1044 The Econometrics of Exchange Rates
Such a regression appears to imply rejection of efficiency. Granger and Joyeux
(1980) illustrate how long memory can arise via aggregation. Alternatively, Granger
and Hyung (2004) and Diebold and Inoue (2001) show that structural breaks or
regime switching can generate spurious long-memory behavior in an observed time
series. Granger and Teräsvirta (1999) provide an abstract example of a nonlinear
model that can generate data with the misleading linear property of long memory.
They suggest that other nonlinear models with this property are worth searching
for. Byers and Peel (2003) show that data generated from an ESTAR can exhibit the
long-memory property whether in raw or temporally aggregated form. That this
might be the case was an early conjecture of Acosta and Granger (1995). In this
respect, recent applied work which has tried to explain the anomaly by nonlinear-
ities induced by transactions costs and other frictions seems promising (see, e.g.,
Coakley and Fuertes, 2001; Leonet al., 2003; Sarnoet al., 2006).^45
We know from analysis of the empirical work on PPP discussed in section 22.2
that fractional processes can appear to parsimoniously explain PPP deviations but
are not as theoretically well motivated as the nonlinear models that also appear to
parsimoniously explain the data. Leonet al.(2003) and Sarnoet al.(2006) provide a
nice rationale for nonlinearity based on arguments by Lyons (2001) as to the limits
to speculation (see Shleifer and Vishny, 1997). The idea is that financial institutions
will only engage in uncovered arbitrage – a currency trading strategy – if the strategy
yields a Sharpe ratio at least equal to an alternative investment strategy, such as
a buy-and-hold equity strategy. The Sharpe ratio is defined as(E[R(s)]−R(f))/σs,
whereE[R(s)]is the expected return on the strategy,R(f)is the risk-free interest
rate, andσsis the standard deviation of the returns to the strategy. In the foreign
exchange market the excess return equalsE[R(s)]−R(f)=E[st+ 1 −st−(ft−st)]
and the denominator is determined by the exchange rate variances. In the case
of multiple-exchange rate strategies, the covariances among the exchange rates
considered in the currency strategy would also be included in the denominator.
The Sharpe ratio can be interpreted as the expected excess return from specu-
lation per unit of risk. Sarnoet al.(2006) point out that the Sharpe ratio for a
buy-and-hold equity strategy has averaged about 0.4 on an annual basis for the US
over the last 50 years or so. Only whenθdeparts from unity does the numerator of
the Sharpe ratio becomes positive. In fact, perhaps surprisingly, only whenθ≤− 1
orθ≥3 is the Sharpe ratio for currency strategies about the same magnitude as
the average from a buy-and-hold equity strategy, i.e., 0.4 (see Lyons, 2001, p.210).
Consequently, there is a band of inaction such that, if− 1 <θ<3, financial insti-
tutions would have no incentive to take up a currency strategy. Deviations from
uncovered interest parity are too small to attract speculative funds so that the spot
exchange rate and forward exchange rate need not move together.
As with PPP, these types of considerations suggest either a threshold or an ESTAR
type of adjustment mechanism. The latter is justifiable by an appeal to heteroge-
neous agents, who face different levels of position limits and the like. Consider the
following ESTAR adjustment mechanism estimated by Sarnoet al.(2006):^46