2. THE TRANSFORMED

PROCESS UNDER MARTINGALE DIFFERENCE ERRORS

This section first discusses the weak convergence of the process

under H₀ when the covariate process has the long memory structure (1.3) and when the errors are the martingale differences (1.1), independent of the covariate process. The distribution of this limiting process depends on the null model and hence is unknown. Then a transformation

is given such that

converges weakly to σ_e B(G), where B is the standard Brownian motion on [0,1]. Consequently tests based on suitably normalized process

will be asymptotically distribution free. A computation formula for the specific test based on the supremum norm is also given.

Recall the notation from (1.6) and (1.8) and let

Assume that

Then, almost surely, the least squares estimator

satisfies

, and we obtain

Under (1.1), (1.3), and the assumed independence between {X_t} and {ε_t}, the covariance terms in the variances of

will be zero so that the long memory aspect of {X_t} is dominated by the martingale structure of the errors. Hence,

Moreover, from Lemma 5.4, proved in the last section of this paper, we obtain that

converges weakly to W_G, where W_G is a Gaussian process on

with mean zero and covariance function

. Because of the complicated nature of this covariance function and the fact G is unknown, the close form of the distribution of this limiting process is unknown. To overcome this we now consider a transformation of

whose limiting null distribution is like that of σ_e B(G). Let

Assume that

Then, H(x) is positive definite for all x ≤ x₀.

For a real valued function f, define the transform, for x ≤ x₀,

Whenever

is applied to a Brownian motion, the inner integral is to be interpreted as a stochastic integral. Because

is a linear functional,

is a centered Gaussian process. In fact Stute et al. (1998) observe that

in distribution. They also prove that in the case of the i.i.d. errors and i.i.d. design,

converges weakly to σ_e B(G), on D[−∞,x₀] , so that the asymptotic null distribution of the test based on

is known, where

. A similar result is established when fitting an autoregressive model of order one in Koul and Stute (1999).

This continues to be true when the errors form martingale differences as in (1.1) and the process X_t has long memory as in (1.3), independent of the errors. In view of Lemma 5.4 in Section 5, the proof of this claim is similar to that in Stute et al. (1998).

A computational formula for

is as follows. Let X_(j), 1 ≤ j ≤ T, denote the ordered X_t's in ascending order and

denote the corresponding

's. Also, let, for 1 ≤ s, j ≤ T − 1,

Then, for all x < X_(T),

From (2.4), one sees that if for some 1 ≤ j ≤ T − 1, X_(j) ≤ x < X_(j+1), then

, where

We shall now give a further simplification of this formula in the case of fitting a simple linear regression model: q = 2, h₁(x) ≡ 1, h₂(x) ≡ x. Let

Note that τ_T² = 0. Thus, in calculating the preceding entities we must stay away from the last observation. We will compute our test statistics with x₀ = X_(T−1). Then

is well defined for 1 ≤ j ≤ T − 1. Thus the proposed test of the asymptotic level 0 < α < 1 is to reject the null hypothesis of a linear fit if the statistic

is larger than b_α, the upper 100(1 − α)% quantile of the sup{|B(u)|; 0 ≤ u ≤ 1}.

3. APPLICATION TO THE FORWARD PREMIUM ANOMALY

This section considers some applications of the tests (1.7) and (2.6) to the well-known forward premium anomaly in financial economics and international finance. The anomaly refers to the widespread empirical finding that the returns on most freely floating nominal exchange rates up until the early 1990s appear to be negatively correlated with the lagged forward premium or forward discount. This implies that a nonnegative interest rate differential would on average result in an appreciating currency for the country with the higher rate of interest, which violates the important theory of uncovered interest parity. This rejection has been found to create considerable problems for subsequent modeling and understanding in international finance. A good survey of the literature on the forward premium anomaly is provided by Engel (1996). The papers by Baillie and Bollerslev (1994, 2000) and Maynard and Phillips (2001) emphasize the role of long memory processes in this context and how inappropriate inference that ignores the nature of the long memory aspects may have been partly responsible for the anomaly.

