International Macroeconomics and Finance: Theory and Empirical Methods Phần 5 docx

Consumption implied by the model is morevolatile than output, which is counterfactual... In the market competitive equilibrium interpreta-tion, the excess absorption is Þnanced by intere

A second but equivalent option is to take a second—order Taylor

approximation to the objective function around the steady state and to

solve the resulting quadratic optimization problem The second option

is equivalent to the Þrst because it yields linear Þrst—order conditions

around the steady state To pursue the second option, recall that λt=

(kt+1, kt, At)0 Write the period utility function in the unconstrained

optimization problem as

R(λt) = U [g(λt)] (5.20)Let Rj = ∂R(λt)/∂λjt be the partial derivative of R(λt) with respect

to the j−th element of λt and Rij = ∂2R(λt)/(∂λit∂λjt) be the second

cross-partial derivative Since Rij = Rji the relevant derivatives are,

R(λt) = R(λ) + R1(kt+1− k) + R2(kt− k) + R3(At− A) + 1

2R11(kt+1− k)2

2R22(kt− k)2+ 1

2R33(At− A)2+ R12(kt+1− k)(kt− k)+R13(kt+1− k)(At− A) + R23(kt− k)(At− A)

Suppose we let q = (R1, R2, R3)0 be the 3 × 1 row vector of partialderivatives (the gradient) of R, and Q be the 3× 3 matrix of secondpartial derivatives (the Hessian) multiplied by 1/2 where Qij = Rij/2.Then the approximate period utility function can be compactly written

in matrix form as

R(λt) = R(λ) + [q + (λt− λ)0Q](λt− λ) (5.21)The problem is now to maximize

Et

∞ X j=0

Now to solve the linearized Þrst-order conditions, work with (5.19).Since the data that we want to explain are in logarithms, you can con-vert the Þrst-order conditions into near logarithmic form Let

˜i = kai for i = 1, 2, 3, and let a “hat” denote the approximate logdifference from the steady state so that ˆkt = (kt − k)/k ' ln(kt/k)and ˆAt = At − 1 (since the steady state value of A = 1) Now let

b1 =−˜a2/˜a1, b2 =−˜a3/˜a1, b3 =−a4/˜a1, and b4 =−a4/˜a1

The second—order stochastic difference equation (5.19) can be ten as

writ-(1− b1L− b2L2)ˆkt+1= Wt, (5.24)where

Wt = b3Aˆt+1+ b4Aˆt.

The roots of the polynomial (1− b1z− b2z2) = (1− ω1L)(1− ω2L)satisfy b1 = ω1 + ω2 and b2 = −ω1ω2 Using the quadratic formulaand evaluating at the parameter values that we used to calibrate themodel, the roots are, z1 = (1/ω1) = [−b1−

1+ 4b2]/(2b2)' 0.81 There is a stable root,

|z1| > 1 which lies outside the unit circle, and an unstable root, |z2| < 1which lies inside the unit circle The presence of an unstable root meansthat the solution is a saddle path If you try to simulate (5.24) directly,the capital stock will diverge

To solve the difference equation, exploit the certainty equivalenceproperty of quadratic optimization problems That is, you Þrst getthe perfect foresight solution to the problem by solving the stable rootbackwards and the unstable root forwards Then, replace future ran-dom variables with their expected values conditional upon the time-tinformation set Begin by rewriting (5.24) as

Wt = (1− ω1L)(1− ω2L)ˆkt+1

= (−ω2L)(−ω2−1L−1)(1− ω2L)(1− ω1L)ˆkt+1

= (−ω2L)(1− ω−12 L−1)(1− ω1L)ˆkt+1,and rearrange to get

It follows that the solution for the capital stock is

Simulating the Model

We’ll use the calibrated model to generate 96 time-series observationscorresponding to the number of observations in the data From thesepseudo-observations, recover the implied log-levels and pass them throughthe Hodrick-Prescott Þlter The steady state values are

y = 1.717, k = 5.147, c = 1.201, i/k = 0.10

Plots of the Þltered log income, consumption, and investment tions are given in Figure 5.3 and the associated descriptive statistics aregiven in Table 5.2 The autoregressive coefficient and the error variance

observa-of the technology shock were selected to match the volatility observa-of outputexactly From the Þgure, you can see that both consumption and in-vestment exhibit high co-movements with output, and all three seriesdisplay persistence However from Table 5.2 the implied investmentseries is seen to be more volatile than output but is less volatile thanthat found in the data Consumption implied by the model is morevolatile than output, which is counterfactual

