Financial Data Analysis

Transcription

1 Financial Data Analysis Forecasting Asset Returns and Market Eciency (ME) Roman Liesenfeld (University of Kiel) October 7, 008

2 Contents Forecasting Asset Returns and Market Eciency (ME) 3.1 Martingale Random Walk (RW) Hypotheses Tests for Randomwalk I Test for Randomwalk III Tests for LongRange Dependence

3 Forecasting Asset Returns and Market Eciency (ME).1 Martingale The possibility to forecast asset returns is closely related to the concept of market eciency. The classical denition of market eciency was introduced by Fama (1970): Implications: A market in which prices always fully reect available information is called [informationally] ecient. 1) Prices react instantaneously to new information and the best prediction of tomorrow's price, is today's price. ) It is not possible to make prots by trading on the available information. Using dierent information sets, which should be reected in prices, leads to dierent forms of ME: 1) Weakform of ME: Information set includes only past prices. ) Semistrong form of ME: Information set includes publicly available information. 3) Strong form of ME: Information set includes all information, including private information. A stochastic specication which is often associated with the hypotheses of ecient market (in the weak form) is so called martingale model: A martingale is a stochastic process {p t } which satises the following conditions: or equivalently E [p t+1 p t, p t 1,...] = p t E [p t+1 p t p t, p t 1,...] = 0. This implies: The best forecast (in the sense of minimizing the MSE) of tomorrow's price is today's price. 3

4 In other words, all information contained in past prices is instantly and fully reected in asset's current price. Formally, this implies the absence of any linear relationship between p t+1 p t and past prices: if g( ) denotes an arbitrary function and I t = {p t, p t 1,...}, then we have cov [(p t+1 p t ), g (I t )] = E [(p t+1 p t ) g (I t )] = E [E (p t+1 p t I t ) g (I t )] = 0. A problem of associating a martingale with market eciency is the fact, that the martingale ignores the tradeo between expected return and risk, which is supposed in modern nancial economics: According to the CAPM, e.g., we have E [p t 1 p t ] = linear function of the risk premium > 0, where the risk premium is necessary to attract investors to bear the risk. In particular, the expected return in a CAPMmodel is: E [r t+1 I t ] = r f,t+1 + β (E [r m,t+1 I t ] r f,t+1 ), with: r f,t = risk free rate r m,t = return of the market portfolio β = beta factor ( ) = risk premium for bearing the systematic risk associated with the asset = information set I t Hence, the asset returns should be properly adjusted for risk before applying the martingale specication. Another statistical model associated with the hypothesis of ecient market is the random walk (RW) model: closely related to the martingale; often used to test the implication of the EMhypothesis that prices changes should not be predictable; will be discussed in more detail in the next section: 1) Three versions of the RW hypothesis will be introduced; ) tests will be developed for each of these versions. 4

5 . Random Walk (RW) Hypotheses RW I: iid increments The simplest version of a RW for a price process is given by p t = µ + p t 1 + ε t, ε t iid ( 0, σε ) p t p t 1 = µ + ε t, where p t is the logprice, µ is the expected price change (drift), and ε t is a White Noise process. The implications of this RWmodel are: The average increment in prices is given by µ, representing the deterministic time trend of the price process. Increments are serially independent and identically distributed. For µ = 0, the price process is a martingale, since E [p t p t 1 I t 1 ] = µ. However, note that under the RW I increments/returns are not only uncorrelated with past prices (or functions of them) but also independent. To interpret the properties of a RW I, consider the rst two moments of p t p 0 : E [p t p 0 ] = µt + p 0, p 0 = starting value var [p t p 0 ] = σ t. Hence, mean and variance are increasing without bounds, reecting the nonstationarity of the a RW. (This also holds for the two other forms of the RW below). The problem of the RW I applied to model prices over a longer time span: The assumption of iid increments is typically not plausible: Structural breaks due to changes in the institutional and/or economic environment. Volatility clustering. (see, Figure 1) 5

6 Hence, we would expect changes in the distributional behavior of prices and returns. This leads to the following weaker forms of the RWhypotheses: 6

7 Figure 1: Time plots of monthly log returns for IBM stock and S&P500 index from January 196 to December 1999 (T0, p.115). 7

