Structural Change Identification and Mean Shift Detection for ARFIMA Processes

Transcription

1 Structural Change Identification and Mean Shift Detection for ARFIMA Processes Fangyi He School of Finance, Southwestern University of Finance and Economics Chengdu, Sichuan , P. R. China fangyi Wei Jiang Antai College of Economics and Management Shanghai Jiaotong University, Shanghai, China Lianjie Shu Faculty of Business Administration University of Macau, Macau, China Abstract In this paper we develop a general method to detect structural changes in a long memory process. This method first uses inverse filters of an ARFIMA model to remove the autocorrelations, and then use generalized fluctuation tests to monitor the residuals. In particular, this paper analyzes the method s performance for detecting mean shifts of a long memory process. Three types of structural changes of the residuals are identified for an ARFIMA(1,d,1) process when its mean incurs a shift. The performance of recursive-estimates and moving-estimates tests for detecting the structural changes of the residuals are analyzed by simulations. Keywords: ARFIMA; Change point; Long memory; Structural change JEL-codes: C15 C44 C63 1

2 1 Introduction Long memory series models have been found useful in economics, hydrology, geophysics, and many other fields. For long memory series, correlations between distant observations decay at a hyperbolic rate. Empirical analysis has shown that many economic or financial data series possess this long memory property, such as foreign-exchange rates, volatility of asset price returns, future interest rates, inflation rates and consumer price index (Cheung, 1993; Bhansali et al., 2007; Tsay, 2000; Baillie and Chung, 1996; Man, 2003). Recently there has been a considerable interest in long memory and structural changes in series. One important problem is to detect a change point in the mean of a longmemory series. If the change point cannot be detected on, forecasts will lose accuracy and decisions based on the forecasts will be wrong. Gabriel and Martins (2004) show that the forecasting performance of long memory models is relatively poor when shifts occur in the series, compared to simple linear and Markov switching models. Bisaglia and Gerolimetto (2008) compare the forecast accuracy of both long memory and regime switching models with an extensive Monte Carlo experiment. The authors find that forecasting from long memory models in the presence of structural changes can be dangerous. The purpose of this paper is to develop a new method to monitor structural changes of long memory processes. In contrast with the large body of statistical and econometric literature on testing parameter instability and structural changes, such as Ray and Tsay (2002), Lazarová (2005), Wang (2008), Qu (2010) and Shao (2011), our approach does not intend to analyze the original process directly. Instead, it uses series modelling to remove autocorrelation of the long memory process first, and then to apply real- monitoring procedures to test structural stability of the residuals. This new method could be used to test the null hypothesis that all parameters in the long memory process remain constant against the alternative that at least one of the parameters changes at some unknown point in the future. For simplicity, in this paper we only investigate the method s performance of detecting a mean shift in the long memory process. Section 2 gives a brief introduction of this technique. In particular, the change patterns of residuals for ARFIMA(0, d, 0) and ARFIMA(1, d, 1) processes are identified in Section 3 when the process incurs a mean shift. By monitoring the change patterns of residuals using two types of generalized fluctuation tests, Section 4 analyzes the new method s performance for detecting a mean shift of an ARFIMA(0, d, 0) or ARFIMA(1, d, 1) process through simulations. We give conclusions in Section 5. 2

