Monitoring Shifts in Mean: Asymptotic Normality of Stopping Times 1

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1 Monitoring Shifts in Mean: Asymptotic Normality of Stopping Times Alexander Aue 2 Lajos Horváth 2 Piotr Kokoszka 3 Josef Steinebach 4 Abstract: We consider a sequential procedure designed to detect a possible change in the mean of a random sequence. The procedure is motivated by the problem of detecting an early change in the mean of returns and is based on the CUSUM (cumulative sum) statistic. It terminates when the CUSUM crosses a boundary function or the number of collected observations reaches a prescribed number. We show that the stopping time is asymptotically normal. Simulations using models derived from returns on indexes and individual stocks show that the normal approximation holds in finite samples. Keywords and Phrases: Asymptotic normality; change in the mean; CUSUM; Sequential detection. Research partially supported by NATO grant PST.EAP.CLG , NSF grants INT and DMS and grant MSM Department of Mathematics, University of Utah, 55 South 440 East, Salt Lake City, UT , USA, s: aue@math.utah.edu and horvath@math.utah.edu 3 Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT , USA, Piotr.Kokoszka@usu.edu 4 Mathematisches Institut, Universität zu Köln, Weyertal 86 90, D 5093 Köln, Germany, jost@math.uni-koeln.de

2 Introduction We study the change in the mean model defined as i + µ, i < m + k, X i = i + µ + m, m + k i <, where k and i } is an error sequence to be specified below. If m 0, then a change in mean occurs at time k. The means before and after the possible change are unknown and k, the time of the possible change, is also not given. We are interested in on line monitoring for a change point k. We consider the case of small k, i.e. we assume that a change occurs soon after the monitoring has commenced. Such a formulation is, for example, relevant whenever there is a need to quickly detect a change in the mean returns which corresponds to a change in the behavior of the stock price from growth to a sudden decline, or vice versa. Building on schemes of Chu et al. (996), a family of monitoring procedures was developed by Horváth et al. (2004). In these procedures, the first m observations are used as a training sample and asymptotics are established as m tends to infinity. In this paper we further study a modification proposed in Horváth et al. (2005), which was specifically designed to detect early changes. We show that the detection time is asymptotically normal, as m. We are not aware of similar results formulated in the context of traditional sequential monitoring; typically only the expected value of the detection time, known as the Average Run Length, can be computed, see Basseville and Nikiforov (993) for an extensive overview. The paradigm of Chu et al. (996) and Horváth et al. (2004, 2005) allows however to derive more complex distributional properties of the detection time, the asymptotic normality being an important result. Our procedure is based on the CUSUM (cumulative sum) statistic, because this statistic leads to procedures with good power when k is small. Leisch et al. (2000) and Zeileis et al. (2005) proposed MOSUM (moving sum) procedures to detect changes. Their method works better if the change occurs later in the sequence. Leisch et al. (2000) and Zeileis et al. (2005) defined closed end stopping rules, i.e. procedures that will terminate after a prescribed number of steps even under the no change null hypothesis. The procedure we study here is a closed end procedure. While Leisch et al. (2000) and Zeileis et al. (2005) concentrated on empirical adjustments, our focus is on rigorous mathematical justification. In the remainder of this section we provide a more detailed background. Horváth et al. (2004) studied the stopping time τ m = infk : Γ(m, k) g(m, k)} with the understanding τ m = if Γ(m, k) < g(m, k) for all k =, 2,.... The detector Γ(m, k) and the boundary function g(m, k) were chosen so that under the no change null hypothesis ( m = 0), lim P τ m < } = α, (.) 2

3 where 0 < α < is a prescribed number, and under the alternative ( m 0), lim P τ m < } =. (.2) By (.), as in the Neyman-Pearson test, the probability of a false alarm and hence stopping when we should not, is asymptotically α. Horváth et al. (2004) used the detector where Γ(m, k) = ˆQ(m, k) = ˆσ m X m = m i m X i (X i X m ), (.3) and where ˆσ m 2 is an asymptotically consistent estimator for σ 2 = lim n Var( i n i)/n. The boundary function was chosen as ( g(m, k) = cm /2 + k ) ( ) γ k, 0 γ < /2. (.4) m m + k Under regularity conditions and different assumptions on the errors i }, Horváth et al. (2004) and Aue et al. (2005) proved that under H 0 ( m = 0), } lim P τ W(t) m < } = P c, (.5) t γ sup 0<t where W(t), 0 t < } is a standard Wiener process (Brownian motion). Aue and Horváth (2004) showed that } lim P τm a m x = Φ(x), (.6) b m where Φ denotes the standard normal distribution function, a m = ( ) cm 2 γ γ m Relation (.6) implies that τ m ( ) c γ 2γ m 2( γ), m and b m = σ γ m a/2 m. so the reaction time to the change is (asymptotically) small, if γ is close to /2. Naturally, we want to stop as early as possible after the change in the mean, so we might want to 3