We now add a second country This two-country model is a specialcase of Backus et al [5] Each county produces the same good so wewill not be able to study terms of trade or real exchange rate issues.The presence of country-speciÞc idiosyncratic shocks give an incentive

to individuals in the two countries to trade as a means to insure each

Table 5.2: Calibrated Closed-Economy Model

Table 5.3 displays the features of the data that we will attempt toexplain–their volatility, persistence (characterized by their autocorre-lations) and their co-movements (characterized by cross correlations).Notice that US and European consumption correlation is lower thanthe their output correlation

The Two-Country Model

Both countries experience identical rates of depreciation of physicalcapital, long-run technological growth Xt+1/Xt = X∗

t+1/X∗

t = γ, have

Table 5.3: Open-Economy Measurements

to 1996.4 and have been passed through the Hodrick—Prescott Þlter with λ = 1600.

the same capital shares and Cobb-Douglas form of the production tion, and identical utility Let the social planner attach a weight of ω tothe domestic agent and a weight of 1− ω to the foreign agent In terms

func-of efficiency units, the social planner’s problem is now to maximize

Et

∞ X j=0

βj[ωU (ct+j) + (1− ω)U(c∗t+j)], (5.27)subject to,

Cobb—Douglas production functions for the home and foreign counties,with normalized labor input N = N∗ = 1 (5.30) and (5.31) are thedomestic and foreign capital accumulation equations, and (5.31) is thenew form of the resource constraint Both countries have the sametechnology but are subject to heterogeneous transient shocks to totalproductivity according to

Next, transform the constrained optimization problem into an constrained problem by substituting (5.34) and (5.35) into (5.27) Theproblem is now to maximize

un-ωEt

³

u[g(λt)] + βU [g(λt+1)] + β2U [g(λt+2)] +· · ·´ (5.36)+(1− ω)Et

³

u[h(λt)] + βU [h(λt+1)] + β2U [h(λt+2)] +· · ·´

At date t, the choice variables available to the planner are kt+1, k∗

t+1,and c∗

t Differentiating (5.36) with respect to these variables and arranging results in the Euler equations

re-γUc(ct) = βEtUc(ct+1)[g3(λt+1)], (5.37)

γUc(ct) = βEtUc(ct+1)[g4(λt+1)], (5.38)

Uc(ct) = [(1− ω)/ω]Uc(c∗t) (5.39)(5.39) is the Pareto—Optimal risk sharing rule which sets home marginalutility proportional to foreign marginal utility Under log utility, homeand foreign per capita consumption are perfectly correlated,

ct= [ω/(1− ω)]c∗

t.The Two-Country Steady State

From (5.37) and (5.38) we obtain y/k = y∗/k∗ = (γ/β +δ−1)/α We’vealready determined that c = [ω/(1− ω)]c∗ = ωcw where cw = c + c∗

is world consumption From the production functions (5.28)—(5.29) weget k = (y/k)1/(α−1) and k∗ = (y∗/k∗)1/(α−1) From (5.30)—(5.31) weget i = i∗ = (γ + δ− 1)k It follows that c = ωcw = ω[y + y∗− (i + i∗)]

= 2ω[y− i]

Thus y− c − i = (1 − 2ω)(y − i) and unless ω = 1/2, the currentaccount will not be balanced in the steady state If ω > 1/2 the homecountry spends in excess of GDP and runs a current account deÞcit.How can this be? In the market (competitive equilibrium) interpreta-tion, the excess absorption is Þnanced by interest income earned on pastlending to the foreign country Foreigners need to produce in excess oftheir consumption and investment to service the debt In a sense, theyhave ‘over-invested’ in physical capital

In the planning problem, the social planner simply takes away some

of the foreign output and gives it to domestic agents Due to the

concavity of the production function, optimality requires that the worldcapital stock be split up between the two countries so as to equate themarginal product of capital at home and abroad Since technology isidentical in the 2 countries, this implies equalization of national capitalstocks, k = k∗, and income levels y = y∗, even if consumption differs,

c6= c∗

Quadratic Approximation

You can solve the model by taking the quadratic approximation of theunconstrained objective function about the steady state Let R be theperiod weighted average of home and foreign utility

R(λt) = ωU [g(λt)] + (1− ω)U[h(λt)]

Let Rj = ωUc(c)gj + (1− ω)Uc(c∗)hj, j = 1, , 7 be the Þrst partialderivative of R with respect to the j−the element of λt Denote thesecond partial derivative of R by

Rjk = ∂R(λ)

∂λj∂λk

= ω[Uc(c)gjk+Uccgjgk]+(1−ω)[Uc(c∗)hjk+Ucc(c∗)hjhk]

(5.40)Let q = (R1, , R7)0 be the gradient vector, Q be the Hessian matrix

of second partial derivatives whose j, k−th element is Qjk = (1/2)Rj,k.Then the second-order Taylor approximation to the period utility func-tion is