3 2 ARFIMA(p, d, q) Process and Inverse Filtering One popular approach to model long-memory processes of series is to use various forms of autoregressive fractionally integrated moving average (ARFIMA) models. A series X t is said to follow an ARFIMA(p, d, q) process if Φ(B)(1 B) d (X t m X ) = Θ(B)a t, (1) where m X is the population mean of X t ; a t is a white noise with mean 0 and variance σ 2 a; B is the back-shift operator such that BX t = X t 1 ; Φ(B) = 1 φ 1 B φ p B p and Θ(B) = 1 + θ 1 B + + θ q B q. All roots of Φ(B) and Θ(B) are outside of unit circle, and (1 B) d is the fractional differencing operator defined by (1 B) d = k=0 Γ(k d)b k Γ(k + 1)Γ( d), (2) with Γ( ) being the gamma function. In particular, a fractional white noise process is defined as (1 B) d X t = a t, which is the simplest case of the ARFIMA model. This process is first introduced by Adenstedt (1974), Granger and Joyeux (1980), Granger (1980, 1981), and Hosking (1981). They show that X t is stationary when d < 0.5 and is invertible when d > 0.5. From Model (1) it is easy to get that a t = Φ(B) Θ(B) (1 B)d (X t m X ). (3) It means that if we have a series signal X t which follows (1), we can inverse filter the data to remove the autocorrelation and get a white noise series a t. Here a t can be viewed as the error between the measured signal value X t and the predicted value based on the ARFIMA model. Under the null hypothesis, it is well known that a t satisfies a functional central limit theorem (FCLT): [T r] 1 a i W (r), (4) T σa i=1 as T, where [T r] is the integer part of T r, denotes weak convergence of associated probability measures, and W is a standard Wiener process. If X t incurs structural breaks at some point τ, then a t derived from (3) will not be white noise starting from point τ. The behavior of (4) is not completely characterized by the FCLT but will fluctuate. Hence, generalized fluctuation tests can be constructed by evaluating the fluctuation of the residuals. The similar idea can also be found from the residual chart in the quality control field. One can refer to Alwan and Roberts (1988), Lin and Adams (1996), and Lu and Reynolds (1999, 2001), among others. 3

4 3 Structural Change Identification In this section, we will identify the structural changes of residuals after inverse filtering an ARFIMA process X t with a mean shift. Suppose that X t has a sudden mean change of size δ occurring at τ. Let s denote the new series as X t. Then X t = X t + δ t, where δ t = { 0 t = 1, 2, τ 1 δ t = τ, τ + 1, From (3) we know that the output a t from inverse filtering X t can be expressed as (5) where a t = a t + µ t (6) µ t = Φ(B) Θ(B) (1 B)d δ t (7) That is to say, a t is the summation of the response due to the ARFIMA process X t, which is a t, and the response due to the change function δ t, which is µ t. Note that µ t = E(a t ). Inverse filtering a level shift in X t will likely result in a level shift plus a transient response in a t. However, the magnitude of this mean shift in a t depends on the parameters of the ARFIMA model. Also, the magnitude and the pattern of the transient response of a t depend on the structure of the ARFIMA model and the model parameters. Identifying the change pattern in a t is very useful because knowledge of the change pattern can help us to develop efficient methods to detect level shifts in X t. Hu and Roan (1996) investigate the effects of ARMA model parameters on the change pattern of residuals from inverse filtering. We will perform a similar analysis for ARFIMA model. In addition, we will only focus on ARFIMA(0, d, 0) and ARFIMA(1, d, 1) models since large values of p or q demand models with many parameters, which contradicts the parsimony principle of univariate series analysis. Without loss of generality, δ is set to 1 in the rest of this paper. 3.1 ARFIMA(0, d, 0) Processes For the case that X t is the fractional white noise process, we investigate the relationship between d and µ t. At this, Φ(B) = Θ(B) 1 in (7). Then, it is easy to know from (2) and (5) that µ t = (1 B) d Γ(k d) δ t = Γ(k + 1)Γ( d) δ t k = = k=0 t τ k=0 t τ k=0 Γ(k d) Γ(k + 1)Γ( d) δ t k + k=t τ+1 Γ(k d) Γ(k + 1)Γ( d) δ t k Γ(k d) Γ(k + 1)Γ( d). (8) 4

