Testing the martingale hypothesis for gross returns

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Testing the martingale hypothesis for gross returns✩ Oliver Linton, Ekaterina Smetanina* Faculty of Economics, University of Cambridge, Sidgwick Avenue, Cambridge CB3 9DD, United Kingdom

A R T I C L E

I N F O

Article history: Received 21 December 2014 Received in revised form 21 February 2016 Accepted 25 February 2016 Available online xxxx JEL classification: C10 C22 G10 G14

A B S T R A C T We propose an alternative ratio statistic for measuring predictability of stock prices. Our statistic is based on actual returns rather than logarithmic returns and is therefore better suited to capturing price predictability. It captures not only linear dependence in the same way as the variance ratio statistics of Lo and MacKinlay (1988) but also some nonlinear dependencies. We derive the asymptotic distribution of the statistics under the null hypothesis that simple gross returns are unpredictable after a constant mean adjustment. This represents a test of the weak form of the Efficient Market Hypothesis. We also consider the multivariate extension, in particular, we derive the restrictions implied by the EMH on multiperiod portfolio gross returns. We apply our methodology to test the gross return predictability of various financial series. © 2016 Elsevier B.V. All rights reserved.

Keywords: Variance ratio tests Martingale Predictability

1. Introduction Variance ratio tests (Cochrane (1988); Lo and MacKinlay (1988); Poterba and Summers (1988)) are widely used to test the (weak form of) Efficient Market Hypothesis (EMH) of no predictability of asset returns. One particular advantage of the variance ratio test over the alternatives, such as the standard Box-Pierce statistic, is that the direction of the ratio depends on all the first K autocorrelations and their relative magnitudes, thus providing the direction of the predictability. The original variance ratio test, developed by Lo and MacKinlay (1988) and all other modifications thereof focus on the log return predictability, where the log return is defined to be the first difference of the log prices, i.e., rt := logPt −logPt−1 . Although very convenient, log returns are just an approximation of the actual return defined by Rt := PPt − 1, which is much harder to work with. Due to its convenience, t−1

most tests of the EMH were developed for the log returns. Here, we focus directly on the simple gross return Rt := PPt and t−1 derive alternative ratio statistics to test the hypothesis that risk adjusted gross returns are a martingale difference sequence. There are many discussions around which choice of return to use to measure performance and to fit asset pricing models. The main argument for logarithmic returns is mathematical simplicity and their continuously compounded interpretation that fits with continuous time models. From the point of view of the buy and hold investor though, the actual return over the relevant

✩ We thank Dario Bonciani, Steve Thiele and Mark Salmon for the helpful comments and suggestions and the Cambridge INET Institute for the financial support. * Corresponding author at: Queens College, Silver Street, Cambridge CB3 9ET, UK. E-mail addresses: [email protected] (O. Linton), [email protected] (E. Smetanina).

http://dx.doi.org/10.1016/j.jempfin.2016.02.010 0927-5398/© 2016 Elsevier B.V. All rights reserved.

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horizon is what matters. In fact, standard economic theory requires that the appropriate measure of return used in deriving the cost of capital for one period, for example, should be E(Rt ), i.e., the true arithmetic mean. This requirement holds whatever the nature of the process that generates Rt . See, for example, the treatment in Copeland and Weston (1988) Chapters 7 and 13. The difference between these return measures can lead to substantial differences in practice for longer horizons, see for example Roll (1983) who argues that using buy and hold returns produces an estimated small firm premium only one half as large as that based on alternative methods. Mindin (2011) also investigates the difference between arithmetic and geometric returns. We exploit the implied scaling of gross returns that follows from a martingale assumption on prices to derive our test statistic. Under our null hypothesis and some mild additional conditions it satisfies a Central Limit Theorem, and we show how to conduct inference under the null hypothesis. In Section 2, we describe our null hypothesis and test statistic. In Section 3 we derive the limiting null distribution under two alternative sets of regularity conditions. In Section 4 we define asymptotic standard errors and and a bias correction based on asymptotic expansion. In Section 5 we define critical values based on subsampling method. In Section 6 we discuss two alternative hypotheses and how they influence the test statistic. In Section 7 we provide the theory for multivariate version of our test statistic. In Section 8 we present an application as well as size and power analysis of our univariate and multivariate test statistics. Section 9 concludes. Throughout the paper “⇒” denotes convergence in distribution. 2. The null hypothesis and test statistic Suppose that stock prices Pt obey the martingale hypothesis (after a constant risk adjustment which we take to be represented by l), or more precisely suppose that the gross return series satisfies  E [Rt+1 |Ft ] = E

Pt+1 |Ft Pt

 = (1 + l )

(1)

for each t, where Ft = s(Pk , k ≤ t) is a sigma-algebra, containing current and past prices and l is a constant. The gross return over the horizon t to t + j can be written as Rt+j ( j) =

Pt+j Pt+j Pt+j−1 Pt+1 = × × ··· × = Rt+1 × Rt+2 × . . . × Rt+j , Pt Pt+j−1 Pt+j−2 Pt

(2)

which is also the buy and hold return for horizon j, Roll (1983). By the law of iterated expectations it follows that   E Rt+j ( j) |Ft = (1 + l )j ≡ l j

(3)

  for all j ∈ Z and all t. Hence, the unconditional means satisfy E [Rt+1 ] = (1 + l ) and E Rt+j ( j) = (1 + l )j . We consider the following ratio tK =

E [Rt+K (K)] = 1. EK [Rt+1 ]

(4)

This ratio is the basis of our testing strategy. Unlike the usual variance ratio statistics, this quantity only depends on the first moments of gross returns, but we show below how this quantity captures linear dependence under the alternative hypothesis. In fact there is a more general class of statistics tK,L,a,b , which can be written as tK,L,a,b ≡

(E [Rt+K (K)])a = 1, (E [Rt+L (L)])b

(5)

where b/a = K/L. We mostly focus on tK and tK,1,1/K,1 . We next turn to estimation. Suppose that we observe a sample of prices on an unequally spaced grid {t1 , . . . , tT }, Pti , i = 1, . . . , T. Define the spacing of the observations di = ti+1 − ti ∈ Z + , for i = 2, . . . , T; regular sampling would have di = 1 for all i, but other structures are encountered in practice. Then define for j = 1, 2, . . . lˆ j =

1  Pt(i+1) 1  Pti +j = , Tj Pti Tj Pt {i:di =j} {i:di =j} i

(6)

  where Tj = T−1 i=1 1 di = j is the number of observations available to compute the j period return. In the special case that the observations are equally spaced, the spacing is di = ti+1 − ti = 1. Then define for j = 1, 2, . . .) lˆ j =

T−j 1  Pt+j . T −j Pt t=1

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We then let lˆ K tˆ K = K . ˆl1

(7)

Although we focus here on this particular statistic, alternative statistics and their limiting distributions are discussed in the Appendix B.1 Many studies use the logarithmic return. The scaling law for logarithmic returns follows from the time additivity they obey, namely rt (K) = log Pt+K − log Pt =

K 

K   log Pt+j − log Pt+j−1 = rt+j ,

j=1

j=1

so that Ert (K) = KE(rt ) and var(rt (K)) = Kvar(rt ). In this case, the mean scaling property is not testable, it holds automatically for the sample average, but the variance scaling law is testable and indeed is widely tested. It is very commonly assumed that returns are lognormal, i.e., letting lower case letters define log returns, rt = log Rt ∼ N(lr , sr2 ),

(8)

where l r = E(rt ) and sr2 = var(rt ). Standard properties of the lognormal distribution imply that

s2 1 + E(Rt ) = exp lr + r 2

implying, to a linear approximation the following relationship between the mean return, and the mean log return E(Rt ) ≈ log(1 + E(Rt )) = lr +

sr2 . 2

So one could understand our testing strategy as capturing the variance effect indirectly, although we do not assume lognormality or even i.i.d. We explore this below in the section on alternatives. We next provide an example when the null hypothesis (1) holds true for gross returns but is rejected for log returns. Suppose that gross returns follow the process Rt+1 =

Pt+1 = (1 + l)(1 + zt+1 vt ), Pt

where zt ∈ (−1, 1) continuously distributed with mean zero and vt ∈ (0, 1) a conditional scale process, for example vt =

1 1 + exp(−st2 )

∈ Ft

with st2 say a GARCH-type process. It follows that E [Rt+1 |Ft ] = (1 + l) which is consistent with our null hypothesis. However, for the log returns we have rt+1 = ln(Pt+1 ) − ln(Pt ) = ln(1 + l) + ln(1 + zt+1 vt ), which is no longer a constant plus a martingale difference sequence. Suppose, for example, that zt is uniformly distributed, then we have E [rt+1 |Ft ] = ln(1 + l) + E [(1 + zt+1 vt )|Ft−1 ]     1 + vt 1 ln − 2vt + vt ln(1 − v2t ) , = ln(1 + l) + 2vt 1 − vt

1

Note that we could also analagously to CLM 2.4.22 calculate lˆ j using non-overlapping observations so that lˆ jno =

1 M

M

Pjk+1 k=1 Pjk+1−j

, where Mj + 1 = T.