8 RW II: independent increments The RW II has the form p t = µ + p t 1 + ε t ; ε t = serially independent process with zero mean This form allows for dierent distributional forms for ε t including heteroscedasticity. An even less restrictive and weaker form of the RWhypothesis is given by: RW III: uncorrelated increments The RW III has the form p t = µ + p t 1 + ε t ; E [ε t ] = 0, E [ε t ε s ] = { σ ε, if s = t 0, else. The RW III represents the weakest form of the RWhypothesis allowing for nonlinear serial dependencies such as serial correlation in ε t or ε t, ε 3 t. 8

9 .3 Tests for Randomwalk I Based on the assumption of iid increments one has the choice between parametric and nonparametric procedures to test the RW. Nonparametric tests are in general more robust than parametric test, but often more dicult to construct. Here, I'll discuss two nonparametric tests, the CowlesJonestestand the runstest. CowlesJones (CJ)Test The basic idea is to compare the observed number of sequences and reversals in the time series of prices with the expected number of reversals and sequences under a RW I. A sequence is dened as two consecutive returns with the same sign, i.e.: {r t, r t+1 } > 0 or {r t, r t+1 } < 0. A reversal is dened as two consecutive returns with opposite signs Construction of the CJtest statistic: The model to be tested, is the RW I: with a symmetric pdf for ε t. {r t > 0, r t+1 < 0} or {r t < 0, r t+1 > 0}. r t = p t p t 1 = µ + ε t ; ε t iid ( 0, σ ε), Dene the indicator (representing the sign of the return): I t = { 1, if rt 0 0, else, and note that under the RW I, I t is a iid Bernoulli random variable with π P (I t = 1) = P (r t 0) 1 π = P (I t = 0) = P (r t < 0). Furthermore, note that under the RW I, the probability π is given by: > 0.5, if µ > 0 π = 0.5, if µ = 0. < 0.5, if µ < 0 Now, the number of sequences N s in a time series with n + 1 return observations {r t } n+1 t=1 can be obtained as n N s = Y t with Y t = I t I t+1 + (1 I t ) (1 I t+1 ), t=1 9

10 where Y t = { 1, if It = I t+1 = 1 or I t = I t+1 = 0 (sequence) 0, if I t = 1, I t+1 = 0 or I t = 0, I t+1 = 1 (reversal). Note, that the indicator Y t (indicating a sequence) is a Bernoulli random variable with π s P (Y t = 1) = P (I t+1 = 1, I t = 1) + P (I t+1 = 0, I t = 0). Hence, under the RW I which implies serially independent r t 's and I t 's, this probability for a sequence has the form: π s = P (I t+1 = 1) P (I t = 1) + P (I t+1 = 0) P (I t = 0) = π + (1 π) ; and for a reversal: 1 π s = π(1 π). Furthermore observe that π s { = 0.5, if µ = 0 > 0.5, if µ 0. This reects the fact that under a RW without a drift, sequences and reversals are equally likely, and under a RW with a drift, sequences are more likely than reversals (due to the deterministic trend). The CJStatistic is given by ĈJ = # sequences # reversals = N s n N s = 1 n n t=1 Y t 1 1 n n t=1 Y. t In order to use ĈJ for a test of the RW I, it is necessary to know the properties of ĈJ under the RW I: According to the weak law of large numbers (WLLN) the probability limit (for n ) of 1 n T t=1 Y t is given by plim 1 n n Y t = E [Y t ] = 1 π s + 0 (1 π s ) = π s = π + (1 π). t=1 Hence, the CJstatistic has the following probability limit: plim ĈJ = π s = π + (1 π). 1 π s π (1 π) This implies that under a RW without drift, one would expect ĈJ 1, since π s =