5 Using the facts that Γ(z + 1) = z Γ(z) and Γ(n) = (n 1)!, we can transform (8) into { 0 t < τ µ t = t τ j=1 (1 d/j) t τ (9) Theorem 1. For the case that X t is the fractional white noise process, µ t is divergent with t if d < 0 and convergent with t if 0 d < 1. Proof: For t τ + 1, note the equality that t τ (1 d t τ 1 j ) (1 d j j=1 So with t big enough, µ t = 1 + If d < 0, we can get that µ t > 1 + j=1 t k=τ+1 t k=τ+1 ) = ( d)(1 d)(2 d) (t τ 1 d) (t τ)! ( d)(1 d)(2 d) (k τ 1 d) (k τ)! ( d)(k τ 1)! (k τ)! = 1 d t k=τ+1 (10). (11) 1 k τ. (12) Since lim t 1 t k=τ+1 k τ =, we prove that µ t increases to infinity as t. If 0 < d < 1, it is easy to know that µ t is decreasing when t τ and µ t 1 from (9), so µ t is convergent as t. For the case that d = 0, µ t 1 as t and the ARFIMA(0, d, 0) becomes to a white noise process. Figure 1 shows the curves of µ t changing with t for different values of d. Without loss of generality, we set τ equal 5. It is clear from the figure that: at τ, µ t has a level shift with the same size as that of X t ; from then on, µ t is increasing and concave with t if 0.5 d < 0, decreasing and convex with t if 0 < d 0.5, and keeps constant if d = ARFIMA(1, d, 1) Processes When X t is an ARFIMA(1, d, 1) process, from (7) we know Then it can be transformed based on (9) that µ t = 1 φb 1 + θb (1 B)d δ t θ < 1, φ < 1 (13) µ t + θµ t 1 = t τ (1 d t τ 1 j ) φ (1 d j ) j=1 j=1 = (1 d t τ 1 t τ φ) (1 d ). (14) j j=1 5

6 µ t d= 0.5 d= 0.3 d= 0.1 d=0 d=0.1 d=0.3 d= Figure 1: The changes of µ t for different values of d after X t has a level shift at τ = 5. Theorem 2. For the case that X t is an ARFIMA(1, d, 1) process with θ < 1 and φ < 1, µ t is divergent with t if d < 0 and convergent with t if 0 d < 1. Proof: Let s denote C(t) = (1 d t τ 1 t τ φ) (1 d ). (15) j If d < 0, from the proof of Theorem 1 we can get that lim t C(t) =. Suppose lim t µ t = µ 0, we can make t in (14) and get that µ 0 =. That is to say, µ t is divergent. If 0 < d < 1, from the proof of Theorem 1 we know C(t) is convergent. Then make t in (14), we prove that µ t is convergent. It is easy to know lim t µ t = 1 φ 1+θ ARFIMA(1, d, 1) becomes to an ARMA(1, 1) process. j=1 if d = 0. At this, the Same as Hu and Roan (1996), we divide the stability region for φ and θ into three zones and pick up several representative values of (φ, θ) in each zone, see Figure 2. For the representative pairs of (φ, θ), we show the change patterns of µ t when d = 0.3, d = 0, and d = 0.3 respectively. As can be seen from Figure 3, we get the same change pattern as Hu and Roan (1996) when X t is an ARMA(1,1) process (i.e., d = 0). That is, µ t consists of a mean shift of increased magnitude in Zone 1; consists of a spike and a steady state shift of a smaller magnitude in Zone 2; and is decaying oscillations in Zone 3. When d 0, it is shown that positive d lessens the 6

7 magnitude of mean shift in µ t while negative d magnifies the magnitude. Specially, in Zone 3 positive d magnifies the amplitude while negative d lessens the amplitude. Figure 2: Three Zones of the Stability Region for φ and θ. 4 Structural Change Detection Kuan and Hornik (1995) introduce a general principle of constructing tests for parameter constancy without assuming a specific alternative. The class of tests is called the generalized fluctuation test. Roughly speaking, this class of tests could be classified into two groups. One is based on parameter estimates, such as the recursive-estimates test (denoted as RE test hereafter) of Ploberger et al. (1989) and the moving-estimates test (denoted as ME test hereafter) of Chu et al. (1995a). The other is based on regression residuals, such as the CUSUM test of Brown et al. (1975) and the MOSUM test of Bauer and Hackl (1978) in which recursive residuals are used; and the OLS-CUSUM test of Ploberger and Kramer (1992) and the OLS-MOSUM test of Chu et al. (1995b) in which OLS residuals are used. Leisch et al. (2000) extend the tests in the first group so that they can be applied for monitoring future structural changes. In this section, we will analyze the performance of RE test and ME test through simulations to monitor the residuals with the change patterns discussed in Section 3. In this way, we could hopefully detect mean shifts on an ARFIMA(0,d,0) or ARFIMA(1,d,1) process when its model parameters are in Zones 1, 2, or 3 of the stability region in Figure 2. 7