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which is not a constant and in general will correlate with rt . That is, we may find evidence against (our version of) the martingale null hypothesis by working with logarithmic returns instead of the actual returns. By the same token, if actually Eq. (8) held with sr2 time varying, we may reject the null based on our statistics, whereas variance ratio statistics with correct critical values would not reject. 3. Distribution theory We now turn to the distribution theory of tˆ K under the null hypothesis. We shall assume that the observations are recorded at equally spaced intervals. Define the sequence ut:t+K ≡

Pt+K − (1 + l )K = Rt+K (K) − (1 + l )K , Pt

(9)

which determines the estimation error in lˆ K . We consider two different cases, namely the “M-dependent” case where we take ut:t+K to be the 2(K − 1)- dependent sequence, and the mixing case, where we allow ut:t+K to be an a−mixing process.2 3.1. M(T)-dependent case Let Zt,K be the following 2 × 1 vector  Zt,K =

ut:t+K ut:t+1



 =

Rt+K (K) − (1 + l)K Rt+1 − (1 + l)

 (10)

Under the null hypothesis (1), the autocorrelation function of Zt,K is zero for all lags bigger than K − 1. Furthermore, if Rt+1 are independent then Zt,K is a 2(K − 1)-dependent sequence, i.e., Zt,K , Zs,K are independent when |t − s| > K − 1. We will not assume that underlying returns are independent over time, but allow them to be “M-dependent” where the order, say L(T), may increase with T. In fact, we will make the high level assumption that Zt,K is M(T) dependent sequence, which is consistent with the underlying return series being L(T) dependent for some L(T). In this case we can apply Berk’s (1973) CLT for finitely dependent triangular array of random variables. Sufficient conditions (which we call MD to denote M-dependence) for a given fixed value of K are as follows. Assumption (MD). 2+d ] ≤ C < ∞ where l = 1, . . . , L with L being the row rank of Z MD1 For some d > 0, for all t, l E[|Z t,K and C is a constant.   lt,K | j MD2 For all i, j, var Z ≤ ( j − i)C . t=i+1 lt,K MD3 The limit below exists and is positive and finite

lim

T→∞

1 var T

T−K 

Zt,K

≡ Y.

t=1

MD4 As T → ∞, M(T)2+2/d /T → 0. For a stationary process, condition MD2 obviously holds; for nonstationary process maybe a further explanation is required. The moment condition MD1 seems natural. A sufficient condition would be: for some d > 0, for all t,     E Rt+K (K)2+d + E R2+d ≤C<∞ t

We can apply this to the case where Rt+1 is an independent sequence, in which case M(T) = K−1, where K is fixed so MD4 is automatically satisfied. However, we allow the gross one period returns to be L(T) dependent, where M(T) = (K − 1)L(T) satisfies condition MD4.

2

The special case of Rt +1 being an i.i.d.sequence is considered in the Appendix A.

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3.2. Mixing case We next allow Zt,K , defined by Eq. (10), to be the a− mixing sequence. Let Ft be the natural filtration {Rt , Rt−1 , . . .}. For this case the sufficient conditions (which we denote by M to denote mixing) for a given fixed value of K for applying the CLT are as follows Assumption M. M1. For a given K the following limit exists T−K 

1 lim var T→∞ T

Zt,K .

t=1

Denote the above limit by Y. M2. Zt,K is a-missing with coefficient a(m) of size r/(r − 1), where r > 1, such that for all t and for any j ≥ 0, there exists some d > 0 for which E|Zit,K Zl(t−j),K |(r+d) < D < ∞ for all i, l = 1, 2. Under Assumptions M1 and M2 the result in Eq. (11) holds. Note that one can replace the mixing condition on Zt,K by the same condition on Rt . These conditions do not require stationarity but do require some uniform bound on moments and mixing. 3.3. Central limit theorem Define  AK =

1 , (1 + l)K

−K (1 + l)



.

Theorem 1. Suppose that Assumptions MD1–MD4 or Assumptions M1–M2 are satisfied. Then under null hypothesis (1) the following holds √

T−K 1   T(tˆ K − 1) = √ AK Zt,K + op (1) ⇒ N(0, WK ), T t=1

(11)



where WK ≡ AK YAK and Y is defined by Eq. (12). The proof of Theorem 1 is provided in Appendix A. In our case, Zt,K and Zt+j,K are uncorrelated for |j| ≥ K, and the form of the asymptotic variance is simpler ⎛ ⎞  ±(K−1)  T−K  j 1 ⎝   Y = lim EZt,K Z(t+j),K ⎠ . 1− EZt,K Zt,K + T −K T→∞ T t=1

(12)

j=±1

Note that the Theorem 1 and the corresponding assumptions are stated for a given fixed value of K. The optimal value of K can be chosen with the data-dependent methods of Andrews (1991), see e.g. Choi (1999) who employs this methodology for the variance ratio tests.

4. Standard errors and bias correction This section is aimed at providing empirical implementation of the mean ratio statistics tˆ K , including the bias correction, and inference based on the asymptotic result stated in Theorem 1. First note that there is a simple expression for the asymptotic variance, namely 

WK ≡ AK YAK =

1 2K

(1 + l )

UK +

K2

(1 + l )2

U1 −

2K UK1 , (1 + l)K+1

(13)

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where U1 and UK are given by:

U1 ≡ lim

T→∞

1 E [(Rt+1 − (1 + l)) (Rt+1 − (1 + l))] = c1 (0) T t

±(K−1)    1    UK ≡ lim E Rt+K (K) − (1 + l)K Rs+K (K) − (1 + l)K = cK (0) + cK ( j) T→∞ T s t

UK,1

j=±1

K−1    1    ≡ lim E Rt+K (K) − (1 + l)K (Rs+1 − (1 + l)) = cK,1 (0) + cK,1 ( j). T→∞ T s t

j=1

The detailed derivation of this result is provided in the proof of Theorem 1 in Appendix A. Empirically, we should estimate WK as follows 2 2K ˆ K + K − 2K ˆ1 − ˆ K1 U 2K U 2 U (1 + l)K+1 1 + lˆ 1 + lˆ

1

ˆK = W

ˆ K ≡ cˆ K (0) + ˆ 1 ≡ cˆ 1 (0) and U U

(14)

±(K−1)



cˆ K ( j)

(15)

j=±1

with T−j−K  1 ût:t+K ût+j:t+j+K , T −j−K

cˆ K ( j) =

(16)

t=j+1

K ˆ K → WK with probability one. The standard errors can be then where ût:t+K = Rt+K (K) − 1 + lˆ and lˆ = lˆ 1 − 1. Then, W ˆ K. easily derived from Eq. (13) as the square root of the corresponding variance W Recall, however, that tˆ K is formed as a ratio of two estimated means: tˆ K =

lˆ K , (lˆ 1 )K

which means that it may be biased in finite samples, just like the sample autocorrelations. Forming the second-order Taylor expansion of tˆ K we have

 2 K(K + 1)lK

lK lˆ 1 − l1 + lˆ 1 − l1 K+1 K+2 l 2 l ( 1) ( 1)

 2 K(K + 1)

K lˆ 1 − l1 + lˆ 1 − l1 . − K 2 1 + l ( ) 2(1 + l ) (1 + l )

tˆ K − 1

lˆ K − lK

(l 1 )K lˆ K − lK

−K

Taking expectations of the above expression we can deduce that the bias corrected estimator of tˆ Kbc is given by K(K + 1)Vˆ 1 , tˆ Kbc = tˆ K − 2 2 lˆ 1 (T − 1)