R(λt) = [q + (λt− λ)0Q](λt− λ),and you can rewrite (5.36) as

max Et

∞ X j=0

βj[q + (λt+j− λ)0Q](λt+j− λ) (5.41)

Let Qj• be the j−th row of the matrix Q The Þrst-order conditionsare

(kt+1) : 0 = R1+ βR3+ Q1•(λt− λ) + βQ3•(λt+1− λ), (5.42)(kt+1∗ ) : 0 = R2+ βR4+ Q2•(λt− λ) + βQ4•(λt+1λ), (5.43)(c∗t) : 0 = R7+ Q7•(λt− λ) (5.44)

Now let a ‘tilde’ denote the deviation of a variable from its steady statevalue so that ˜kt= kt− k and write these equations out as

0 = a1˜t+2+ a2˜∗

t+2+ a3˜t+1+ a4˜∗

t+1+ a5˜t+ a6˜∗

t + a7A˜t+1+a8A˜∗

t+1+ ˜b9A˜t+ ˜b10A˜∗

At this point, the marginal beneÞt from looking at analytic expressionsfor the coefficients is probably negative For the speciÞc calibration ofthe model the numerical values of the coefficients are,

t +˜7+ ˜b7

2 A˜

w t+1+˜a9A˜w

t+˜11+ ˜b11

2 = 0 (5.50)(5.50) is a second—order stochastic difference equation in ˜kw

t = ˜kt+ ˜k∗

t,which can be rewritten compactly as4

˜w t+2− m1˜w

t+1− m2˜w

t = Wt+1w , (5.51)where Ww

4 Unlike the one-country model, we don’t want to write the model in logs because

we have to be able to recover ˜ k and ˜ k ∗ separately.

You can write second—order stochastic difference equation (5.51) as(1 − m1L − m2L2)ˆkw

us-1+ 4m2]/(2m2) ' 1.17, and z2 = (1/ω2) =[−m1+qm2

1+ 4m2]/(2m2)' 0.84 The stable root |z1| > 1 lies outsidethe unit circle, and the unstable root |z2| < 1 lies inside the unit circle.From the law of motion governing the technology shocks (5.33), youhave

˜

Awt+1 = (ρ + δ) ˜Awt + ²wt, (5.52)where ²wt = ²t + ²∗t Now EtWt+k = m3A˜w

t+1 + m4A˜w

t + m5 =[m3(ρ + δ) + m4](ρ + δ)kA˜w

t + m5 As in the one-country model, usethese forecasting formulae to solve the unstable root forwards and thestable root backwards The solution for the world capital stock is

˜t+1 = 1

2[˜k

w t+1+ (˜kt+1− ˜kt+1∗ )] (5.55)The date t + 1 world capital stock is predetermined at date t How thatcapital is allocated between the home and foreign country depends onthe realization of the idiosyncratic shocks ˜At+1 and ˜A∗t+1

and investment rates are

˜it = γ˜kt+1− (1 − δ)˜kt, (5.58)

˜i∗

t = γ˜k∗t+1− (1 − δ)˜kt∗ (5.59)Let world consumption be ˜cw

t = ˜ct+ ˜c∗

t = ˜yt+ ˜y∗

t − (˜it+ ˜i∗

t) By theoptimal risk-sharing rule (5.39) ˜c∗t = [(1− ω)/ω]˜ct, which can be used

man-Simulating the Two-Country Model

The steady state values are

y = y∗ = 1.53, k = k∗ = 3.66, i = i∗ = 0.42, c = c∗ = 1.11.The model is used to generate 96 time-series observations Descriptivestatistics calculated using the Hodrick—Prescott Þltered cyclical parts ofthe log-levels of the simulated observations and are displayed in Table5.4 and Figure 5.4 shows the simulated current account balance.The simple model of this chapter makes many realistic predictions

It produces time-series that are persistent and that display coarse movements that are broadly consistent with the data But there arealso several features of the model that are inconsistent with the data.First, consumption in the two-country model is smoother than output.Second, domestic and foreign consumption are perfectly correlated due

co-to the perfect risk-sharing whereas the correlation in the data is muchlower than 1 A related point is that home and foreign output arepredicted to display a lower degree of co-movement than home andforeign consumption which also is not borne out in the data

Figure 5.4: Simulated current account to GDP ratio.

Table 5.4: Calibrated Open-Economy Model

International Real Business Cycles Summary

1 The workhorse of real business cycle research is the dynamicstochastic general equilibrium model These can be viewed asArrow-Debreu models and solved by exploiting the social plan-ner’s problem They feature perfect markets and completelyfully ßexible prices The models are fully articulated and arehave solidly grounded micro foundations

2 Real business cycle researchers employ the calibration method toquantitatively evaluate their models Typically, the researchertakes a set of moments such as correlations between actual timeseries, and asks if the theory is capable of replicating these co-movements The calibration style of research stands in contrastwith econometric methodology as articulated in the Cowles com-mission tradition In standard econometric practice one begins

by achieving model identiÞcation, progressing to estimation ofthe structural parameters, and Þnally by conducting hypothesistests of the model’s overidentifying restrictions but how one de-termines whether the model is successful or not in the calibrationtradition is not entirely clear

Foreign Exchange Market

sim-He goes on to say,

“ , market efficiency per se is not testable It must

be tested jointly with some model of equilibrium, an pricing model.”

asset-Market efficiency does not mean that asset returns are serially correlated, nor does it mean that the Þnancial markets present zeroexpected proÞts The crux of market efficiency is that there are nounexploited excess proÞt opportunities What is considered to be ex-cessive depends on the model of market equilibrium

un-This chapter is an introduction to the economics of foreign exchangemarket efficiency We begin with an evaluation of the simplest model ofinternational currency and money-market equilibrium–uncovered in-terest parity Econometric analyses show that it is strongly rejected by

161

the data The ensuing challenge is then to understand why uncoveredinterest parity fails.

We cover three possible explanations The Þrst is that the ward foreign exchange rate contains a risk premium This argument

for-is developed using the Lucas model of chapter 4 The second tion is that the true underlying structure of the economy is subject tochange occasionally but economic agents only learn about these struc-tural changes over time During this transitional learning period inwhich market participants have an incomplete understanding of theeconomy and make systematic prediction errors even though they arebehaving rationally This is called the ‘peso-problem’ approach Thethird explanation is that some market participants are actually irra-tional in the sense that they believe that the value of an asset depends

explana-on extraneous informatiexplana-on in additiexplana-on to the ecexplana-onomic fundamentals.The individuals who take actions based on these pseudo signals arecalled ‘noise’ traders

The notational convention followed in this chapter is to let uppercase letters denote variables in levels and lower case letters denote theirlogarithms, with the exception of interest rates, which are always de-noted in lower case As usual, stars are used to denote foreign countryvariables

Let s be the log spot exchange rate, f be the log one-period forwardrate, i be the one-period nominal interest rate on a domestic currency(dollar) asset and i∗ is the nominal interest rate on the foreign currency(euro) asset If uncovered interest parity holds, it− i∗t = Et(st+1)− st,but by covered interest parity, it− i∗t = ft− st Therefore, unbiasedness

of the forward exchange rate as a predictor of the future spot rate

ft= Et(st+1) is equivalent to uncovered interest parity

We begin by covering the basic econometric analyses used to detectthese deviations

Hansen and Hodrick’s Tests of UIP

Hansen and Hodrick [71] use generalized method of moments (GMM)

to test uncovered interest parity The GMM method is covered in

chapter 2.2 The Hansen—Hodrick problem is that a moving-average

se-rial correlation is induced into the regression error when the prediction

horizon exceeds the sampling interval of the data

The Hansen—Hodrick Problem

To see how the problem arises, let ft,3 be the log 3-month forward

ex-change rate at time t, stbe the log spot rate, Itbe the time t information ⇐(102)set available to market participants, and Jt be the time t information

set available to you, the econometrician Even though you are working

with 3-month forward rates, you will sample the data monthly You

want to test the hypothesis

H0 : E(st+3|It) = ft,3

In setting up the test, you note that It is not observable but since Jt is

a subset of It and since ft,3 is contained in Jt, you can use the law of

iterated expectations to test

H00 : E(st+3|Jt) = ft,3,which is implied by H0 You do this by taking a vector of economic

variables zt−3 in Jt−3, running the regression

st− ft−3,3 = z0t−3β + ²t,3,and doing a joint test that the slope coefficients are zero

Under the null hypothesis, the forward rate is the market’s forecast

of the spot rate 3 months ahead ft−3,3 = E(st|Jt−3) The observations,

however, are collected every month Let Jt= (²t, ²t−1, , zt, zt−1, )

The regression error formed at time t− 3 is ²t = st− E(s|Jt−3) At

t − 3, E(²t|Jt−3) = E(st − E(st|Jt−3)) = 0 so the error term is un- ⇐(103)predictable at time t− 3 when it is formed But at time t − 2 and

t− 1 you get new information and you cannot say that E(²t|Jt−1) =

E(st|Jt−1)−E[E(st|Jt−3)|Jt−1] is zero Using the law of iterated

expecta-tions, the Þrst autocovariance of the error E(²t²t−1) = E(²t−1E(²t|Jt−1))