- •The Balassa—Samuelson Model
- •Size Distortion in Unit-Root Tests
- •Problems
- •Chapter 8
- •The Mundell-Fleming Model
- •Steady-State Equilibrium
- •Exchange rate dynamics
- •8.3 A Stochastic Mundell—Fleming Model
- •8.4 VAR analysis of Mundell—Fleming
- •The Eichenbaum and Evans VAR
- •Clarida-Gali Structural VAR
- •Appendix: Solving the Dornbusch Model
- •Problems
- •Chapter 9
- •9.1 The Redux Model
- •9.2 Pricing to Market
- •Full Pricing-To-Market
- •Problems
- •Chapter 10
- •Target-Zone Models
- •10.1 Fundamentals of Stochastic Calculus
- •Ito’s Lemma
- •10.3 InÞnitesimal Marginal Intervention
- •Estimating and Testing the Krugman Model
- •10.4 Discrete Intervention
- •10.5 Eventual Collapse
- •Chapter 11
- •Balance of Payments Crises
- •Flood—Garber Deterministic Crises
- •11.2 A Second Generation Model
- •Obstfeld’s Multiple Devaluation Threshold Model
- •Bibliography
- •Author Index
- •Subject Index
10.3. INFINITESIMAL MARGINAL INTERVENTION |
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which follows because the integral in term (a) is Rt∞ e−x/αdx = αe−t/α and the integral in term (b) is Rt∞ xe−x/αdx = α2e−t/α(αt +1). (10.26) is the no bubbles solution for the exchange rate under a permanent freeßoat regime where the fundamentals follow the (η, σ)—di usion process (10.23) and are expected to do so forever on. This is the continuoustime analog to the solution obtained in chapter 3 when the fundamentals followed a random walk.
10.3InÞnitesimal Marginal Intervention
Consider now a small-open economy whose central bank is committed to keeping the nominal exchange rate s within the target zone, s < s < s¯. The credibility of the Þx is not in question. Krugman [88] assumes that the monetary authorities intervene whenever the exchange rate touches one of the bands in a way to prevent the exchange rate from ever moving out of the bands. In order to be e ective, the authorities must engage in unsterilized intervention, by adjusting the fundamentals f(t). As long as the exchange rate lies within the target zone, the authorities do nothing and allow the fundamentals to follow the di usion process df(t) = ηdt + σdz(t). But at those instants that the exchange rate touches one of the bands, the authorities intervene to an extent necessary to prevent the exchange rate from moving out of the band.
During times of intervention, the fundamentals do not obey the diffusion process but are following some other process. Since the forecasting rule (10.25) was derived by assuming that the fundamentals always follows the di usion it cannot be used here. To solve the model using the same technique, you need to modify the forecasting rule to account for the fact that the process governing the fundamentals switches from the di usion to the alternative process during intervention periods.
Instead, we will obtain the solution by the method of undetermined coe cients. Begin by conjecturing a solution in which the exchange rate is a time-invariant function G(·) of the current fundamentals
s(t) = G[f(t)]. |
(10.27) |
Now to Þgure out what the function G looks like, you know by Ito’s
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CHAPTER 10. TARGET-ZONE MODELS |
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ds(t) = dG[f(t)] |
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Et[ds(t)] = G0[f(t)]ηdt + |
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Et[sú(t)] = ηG0[f(t)] + |
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Now substitute (10.27) and (10.29) into the monetary model (10.21) and re-arrange to get the second-order di erential equation in G
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Digression on second-order di erential equations. Consider the secondorder di erential equation,
y00 + a1y0 + a2y = bt |
(10.31) |
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A trial solution to the homogeneous part (y00 |
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y = Aeλt, which implies y0 = λAeλt and |
y00 = λ2Aeλt, |
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10.3. INFINITESIMAL MARGINAL INTERVENTION |
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Solution under Krugman intervention. To solve (10.30), replace y(t) in
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To solve for the constants A and B, you need two additional pieces of information. These are provided by the intervention rules.4 From (10.33), you can see that the function mapping f(t) into s(t) is one-to-one. This means that there is a lower and upper band on the fundamentals, [f, f¯] that corresponds to the lower and upper bands for the exchange rate [s, s¯]. When s(t) hits the upper band s¯, the authorities intervene to prevent s(t) from moving out of the band. Only inÞnitesimally small interventions are required. During instants of intervention, ds = 0 from which it follows that
G0(f¯) = 1 + λ1Aeλ1f¯ + λ2Beλ2f¯ = 0. |
(10.36) |
Similarly, at the instant that s touches the lower band s, ds = 0 and
G0(f) = 1 + λ1Aeλ1 |
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4In the case of a pure ßoat and in the absence of bubbles, you know that A = B = 0.
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CHAPTER 10. TARGET-ZONE MODELS |
The signs of A and B follow from noting that λ1 is positive and λ2 is negative so that eλ1(f¯−f) > eλ2(f¯−f). It follows that the square bracketed term in the denominator is positive.
The solution becomes simpler if you make two symmetry assumptions. First, assume that there is no drift in the fundamentals η = 0. Setting the drift to zero implies λ1 = −λ2 = λ > 0. Second, center the admissible region for the fundamentals around zero with f¯ = −f so that B = −A > 0. The solution becomes
G[f(t)] = f(t) + B[e−λf(t) − eλf(t)], |
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Figure 10.1 shows the relation between the exchange rate and the fundamentals under Krugman-style intervention. The free ßoat solution s(t) = f(t) serves as a reference point and is given by the dotted 45degree line. First, notice that G[f(t)] has the shape of an ‘S.’ The S-curve lies below the s(t) = f(t) line for positive values of f(t) and vice-versa for negative values of f(t). This means that under the targetzone arrangement, the exchange rate varies by a smaller amount in response to a given change in f(t) within [f, f¯] than it would under a free ßoat.
Second, note that by (10.21), we know that E(sú) < 0 when f > 0, and vice-versa. This means that market participants expect the exchange rate to decline when it lies above its central parity and they expect the exchange rate to rise when it lies below the central parity. The exchange rate displays mean reversion. This is potentially the explanation for why exchange rates are less volatile under a managed ßoat than they are under a free ßoat. Since market participants expect the authorities to intervene when the exchange rate heads toward the bands, the expectation of the future intervention dampens current exchange rate movements. This dampening result is called the
Honeymoon e ect.