In the following discussion, S_t denotes the spot exchange rate at time t, whereas F_t refers to the forward exchange rate at time t, for delivery at time t + 1. Corresponding logarithmic values are denoted by the lowercase variables s_t and f_t, respectively, and all the rates are denominated with the U.S. dollar as the numeraire currency. The uncovered interest rate parity theory assumes rational expectations, risk neutrality, free capital mobility, and the absence of taxes on capital transfers. A common test of the theory is to estimate the regression

and to test the hypothesis that α = 0, β = 1, and u_t+1 are serially uncorrelated. However, the uncovered interest rate parity condition can be derived from a discrete time, consumption-based asset pricing model, which includes a time dependent risk premium. The risk adjusted real returns over current and future consumption streams of the representative investor are given by E_t [(F_t − S_t+1)/P_t+1] × U′(C_t+1)/U′(C_t) = 0, where U′(C_t+1)/U′(C_t) equals the marginal rate of substitution in terms of utility derived from current and future consumption and P_t represents the price level. A Taylor series approximation to second-order terms implies the alternative regression function

where q_t+1 denotes the logarithm of the intertemporal marginal rate of substitution. Under the null hypothesis of uncovered interest parity, α = 0, β = 1, and u_t+1 is serially uncorrelated. Usually, the Jensen inequality terms Var_t(s_t+1) and Cov_t(s_t+1,p_t+1) are considered statistically negligible and are ignored. Important, the covariance term Cov_t(s_t+1,q_t+1) has the interpretation of being a time dependent risk premium and may have long memory.

To illustrate the preceding testing procedures, we now consider instances where both the forward premium and spot returns appear to have long memory characteristics. Baillie and Bollerslev (2000) and Maynard and Phillips (2001) discuss the case of an unbalanced regression when spot returns are close to being uncorrelated, whereas the forward premium is a long memory process. However, a number of authors, including Booth, Kaen, and Koveos (1982), Cheung (1993), and Lo (1991), have argued that asset and exchange rate returns in general may exhibit long memory behavior, in which case the regression may not be unbalanced. In particular, Cheung (1993) provides convincing evidence for the presence of long memory in spot returns for certain currencies in the initial part of the post–Bretton Woods era of floating exchange rates. He argues that slow adjustments to purchasing power parity can cause the long memory property of spot returns.

A more general motivation is to use a model where both the spot and forward rates have a common stochastic trend term z_t = z_t−1 + v_t, where v_t is a long memory process. Then s_t = z_t + u_1t and f_t = z_t + u_2t, where both u_1t and u_2t are short memory processes. Hence this type of model defines the spot exchange rate as being determined by fundamentals, or long run equilibrium value, z_t, that follow an integrated process with a long memory noise component. This is an empirically very reasonable model because it is well known that price levels and money supplies follow this type of process (e.g., see Baillie, Chung, and Tieslau, 1996). The model is also similar to the theoretical formulation of Mussa (1982), with the v_t process partly determining the state of disequilibrium in the goods market and speed of adjustment coefficient. Hai, Mark, and Wu (1987) further consider this model in an unobserved components framework. The spot returns are then the sum of a short memory, possibly white noise component and a long memory component. It should be noted that if the variance of the u_1t component dominates that of the v_t component, the spot returns may appear approximately uncorrelated in small sample sizes. This would explain the mixed findings on the order of integration of the spot returns that have been reported in the literature. However, there is very widespread empirical evidence that forward premiums are invariably long memory processes and Baillie and Bollerslev (1994, 2000) and Maynard and Phillips (2001) find them well described by ARFIMA(p,d,q) processes, so that (1 − L)^d(f_t − s_t) is a stationary and invertible I(0) process. The presence of long memory components in both the spot returns and forward premium will then suggest the possibility of a balanced regression.