11 and under a RW with drift ĈJ > 1, since π s > 0.5. In order to construct a formal test for the RW Ihypothesis based upon the CJ statistic the sampling distribution of ĈJ must be known. It can be shown that the asymptotic distribution for ĈJ is given by: ( ĈJ a πs ) N, asy.var (ĈJ ) 1 π s with [ ] ) π s (1 π s ) + π asy.var (ĈJ 3 + (1 π) 3 πs = n (1 π s ) 4. The unknown parameters can be estimated consistently by ˆπ = 1 n n I t and ˆπ s = ˆπ + (1 ˆπ). t=1 This asymptotic distribution of the CJstatistic is obtained as follows: First, observe that under the RW I the sequence of indicators {Y t } with Y t = I t I t 1 + (1 I t ) (1 I t 1 ) constitute a homogeneous Markov Chain with states Y t = 1 and Y t = 0. In particular, under the RW I we have Y t, Y t 1 Y t, Y t k : are stochastically dependend : k > 1 are stochastically independent with the following homogeneous transition probabilities: P t (Y t = y t Y t 1 = y t 1, Y t = y t, ) = P (Y t = y t Y t=1 = y t 1 ). Note, that the Y t 's are neither iid nor serially uncorrelated. To establish an asymptotic distribution for 1 n t Y t (and hence for ĈJ), one can use the Central Limit Theorem for homogeneous Markov Chains provided by Fish (1976) 1 : 1 see, Fish, M. Wahrscheinlichkeitsrechnung und mathematische Statistik, VEB Deutscher Verlag der Wissenschaften, Berlin, p

12 Theorem: Let {x n } be a sequence of random variables constituting a homogeneous Markov Chain with a nite number of states and E[x j ] = µ < i. Let the states be irreducible with transition probabilities being strictly positive, and [ n ] 1 lim n n var x i = σ > 0. Then n ( 1 n i=1 i=1 ) n d x i µ N ( 0, σ ). In order to apply this result to 1 n First, note that with [ ] var Y t = t t t Y t, we need to calculate σ y = lim n 1 n var[ t Y t]: var [Y t ] + t>s cov (Y t, Y s ), var (Y t ) = 1 P (Y t = 1) + 0 P (Y t = 0) [1P (y t = 1) 0P (Y t = 0)] = π s π s = (1 π s ) π s, cov (Y t, Y t j ) = 0 j > 1, cov (Y t, Y t 1 ) = y t y t 1 Y t Y t 1 P (Y t = y t, Y t 1 = y t 1 ) E [Y t ] E [Y t 1 ] = 1 1 P (Y t = 1; Y t 1 = 1) π s t = π 3 + (1 π) 3 π s. Hence, we obtain: [ ] [ ] var Y t = n (1 π s ) π s + (n 1) π 3 + (1 π) 3 πs and σ y = lim n [ ] 1 [ ] n var Y t = (1 π s ) π s + π 3 (1 π) 3 πs. t This yields according to the Theorem given above the following limiting distribution for stabilizing transformation of 1 n t Y t: [ ] 1 d n Y t π s N ( 0; σ n y). t 1

13 Now, note that the CJstatistic is a continuous function in Z = 1 n t Y t: ĈJ = f(z) = Z df( ), with 1 Z dz = 1 (1 Z). Applying a rstorder Taylor expansion to ĈJ and using the limit distribution for n (Z π s ) then yields : ( ) ( ) ) d df (πs ) n (ĈJ f (πs ) N 0, σy dz and hence n (ĈJ π s 1 π s ) ( d 1 [ ( N 0, (1 π s ) 4 (1 π s ) π s + π 3 + (1 π) 3 πs)] ). Based on this asymptotic result, one can construct the following tstatistic: tĉj = ĈJ ˆπ s/ (1 ˆπ s ) ( ) 1/, asy.var ĈJ which is under the RW I approximately N(0, 1)distributed. Based on this statistic the RW I is rejected at signicance level α, if tĉj > Z1 α/ (twosided test), where Z 1 α/ is the (1 α/)quantile of a N(0, 1)distribution. Figure shows the daily closing prices of the Daimler-Benz stock and the german stock index DAX, and the daily spot exchange rates for the US-Dollar/Deutsche Mark together with the corresponding CJ-statistic. As to the quality of this test, one can show, that it can have a rather low power w.r.t. certain alternatives. In particular, CLM97 construct a twostate Markov Chain for I t implying serially correlated I t 's and r t 's, (and hence violating the RW I hypothesis) such that ĈJ converges to the same probability limit as under a RW I. Hence, the test based on ĈJ can have the tendency to accept the RW I model, even if it is false, too often. Also, note that the test is based upon the assumption of symmetric return distributions, which might be not appropriate. see, e.g., M96, Theorem 5.39, p

14 Figure : The Cowles-Jones statistic for the return series of the Daimler-Benz stock, the german stock index DAX, and the spot exchange rate for the US-Dollar/Deutsche Mark. 14

15 RunsTest Another test for RW I is the (WaldWolfowitz) runs test. 3 This test compares the number of runs which is expected under the RW I with the observed number of runs. A runs is dened as a sequence of consecutive realized values with the same sign. Consider, for example, the following sequence of returns {r t }: t r t sign run no Hence, this particular series contains 5 runs. The runs test is a test which can be used to test the iid assumption. Note that under the RW I r t = p t p t 1 = µ + ε t, ε t iid ( 0, σε ) we have r t iid ( µ, σ ε). Hence, the runs test testing the iid property of the returns can be used as a test for RW I. The basic idea of the runs test of the iid property is that if the random variables under consideration are truly iid, then neither too few nor too many runs should be observed in the sample: Too few runs could indicate clustering or trending. To many runs could indicate a systematic alternating pattern. To perform the runs test, we require the sampling distribution of the total number of runs under the iid property of returns: Let W = number of runs observed in the sequence {r t } n t=1 n 1 = number of positive returns = (n n 1 ) = number of negative returns. n 0 Then, it can be shown that, for given n 0, n 1, the sampling distribution of W under 3 The essential sampling distributions, which are necessary for the construction of runs test were derived by Wald and Wolfowitz (1940) (Annals of Mathematical Statistics, vol. 11.) 15

16 iid returns r t is given by 4 : P (W = w n 1, n 0 ) = with mean and variance: ( n1 1 w/ 1 ) ( ) n0 1 w/ 1 ( n n1 ), for w even ( ) ( n1 1 n0 1 (w 1)/ (w 3)/ E [W ] = n 1n n var [W ] = n 0n 1 (n 0 n 1 n) n. (n 1) ) ( ) ( ) + n1 1 n0 1 (w 3)/ (w 1)/ ( ), for w odd n n1 This sampling distribution is obtained using combinatorial arguments. The computations involved in calculating P (W = w n 1, n 0 ) can become quite tedious. Fortunately, the distribution of runs W is approximately Normal for large samples 5. In particular, for n 0 and n 1, it follows that under H 0 : r t iid Z = W E [W ] = W n 1n 0 n 1 var [W ] n0 n 1 (n 0 n 1 n) n (n 1) d N (0, 1). Hence, based on the runs test statistic Z, the RW I hypothesis is rejected at a significance level α, if Z Z 1 α/ (twosidedtest), where Z 1 α/ is the (1 α/)quantile of a N(0, 1)distribution. A slight adjustment to Z is often made to account for the fact that while a Normal random variable is continuous, W is a discrete random variable: Z = W E [W ] + 1 var [W ] (continuity correction). Figure 3 shows the daily closing prices of the Daimler-Benz stock and the german stock index DAX, and the daily spot exchange rates for the US-Dollar/Deutsche Mark together with the corresponding run statistic. In order to construct runs, we have transformed the returns into a dichotomous variable using two types of returns: positive and negative, { 0 if rt 0 I t = 1 if r t > 0. 4 See, e.g., M96, p See, Mood (1940), The distribution theory of Runs, Annals of Mathematical Statistics, vol. 11,

17 This approach can be generalized by using more than classes to transform the variable r t, e.g., 1 if c 0 < r t c 1 if c It 1 < r t c =.. 1 if c q 1 < r t c q with the number of observations of type i n i = Σ t I (ci 1,c i ] (r t ), i : 1 q and w i w For further details, see CLM97. = number of runs of type i = q i=1 w i = total number of runs. 17

18 Figure 3: The run statistic for the return series of the Daimler-Benz stock, the german stock index DAX, and the spot exchange rate for the US-Dollar/Deutsche Mark. 18

19 As to the quality of this test: The runs test is independent of distributional assumptions. In particular, it does not require symmetric distributions for r t, which is a requirement for the CJtest. A disadvantage of the runs test is that it exhibits low power against higher serial dependence (e.g. serial correlation in r t or r t ). Tests for Randomwalk II The assumption of the RW I of identical return distribution is clearly implausible, especially, when applied to data that span a long horizon (see, e.g., the volatility clustering in Figure 9). Hence, a RW I with a serially independent error process and a varying distributional forms for ε t might be more adequate. However, testing for independence without assuming identical distributions is quite dicult in a time series context; and if we place no restriction on how the distribution of ε t varies over time, it is completely impossible to conduct a test for independence. Hence, we immediately turn to tests for the RW III. 19

20 .4 Test for Randomwalk III Autocorrelation tests The most direct test of the RWhypothesis is to check for serial correlation in the return series. Under the RW III: r t = p t p t 1 = µ + ε t ; E [ε t ] = 0, E [ε t ε s ] = { σ ε, if s = t 0, else (and under RW I and RW II) the returns r t should be serially uncorrelated at all lags. Hence, the RW III can be tested by testing H 0 : r t 's are serially uncorrelated, i.e., H 0 : ρ l = 0, l, where ρ l is the lagl autocorrelation for r t. In order to use the sample autocorrelation ˆρ l = ˆγ l ˆγ 0, ˆγ l = lagl sample autocovariance for {r t } to test H 0 : ρ l = 0, the sampling distribution of ˆρ l must be known: It can be shown that if {r t } T t=1 and E [ rt 6 ] σ6, then 6 : is a sequence of iid variables with nite variance σ E [ˆρ l ] = T l T (T 1) + O ( T ). (The expression O(T ) represents a term which when divided by T converges to a constant as T.) Hence, under the RW I, the sample antocorrelation is a biased estimate for ρ l = 0. However, this bias vanishes for T. Furthermore, it can be shown that under the RW I with uniformly bounded sixth moment: T ˆρl a N (0, 1). 6 See, Fuller, W. 1996, Introduction to statistical Time Series, Chapter 6. 0

21 This last result can be used to perform a test of H 0 : ρ l = 0 and hence of the RW I and RW III hypotheses. H 0 is rejected at a signicance level α if T ˆρl > Z 1 α/ (twosided test), where Z 1 α/ is the (1 α/) quantile of a N(0, 1)distribution. Note that the RW III implies that all ρ l 's are zero. A test for the joint hypothesis H 0 : ρ 1 = ρ = = ρ m = 0 is the BoxPierce test which is based upon the previous results. The test statistic is given by m m Q m = T ˆρ l = ( ) T ρl. Under the RW Ihypothesis l=1 l=1 Q m a χ (m), which results from the fact that Q m a sum of squared random variables which are asymptotically N(0, 1)distributed. In practice, the selection of m may aect the performance of the test: If m is too small: If m is too large: the presence of higher order autocorrelation may be missed; the power may be low due to many insignicant higherorder autocorrelation. To increase the power of the test in nite samples, Q m can be modied as follows (LjungBox statistic): m Q ˆρ l m = T (T + ) T l. Note, that this modication has no impact on the asymptotic properties as (T + )/(T l) 1 for T. l=1 Figure 4 shows the daily closing prices of the Daimler-Benz stock and the german stock index DAX, and the daily spot exchange rates for the US-Dollar/Deutsche Mark together with the corresponding Ljung-Box statistic. One issue which requires a careful interpretation of test results is: the asymptotic distribution of ˆρ l, Q m, Q m are obtained under the assumption that the r t 's are iid However, suppose that r t is uncorrelated but not serially independent (due to, e.g., volatility clustering eects) the tests of H 0 :r t is serially uncorrelated discussed above might lead to biased results. 1

22 Figure 4: The ACF and Ljung-Box statistic for the return series of the Daimler-Benz stock, the german stock index DAX, and the spot exchange rate for the US-Dollar/Deutsche Mark.

23 Varianceratio test The varianceratio (VR) test exploits the fact that under the RW IIII the variance of kperiod log returns must be linear in the number of periods k. In particular, for k = we obtain: var (r t []) = var (r t + r t 1 ) = var (r t ) + var (r t 1 ) + cov (r t, r t 1 ). Hence, under a RW for logprices, we have var (r t []) = var [r t ]. Now, in order to check the RWhypothesis, one can use the varianceratio: V R = var (r t []) var [r t ], which, under stationarity, can be expressed as: V R = var [r t] + cov [r t, r t 1 ] var [r t ] = 1 + ρ 1. Note in particular, that under a RW with ρ l = 0, k 1, the variance-ratio is: V R = 1. In order to construct a statistical test of the RWhypothesis based on VR, we need the corresponding sampling distribution of VR, which will be derived in the following: Let's start with the Null under which the distribution of VR is derived: 7 H 0 : r t = µ + ε t, ε t iid N ( 0, σ ), implying that var [r t ] = σ, var (r t []) = σ. Furthermore, let the data be {p k } n k=0 : time series with n + 1 log-prices. 7 The normality assumption is imposed just for expositional convenience. 3

24 These data can be used to estimate as follows: Note, that ˆµ = 1 n E [r t ] = µ and var [r t ] = σ n k=1 (p k p k 1 ) n ˆσ a = var [r t ] = 1 (p k p k 1 ˆµ) n k=1 ˆσ b = 1 var (r t []) = 1 1 n (p k p k ˆµ). n k=1 }{{} contains only {p 0,p,p 4,...} ˆµ, ˆσ a: conventional ML estimators ˆσ b : estimator, which makes use of the fact that under the Null 1 var (r t []) = σ. According to standard asymptotic theory, these estimators exhibit under H 0 the following properties 8 : plim ˆσ a = σ ; plimˆσ b = σ (consistency) and: n (ˆσ a σ ) a N ( 0, σ 4 ) n (ˆσb σ ) a N ( 0, 4σ 4 ) (asymptotic normality) Furthermore, note that ˆσ a, which is the MLestimator based on the complete data set, is asymptotically ecient (i.e., it exhibits an asymptotic normal distribution with the smallest possible variance, see, e.g., M96.). Accordingly: asy.var [ˆσ a] = σ 4 < asy.var [ˆσ b ] = 4σ 4. Also note that under the Null of a RW the distance between the two variance estimates ˆσ a and ˆσ b should be small. In particular, since ˆσ b and ˆσ a are under the Null asymptotically normal, the same holds for the distance: n ([ˆσ b ˆσ a] 0 ) a N ( 0, σ 4 ), where the asymptotic variance asy.var [ˆσ b ˆσ a] = σ4 is obtained as follows 9 : 8 See, e.g., M96. 9 See, CLM97, p. 51 4

25 First note that asy.var (ˆσ b ) = asy.var (ˆσ a + ˆσ b ) ˆσ a = asy.var (ˆσ a) + asy.var (ˆσ b ˆσ a) + asy.cov ([ˆσ b ˆσ a] ), ˆσ a. Now, observe that the covariance between ˆσ a and ˆσ b ˆσ a must be zero, since ˆσ a is asymptotically ecient (otherwise it would be possible to construct a linear combination of ˆσ a and ˆσ b ˆσ a with lower asymptotic variance than ˆσ a, which would be a contradiction). Hence, we obtain asy.var (ˆσ b ) ˆσ a = ) ) asy.var (ˆσ b asy.var (ˆσ a = 4σ 4 σ 4 = σ 4. In order to construct a test statistic based on the distance of the variance estimators, we need a consistent estimation of the asymptotic variance σ 4. For this purpose, e.g., σ 4 = (ˆσ a) can be used. Then, we obtain under the Null n (ˆσ d 1 = b ˆσ a) = ˆσ b ˆσ a a N (0, 1), (ˆσ a) 1 n (ˆσ a) which can be used to test the RW. In particular, the RW III and RW I is rejected at a signicance level α, if d 1 Z 1 α/ = (1 α/) quantile of a N (0, 1)distribution Another possible test statistic is based on the estimated varianceratio exploiting the fact that under H 0 the variance is linear in the periods: V R = which can be transformed to: var (r t []) var [r t ] = 1 V R 1 = ˆσ b ˆσ a ˆσ a var (r t []) = ˆσ b var [r t ] ˆσ a, 5

26 Under a RW I model: d = n [ V R 1 ] a N (0, 1), which results from the fact that d 1 = d : d = [ ˆσ n b ˆσ a ] ˆσ a = ˆσ b ˆσ a = d 1. 1 n (ˆσ a) However, observe that this equality holds only if (ˆσ a) is used as an estimate of σ 4 in order to construct d 1. For other estimators d 1 d. Figure 5 shows the daily closing prices of the Daimler-Benz stock and the german stock index DAX, and the daily spot exchange rates for the US-Dollar/Deutsche Mark together with the corresponding VR test statistic. Comments on the VRtest The VRtest for period returns can be easily generalized to multiperiod returns. In particular, note that under the RW var (r t [q]) = var [r t + r t h t q+1 ] = q var [r t ], which can be exploited to construct a corresponding test statistic (for details, see, CLM97). The discussion of the asymptotic distributions of the test statistics was based on the iid assumption of the returns. However, this strong assumptions is typically violated since we observe volatility changes and volatility clustering over time. In order to account for this fact, CLM 97 derive a VR test which allows for special forms of serial dependence in the return process (ARCHeects). This modied test then can be used to check the RW III hypothesis, of uncorrelated returns. 6