4 use the boundary function g(m, k) with γ = /2. Note however, that γ = /2 is not allowed in definition (.4). This is because by the law of the iterated logarithm at zero, (.5) cannot hold when γ = /2. It can be readily verified that if we used γ = /2 in the definition of τ m, the procedure would stop with probability one even under H 0. Horváth et al. (2005) therefore proposed the following modification of τ m : τ m = infk : k N, ˆQ(m, k) c(m; t)g (m, k)}, with τ m = N if ˆQ(m, k) < c(m; t)g (m, k) for all k N, where N = N(m), ( g (m, k) = m /2 + k ) ( ) /2 k (.7) m m + k and with c(m; t) = t + D(log m) A(log m) (.8) A(x) = (2 logx) /2, D(x) = 2 log x + 2 log log x log π. 2 Comparing τ m to τ m, one can observe that, apart from using a finite monitoring horizon and γ = /2 instead of γ < /2, the constant c in (.5) is replaced with a sequence c(m; t). The asymptotic properties of τ m were studied by Horváth et al. (2005) under the no change null hypothesis and the assumptions of independent, identically distributed errors. For ease of reference, we state here the main result of Horváth et al. (2005). Theorem. If i, i < } are independent, identically distributed random variables with E i = 0, E 2 i = σ2 > 0, E i ν < with some ν > 2, m = 0 and then c m N c 2 m λ with some c, c 2 > 0 and λ <, (.9) lim P ˆQ(m, k) < c(m; t)g (m, k) for all k N} = exp( e t ). (.0) The limit result in Theorem. is a Darling Erdős type extreme value limit theorem and it is related to the distribution of W(t)/t /2. To apply it, we choose t in (.8) such that exp ( e t ) = α. This implies that under the null hypothesis lim P τ m < } = lim P stopping in N steps } = α. In the present paper we focus on the behavior of τm under the alternative. Moreover, we allow the errors i to be dependent. We show that under weak assumptions on the errors, which permit serial dependence and conditional heteroskedasticity, the stopping 4

5 time τm is asymptotically normally distributed. We use this result to conclude that the delay time of τm is asymptotically smaller than the delay time of any τ m with γ < /2. This provides a theoretical justification for the simulation results reported in Horváth et al. (2005). These asymptotic results are stated in Section 2. Section 3 examines the normality of τm for finite values of m and conditionally heteroskedastic models derived from estimating returns on market indexes and individual stocks. The proof of our main result, Theorem 2.2, is presented in Section 4. 2 Main results In order to accomodate applications, we require much weaker conditions than the independence of the innovations, 2,.... Namely, we assume that, 2,... form a strictly stationary sequence with E = 0 and E 2 <, (2.) k ( k /2 i = O ) P (log log n) /2 (n ), (2.2) k n σn /2 i nt i= i D[0,] W(t) (n ) with some σ > 0, (2.3) where W(t), 0 t < } denotes a standard Wiener process. Condition (2.3) means that the innovations satisfy the functional central limit theorem, while (2.2) is a weak version of the law of the iterated logarithm. If the i } are independent and identically distributed, then (2.2) and (2.3) hold. If i } are homoskedastic and serially dependent, e.g. if they satisfy ARMA equations, then (2.2) and (2.3) are also satisfied due to the mixing properties of these sequences. Gorodetskii (977) and Withers (98) established the mixing properties of linear processes assuming a smoothness condition on the density of the summands. Mixing properties of augmented GARCH sequences were established by Carrasco and Chen (2002) assuming moment and smoothness conditions on the errors. Aue et al. (2005) provides another justification for (2.2) and (2.3) in case of augmented GARCH sequences under very mild conditions. The strong approximation result for errors following a random coefficient autoregressive model established in Aue (2006) implies (2.2) and (2.3) too. The parameter σ in (2.3) is equal to the square root of the long run variance σ 2 = lim n Var( i n i)/n. If the i are uncorrelated, e.g. if they follow a GARCH type model, σ 2 can be estimated by the sample variance of the first m residuals. If the i are serially correlated, different estimators are needed, but discussing them would distract us from the focus of this paper, see Andrews and Monahan (992), Giraitis et al. (2003), Berkes et al. (2005, 2006) and Sul et al. (2005) for some recent contributions which contain a wealth of references. We first state an extension of Theorem. 5