(17)

where Vˆ 1 estimates consistently the asymptotic variance of

Vˆ 1 =

T−1 2 1 

Rt+1 − R , T −1

R=

t=1

√

 T lˆ 1 − l1 , specifically,

T−1 1  Rt+1 . T −1 t=1

Define:  seK ≡

 WK T

and

ˆK ≡ se

ˆK W . T −K

(18)

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Accounting for the bias correction and estimated standard errors, we obtain the following. Theorem 2. Suppose that Assumptions MD1–MD4 or Assumptions M1–M2 are satisfied. Then

tˆ Kbc − 1 ⇒ N(0, 1) ˆK se

(19)

ˆ K are defined by Eqs. (18) and (17). where tˆ Kbc and se

This version of the CLT for our mean ratio statistic is particularly convenient for empirical implementations. Remark 1. Note that expression (13) invokes 2(K−1) autocovariances of Rt+K and K−1 covariances between Rt+K and Rt+1 , thus WK may or may not be a positive number. This is a well-known problem in long-run variance estimation and different methods exist to ensure that the limiting variance is positive-definite (see Andrews (1991), Newey and West (1987) among others). The majority of the methods are based on the proper scaling of the autocovariances such that the variance terms dominate, which, however, may come at the price of the distorted empirical size of the test statistic. As an alternative, Kiefer and Vogelsang (2005) developed nonparametric covariance matrix estimators relying on fixed-b asymptotics. Finally, the most popular method in the literature is to use the subsampling procedure to directly approximate the limiting distribution of the test statistics, see Politis et al. (1999). We proceed by using the subsampling method to approximate the distribution of our test statistic and calculating its variance.

5. Subsampling With some abuse of notation, the centered and properly scaled test statistic, call it TK , can be re-written as a function of the data {Rt : t = 1, . . . , T}: TK =

√   T tˆ K (R1 , . . . , RT ) − 1 .

Let GT (x) = Pr

√    T tˆ K (R1 , . . . , RT ) − 1 ≤ x

(20)

denote the distribution function of WK . Let tˆ K,b,t be equal to the statistic tˆ K but evaluated at the subsample {Rt , . . . , Rt+b−1 } of size b, i.e.,

tˆ K,b,t = tˆ K (Rt , Rt+1 , . . . , Rt+b−1 )

for

t = 1, . . . , T − b + 1.

We note that each subsample of size b (taken without replacement from the original data) is indeed a sample of size b from the true sampling distribution of the original data. Hence, it is clear that one can approximate the sampling distribution of TK using the distribution of the values of tˆ K,b,t computed over T − b + 1 different subsamples of size b. That is, we approximate the sampling distribution GT of TK by

Gˆ T,b (x) =

T−b+1   √

 1 1 b tˆ K,b,t − tˆ K ≤ x . T −b+1 t=1

Let gT,b (1 − a) denote the (1 − a)-th sample quantile of Gˆ T,b ( • ), i.e.,

gT,b (1 − a) = inf{w : Gˆ T,b (w) ≥ 1 − a}. Please cite this article as: O. Linton, E. Smetanina, Testing the martingale hypothesis for gross returns, Journal of Empirical Finance (2016), http://dx.doi.org/10.1016/j.jempfin.2016.02.010

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We call it the subsample critical value of significance level a. Thus, we reject the null hypothesis at the significance level a if TK > gT,b (1 − a). The computation of this critical value is not particularly onerous, although it depends on how big b is. The subsampling method has been proposed in Politis and Romano (1994) and is thoroughly reviewed in Politis et al. (1999). It works in many cases where the standard bootstrap fails: in heavy tailed distributions, in unit root cases, in cases where the parameter is on the boundary of its space, etc. The following theorem shows that our test based on the subsample critical value has asymptotically correct size: Theorem 3. Suppose that Assumptions MD1–MD4 or Assumptions M1–M2 are satisfied. Suppose that b(T)/M(T) → ∞ and b(T)/ T → 0. Then, under the null hypothesis H0 , lim Pr[WK > gT,b (1 − a)] ≤ a .

T→∞

The proof of this result is standard and goes as follows. The finite sample subsampling distribution of  the statistics W K √ test 

T−b+1 1 ˆK ≤ x ˆ that we denote Gˆ T,b (x) → G(x) in probability as the variance of the U-statistics Gˆ T,b (x) = T−b+1 1 b t K,b,t − t t=1 goes to zero as one can bound the difference between the true sampling distribution and the distribution of the U statistics by the exponential inequality of Hoeffding (1963) (the exact inequality can be found in Theorem 2.2.1 p. 43–44 of Politis et al. (1999)). It follows then that the (1 − a)th quantile of Gˆ T,b (x), gT,b (1 − a) → g(1 − a) in probability, where g(1 − a) is the (1 − a) th quantile of G(x). It then follows by Slutsky’s theorem that the asymptotic rejection probability of the event {WK > gT,b (1 − a)} is exactly a. Theorem 3 shows that our test based on the subsampling critical values has an asymptotically valid size under the null hypothesis. Under additional regularity conditions, we can extend this pointwise result to establish that our test has asymptotically correct size uniformly over the distributions under the null hypothesis, using the arguments of Andrews and Shi (2013) and Linton et al. (2010). For brevity, we do not discuss the details of this issue in this paper. 6. Behaviour under the alternative 6.1. Generic stationary alternative In this section we discuss the behaviour of the population statistics under the generic stationary alternative to Eq. (1). For illustration consider the special case K = 2, when

E



E2

Pt+2 Pt





Pt+1 Pt

 =

=

E [(1 + l + R1 − (1 + l)) (1 + l + R2 − (1 + l))]

(1 + l )2 (1 + l )2 + cov(R1 , R2 ) (1 + l )2

=1+

c1 (0) . (1 + l)2

The second term captures the linear autocovariances and follows the same direction as the usual variance ratio statistics applied to log returns. This shows that our ratio will be one if and only if c1 (0) = 0, where c1 (0) is the first order autocovariance of the net/gross return series. In general, the following formula holds

 K−1  E [Rt+K (K)] j K c( j) 1 − = 1 + K EK [Rt+1 ] (1 + l )2 j=1 +

(K − 2)E[(Rt+1 − ERt+1 ) (Rt+2 − ERt+2 ) (ERt+3 − ERt+3 )]

(1 + l )3 (K − 3)E[(Rt+1 − ERt+1 ) (Rt+2 − ERt+2 ) (ERt+4 − ERt+4 )] + (1 + l )4 E[(Rt+1 − ERt+1 ) (Rt+2 − ERt+2 ) . . . (ERt+K − ERt+K )] + ... + . (1 + l )K Please cite this article as: O. Linton, E. Smetanina, Testing the martingale hypothesis for gross returns, Journal of Empirical Finance (2016), http://dx.doi.org/10.1016/j.jempfin.2016.02.010

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We should associate values of the ratio greater than one with positive dependence/momentum in stock prices and likewise a value of the ratio less than one is associated with negative dependence/contrarian movements in stock prices. Note that under the generic alternative we have that ⎡ E[Rt+K (K)] = E ⎣

K 

⎤

⎡

Rt+j (1)⎦ = E ⎣

j=1

K  



⎤

(1 + l) + Rt+j (1) − (1 + l) ⎦

j=1

= (1 + l)K + (1 + l)K−1

K 

 E Rt+j (1) − (1 + l) j=1

   j E (Rt+j (1) − (1 + l)(Rt+j+1 (1) − (1 + l)) K j=1 ⎡ ⎤ j+1 K−1   K−j ⎣ (K − j)(1 + l) E + (Rt+i − (1 + l))⎦ + (1 + l)K−2 K

K−1 

1−

j=2

i=1

Since the last term on the RHS of the above equation is of a smaller order, we have that  K−1    E[Rt+K (K)] p K j E (Rt+j (1) − (1 + l)(Rt+j+1 (1) − (1 + l)) − → 1 + K 1 − K EK [Rt+1 ] (1 + l)2 j=1

=1+

K K (1 + l)2

K−1 

1−

j=1

j K

 cR ( j),

which in general will not be equal to 1, but will depend on all the autocovariances and their relative magnitudes. In the high frequency situation we might take l 0 and the ratio is approximately E [Rt+K ] E[Rt+1 ]K

1+K

K−1 

1−

j=1

j K

 cR ( j),

which is similar to the usual variance ratio and is likewise depending on all the autocovariances (and their relative magnitudes) in a linear fashion. The above ratio shares a similar advantage of providing the direction of the predictability (in comparison to Box-Pierce Q statistic) and provides an additional advantage of dealing directly with gross returns rather than with log returns. 6.2. Fads model Suppose that the true efficient price obeys ∗ Pt+1

Pt∗

= (1 + l ) Zt+1 ,

(21)

where Zt > 0 is i.i.d. with mean one. Suppose however that the observed price is Pt = Pt∗ gt , where gt > 0 is an i.i.d. (or more generally stationary) misspricing error that has mean one. This model allows actual prices P to deviate from fundamental prices P∗ but only in the short run through the fad process gt . This process is a plausible alternative to the efficient markets hypothesis. It follows that the observed gross return satisfies Rt+1 =

Pt+1 gt+1 = (1 + l ) Zt+1 . Pt gt

(22)

Taking logs we have rt+1 = ln (1 + l ) + ln(Zt+1 ) + ln(gt+1 ) − ln(gt ) ≡ l ∗ + zt+1 + et+1 − et , for some l ∗ , where zt =ln(Zt ) − Eln(Zt ) and et =ln(gt ) − Eln(gt ) are i.i.d. and mean zero; this is just like the Muth (1960) fads model so that observed log returns are autocorrelated. Please cite this article as: O. Linton, E. Smetanina, Testing the martingale hypothesis for gross returns, Journal of Empirical Finance (2016), http://dx.doi.org/10.1016/j.jempfin.2016.02.010

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For gross returns, we have  E

Pt+1 |Ft Pt

 = (1 + l )

1 , gt

so that observed gross returns do not obey the martingale structure — they have time varying conditional mean. In terms of the unconditional means we have  E

Pt+1 Pt



 = (1 + l ) E

 1 , gt

whereby the Cauchy–Schwarz inequality  (1 + dg ) ≡ E

1 gt

 ≥ 1.

Furthermore, we have gt+K Pt+K = (1 + l )K Zt+1 × . . . × Zt+K , Pt gt since we obtain cancellations of the misspricing errors. It follows that  E

Pt+K Pt



= (1 + l )K E



1 gt



and so   P E t+K 1−K

Pt   ,   = 1 + dg Pt+1 K E Pt which tends to be zero as K → ∞. Likewise, we have   1/K P E t+K 1/K−1

Pt 1   → < 1. = 1 + dg Pt+1 1 + dg E Pt In this case the long run value has an interpretation as representing the magnitude of departure from the martingale hypothesis. Theorem 4. Suppose that Assumptions MD1–MD4 or Assumptions M1–M2 are satisfied. Then under the alternative H1 :

E [Rt+K (K)] E[Rt+1 ]K

= 1

it follows that p

tK − →∞.

7. Multivariate case In the multivariate case, we consider directly portfolios. Similarly to the univariate case, for assets j = 1, . . . , J define  E

Pj,t+K |Ft Pj,t





K  = E Rj,t+K (K) |Ft = 1 + lj ≡ lj,K ,

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where Ft = s(Rj,k : k ≤ t, ∀j ∈ J). Let portfolio p be constructed from the assets with weights {wj , j = 1, . . . J} such that J w = 1. It follows that for K = 1, 2, . . . the expected gross return on the portfolio is j=1 j ⎡ lp,K

= E⎣

J 

⎤ wj Rj,t+K (K)⎦ =

j=1

J 



K wj 1 + lj = wj lj,K . J

j=1

j=1

By the binomial theorem, we have

lp,K =

J 

wj

j=1

where lp =

tp,K

J K   K     K l K lj = 1 + Klp + wj ljl , l l l=0

J j=1

l=2

j=1

wj lj . Define the portfolio ratio statistic

  J  J E wR (K) 1 + Klp + Kl=2 (Kl ) j=1 wj ljl lp,K j=1 j j,t+K =   = = .  K J (1 + lp )K 1 + Klp + Kl=2 (Kl )lpl w R E j j,t+1 j=1

(23)

If l 1 = · · · = l J = l, then tp,K = 1. If l j = l + cj /Ja for some cj with |cj | ≤ c < ∞, a > 0, and J is large, then tp,K 1. However, in general this is not the case, and tp,K doesn’t have a simple limit under the martingale hypothesis, even in the equally weighted case where wj = 1/J. We give a further interpretation of tp,K . In the case where wj ≥ 0, we may think of l p,K as an expectation, specifically E*XK , where X is the random variable with outcome 1 + l j with probability wj . Then tp,K = E*XK /(E*X)K . By Liapunov’s inequality, tp,K ≥ 1 for all K. For K = 2, we have explicitly tp,K =

E∗ X 2 var∗ (X) =1+ ≥ 1, (E∗ X)2 (E∗ X)2

with equality on the right hand side only when the assets have the same mean. This suggests that if one forms portfolios from assets that have similar means, i.e., sets of assets within which the cross-sectional variability of one period gross expected return is small, then we should find tp,K 1. We now turn to estimation of tp,K . We can form the sample analogue of l j,K , ∀j ∈ J and ∀K = 1, 2, . . . < T lˆ j,K =

T−K 1  Rj,t+K (K). T −K t=1

Then the sample analogue of tp,K is given by: J w lˆ j=1 j j,K tˆ p,K =  K . J w lˆ j=1 j j,1

(24)

Define uj,t:t+K ≡ Rj,t+K (K) − lj,K . Making use of the first order Taylor expansion we get:  

 J ˆ j,1 − lj,1 ) w l ( l ˆ l w − l j j,K j j,K j,K j=1 j=1 −K =  K  K+1 J J wl wl j=1 j j,1 j=1 j j,1 ⎡ ⎛

J tˆ p,K − tp,K ≈

(25) ⎞⎤

⎢ ⎟⎥ T−K ⎜ ⎢ J ⎟⎥ l u 1 ⎜ ⎜uj,t:t+K − K j,K j,t:t+1 ⎟⎥ . w j K ⎢ ⎢ ⎜ ⎟⎥ J T − K  ⎣j=1 ⎠⎦ t=1 ⎝ wl wj lj,1 j=1 j j,1

=  J

1

j=1

Since Eq. (24) is just a linear combination of martingales adapted to the same filtration Ft , we still have asymptotic normality by the CLT for stationary ergodic martingale difference sequences. Define: ±(K−1)

CK ( j, i) ≡ cK (j, i) (0) +



cK (j, i) (l)

(26)

l=±1

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Fig. 1. The above figure shows the shape of the ratio statistics tˆ Kbc for S&P500 daily prices separately for each decade starting from 1954.

C1 ( j, i) ≡ c1 ( j, i) (0)

(27)

to be the longish-run variances of uj,t:t+K and uj,t:t+1 respectively, where:   cK ( j, i)(l) = E Rj,t+K (K)Ri,t−l+K (K) − lj,K li,K   c1 ( j, i)(l) = E Rj,t+1 Ri,t−l+1 − lj,1 li,1

for

for

l = 0, 1, 2, . . .

l = 0, 1, 2, . . .

(28) (29)

Define also (K−1)

CK,1 ( j, i) ≡ cK,1 ( j, i)(0) +



cK,1 ( j, i)(l),

(30)

l=1

  cK,1 ( j, i)(l) = E Rj,t+K (K)Ri,t−l+1 − lj,K li,1

for

l = 0, 1, 2, . . .

(31)

Fig. 2. The above figure shows the shape of the ratio statistics tˆ Kbc for Dow Jones 30 stocks.

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Fig. 3. The figure plots ratio tˆ Kbc for S&P500 daily prices. The dotted lines represent 95% confidence bands obtained with subsampling.

Fig. 4. The figure plots variance ratio statistic for S&P500 daily log returns. The dotted lines represent 95% confidence bands obtained with subsampling.



  Theorem 5. Suppose that the gross return process is stationary, ergodic and square-integrable ∀j ∈ J, and E Rj,t+1 |Ft = 1 + lj = lj,1 ∀j ∈ J , where Ft = s(Rj,k : k ≤ t, for all j ∈ J) . Then √

 T tˆ p,K − tp,K ⇒ N(0, MVK ) with variance MVK given by

MVK =

J J   j=1 i=1

wj wi CK ( j, i) −

2Klj,K K 2 lj,K li,K CK,1 ( j, i) + C ( j, i) , 1 lp lp2

where CK ( j, i), C1 ( j, i)and CK,1 ( j, i)are defined by Eqs. (26)–(30) and lp ≡

J j=1

wj (1 + lj ) .

Fig. 5. The figure plots ratio tˆ Kbc for IBM daily prices. The dotted lines represent 95% confidence bands obtained with subsampling.

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Fig. 6. The figure plots variance ratio statistic for IBM daily log returns. The dotted lines represent 95% confidence bands obtained with subsampling.

Fig. 7. The figure plots ratio tˆ Kbc for GBP/US daily prices. The dotted lines represent 95% confidence bands obtained with subsampling.

The proof of Theorem 4 can be found in Appendix A. 8. Numerical results 8.1. Applications We next employ our methodology on different datasets: stock market index, high and low-cap stocks, and exchange rate data. We also employ the variance ratio test on the same datasets for comparison. Although, variance ratio tests are widely used in the literature, the results are very mixed and sometimes even conflicting, see e.g. Lim and Brooks (2011). It is also worth mentioning that the results of the variance ratio tests might change depending on the inference method used in application. To make the two tests comparable, we apply them to the same datasets and use subsampling as our inference method for both tests. We first present the graphs for the shape of our test statistics calculated for daily S&P500 and Dow Jones 30 stocks gross returns. (See Figs. 1–2.)

Fig. 8. The figure plots variance ratio statistic for GBP/US daily log returns. The dotted lines represent 95% confidence bands obtained with subsampling.

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Fig. 9. The figure plots ratio tˆ Kbc for INFN daily prices. The dotted lines represent 95% confidence bands obtained with subsampling.

Fig. 10. The figure plots variance ratio statistic for INFN daily log returns. The dotted lines represent 95% confidence bands obtained with subsampling.

In fact, empirically we find that it is not possible to reject the null (of martingale hypothesis) for S&P500 gross returns for small K (Figs. 3–4). The rejection occurs only at K = 90 days, i.e. approximately 3 months horizon. In comparison, for the variance ratio test, the martingale hypothesis is not rejected at any horizon. Similar picture can be seen for the high-cap stock, which we choose to be IBM. The prices are spanned from January, 1997 till August, 2014 (Figs. 5–6). Similar to S&P500 gross returns, the null of mean predictability is not rejected at short horizons, the rejection occurs only at K ≈ 100. We next employ out test statistics for the exchange rate data, namely GBP/US daily prices, spanned from April, 1971 till August, 2014 (Figs. 7–8). For daily GBP/US gross returns we do not reject the null for short horizons, but do so for K ≈ 175. In comparison to S&P500 it takes almost double the time (i.e. horizon) for the null to be rejected, providing an evidence that there is even less predictability for exchange rate data even at long horizons such as half a year. The picture is somewhat unclear for the variance ratio test. It appears that the null is rejected at short horizons, while is not rejected at K > 15. And finally we apply the methodology on the low cap stocks, which sometimes exhibit different (from high-cap stocks) behavior. We choose INFN (Informational Technology) to represent the low-cap stock. It turns out, that although, tK decreases

Fig. 11. The figure plots ratio tˆ Kbc for S&P500 weekly prices. The dotted lines represent 95% confidence bands obtained with subsampling.

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Fig. 12. The figure plots variance ratio statistic for S&P500 weekly log returns. The dotted lines represent 95% confidence bands obtained with subsampling.

Fig. 13. The figure plots ratio tˆ Kbc for S&P500 monthly prices. The dotted lines represent 95% confidence bands obtained with subsampling.

more rapidly compared to the high-cap stocks, the overall picture is very similar for our statistic, while the null is rejected by the variance ratio test at all horizons (Figs. 9–10). We also apply our methodology and the variance ratio test on weekly and monthly SP500 daily data, spanned from January, 1954 till July, 2014. The results are very similar to the daily data for each of the tests. (See Figs. 11–14 .)

8.2. Simulation study In this section we present Monte Carlo simulations to investigate a power and a size of the univariate and multivariate versions of the mean ratio statistic. The calculations below show the results for the theoretical ratio in Theorem 1, when asymptotic variance is calculated according to Eq. (13). We also provide the results for the variance ratio test for comparison.

Fig. 14. The figure plots variance ratio statistic for S&P500 monthly log returns. The dotted lines represent 95% confidence bands obtained with subsampling.

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Table 1 Empirical size of the nominal 5% and 10% univariate mean ratio (MR) and variance ratio (VR) statistics. MR statistic

VR statistic

# of periods

a = 5%

a = 10%

a = 5%

a = 10%

K=2

0.0530 (0.0594) 0.0500 (0.0594) 0.0430 (0.0594) 0.0450 (0.0594) 0.0440 (0.0594) 0.0400 (0.0594)

0.1080 (0.0990) 0.0930 (0.0990) 0.0800 (0.0990) 0.0610 (0.0990) 0.0530 (0.0990) 0.0510 (0.0990)

0.0540 (0.0532) 0.0430 (0.0532) 0.0420 (0.0532) 0.0460 (0.0532) 0.0440 (0.0532) 0.0430 (0.0532)

0.0990 (0.0997) 0.1060 (0.0997) 0.0930 (0.0997) 0.1010 (0.0997) 0.0980 (0.0997) 0.0997 (0.0997)

K=4 K=8 K=10 K=12 K=16

Note: Number of replications N = 1000, T = 250.

8.2.1. Size To investigate the size of the test statistics we simulate the data under the H0 as follows H0 : Pt+1 = (1 + l ) Pt ut+1 , where ut ∼ U (0, 2) such that E[ut ] = 1, l = 0.3 and P1 = 1. Then under H0 it holds that  E

Pt+K Pt

 = E[Rt+K ] = (1 + l)K .

Multivariate version uses the same H0 for each of j = 1, . . . , J assets and portfolio is formed with equal weights wj = 1/J ∀ j ∈ J. We calculate the size of the test for T = 250, 500, 1000, 5000 and report the the size calculated with subsampling in parenthesis. Tables 1–4 show that both mean ratio and variance ratio statistics have some size distortions. For the mean ratio statistics there is slight under-rejection,which becomes quite large when K > 15. For the variance ratio test, for relatively small values of K, there is an over-rejection and under-rejection for large values of K. For both tests the size is recovered with subsampling. In fact, this will hold not only for the univariate version but for the multivariate test statistics as well. As in Tables 1–4 for the univariate test statistics the multivariate analogue has a proper empirical size for K < 10. This result does not depend on the number of assets, rather the number of periods is what matters. For large values of K the test, similar to the univariate case, has severe size distortions, but can be corrected with subsampling. (See Tables 5–7.)

Table 2 Empirical size of the nominal Empirical size of the nominal 5% and 10% univariate mean ratio (MR) and variance ratio (VR) statistics. MR statistic

VR statistic

# of periods

a = 5%

a = 10%

a = 5%

a = 10%

K=2

0.0500 (0.0513) 0.0500 (0.0513) 0.0550 (0.0513) 0.0450 (0.0513) 0.0490 (0.0513) 0.0420 (0.0513)

0.1020 (0.1016) 0.1020 (0.1016) 0.0850 (0.1016) 0.0740 (0.1016) 0.0640 (0.1016) 0.0580 (0.1016)

0.0490 (0.0532) 0.0440 (0.0532) 0.0490 (0.0532) 0.0490 (0.0532) 0.0450 (0.0532) 0.0480 (0.0532)

0.1080 (0.0997) 0.0990 (0.0997) 0.1000 (0.0997) 0.1060 (0.0997) 0.1020 (0.0997) 0.0960 (0.0997)

K=4 K=8 K=10 K=12 K=16

Note: Number of replications N = 1000, T = 500.

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Table 3 Empirical size of the nominal 5% and 10% univariate mean ratio (MR) and variance ratio (VR) statistics. MR statistic

VR statistic

# of periods

a = 5%

a = 10%

a = 5%

a = 10%

K=2

0.0478 (0.0500) 0.0494 (0.0500) 0.0484 (0.0501) 0.0466 (0.0501) 0.0476 (0.0500) 0.0400 (0.0500)

0.0930 (0.1000) 0.0948 (0.1000) 0.0826 (0.1001) 0.0782 (0.1001) 0.0768 (0.1001) 0.0550 (0.1001)

0.0488 (0.0502) 0.0428 (0.0502) 0.0530 (0.0502) 0.0494 (0.0502) 0.0490 (0.0502) 0.0460 (0.0502)

0.0936 (0.1001) 0.0950 (0.1001) 0.0966 (0.1001) 0.0922 (0.1001) 0.0920 (0.1001) 0.0948 (0.1001)

K=4 K=8 K=10 K=12 K=16

Note: Number of replications N = 5000, T = 1000.

8.2.2. Power In order to investigate how powerful is our mean ratio statistic against different alternatives, we consider two alternatives: (1) (2) H1 , representing slowly varying mean and thus being close to H0 ; and H1 under which prices follow stationary AR(2) process — an alternative quite different to H0 . Consider first the fist alternative: (1)

H1 : Pt+1 = (1 + lt ) Pt vt ,

Table 4 Empirical size of the nominal 5% and 10% univariate mean ratio (MR) and variance ratio (VR) statistics. MR statistic

VR statistic

# of periods

a = 5%

a = 10%

a = 5%

a = 10%

K=2

0.0500 (0.0500) 0.0500 (0.0500) 0.0498 (0.0501) 0.0496 (0.0501) 0.0460 (0.0500) 0.0442 (0.0500)

0.0926 (0.1000) 0.0914 (0.1000) 0.0918 (0.1001) 0.0840 (0.1001) 0.0790 (0.1001) 0.0761 (0.1001)

0.0584 (0.0502) 0.0570 (0.0502) 0.0530 (0.0502) 0.0514 (0.0502) 0.0470 (0.0502) 0.0582 (0.0502)

0.1018 (0.1002) 0.0948 (0.1002) 0.0898 (0.1002) 0.0956 (0.1002) 0.0972 (0.1002) 0.0978 (0.1002)

K=4 K=8 K=10 K=12 K=16

Note: Number of replications N = 5000, T = 5000.

Table 5 Empirical size of the nominal 5% multivariate mean ratio statistics. # of periods/assets

J=2

J=4

J=8

J=16

J=24

K=2

0.0522 (0.0580) 0.0480 (0.0580) 0.0456 (0.0580) 0.0418 (0.0580) 0.0426 (0.0583) 0.0320 (0.0587) 0.0256 (0.0602)

0.0494 (0.0580) 0.0542 (0.0580) 0.0430 (0.0580) 0.0436 (0.0580) 0.0436 (0.0583) 0.0284 (0.0587) 0.0210 (0.0602)

0.0542 (0.0580) 0.0498 (0.0580) 0.0490 (0.0580) 0.0450 (0.0580) 0.0446 (0.0583) 0.0358 (0.0587) 0.0274 (0.0602)

0.0516 (0.0580) 0.0490 (0.0580) 0.0506 (0.0580) 0.0450 (0.0580) 0.0414 (0.0583) 0.0358 (0.0587) 0.0284 (0.0602)

0.0486 (0.0580) 0.0536 (0.0580) 0.0492 (0.0580) 0.0478 (0.0580) 0.0420 (0.0583) 0.0394 (0.0587) 0.0316 (0.0602)

K=4 K=8 K=10 K=12 K=16 K=20

Note: Simulations are based on N = 5000 replications, T = 250.

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Table 6 Empirical size of the nominal 5% multivariate mean ratio statistics. # of periods/assets

J=2

J=4

J=8

J=16

J=24

K=2

0.0506 (0.0501) 0.0505 (0.0501) 0.0467 (0.0501) 0.0454 (0.0501) 0.0445 (0.0501) 0.0370 (0.0501) 0.0205 (0.0501)

0.0504 (0.0501) 0.0516 (0.0501) 0.0508 (0.0501) 0.0473 (0.0501) 0.0486 (0.0501) 0.422 (0.0501) 0.0300 (0.0501)

0.0492 (0.0501) 0.0488 (0.0501) 0.0503 (0.0500) 0.0483 (0.0501) 0.0477 (0.0501) 0.0377 (0.0501) 0.0247 (0.0501)

0.0510 (0.0501) 0.0496 (0.0501) 0.0452 (0.0501) 0.0508 (0.0501) 0.0490 (0.0501) 0.0410 (0.0501) 0.0398 (0.0501)

0.0496 (0.0501) 0.0518 (0.0501) 0.0528 (0.0501) 0.0508 (0.0501) 0.0432 (0.0501) 0.0430 (0.0501) 0.0372 (0.0501)

K=4 K=8 K=10 K=12 K=16 K=20

Note: Simulations are based on N = 10000 replications, T = 1000.

where vt ∼ ln N (1, sv ), where s v = ds l , and s l is the variance of the time-varying mean l t . We will vary the value of d to see (1) (1) how the power under H1 changes. Under alternative H1 we have:  E

Pt+K Pt

 = E[Rt+K ] = (1 + lt )K .

In order to simulate l t we simulate returns rt according to the GARCH(1,1) model and define lt = ert − 1, rt = st zt 2 2 st2 = a + bst−1 + crt−1 ,

with zt ∼ N (0, 1) and [a, b, c] = [0.01, 0.95, 0.03] and s12 = 0. The other alternative we consider is that prices follow a stationary AR(2) process. (2)

H1 : Pt+1 = a0 + a1 Pt + a2 Pt−1 + gt , Table 7 (1) (2) Power of mean ratio (MR) and variance ratio (VR) tests against H1 and H1 . # of periods

(1)

(2)

H1

H1 d = 1 .5

d=1 K 2 4 8 16

MR 0.2365 0.5315 0.8340 0.9795

VR 0.0625 0.0535 0.0480 0.0505

MR 0.3100 0.6890 0.9375 0.9985

VR 0.0530 0.0450 0.0485 0.0640

MR 1 0.9998 0.5196 1

VR 0.0718 0.4600 0.4624 0.4294

Note: Simulations are based on N = 5000 replications, T = 1000. Nominal test size is 5%. MR= mean ratio statistics, VR = variance ratio statistic and d = s v /s l = 1.

Table 8 Power of the multivariate test statistics against H12 . # of lags/assets

J=2

J=4

J=8

J=16

J=24

K=2 K=4 K=6 K=10 K=12 K=16

0.1880 0.7070 0.9998 0.9998 1.0000 0.9936

0.5880 0.9450 1.0000 1.0000 0.9996 0.9994

0.9230 0.9960 0.9972 0.9998 1.0000 0.9996

0.9890 0.9940 0.9680 1.0000 1.0000 1.0000

0.9900 0.9960 0.9702 1.0000 1.0000 1.0000

Note: Simulations are based on N = 5000 replications, T = 1000. Nominal test size is 5%.

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Table 9 Power of the multivariate test statistics against H11 . # of lags/assets

J=2

J=4

J=8

J=16

J=24

K=2 K=4 K=6 K=10 K=12 K=16 K=20

1.0000 1.0000 0.2656 0 1.0000 1.0000 1.0000

1.0000 1.0000 0.8522 0.9990 1.0000 1.0000 1.0000

1.0000 1.0000 0.9998 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Note: Simulations are based on N = 5000 replications, T = 5000. Nominal test size is 5%.

where gt ∼ N (0, 1) and a 0 = 0.1; a 1 = 0.9; and a 2 = 0.8. (1) (2) As before for the multivariate version we simulate prices according to the same H1 and H1 for each of j = 1, . . . , J assets and portfolio is formed with equal weights wj = 1/J∀j ∈ J. (2) (1) From Tables 7–9 the power of the mean ratio statistic is quite good against H1 and it also has power against H1 , i.e. against conditional heteroskedasticity, which increases whenever s v /s l > 1. In comparison, the variance ratio test has less (2) power against H1 and has almost no power against heteroskedasticity. However, Charles et al. (2011) conduct a Monte Carlo study, where they show that the Automatic variance ratio (AVR) test of Kim (2009) performs very well against conditional heteroskedasticity. For the multivariate mean ratio test the power against H11 , similarly to the univariate test is increasing with the number of periods and with the number of assets. 9. Conclusion We propose an alternative ratio statistic for measuring mean predictability, which represents the test of the weak form of the EMH. We propose different versions the statistics can be stated and derive their limiting distributions. Applying our methodology to different financial series we do not find evidence against the martingale hypothesis at short horizons, however the null of the mean predictability is rejected at longer (K > 80 days) horizons. We also compare our test with the widely used variance ratio test and show that applied to the same data mean and variance ratio statistics the test may deliver quite different results. Appendix A Proof of Theorem 1. First observe that % #√  $ T lˆ K − (1 + l )K = Var (T − K )0.5