27 Figure 5: The ACF and Ljung-Box statistic for the return series of the Daimler-Benz stock, the german stock index DAX, and the spot exchange rate for the US-Dollar/Deutsche Mark. 7

28 .5 Tests for LongRange Dependence The test procedures for the RWhypotheses discussed so far, test the Null: r t = p t p t 1 = µ + ε t, ε t iid or ε t WN against the alternative r t is serially dependent or correlated. In particular, these tests are designed to reveal shortrange serial dependence in the return process, i.e., correlation between r t and r t τ, where τ is comparably small. However, a departure from the RW which might be important and which cannot be detected by the tests discussed so far, is the longrange dependence, i.e., correlation between r t and r t τ, where τ is comparably large. The phenomenon of longrange dependence, implying as unusually high degree of persistence and a very long memory in the time series behavior is an important issue discussed in hydrology and geophysics 10. In the following, I provide a short review of the relevant instruments which are necessary to analyze longrange dependence. Fractionally integrated processes Fractionally integrated processes are typical examples for processes with a longrange dependence. Recall that the RWmodel for the log prices can be written as r t = p t p t 1 = ε t, ε t WN (1 L) k p t = k p t = ε t with k = 1, where k (1 L) k is the dierence operator. Hence, the k-th dierence (k = 1) of the p t series is stationary. The typical assumption is that k is an integer, typically k = See, e.g., CLM97. 8

29 Granger and Joyeux (1980) 11 suggested that noninteger values for k might also be useful, and introduced the fractionally integrated processes characterized by non-integer values for k. In order to express (1 L) k is terms of a sum, which is helpful for interpretation, one can use the binomial theorem for noninteger powers k 1 : (1 L) k = ( ( 1) j k j j=0 ) L j, where ( ) k = j ( ) ( 1) j k = j Γ(x) = with Γ(x + 1) = xγ(x). k(k 1)(k ) (k j + 1) j! Γ(j k) Γ( k)γ(j + 1) 0 t x 1 e t dt, (Gamma function), (binomial coecient) Applying this expansion to p t as dened above yields: which can be solved for p t : p t = (1 L) k p t = Γ( k) Γ( k)γ(1) p t + j=0 j=1 Γ(j k) Γ( k)γ(j + 1) p t j = ε t, Γ(j k) Γ( k)γ(j + 1) p t j = ε t Γ(j k) φ j P t j + ε t, with φ j = Γ( k)γ(j + 1). j=1 Hence, a fractionally integrated process can be expressed as an AR( ) process with a particular sequence of coecient {φ j }. p t may also be viewed as a MA( ) process. In particular, assume that (1 L) k is invertible (which is the case for k < 1/). Then, we obtain p t = (1 L) k ε t ( ) = ψ j ε t j, with ψ j = ( 1) j k = j j=0 Γ(j + k) Γ(k)Γ(j + 1). 11 See, Granger and Joyeux (1980), An Introduction to Long Memory Time Series Models and Fractional Dierencing, Journal of Time Series Analysis, Vol 1. 1 Another possibility is to apply a Taylorseries expansion to f(l) = (1 L) k around L = 0 (see, H94, p. 448). 9

30 It can be shown by a corresponding Taylor series expansion that 13 : ψ j = Γ(j + k) Γ(k)Γ(j + 1) (j + 1)k 1, as j. Hence, the impulse response function of the MArepresentation of an invertible fractionally integrated process decays (for large j) hyperbolically with j. Remember, that for a stationary AR(1) p t = (1 φ 1 L) 1 ε t = j φj 1 ε t, e.g., the impluse response function is ψ j = φ j 1, which decays geometrically with j, and hence much faster. This slow decay of the impulse function of fractionally integrated processes reects the longmemory and strong persistence of these processes. Closely related to this result is the fact that for 1 < k < 1 integrated process can be approximated by the ACF of a fractionally where: ρ l Γ(1 k) l k 1, as l, Γ(k) ρ l > 0, if k > 0. (<) (<) Hence, for large l and k ( 1 ; 1 ) the ACF decays hyperbolically with lag l. Remember that the ACF of an AR(1) decays geometrically. Figure 6 shows the ACF of fractionally intergrated processes and of a stationary AR(1). 13 See, H94, p