6 Theorem 2. Suppose i, i < } satisfy assumptions (2.) (2.3). If m = 0 and condition (.9) is satisfied, then relation (.0) holds. Theorem 2. can be established by modifying the technique developed in Horváth et al. (2005). For this reason we omit its proof. We now turn to the main result of the paper, the asymptotic normality of τ m. Simulations in Horváth et al. (2005) showed that the monitoring scheme works well if k is small ( early change ) and m, the size of the change is not too small. Thus we assume and c 3 (log m) β m c 4 with some c 3, c 4 > 0 and β > 0 (2.4) ( ) (log log m) k /2 = o. (2.5) 2 m Condition (2.5) means that we are trying to detect an early change. This assumption is also justified by the observation in Chu et al. (996) and Zeileis et al. (2005) that CUSUM based procedures should be used to detect early changes. Theorem 2.2 If (.9) and (2.) (2.5) are satisfied, then for all x ( ) } lim P 2 m τ m 2 log log m x = Φ(x). (2.6) 2(2 loglog m) /2 σ2 2 m If, in addition, ˆσ m σ = o P (), as m, (2.7) then ( lim P 2 m τ m 2(2 loglog m) /2 ˆσ 2 m Theorem 2.2 is proved in Section 4. ) } 2 log log m x = Φ(x). (2.8) 2 m Observe that under the conditions of Theorem 2.2, we immediately have τm 2σ2 log log m. (2.9) 2 m This means that we stop in 2σ 2 log log m/ 2 m steps using τm, while (c/ m ) /( γ) m ( 2γ)/(2( γ)) steps are needed with τ m. In particular, if m is constant, we have an iterated logarithmic order of the detection time using τm compared to a polynomial growth of τ m. We also note that relations (2.6) and (2.8), and so (2.9), do not involve the change point k, and one would expect the mean detection time to increase with k. This is because the goal of Theorem 2.2 is to establish the asymptotic normality of k and its rate as a function of the size m of the training sample, rather than the precise parameters of the limiting normal distribution. Observe that by (2.5), k can be added to the mean without changing the asymptotic distribution. For this reason, we investigate in Section 3 whether the distribution of k is normal and how the approximation to normality depends on m for finite values of m without using the specific mean and variance of Theorem

7 3 Numerical examples The objective of this section is to explore the approximate normality of τ = τm for finite values of m. We do not attempt to provide a definite guide for practitioners, but merely illustrate the potential applicability of the theory developed in this paper. We first consider the simplest case of independent identically distributed standard normal errors i. Figure shows the histograms of 2500 simulated values of τ for m = 200, 400, 600 and k = 3 and k = 5. The choice of the values of m is motivated by reasonable sizes of training samples encountered in financial time series to be discussed below. The values of k = 3 and k = 5 were chosen as reasonable early changes. For series of daily returns, they correspond to a change in mean within the first week after the monitoring has commenced. The histograms look roughly normal but are visibly skewed to the right and lack some probability mass in the left tail. This is to be expected, as the detection delay can in principle be arbitrarily long, whereas a detection before the change point is unlikely and τ cannot be smaller than. These observations are illustrated in a different way in Figure 2 which shows QQ-plots against the quantiles of the standard normal distribution. If the distribution of τ were normal, these plots would be approximately straight lines. Figure 2 shows that the central part of the QQ-plots becomes closer to a straight line as m increases. There is a pronounced step at the left tail due the the aforementioned boundedness of τ. Asymptotic normality of the distribution of a statistic is typically used to construct confidence intervals and to test statistical hypotheses. In such applications, it is only important that the empirical quantiles are close to the corresponding critical values derived from the normal approximation. It is thus useful to know if [ ] P τ Eτ z α/2 Var[τ ] α, where the standard normal critical value z α/2 is defined by Φ(z α/2 ) = α/2. Table shows the empirical probabilities that τ Eτ z α/2 Var[τ ] for α =.05 and α =.0 for the simulated values of τ used to construct Figures and 2. The expected value Eτ was replaced by the sample mean and Var[τ ] by the sample variance. It is seen that in this sense the normal approximation is excellent for α =.05 and reasonably good for α =.0 in which case the tails of the empirical distribution are slightly heavier (by about % of the total probability mass) than implied by the normal approximation. The empirical probabilities in Table (and in Table 2 below) are based on R = 2500 replications, so they are unlikely to differ from the probabilities based on R = replications by more than 0.0 (standard errors are about 0.005). As explained in Section 2, the theory developed in this paper allows conditionally heteroskedastic errors i. We now investigate if the normal approximation holds in case of GARCH(,) errors and values of m relevant for the series of daily returns. To conserve space, we focus on the analysis analogous to that presented in Table ; the histograms and QQ-plots are very similar to those in Figures and 2. We present the results only 7

8 m = 200 m = 400 m = 600 Nominal Empirical k = Empirical k = Table : Percentage of stopping times τ within.960 and standard deviations away from the sample mean. Standard normal errors. =.2. for k = 5, what corresponds to a change a week after the monitoring has commenced. We however consider four different GARCH(,) models estimated on index and stock returns with different dynamics and five values of the shift = m. The values of m are equal to those determined in Kokoszka and Zhang (2005) and reflect periods of time over which the models are believed to be stable. In the simulations we used the following four models which well represent the various dynamics of return series: DJIA, m = 06 trading days starting //992: µ =.076; ω =.025; α =.064; β =.879. Implied variance NASDAQ, m = 775 trading days starting 7//994: µ =.29; ω =.33; α =.32; β =.750. Implied variance.27. GE, m = 473 trading days starting //200: µ =.30; ω = 4.345; α =.094; β =.28. Implied variance WALMART, m = 457 trading days starting //200: µ =.027; ω =.407; α =.72; β =.499. Implied variance The parameter µ is equal to the mean return over the indicated period. The parameters ω, α and β are the usual GARCH(,) parameters in the specification σt 2 = ω + αrt 2 + βσt 2 estimated on the demeaned returns. The implied variance is equal to ω/( α β). The goal of the monitoring is to detect a relatively large change in the mean returns which occurs not long after the monitoring has commenced. In practice, this means that the GARCH model would have to be updated every week or so, and an a posteriori test of the stability of this model would have to be performed. These issues are however not the focus of the present paper. Table 2 shows results similar to those presented in Table. We also included nominal coverage of 0.90, in which case the empirical distribution tends to have slightly too much mass (about 2%). This is also the case for independent errors. For the nominal coverage of 0.99, some mass is missing (about %). As for independent errors, this is due to slightly too heavy right tails. The empirical coverage corresponding to the nominal coverage of 0.95 continues to be remarkably good except in the cases of very small or very large changes in mean returns in models corresponding to individual stocks. This is in accordance with condition (2.4) which requires that the shift cannot be too small or two large. For all 8

9 DJIA m = 06 NASDAQ m = 775 Nominal Empirical = Empirical = Empirical = Empirical = Empirical = GE m = 473 WALMART m = 457 Nominal Empirical = Empirical = Empirical = Empirical = Empirical = Table 2: Percentage of stopping times τ within.645,.960 and standard deviations away from the sample mean. GARCH(,) errors. The shift is expressed in units equal to the implied standard deviations of the GARCH(.) errors, i.e. to ω/( α β). models considered in this simulation study, the empirical coverage is excellent provided is larger than 0.8 and smaller than.6 standard deviations of the returns (for all models the implied standard deviation is practically equal to the sample standard deviation). In most cases, the quantiles computed using the normal approximations are quite close to the empirical quantiles. 4 Proof of Theorem 2.2 Recall the definitions of the detector ˆQ(m, k) and the threshold g (m, k) in (.3) and (.7), respectively. It is easy to see that (X i X m ) = ( i m ) + (k k + ) m Ik k }, (4.) where m = m i m i. Furthermore, we can assume without loss of generality that m > 0. The proof of Theorem 2.2 is based on a series of lemmas, in which the following quantity is frequently used. For any x and t, we introduce ) 2 (ˆσm c(m; t) σx K(m) = K(m, x, t) =. m 9

10 In the following, we focus on the verification of (2.8). The verification of (2.6) follows exactly the same lines except that ˆσ m in the definition of K(m) is replaced by σ and condition (2.7) is then not needed. It follows from the definition of c(m; t) in (.8) and from condition (2.7) that ( ) 2σ 2 log log m P K(m) (m ). (4.2) 2 m In particular, K(m) is of lower order than N whose range is determined by (.9). The proof steps can be outlined as follows. In Lemma 4. we show that the threshold function g (m, k) can be replaced asymptotically by the simpler expression k /2. Lemma 4.2 simplifies the CUSUM type expression in (4.) by dropping the average m. Then, Lemma 4.3 determines the range of those k which contribute asymptotically, while the normal limit distribution is eventually established in Lemmas 4.4 and 4.5. Theorem 2.2 follows readily from these auxiliary results. Lemma 4. If the conditions of Theorem 2.2 are satisfied, then, as m, (X k K(m) g i X m ) (m, k) = (X k K(m) k /2 i X m ) +o P(). Proof On applying (4.) and the mean value theorem, it is clear that ( (X k K(m) k /2 i X m ) + k /2 m) k /2 = O() k K(m) m ( i m ) + (k k + ) m Ik k }. It follows from the stationarity of i } and the weak convergence in (2.3) that, as m, k /2 D k /2 k K(m) m i = k K(m) m i i k K(m) m K(m) /2 i k K(m) log log m = O P () m 2 m = o P () i k on account of (4.2), since the order of magnitude of m is controlled by (2.4). The same arguments imply similarly, as m, k /2 k K(m) m k m = O P () K(m)3/2 = o m 3/2 P () 0

11 and k /2 k K(m) m (k k + ) m Ik k } = O() K(m)3/2 m (log log m)3/2 = O P () m 2 m completing the proof of Lemma 4.. = o P (), Lemma 4.2 If the conditions of Theorem 2.2 are satisfied, then, as m, (X k K(m) k /2 i X m ) = k K(m) k /2 i + (k k + ) m Ik k } + o P(). Proof Conditions (2.3) and (2.4) give ( (K(m) ) ) /2 k K(m) k /2k m = O P = o P () (m ), m so that the result is readily proved. Lemma 4.3 If the conditions of Theorem 2.2 are satisfied, then, for any 0 < δ < as m, k ( δ)k(m) k /2 i + (k k + ) m Ik k } K(m)/2 P m. Proof Using (2.2) and the stationarity of i } we conclude ( k K(m) k /2 i = O ) ( P (log log K(m)) /2 = O ) P (log log log m) /2, where relation (2.4) has been used to obtain the last estimate. Also, k ( δ)k(m) m k /2(k k + ) m Ik k } ( ( δ)k(m) ) /2 m. P On observing that, by (4.2), K(m) /2 m as m, the proof of Lemma 4.3 is complete. Lemma 4.4 If the conditions of Theorem 2.2 are satisfied, then ( ) } lim lim P δ 0 ( δ)k(m) k K(m) k /2 i + k m K(m) /2 m σx = Φ(x) for all x.

12 Proof By the stationarity of i } it is enough to prove ( ) } lim lim P δ 0 ( δ)k(m) k K(m) k /2 i + k m K(m) /2 m σx = Φ(x). i k Let denote integer part. Clearly, if K(m), ( ) i + k m K(m) /2 m ( δ)k(m) k K(m) k /2 i k K(m) /2 i K(m) i ( K(m) /2 K(m) /2) m. ( Since, by the mean value theorem, K(m) /2 K(m) /2 = O ) P K(m) /2 = o P (), we get from (2.3) and (2.4), lim P σ K(m) /2 i ( K(m) /2 K(m) /2) m x = Φ(x). On the other hand, ( δ)k(m) k K(m) k /2 i K(m) ( i k ( δ)k(m) k K(m) i + k m ) K(m) /2 m k /2 i k ( δ) /2 ( δ)k(m) k K(m) K(m) /2 Condition (2.3) yields that, for any 0 < δ <, ( δ)k(m) k K(m) K(m) /2 i k i D σ i sup ( δ) t W(t) i k i. (m ). Since sup ( δ) t W(t) lim lim P δ 0 D W() as δ 0, we get that ( δ) /2 ( δ)k(m) k K(m) The proof of Lemma 4.4 is complete. K(m) /2 i k i σx Lemma 4.5 If the conditions of Theorem 2.2 are satisfied, then } = Φ(x). lim P τ m K(m, x, t)} = Φ(x) for all x and t. 2