T−K 1  Rt+K (K) − (1 + l )K T −K t=1 &

% T−K T−K    −0.5 K = Var (T − K )−0.5 Rt+K (K) − (1 + l ) ut:t+K = Var (T − K )

VT,K ≡ Var

=

T−K  T−K  t=1 s=1

t=1

1 T −K



cov (ut:t+K , us:s+K ) =

T−K  T−K  t=1 s=1

1 T −K

&

t=1



E (ut:t+K • us:s+K ) =

 T−K T−K T−K−1 T−K−1  T − K − j   1  j c j = c0 + 2 1− cj ct−s = c0 + 2 T −K T −K T −K t=1 s=1

VK ≡ lim VT,K = c0 + T→∞

j=1

∞  j=−∞

j=1

±(K−1)

cj = c 0 +



cj

j=±1

We have for K = 1, 2, . . . and j = 0, 1, 2, . . . , K − 1

 cK ( j) = cov Rt+K (K), Rt+j+K (K)   = E Rt+K (K)Rt+j+K (K) − (1 + l )2K ⎧   2K 2 ⎨ (1 + l )j E R2 R2 ifj > 0 t+K t+K−1 · · · Rt+j Rt+j−1 . . . Rt+1 − (1 + l )  = 2K 2 2 ⎩ (1 + l )−j E R2 R · · · Rt Rt−1 . . . Rt+j − (1 + l ) ifj < 0 t+j+K t+j+K−1 Please cite this article as: O. Linton, E. Smetanina, Testing the martingale hypothesis for gross returns, Journal of Empirical Finance (2016), http://dx.doi.org/10.1016/j.jempfin.2016.02.010

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21

Furthermore c1 (0) = var

1 Rt+1 T

=

t

1  var T2 t

(Rt+1 )

because Rt+1 − E[Rt+1 ] is a martingale difference sequence. We start by performing the first-order Taylor expansion of tˆ K − 1 around the point (l K , l 1 ) is given by:



 lˆ − (1 + l )K K ˆ 1 − l1 = K − l lˆ 1 − (1 + l) K K K+1 1 + l ( ) (l 1 ) (1 + l ) (l 1 ) T−K $  1  # 1 K K−1 R (K) − 1 + l 1 + l R − (1 + l)] = − K [ ( ) ( ) t+K t+1 (1 + l )K T − K t=1 lˆ K − lK

tˆ K − 1

=

−K

lK

T−K $ 1 # ut:t+K − K (1 + l )K−1 ut:t+1 . K T −K (1 + l ) t=1

1

Define U1 ≡ lim

T→∞

1  E [(Rt+1 − (1 + l)) (Rs+1 − (1 + l))] = c1 (0) T s t

and UK ≡ lim

T→∞

±(K−1)    1    E Rt+K (K) − (1 + l)K Rs+K (K) − (1 + l)K = cK (0) + cK ( j) T s t

j=±1

to be the “long-runish” variances of ut:t+1 and ut:t+K respectively, where cK (j) are defined above. Define also UK,1 ≡ lim

T→∞

K−1    1    E Rt+K (K) − (1 + l)K (Rs+1 − (1 + l)) = cK,1 (0) + cK,1 ( j) T s t

j=1

 √ √ Since E T(tˆ K − 1) = 0, then the asymptotic variance of T(tˆ K − 1) is given by

WT,K

& %T−K   1 K−1 Var ut:t+1 = ut:t+K − K (1 + l ) (1 + l )2K (T − K ) t=1 % *T−K +& % *T−K +&   1 K 2 (1 + l )2(K−1) 1 1 var + var − u u t:t+K t:t+1 T −K (1 + l)2K (1 + l )2K (T − K ) t=1 t=1 % T−K

& T−K   1 2K (1 + l )K−1 Cov = ut:t+K , ut:t+1 − T − K (1 + l)2K

#√

$ ≡ Var T tˆ K − 1 =

1

t=1

T−K  T−K 

1

(1 + l )2K (T − K ) t=1 s=1 −

2K

(1 + l )K+1 (T − K )

E

t=1

E (ut:t+K • us:s+K ) +

*T−K 

ut:t+K •

t=1

T−K 

K2

T−K  T−K 

(1 + l )2 (T − K ) t=1 s=1

E (ut:t+1 • us:s+1 ) −

+

ut:t+1 =

t=1

T−K  T−K  1 K2 1 •u E E (ut:t+1 • us:s+1 ) − u + ( ) t:t+K s:s+K 2K 2 T − K 1 + l 1 + l ( ) t=1 s=1 ( ) t=1 s=1 T − K + *T−K T−K   1 2K • = E u u − t:t+K t:t+1 (1 + l )K+1 T − K t=1 t=1 ⎡ ⎤ ⎡ ⎤   T−K−1 T−K−1     1 j j K2 ⎣cK (0) + 2 ⎦ ⎣ c K (j) + c1 (j)⎦ 1− 1− c1 (0) + 2 T −K T −K (1 + l )2K (1 + l )2 j=1 j=1 ⎤ ⎡  T−K−1   2K j ⎣ − 1− cK,1 ( j)⎦ cK,1 (0) + 2 T −K (1 + l )K+1

1

T−K  T−K 

j=1

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Taking the limit, we get the asymptotic variance: ⎡ WK ≡ lim WT,K = T→∞

1

(1 + l )2K

⎣cK (0) +

−

(1 + l )K+1



⎤

⎡

cK (j)⎦ +

j=±1

⎡

2K

±(K−1)

⎣cK (K − 1) +

±(K−1)



K2

(1 + l )2

⎣c1 (0) +

±(K−1)



⎤ c1 (j)⎦ −

j=±1

⎤

cK (K − 1 + j)⎦ .

j=±1

Making use of U1 , UK and UK,1 , the asymptotic variance WK can be expressed as

WK =

1 2K

(1 + l )

UK +

K2

(1 + l )

2

U1 −

2K

(1 + l )K+1

UK,1

This completes the proof. Remark 2. If we assume that Rt+1 is an i.i.d. sequence, then the asymptotic variance of the statistics tˆ K is given by the following simplified formula

var (tˆ K ) =

1 2K

(1 + l )

var

(lˆ K ) +

K2

(1 + l )

2

var(lˆ 1 ) −

2K

(1 + l )K+1

cov (lˆ K , lˆ 1 ),

where ⎡

2 ⎤ T−1  1 1 var (lˆ 1 ) = E ⎣ Rt+1 − (1 + l ) ⎦ = var T T

(Rt+1 ).

t=1

and var (lˆ K ) =

1 T

 var

+2

K−1 

(Rt+1 ) + (1 + l)2

(1 + l )

2(K−j)

 var

K

 − (1 + l)2K + 2

(Rt+1 ) + (1 + l)

j

 − (1 + l)

2K

⎞ ⎠.

j=1

Finally, cov (lˆ K , lˆ 1 ) =

   1  K (1 + l)K−1 var (Rt+1 ) + (1 + l)2 − (1 + l)K+1 . T

Proof of Theorem 3. The proof follows from standard arguments, see Linton et al. (2005). Proof of Theorem 5. The first-order Taylor expansion of tˆ p,K is given by:



 J J wj lj,K w lˆ j,K − lj,K wl lˆ j,1 − lj,1 j=1 j j=1 j j,K tˆ p,K −  −K  = K ≈  K K+1 J J J wl wl wl j=1 j j,1 j=1 j j,1 j=1 j j,1 ⎡ ⎤ J 1 T−K J T−K  wl 1  1 t=1 uj,t:t+1 j=1 j j,K T−K ⎣ ⎦ wj uj,t:t+K − K =  K J J T −K wj lj,1 t=1 j=1 w l j=1 j=1 j j,1 J

j=1

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23

⎡

MVT,K

⎛ ⎞⎤ J w l √ j j,K j=1 ⎢ ⎜ ⎟⎥ = Var ⎣ T ⎝tˆ p,K −  K ⎠⎦ = J wl j=1 j j,1 ⎡ ⎛ ⎞⎤ J √ 1 T−K J T−K   wj lj,K T−K T 1 t=1 uj,t:t+1 ⎥ j=1 ⎢ ⎠⎦ = = Var ⎣  wj uj,t:t+K − K K ⎝ J J T −K wj lj,1 t=1 j=1 w l j=1 j j,1 j=1 ⎧⎛ * +⎞⎫ J T−K ⎨  ⎬ lj,K uj,t:t+1 wj  T ⎠ = uj,t:t+K − K J  2K Var ⎩⎝ ⎭ T −K J wl t=1 j=1 wl l=1 l l,1 j=1 j j,1 ⎧ ⎡