31 Figure 6: Autocorrelationfunctions ρ l of fractionally integrated processes 31

32 Rescaled Range (R/S) statistic The original statistical measure of long memory proposed by the hydrologist Hurst (1951) 14 is the R/S statistic. Mandelbrot (1971) 15 was the rst who used this measure to analyze longrange dependence in asset markets. Let {r t } T t=1, with r T = 1 T T r t, s T = 1 T denote a time series of returns. Then, the R/Sstatistic is given by: where: Q T = 1 s T max 1 t T t=1 t (r j r T ) j=1 } {{ } =w 1t min 1 t T T (r t r T ), t=1 t (r j r T ) j=1 }{{} =w t w 1t = maximum of the partial sums of the rst t deviations of r j from its mean r T t. Since T j=1 (r j r T ) = 0, w 1t is nonnegative. w t = corresponding minimum of the partial sums, which is nonpositive. Hence, the dierence between w 1t and w t is nonnegative and hence: Q T 0. The R/SStatistic was originally employed by Hurst (1951) to study reservoir storage where the r's are the variable inows from a river in successive time periods and r T was the constant outow. To illustrate the basic principle of this measure consider the following example t r t partial sum t j=1 r j t r T = t t j=1 r j t r T Hence, the maximum of partial sums is w 1t = 0 and the minimum w t = such that Q T =. 14 Hurst (1951), Long Term Storage Capacity of Reservoirs, Transactions of the American Society of Civil Engineers, vol. 116, Mandelbrot (1971), When can Price be Arbitraged Eciently? A Limit to the Validity of the Random Walk and Martingale Models, Review of Economics and Statistics, vol. 53,

33 Feller (1951) 16 has shown that the asymptotic distribution of the standardized R/Sstatistic Q T T 1/ for iid variables is that of the range of a Brownian Bridge. The expectation of this asymptotic distribution is [ ] 1 Asy.E Q T = T π. Hence, under the iidassumption we obtain approximately π E [Q T ] T 1/, as T, and a R/Sstatistic which signicantly exceeds this expectation: π T 1/ Q T is interpreted as evidence of long memory. Since ln E [Q T ] 1 ln ( π ) + 1 ln(t ), as T, a test of the Null of iid returns against the alternative of longrange dependent returns can be implemented using the following auxiliary regression: ln Q τ = α + β ln τ + η τ ; τ = T k,..., T 1, T. If the LSestimate of β signicantly exceeds 1/ one can reject the RWhypothesis for the logprice in favor of longrange dependent returns. Figure 7 plots Q τ and E [Q τ ] π τ 1/ against τ using a log scale for three return series (Dollar/DM, Daimler, DAX). The asymptotic approximation appears to be valid for τ > 30 (Dollar/DM) and for τ > 40 (Daimler, DAX). However, for the Dollar/DMseries, we obtain for τ > 1000 signicant deviations between Q τ and its expected values under the iid assumption, indicating some evidence of long memory. Properties of the R/Sstatistic (see, CLM97): 16 Feller (1951), The Asymptotic Distribution of the Range of Sums of Independent Random Variables, Annals of Mathematical Statistics, vol.,

34 The R/Sstatistic can detect longrange dependence in highly nongaussian time series with large skewness and/or kurtosis, or with nonexisting variance, observed for many return series. (The autocorrelation and varianceratio need not be well dened for a process with innite variance.) An important shortcoming of the R/Sstatistic is its sensitivity to shortrange dependence: symptoms of shortrange memory are sometimes intepreted as longrange dependence. 34

35 Figure 7: Observed and Predicted R/S-statistic for the return series of the Daimler-Benz stock, the german stock index DAX, and the spot exchange rate for the US-Dollar/Deutsche Mark using a log scale. 35