Uncovered Interest Parity and Monetary Policy Near and Far from the Zero Lower Bound

Abstract

Relying upon a standard New Keynesian DSGE, we propose an explanation for two empirical findings in the international finance literature. First, the unbiasedness hypothesis — the proposition that expost exchange rate depreciation matches interest differentials — is rejected much more strongly at short horizons than at long. Second, even at long horizons, the unbiasedness hypothesis tends to be rejected when one of the currencies has experienced a long period of low interest rates, such as in Japan and Switzerland. Using a calibrated New Keynesian dynamic stochastic general equilibrium model, we show how a monetary policy rule can induce the negative (positive) correlation between depreciation and interest differentials at short (long) horizons. The tendency to reject unbiasedness for Japan and Switzerland even at long horizons we attribute to the interaction of the monetary reaction function and the zero lower bound.

Appendix

1.1 A.1 Households

Consumption preferences in the Home economy are described by the following composite index of domestic and imported bundles of goods:

$$ C_{t}\equiv[(1-\gamma)^{\frac{1}{a}}({C_{t}^{h}})^{\frac{a-1}{a}}+\gamma^{\frac{1}{a}}({C_{t}^{f}})^{\frac{a-1}{a}}]^{\frac{a}{a-1}} $$

(49)

where a > 1 is the elasticity of substitution between domestically produced and imported goods. ${C_{t}^{h}}$ (and ${C_{t}^{f}}$) are Dixit-Stiglitz aggregates of domestically produced and imported goods.

$$ {C_{t}^{h}}\equiv\left[{\int}_{0}^{1}{C_{t}^{h}}(j)^{\frac{\theta-1}{\theta}}dj\right]^{\frac{\theta}{\theta-1}} $$

(50)

where j ∈ [0,1] denotes the good variety. ${C_{t}^{f}}$ is an index of imported goods given by

$$ {C_{t}^{f}}\equiv\left[{\int}_{0}^{1}\left( C_{k,t}\right)^{\frac{\xi-1}{\xi}}dk\right]^{\frac{\xi}{\xi-1}} $$

(51)

where C_k,t is, in turn, an index of the quantity of goods imported from country k and consumed by domestic households. It is given by an analogous CES function

$$ C_{k,t}\equiv\left[{\int}_{0}^{1}C_{k,t}(j)^{\frac{\theta-1}{\theta}}dj\right]^{\frac{\theta}{\theta-1}} $$

(52)

The optimal allocation of any given expenditure within each category of goods yields the demand functions

$$ {C_{t}^{h}}(j)=\left[\frac{{P_{t}^{h}}(j)}{{P_{t}^{h}}}\right]^{-\theta}{C_{t}^{h}} $$

(53)

$$ C_{k,t}(j)=\left[\frac{P_{k,t}(j)}{P_{k,t}}\right]^{-\theta}C_{k,t} $$

(54)

for all j,k ∈ [0,1], where ${P_{t}^{h}}$ is the Home price index (i.e., the index of prices of domestically produced goods) defined as

$$ {P_{t}^{h}}\equiv\left[{\int}_{0}^{1}{P_{t}^{h}}(j)^{1-\theta}dj\right]^{\frac{1}{1-\theta}} $$

(55)

and P_k,t is a price index for goods imported from country k (expressed in Home currency) for all k ∈ [0,1]

$$ P_{k,t}\equiv\left[{\int}_{0}^{1}P_{k,t}(j)^{1-\theta}dj\right]^{\frac{1}{1-\theta}} $$

(56)

We derive the price index for imported goods (also expressed in Home currency)

$$ {P_{t}^{f}}\equiv\left[{\int}_{0}^{1}P_{k,t}^{1-\xi}dk\right]^{\frac{1}{1-\xi}} $$

(57)

The Home households’s relative demand for ${C_{t}^{h}}$ and ${C_{t}^{f}}$ depends on their relative prices ${P_{t}^{h}}$ and ${P_{t}^{f}}$. It is easy to derive:

$$\begin{array}{@{}rcl@{}} {C_{t}^{h}}&=&(1-\gamma)(\frac{{P_{t}^{h}}}{{P_{t}^{c}}})^{-a}C_{t} \end{array} $$

(58)

$$\begin{array}{@{}rcl@{}} {C_{t}^{f}}&=&\gamma(\frac{{P_{t}^{f}}}{{P_{t}^{c}}})^{-a}C_{t} \end{array} $$

(59)

where ${P_{t}^{c}}$ is the composite consumer price index defined as

$$ {P_{t}^{c}}\equiv[(1-\gamma)({P_{t}^{h}})^{1-a}+\gamma({P_{t}^{f}})^{1-a}]^{\frac{1}{1-a}} $$

(60)

The representative household maximizes the following expected discounted sum of utilities over possible paths of consumption and labor:

$$ E_{0}\{\sum\limits_{t = 0}^{\infty}\beta^{t} U(C_{t},N_{t})\} $$

(61)

where the period utility is defined as

$$ U(C_{t},N_{t})=\frac{C_{t}^{1-\sigma}}{1-\sigma}-\frac{N_{t}^{1+\eta}}{1+\eta} $$

(62)

By considering the optimal expenditure conditions (58) and (59), the sequence of budget constraints assumes the following form:

$$ C_{t}+B_{t}=\frac{{P_{t}^{h}}}{{P_{t}^{c}}}Y_{t}+B_{t-1}\frac{1+i_{t-1}}{1+{\pi_{t}^{c}}}+T_{t} $$

(63)

where B _t is the bond holdings of representative home household,²⁰ and T _t is the lump-sum profit transfer from the firms.