⎤⎫ J T−K ⎨ 1 ⎬   lj,K uj,t:t+1 T ⎦ = ⎣ uj,t:t+K − K J wj =  2K Var ⎩ T − K J ⎭ wl t=1 j=1 wl l=1 l l,1 j=1 j j,1 ⎡

   J J T−K   T−K  lj,K uj,t:t+1 1 1 ⎣ × wj wi E uj,t:t+K − K J =  2K T − K J wl t=1 s=1 j=1 i=1 wl l=1 l l,1 j=1 j j,1

+ li,K ui,s:s+1 × ui,s:s+K − K J = wl l=1 l l,1 ⎡    J J T−K   T−K  lj,K uj,t:t+K 1 1 ⎣ =  w w E uj,t:t+K ui,s:s+K − K J ui,s:s+1 − j i 2K T − K J wl ll,1 t=1 s=1 j=1 i=1 w l l=1 j j,1 j=1 ⎞⎤ lj,K li,K uj,t:t+1 ui,s:s+1 ⎟⎥ li,K ui,s:s+1 uj,t:t+K + K 2  − K J 2 ⎠⎦ = J wl wl l=1 l l,1 l=1 l l,1 ⎧ ⎤ ⎛⎡  ±(T−K−1)  J J ⎨   T −K−l 1 ⎝ ⎣ cK ( j, i)(l)⎦ − wj wi cK ( j, i)(0) + =  2K ⎩ J T −K j=1 i=1 l=±1 w l j=1 j j,1 ⎤ ⎡  ±(T−K−l)   lj,K T − K − l ⎣cK,1 ( j, i)(0) + cK,1 ( j, i)(l)⎦ + −2K J T −K l=±1 r=1 wr lr,1

& lj,K li,K 2 +K J c1 ( j, i)(0) . r=1 wr lr,1

Taking the limit as T → ∞ the multivariate asymptotic variance becomes:

MVK ≡ lim MVT,K =  J T→∞ j=1

⎡ −2K J

lj,K

r=1

wr lr,1

1 wj lj,1

⎧ J J ⎨ 

2K ⎩

⎣cK,1 ( j, i)(0) +

j=1 i=1 ±(K−1)



⎛⎡ wj wi ⎝⎣cK ( j, i)(0) +

±(K−1)



⎤ cK ( j, i)(l)⎦ −

l=±1

⎤

cK,1 ( j, i)(l)⎦ + K 2 

l=±1

⎞⎫ ⎬ ⎠ . c ( j, i)(0) 1 J ⎭ r=1 wr lr,1 lj,K li,K

Define ±(K−1)

CK (i, j) ≡ cK (j, i) (0) +



cK (j, i) (l)

l=±1

and C1 (i, j) ≡ c1 (j, i) (0) to be the longish-run variances of uj,t:t+K and uj,t:t+1 respectively, where cK ( j, i)(l) ≡ lim

T→∞

 1   E Rj,t+K (K)Ri,s−l+K (K) − lj,K li,K T s

for

l = 0, 1, 2, . . .

t

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and c1 ( j, i)(l) ≡ lim

T→∞

 1   E Rj,t+1 Ri,s−l+1 − lj,1 li,1 T s

for

l = 0, 1, 2, . . .

t

Define also (K−1)

CK,1 (i, j) ≡ cK,1 ( j, i)(0) +



cK,1 ( j, i)(l),

l=1

where

cK,1 ( j, i)(l) ≡ lim

T→∞

 1   E Rj,t+K (K)Ri,s−l+1 − lj,K li,1 T s

for

l = 0, 1, 2, . . .

t

Making use of the notation above, MVK can be expressed as ⎛ 1

MVK =

J 

2K

wj lj,1

⎞

⎜ ⎟

 ⎜ ⎟ 2K lj,K K 2 lj,K li,K ⎜ ⎟ wj wi ⎜CK (i, j) − CK,1 (i, j) + C (i, j) ⎟ 1

2 J ⎜ ⎟  J  j=1 i=1 ⎝ ⎠ wl ll,1 wl ll,1

J J  

l=1

j=1

l=1

This completes the proof.

Appendix B Alternatively statistic tK can be written as

tK ≡

E/K [Rt+K (K)] =1 E [Rt+ ]

(32)

or taking logs of the above equation we get tK ≡ ln E [Rt+K (K)] − K ln E [Rt+1 ] = 0.

(33)

First, consider statistics tK : tK =

E1/K [Rt+K (K)] = 1. E [Rt+1 ]

Forming the sample analogue of tK and approximating it with the first order Taylor expansion we get:  (1−K )/K ˆ l K − lK

1 lK tˆ K − 1 ≈ K

1/K

−

lK

 lˆ 1 − l1 l12

  + oP T −1/2 =

l1     ˆ (1 + l ) lˆ 1 − (1 + l ) 1 lK − (1 + l )K − + oP T −1/2 = = K 2 K (1 + l ) (1 + l )        lˆ 1 − (1 + l ) 1 lˆ K − (1 + l )K 1

−1/2 ˆ K − 1 + oP T −1/2 . = + o t − T = P K 1+l K (1 + l )K

Thus, by the CLT we have:  √  1 T tˆ K − 1 ⇒ N(0, 2 WK ) K Please cite this article as: O. Linton, E. Smetanina, Testing the martingale hypothesis for gross returns, Journal of Empirical Finance (2016), http://dx.doi.org/10.1016/j.jempfin.2016.02.010

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25

where WK is given in Theorem 1. The standard errors for tˆ K will be different from those of tˆ K by the factor of 1/K. However, due to the slightly different Taylor expansion, the bias correction term will be different. More precisely, the second-order Taylor expansion is given by

    1/K

2 l lˆ K − lK lˆ 1 − l1 11 1 (lK )1/K−2 ˆ l K − lK + −1 − K + 2 l1 2K K l1 l1

(1/K−1)

1 lK tˆ K − 1 ≈ K

1/K

lK

+

l13

+=

 

2 lˆ 1 − l1 + oP T −1/2 =

 

2 (1 + l ) lˆ 1 − (1 + l ) (1 − K) 1 lˆ K − (1 + l )K − + lˆ K − lK + K 2 2 2K K 2K (1 + l) (1 + l ) (1 + l )

2 lˆ 1 − l1 (1 +

l)2

  + oP T −1/2 .

bc

Taking expectations of the above expression we can deduce that the bias corrected estimator of tˆ K is given by bc tˆ K = tˆ K −

2K 2

Vˆ 1 (1 − K)Vˆ K + 2 ,

2K lˆ 1 (T − K) lˆ 1 (T − 1)

(34)

where Vˆ K estimates consistently the asymptotic variance of Vˆ K =

T−K 2 1 

Rt+K (K) − RK , T −K

RK =

t=1

√

 T lˆ K − lK , specifically,

T−K 1  Rt+K (K). T −K t=1

and Vˆ 1 =

T−1 2 1 

Rt+1 − R1 , T −1 t=1

R1 =

T−1 1  Rt+1 . T −1 t=1

For the third alternative ratio statistic, which we denote tK we have: tK = ln E [Rt+K (K)] − K ln E [Rt+1 ] = 0. Forming the sample analogue and making use of the first order Taylor expansion we have:   lˆ 1 − (1 + l ) lˆ K − lK lˆ 1 − l1 lˆ K − (1 + l )K tˆ K ≈ = tˆ K − 1. −K = − K lK l1 1+l (1 + l )K This means that tˆ K has the same Taylor expansion as tˆ K , and thus resulting in the same limiting distribution: √ T tˆ K ⇒ N(0, WK ) where WK is given in Theorem 1. Since tˆ K has the same asymptotic distribution, the standard errors and the bias correction coincide with those of tˆ K .

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