13 Proof Let A m be the event K(m, x, t) N}. By (.9), (2.3) and (4.2), we have Clearly, lim P(A m) =. P τm K(m), A m} = P k K(m) g (m, k) = P k K(m) σg (m, k) } (X i X m ) ˆσ mc(m; t), A m (X i X m ) K(m)/2 m σ It follows from the defintion of K(m) that ˆσ m σ c(m; t) K(m)/2 m = x, σ so that, in view of (4.3), it is enough to prove lim P k K(m) g (m, k) ˆσ m σ c(m; t) K(m)/2 m, A m }. σ (4.3) } (X i X m ) K(m)/2 m σx = Φ(x). (4.4) Next observe that, according to Lemmas , we get (4.4) if lim lim P δ 0 ( δ)k(m) k K(m) k /2 i + (k k + ) m } K(m) /2 m σx = Φ(x). Condition (2.5) together with (4.2) gives k m K(m) /2 = o P () and } lim P i + k m > 0 for all ( δ)k(m) k K(m) =. Hence Lemma 4.4 implies Lemma 4.5 Proof of Theorem 2.2. The result is an immediate consequence of Lemma 4.5. First we write (ˆσm P τm c(m; t) σx ) 2 } m ˆσm 2 ( m τ = P m c(m; ) } t)2 x 2 x +. σ 2c(m; t) ˆσ m 2 2 m 2ˆσ m σc(m; t) 3

14 By (2.7) and the defintion of c(m; t), we have, as m, ˆσ m σ P, x 2 2ˆσ m σc(m; t) P 0, c(m; t) (2 log log m) /2, and (2 log log m) /2 c(m; t) 2 2 log log m = o(). Therefore, lim P 2 ( m τ m 2(2 loglog m) /2 ˆσ m 2 2 log log m ) } x = Φ(x), 2 m which completes the proof of Theorem 2.2. References Andrews, D. W. K., Monahan, J. C. (992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica Aue, A. (2006). Testing for parameter stability in RCA() time series. Journal of Statistical Planning and Inference. To appear. Aue, A., Berkes, I., and Horváth, L. (2006). Strong approximation for the sums of squares of augmented GARCH sequences. Bernoulli. To appear. Aue, A., and Horváth, L. (2004). Delay time in sequential detection of change. Statistics and Probability Letters Aue, A., Horváth, L., Hušková, M., and Kokoszka, P. (2005). Change point monitoring in linear models with conditionally heteroskedastic errors. Preprint. Basseville, M., and Nikiforov, I.V. (993). Detection of Abrupt Changes: Theory and Applications. Prentice Hall. Berkes, I., Horváth, L., Kokoszka, P., and Shao, Q. M. (2005). Almost sure convergence of the Bartlett estimator. Periodica Mathematica Hungarica Berkes, I., Horváth, L., Kokoszka, P., and Shao, Q. M. (2006). On discriminating between long-range dependence and changes in mean. The Annals of Statistics. To appear. Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory Chu, C. S.J., Stinchcombe, M., and White, H. (996). Monitoring structural change. Econometrica

15 Giraitis, L., Kokoszka, P., Leipus, R., and Teyssiere, G. (2003). Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics Gorodetskii, V.V. (977). On the strong mixing property for linear processes. Theory of Probability and its Applications Horváth, L., Hušková, M., Kokoszka, P., and Steinebach, J. (2004). Monitoring changes in linear models. Journal of Statistical Planning and Inference Horváth, L., Kokoszka, P., and Steinebach, J. (2005). On sequential detection of parameter changes in linear regression. Preprint. Kokoszka, P., and Zhang, A. (2005). Discriminating between long memory and volatility shifts. Preprint. Leisch, F., Hornik, K., and Kuan, C. M. (2000). Monitoring structural changes with the generalized fluctuation test. Econometric Theory Sul, D., Phillips P.C.B., Choi C-Y. (2005). Prewhitening bias in HAC estimation Oxford Bulletin of Economics and Statistics Withers, C.S. (98). Conditions for linear processes to be strong mixing. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete Zeileis, A., Leisch, F., Kleiber, C., and Hornik, K. (2005). Monitoring structural change in dynamic econometric models. Journal of Applied Econometrics

16 m=200, k*=3 m=200, k*= m=400, k*=3 m=400, k*= m=600, k*=3 m=600, k*= Figure : Histograms of the detection time τ. 6

17 m=200, k*= m=200, k*= m=400, k*= m=400, k*= m=600, k*= m=600, k*= Figure 2: QQ Plots of the detection time τ. 7