8 Zone 1 Zone 2 Zone φ = 0.7, θ = 0.2 φ = 0.8, θ = 0.3 φ = 0.2, θ = 0.6 φ = 0.5, θ = φ = 0.2, θ = 0.6 φ = 0.7, θ = φ = 0.5, θ = 0.9 φ = 0.8, θ = 0.3 (a) d = φ = 0.7, θ = 0.2 φ = 0.8, θ = 0.3 φ = 0.2, θ = 0.6 φ = 0.5, θ = φ = 0.2, θ = 0.6 φ = 0.7, θ = φ = 0.5, θ = 0.9 φ = 0.8, θ = 0.3 (b) d = φ = 0.7, θ = 0.2 φ = 0.8, θ = 0.3 φ = 0.2, θ = 0.6 φ = 0.5, θ = φ = 0.2, θ = 0.6 φ = 0.7, θ = φ = 0.5, θ = 0.9 φ = 0.8, θ = 0.3 (c) d = 0.3 Figure 3: The changes of µ t for different values of d, φ and θ after X t has a level shift at τ = 5. 8

9 Suppose we are currently at T and have historical data X 1, X 2,, X T, which is an ARFIMA(1,d,1) process with parameters m X, d, φ and θ (for ARFIMA(0,d,0) process, φ=θ=0). We take as given that all the parameters were constant and known during the history period ( noncontamination assumption in Chu et al., 1996). We are interested in testing the null hypothesis that all the parameters remain constant against the alternative that m X has a shift at some unknown point in the future (t = T + 1, ). As analyzed in Section 3, three types of change pattern of a t correspond to the shift of m X for different values of φ and θ. Using the inverse filtering techniques in Section 2, we only need to test structural changes in the residuals a t. Let s denote â t (t = 1, 2, ) as the residuals after inverse filtering the real data X t (t = 1, 2, ). For RE test, a recursive estimate is computed when new data arrive and then compared with â T, the mean estimate based on the historical sample. If their difference is too large, the null hypothesis is rejected and the monitoring procedure stops. The null hypothesis would be rejected if there is at least one statistic exceeding proper boundaries. The test statistic used is RE T (r) = max k=t +1,,[T r] k ˆσ T T âk â T, where the period from T + 1 through [T r], r > 1, is the expected monitoring period; â k is the mean estimate based on all the data up to k; and ˆσ T is some suitable estimator 1 of the standard deviation of the residuals, e.g., ˆσ T = T T 1 i=1 (â i â T ) 2. When r =, the monitoring procedure might last indefinitely. For ME test, the moving estimate used is ã T (k, [T h]) = [T h] 1 k i=k [T h]+1 âi,where 0 < h 1 and k = [T h], [T h]+1,. The monitoring test statistic based on the moving estimate is ME T,h (r) = max k=t +1,,[T r] [T h] ˆσ T T ã T (k, [T h]) â T. The monitoring boundaries for RE and ME tests can be found in Chu et al. (1996) and Leisch et al. (2000) respectively. To model a long memory process, we usually have the parameter 0 < d < 0.5 in an ARFIMA(p, d, q) process. In the following, we show the performance of RE and ME tests for monitoring the three types of structural change pattern of residuals a t when ARFIMA(1,0.3,1) (or ARFIMA(0,0.3,0)) processes incur a mean shift. We consider historical samples of sizes T = 25 and 100, moving window sizes h = 0.5 and 1, and r = 10 for the expected monitoring period [T r]. The data-generating process (DGP) X t follows Model (1) with p = q = 1 or p = q = 0 under the null hypothesis. Under the alternative, the mean of the ARFIMA process changes from m X to m X + 3 σ a at 1.1T or 3T. We use the monitoring boundaries for the significance level 5%. All experiments are repeated by 1, 000 s. 9