The representative household chooses processes $\{C_{t},N_{t}\}_{t = 0}^{\infty }$ and asset portfolio to maximize (61) subject to (63). For any given state of the world, the following set of efficiency conditions must hold:

$$\begin{array}{@{}rcl@{}} C_{t}^{-\sigma}&=&\beta E_{t}[(1+i_{t})(\frac{{P_{t}^{c}}}{P_{t + 1}^{c}})C_{t + 1}^{-\sigma}] \end{array} $$

(64)

$$\begin{array}{@{}rcl@{}} \frac{N_{t}^{\eta}}{C_{t}^{-\sigma}}&=&\frac{W_{t}}{{P_{t}^{c}}} \end{array} $$

(65)

Equation (64) takes the form of a familiar consumption Euler equation. Notice that following large part of the recent literature, we do not introduce money explicitly.

1.2 A.2 Asset Markets

The household is subject to a complete set of assets against all sorts of risks. For any state s of the economy at Period t + 1, there exists a corresponding Arrow security with price v _t that pays one unit of Home currency in period t + 1 if and only if state s occurs.²¹

Assume state s occurs with probability τ(s). Then for the Home households, intertemporal substitution yields:

$$ \frac{v_{t}U^{\prime}(C_{t})}{{P_{t}^{c}}}=\frac{\beta U^{\prime}(C_{t + 1}(s))\tau(s)}{P_{t + 1}^{c}(s)} $$

(66)

where s in the parentheses means it is state dependent.

Substituting U(⋅) with its functional form, and a little tranformation, we can get

$$ \beta\left[\frac{C_{t + 1}(s)^{-\sigma}}{C_{t}^{-\sigma}}\cdot\frac{{P_{t}^{c}}}{P_{t + 1}^{c}(s)}\right]=\frac{v_{t}}{\tau(s)} $$

(67)

The foreign households also have access to this Arrow security, and their intertemporal substitution yields:

$$ \frac{v_{t}U^{\prime}(C_{t}^{*})}{S_{t}P_{t}^{*}}=\frac{\beta U^{\prime}(C_{t + 1}^{*}(s))\tau(s)}{S_{t + 1}(s)P_{t + 1}^{*}(s)} $$

(68)

That is,

$$ \beta\left[\frac{C_{t + 1}^{*}(s)^{-\sigma}}{C_{t}^{*-\sigma}}\cdot\frac{S_{t}P_{t}^{*}}{S_{t + 1}(s)P_{t + 1}^{*}(s)}\right]=\frac{v_{t}}{\tau(s)} $$

(69)

i.e.,

$$ \beta\left[\frac{C_{t + 1}^{*}(s)^{-\sigma}}{C_{t}^{*-\sigma}}\cdot\frac{Q_{t}}{Q_{t + 1}(s)}\cdot\frac{{P_{t}^{c}}}{P_{t + 1}^{c}(s)}\right]=\frac{v_{t}}{\tau(s)} $$

(70)

Comparing (67) and (70), we can get

$$ \frac{C_{t + 1}(s)^{-\sigma}}{C_{t}^{-\sigma}}=\frac{C_{t + 1}^{*}(s)^{-\sigma}}{C_{t}^{*-\sigma}}\cdot\frac{Q_{t}}{Q_{t + 1}(s)} $$

(71)

i.e.,

$$ \frac{C_{t + 1}(s)^{-\sigma}}{C_{t + 1}^{*}(s)^{-\sigma}}Q_{t + 1}(s)=\frac{C_{t}^{-\sigma}}{C_{t}^{*-\sigma}}Q_{t} $$

(72)

The right hand side of the above equation is independent of state s, hence for any state s, we must have

$$ \frac{C_{t + 1}^{-\sigma}}{C_{t + 1}^{*-\sigma}}Q_{t + 1}=\frac{C_{t}^{-\sigma}}{C_{t}^{*-\sigma}}Q_{t} $$

(73)

This equation indicates $\frac {C_{t}^{-\sigma }}{C_{t}^{*-\sigma }}Q_{t}$ is a constant for any t. For convenience, normalize this constant to be unity. This is consistent with a symmetric initial condition with zero net foreign asset holdings. Hence we get

$$ C_{t}=Q_{t}^{\frac{1}{\sigma}}C_{t}^{*} $$

(74)

and its log-linearized version

$$ c_{t}=c_{t}^{*}+\left( \frac{1}{\sigma}\right)q_{t}=c_{t}^{*}+\left( \frac{1-\gamma}{\sigma}\right)\delta_{t} $$

(75)

The last equation employs the fact that

$$\begin{array}{@{}rcl@{}} q_{t}&=&s_{t}+p_{t}^{*}-{p_{t}^{c}}\\ &=&(s_{t}+p_{t}^{*}-{p_{t}^{h}})+({p_{t}^{h}}-{p_{t}^{c}})\\ &=&\delta_{t}+({p_{t}^{h}}-{p_{t}^{c}})\\ &=&\delta_{t}-\gamma\delta_{t}\\ &=&(1-\gamma)\delta_{t} \end{array} $$

(76)