6.6. NOISE-TRADERS

203

Problems

1. (Siegel’s [128] Paradox) Let St be the spot dollar price of the euro and Ft be the 1-period forward rate in dollars per euro. The claim is

if investors are risk-neutral and the forward foreign exchange market (128) is e cient, the forward rate is the rational expectation of the future

spot rate. From the US perspective we write this as

Et(St+1) = Ft.

The risk-neutral, rational-expectations, e cient market statement from an European perspective is

(1/Ft) = Et(1/St+1)

since from the euro-price of the dollar is the reciprocal of the dollareuro rate. Both statements cannot possibly be true. Why not? (Hint: Use Jensen’s inequality).

2.Let the Euler equation for a domestic investor that speculates in forward foreign exchange be

	Ft =	Et[u0(ct+1)(St+1/Pt+1)]	,
		Et[u0(ct)/Pt+1]

where u0(c) is marginal utility of real consumption c and P is the domestic price level. From the foreign perspective, the Euler equation

is

1 = Et[u0(ct+1/(St+1Pt+1)]

Ft Et[u0(ct )/Pt+1]

where c is foreign consumption and P is the foreign price level. Suppose further that both domestic and foreign agents are risk neutral. Show that Siegel’s paradox does not pose a problem now that payo s are stated in real terms.16

3.We saw that the slope coe cient in a regression of st −st−1 on ft −st−1 is negative. McCallum [103] shows regressing st − st−2 on ft − st−2 yields a slope coe cient near 1. How can you explain McCallum’s result?

16Engel’s [43] empirical work showed that regression test results on forward exchange rate unbiasedness done with nominal exchange rates were robust to speciÞcations in real terms so evidently Siegel’s paradox is not economically important.

204CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY

4.(Kaminsky and Peruga [82]). Suppose that the data generating process for observations on consumption growth, inßation, and exchange rates is given by the lognormal distribution, and that the utility function is u(c) = c1−γ. Let lower case letters denote variables in logarithms. We have ∆ct+1 = ln(Ct+1/Ct) be the rate of consumption growth, ∆st+1 = ln(St+1/St) be the depreciation rate, ∆pt+1 = ln(Pt+1/Pt) be the inßation rate, and ft = ln(Ft) be the log one-period forward rate.

If ln(Y ) N(µ, σ2), then Y is said to be log-normally distributed and

h	i
E eln(y)	= E(Y ) = ehµ+ σ22

i

. (6.74)

Let Jt consist of lagged values of ct, st, pt and ft be the date t information set available to the econometrician. Conditional on Jt, let yt+1 = (∆st+1, ∆ct+1, ∆pt+1)0 be normally distributed with condi-

tional mean E(yt+1|Jt) =

(µst, µct, µpt)0 and conditional covariance

σsst

σsct

σspt

−

matrix Σt

σpst

σpct

σppt

. Let at+1

= ∆st+1

∆pt+1 and

=

σcst

σcct

σcpt

b

= f

s

∆p

. Show

t+1

t −

t+1

that

t −

µst − ft = γσcst + σspt −

σsst

.

(6.75)

2

5.Testing the volatility restrictions (Cecchetti et. al. [25]). This exercise develops the volatility bounds analysis so that we can do classical statistical hypothesis tests to compare the implied volatility of the intertemporal marginal rate of substitution and the lower volatility bound. Begin deÞning φ as a vector of parameters that characterize the utility function, and ψ as a vector of parameters associated with the stochastic process governing consumption growth.

Stack the parameters that must be estimated from the data into the

vector θ
		µ
θ =	vec(		rΣr)		,
		ψ

where vec(Σr) is the vector obtained by stacking all of the unique elements of the symmetric matrix, Σr. Let θ0 be the true value of θ,

6.6. NOISE-TRADERS

205

ˆ				such that
and let θ be a consistent estimator of θ0
√		ˆ	D
	T (θ − θ0) → N(0, Σθ).

Assume that consistent estimators of both θ0 and Σθ are available.

Now make explicit the fact that the moments of the intertemporal marginal rate of substitution and the volatility bound depend on sample information. The estimated mean and standard deviation of pre-

ˆ

ˆ

dicted by the model are, µˆµ = µµ(φ; ψ) and σˆµ

= σµ(φ; ψ), while the

estimated volatility bound is

0

Σˆr−1

µˆq

− µµ(φ; ψˆ)ˆµr .

σˆr = σr(φ; θˆ) = r

µˆq

− µµ(φ; ψˆ)ˆµr

Let

³

´

³

´

ˆ

ˆ

ˆ

∆(φ; θ) = σM(φ; ψ) − σr(φ; θ),

be the di erence between the estimated volatility bound and the estimated volatility of the intertemporal marginal rate of substitution. Using the ‘delta method,’ (a Þrst-order Taylor expansion about the true parameter vector), show that

√				ˆ	D	2
	T (∆(φ; θ) − ∆(φ; θ0)) → N(0, σ∆),
where		µ ∂θ0 ¶θ0			(θˆ − θ0)(θˆ − θ0)0	µ ∂θ ¶θ0 .
σ∆2 =
			∂∆				∂∆

How can this result be used to conduct a statistical test of whether a particular model attains the volatility restrictions?

6.(Peso problem). Let the fundamentals, ft = mt −mt −λ(yt −yt ) follow the random walk with drift, ft+1 = δ0 + ft + vt+1, where vt iid with E(vt) = 0 and E(vt2) = σv2. Agents know the fundamentals process with certainty. Forward iteration on (6.33) yields the present value

formula

X∞

st = γ Et(ft+j). j=1

Verify the solution (6.38) by direct substitution of Et(ft+j).

Now let agents believe that the drift may have increased to δ = δ1. Show that Et(ft+j) = ft + j(δ0 − δ1)p0t + jδ1. Use direct substitution of this forecasting formula in the present value formula to verify the solution (6.43) in the text.

206CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY