- •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
134 CHAPTER 4. THE LUCAS MODEL
Appendix—Markov Chains
Let Xt be a random variable and xt be a particular realization of Xt. A Markov chain is a stochastic process {Xt}∞t=0 with the property that the information in the current realized value of Xt = xt summarizes the entire past history of the process. That is,
P[Xt+1 = xt+1|Xt = xt, Xt−1 = xt−1, . . . , X0 = x0] = P[Xt+1 = xt+1|Xt = xt].
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(4.64) |
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A key result that simpliÞes probability calculations of Markov chains is, |
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Property 1 If {Xt}t∞=0 is a Markov chain, then |
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P[Xt = xt ∩ Xt−1 = xt−1 ∩ · · · ∩ X0 = x0] = |
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P[Xt = xt|Xt−1 = xt−1] · · · P[X1 = x1|X0 = x0]P[X0 = x0]. |
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Proof: Let Aj be the event (Xj = xj). You can write the left side of (4.65) as,
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P(At ∩ At−1 ∩ · · · ∩ A0) = |
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P(At| Aj)P( |
Aj) |
(multiplication rule) |
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t−1 |
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j\ |
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= P(At|At−1)P( |
Aj) (Markov chain property) |
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t−2 |
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Aj) (mult. rule) |
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P(At|At−1)P(At−1| |
Aj)P( |
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j=0 |
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P(At|At−1)P(At−1|At−2)P( ) |
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= P(At|At−1)P(At−1|At−2) · · · P(A1|A0)P(A0)
Let λj, j = 1, . . . , N denote the possible states for Xt. A Markov chain has stationary probabilities if the transition probabilities from state λi to λj are time-invariant. That is,
P[Xt+1 = λj|Xt = λi] = pij
4.5. CALIBRATING THE LUCAS MODEL |
135 |
Notice that in Markov chain analysis the Þrst subscript denotes the state that you condition on. For concreteness, consider a Markov chain with two possible states, λ1 and λ2, with transition matrix,
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P = |
p11 |
p12 |
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where the rows of P sum to 1.
Property 2 The transition matrix over k steps is
Pk = PP · · · P
| {z } k
Proof. For the two state process, deÞne
p(2)ij = P[Xt+2 = λj|Xt = λi]
=P[Xt+2 = λj ∩ Xt+1 = λ1|Xt = λi] + P[Xt+1 = λj ∩ Xt+1 = λ2|Xt = λi]
X2
=P[Xt+1 = λj ∩ Xt+1 = λk|Xt = λi]
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P[Xt+1 = λj ∩ Xt+1 = λk ∩ Xt = λi] |
(4.66) |
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P(Xt = λi) |
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Now by property 1, the numerator in last equality can be decomposed as, |
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P[Xt+2 = λj|Xt+1 = λk]P[Xt+1 = λk|Xt = λi]P[Xt = λi] |
(4.67) |
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Substituting (4.67) into (4.66) gives, |
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pij(2) = |
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P[Xt+1 = λj|Xt+1 = λk]P[Xt+1 = λk|Xt = λi] |
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k=1
X2
=pkjpik
k=1
which is seen to be the ij−th element of the matrix PP. |
The extension to |
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any arbitrary number of steps forward is straightforward. |
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136 |
CHAPTER 4. THE LUCAS MODEL |
Problems
1. Risk sharing in the Lucas model [Cole-Obstfeld (1991)]. Let the period utility function be u(cx, cy) = θ ln cx + (1 − θ) ln cy for the home agent and u(cx, cy) = θ ln cx + (1 −θ) ln cy for the foreign agent. Suppose That capital is internationally immobile. The home agent owns all of the x−endowment (φx = 1), the foreign agent owns all of the y−endowment (φy = 1). Show that in the equilibrium under portfolio autarchy, trade in goods alone is su cient to achieve e cient risk sharing.
2.Consider now the single-good model. Let xt be the home endowment and xt be the foreign endowment of the same good. The planner’s problem is to maximize
φ ln ct + (1 − φ) ln ct
subject to ct + ct = xt + xt .
Under zero capital mobility, the home agent’s problem is to maximize ln(ct) subject to ct = xt. The foreign agent maximizes ln(ct ) subject to ct = xt . Show that asset trade is necessary in this case to achieve e cient risk sharing.
3.Nontraded goods. Let x and y be traded as in the model of this chapter. In addition, let N be a nonstorable nontraded domestic good
generated by an exogenous endowment, and let N be a nonstorable (100) nontraded foreign good also generated by exogenous endowment. Let the domestic agent’s utility function be u(cxt, cyt, cN ) = (C1−γ)/(1−γ) where C = cθx1 cθy2 cθN3 with θ1 + θ2 + θ3 = 1. The foreign agent has the same utility function. Show that trade in goods under zero capital
mobility does not achieve e cient risk sharing.
4. Derive the exchange rate in the Lucas model under log utility, U(cxt, cyt) = (101) θ ln(cxt) + (1 − θ) ln(cyt) and compare it with the solution under con-
stant relative risk aversion utility.
5. Use the high and low growth states and the transition matrix given (102) in section 4.5 to solve for the price-dividend ratios for equities. What does the Lucas model have to say about the volatility of stock prices?
How does the behavior of equity prices in the monetary economy di er from the behavior of equity prices in the barter economy?