The regression in equation (3.1) was fitted to a very similar sample period and currencies to those of Cheung (1993), where some spot returns appeared to exhibit long memory. The first example uses monthly observations on the Canadian $–U.S. $ spot and the one-month forward rate from January 1974 through December 1991, which realizes a total of T = 215 observations. As expected there is strong evidence of long memory in the forward premium, with the Ljung–Box statistic on the first 20 lags being Q(20) = 803.52. The first twelve autocorrelation coefficients of the forward premium series are 0.89, 0.74, 0.63, 0.55, 0.52, 0.50, 0.49, 0.47, 0.43, 0.36, and 0.28, respectively. The application of the local Whittle estimator resulted in an estimated value of 0.25 for the long memory parameter in the forward premium.¹

The monthly Canadian $–U.S. $ spot returns over the same period also have evidence of long memory with the d estimate in an ARFIMA(1,d,0) model being 0.12 and the robust t statistic being significant at the 0.05 level. We test the hypothesis H₀ with h(x) = (1,x)′, with x representing the lagged forward premium. The OLS estimates of β were (−2.606, −1.432)′ with standard errors of (0.811, 0.407)′. Hence the regression gives rise to the usual negative slope coefficient that is consistent with the anomaly. Ljung–Box tests on the autocorrelations of the standardized residuals and also on their squares failed to reject the hypothesis that the residuals were serially uncorrelated and without autoregressive conditional heteroskedasticity (ARCH) effects. Hence it seemed reasonable to assume that the errors form a martingale difference sequence. This justifies the application of the test statistic

of (2.6) to fit the model (3.1) to this data. For the Canadian forward premium regression,

, which indicates a clear rejection of the null hypothesis.

A further example of the methodology is provided by the monthly British pound–U.S. dollar over the same time period. The maximum likelihood estimation (MLE) of the long memory parameter from an ARFIMA model on spot returns gives a value of 0.293 with standard error of 0.080, which indicates long memory in spot returns. Consistent with previous studies of Baillie and Bollerslev (1994) and Maynard and Phillips (2001) and the evidence for Canada given previously, the British forward premium also exhibits evidence of long memory, the MLE of the long memory parameter being 0.45 with a standard error of 0.124. Estimation of the traditional forward premium anomaly regression (3.1) confirms evidence of the standard anomalous negative slope coefficient. Moreover, there is also evidence of substantial, persistent autocorrelation in the residuals, which can also be represented as a long memory process. An interpretation of this finding can be seen from equation (3.2), where the last covariance term, Cov_t(s_t+1,q_t+1), is associated with a time dependent risk premium. Hence variables associated with the risk premium may give rise to the long memory process in the residuals. Although there are many possible models for a risk premium, for the sake of illustration of the methodology in this paper, it is worth noting that several authors, such as Giovannini and Jorion (1987) and Hodrick (1989), have used the lagged squared forward premium to represent the risk premium.

This motivates fitting the second degree polynomial to the British pound–U.S. dollar data with long memory errors, i.e., testing for H₀ with h(x) = (x,x²)′, where again x represents the lagged forward premium. The ordinary least squares (OLS) estimates of β were (−2.414, −0.244)′ with standard errors of (0.835, 0.129)′. To test for further misspecification of this regression, the statistic in (1.7), with L(T) ≡ 1, was calculated to be 6.72, based on the MLE of the long memory parameter d in the error process equal to 0.262. Thus, again regression also appears to be misspecified. A possible implication of this result is that further economic variable terms associated with a risk premium should be included. The value of the statistic in (1.7) appeared to be relatively stable to alternative semiparametric estimate of d obtained from local Whittle estimation and still led to a clear rejection of the null hypothesis.

5. PROOF OF THE WEAK CONVERGENCE OF

UNDER MARTINGALE DIFFERENCE ERRORS

This section contains the proof of the weak convergence of

to a Gaussian process in the case of the martingale difference errors. The result is stated and proved in Lemma 5.4. The proof of this lemma is facilitated by Lemmas 5.1–5.3.

Recall (1.3)–(1.5). Let

The following two lemmas are analogous to Lemmas 5.1 and 5.2 of Koul and Surgailis (2002), and therefore their proofs can be deduced from that paper and from Lemma 5.1 of Koul and Surgailis (2001). Here we only give the proof of the second part of Lemma 5.2.

LEMMA 5.1. Under assumptions of (1.4) and (1.5) with r = 3, there exist a positive integer l₀ ≥ 1 and a constant C such that for any

,

LEMMA 5.2. Let

, be a real valued function such that

hold for any

. Then there exists a constant C₁ depending only on C in (5.1), such that for any

,

Moreover, for any x₁ < x₂,

Proof. We only prove the (5.3). First, consider |v| ≤ 1; then by (5.1) and (5.2), the l.h.s. of (5.3) does not exceed

Next, consider |v| > 1. Then the l.h.s. of (5.3) does not exceed

By (5.2), the first term of this bound does not exceed

The second term of (5.4) follows similarly. This proves the lemma. █

To state the next important result we now let

LEMMA 5.3. Under the assumptions (1.3)–(1.5) with r = 3, there exists a constant C such that

for all positive integers t and for all −∞ ≤ x₁ ≤ x₂ ≤ x₃ ≤ ∞.

Proof. Fix a positive integer i. Let G(x,y) := G(y) − G(x), G₀(x); = I(x ≥ 0), and

Then one has the telescoping identity

It thus suffices to show the following, where l₀ is as in Lemma 5.1:

These bounds together with the orthogonality of (5.7) imply

Proof of (5.8). According to the definition, we have

Proof of (5.9). For q = 1,

Apply Lemmas 5.1 and 5.2 and (5.3) to obtain the following inequality:

On the other hand, from (5.6), the integrability of g_l, g_l′, which in turn follows from Lemma 5.1, one also obtains

These bounds together with the fact that (x ∨ x³)(x ∧ 1) ≤ x² + x³, for any x > 0, and the independence of {ξ_i} imply that

Note that here the second inequality follows from

, which in turn follows from the Rosenthal inequality and (1.5) with r = 3:

This proves (5.9) for q = 1. █

For q = 2, apply Lemmas 5.1 and 5.2 to obtain

Again, as |U_i,l⁽²⁾(x₂,x₃)| ≤ 2, we obtain (5.9) for q = 2. This completes the proof of (5.5). █

Now consider the process

and assume that (1.1) holds. Note that

. Thus the process

is well defined in D[−∞,∞] . The next lemma gives the weak convergence of the process

to a continuous Gaussian limit in D[−∞,∞] . Recall that G stands for the distribution function of X₀.

LEMMA 5.4. Assume (1.1) with sup_t Eε_t⁴ < ∞, (1.3), (1.4), and (1.5) with r = 3 hold. Then,

where

is a continuous Brownian motion on

with respect to time G and W_G is a continuous mean zero Gaussian process on

with W_G(−∞) = 0 and the covariance function

Proof. Apply the central limit theorem for martingales (Hall and Heyde, 1980, Corollary 3.1) to show that the finite-dimensional distributions converge weakly to the right limit under the assumed conditions.

To prove the tightness, fix −∞ < x₁ < x₂ < x₃ < ∞. Let

. Then

where Y_s = ε_s I(x₂ < X_s ≤ x₃) and W_s = ε_s I(x₁ < X_s ≤ x₂).

Now, if the largest index among s,t,k,l is not matched by any other, then E {Y_sY_tW_kW_l} = 0. Also, because the two intervals (x₂,x₃] and (x₁,x₂] are disjoint, Y_sW_s ≡ 0 for all s, and because the errors are homoskedastic martingale differences (see (1.1)), independent of X_t's, we have E {Y_sW_tY_k²} = 0, for all s < k, t < k. Hence,

But, for a k fixed,

A similar fact is true for E {W_sW_tY_k²}. Consequently, summing over k from 1 to T, in view of the inequality (5.5), we have

This together with Theorem 12.1 and equations (12.5) and (12.10) of Billingsley (1968) and a chaining argument similar to that of Dehling and Taqqu (1989) and Koul and Mukherjee (1993) yields the tightness of the process V_T. Details are left out for the sake of brevity. This completes the proof of (5.10).

To prove (5.11), first we note that from (2.2) and in view of the ergodic theorem, we obtain that

where u_p(1) is a sequence of stochastic processes on

tending to zero uniformly, in probability. This together with (5.10) and the uniform continuity of G and ν(x) completes the proof of (5.11). █