10 Table 1: Empirical Sizes for Monitoring Tests At 5% Level ME test T RE test h = 1/2 h = Table 1 shows the empirical sizes of the RE and ME tests on a t. It is shown that the empirical sizes of both RE and ME tests are closer to the nominal size 5%, except for the ME test when T = 25 and h = 1, which is consistent with the results in Leisch et al. (2000). When the mean shifts, it is more interesting to know how soon a change can be detected. Table 2 shows the mean and standard deviation of the detection delay, as well as Type I and Type II errors. As we can see, given an early change at 1.1T, RE test s performance is better than ME tests performance to detect the structural changes of the residuals a t. On the other hand, RE test performs more poorly for a late change at 3T than ME tests. For the ME tests, their performance depends on the window sizes. A smaller window (h = 1/2) results in quicker detection as well as more Type II errors than a bigger window (h = 1). When the parameters φ and θ of an ARFIMA(1, 0.3, 1) process are in Zone 1, both RE and ME tests on structural changes of a t perform very well for detecting the process mean shift; When the parameters are in Zone 2, their performance is greatly reduced; When the parameters are in Zone 3, both tests almost fail to detect the residuals structural changes. These patterns can be found in Figure 4 which shows the relationship between the RE test s power and different values of φ and θ analyzed in Table 2 for the case that T = 25 with the change point equal to 28. For other cases in Table 2, the patterns are similar. 5 Conclusions This paper has developed a general method to detect structural changes in long memory processes. Based on the idea of residual charts in the quality control field, we use ARFIMA models to inverse filter the data to remove the autocorrelations, and then use RE and ME tests to monitor residuals. In order to detect mean shifts of a long memory process, the structural changes of the residuals are identified for different values of d, φ and θ when an ARFIMA(0,d,0) or ARFIMA(1,d,1) process incurs a mean shift. The performance of RE and ME tests on the structural changes of residuals is analyzed through simulations. Based on three zones of the stability region for φ and θ of an ARFIMA(1,d,1) process, 10

11 Table 2: Mean (and Standard Deviation) of Detection Delay With Type I and II Error T = 25 Change ME test point φ θ RE test type I type II h=1/2 type I type II h=1 type I type II (6.393) (4.625) (6.365) (2.474) (1.931) (3.025) (0.564) (0.838) (1.159) (1.075) (1.273) (1.784) (0.927) (0.972) (1.099) (10.023) (7.655) (8.336) (20.792) (16.330) (14.903) (27.208) (21.301) (22.046) (27.564) (28.144) (25.639) (39.107) (5.213) (6.434) (29.972) (2.820) (4.686) (6.726) (0.986) (1.844) (14.668) (1.448) (2.759) (3.240) (1.112) (1.662) (41.448) (10.127) (11.123) (51.595) (23.844) (23.236) (56.008) (23.875) (23.292) (47.629) (24.546) (28.075) T = (7.569) (7.852) (12.238) (2.544) (4.556) (6.422) (0.867) (1.794) (2.218) (1.476) (2.598) (3.484) (1.087) (1.383) (1.342) (25.019) (19.714) (24.192) (68.367) (55.551) (48.148) ( ) (55.994) ( ) ( ) ( ) (92.656) ( ) (9.302) (19.045) (65.849) (5.785) (11.063) (13.678) (2.525) (4.707) (23.827) (3.649) (6.734) (5.524) (1.940) (2.914) ( ) (21.461) (30.382) ( ) (50.749) (62.379) ( ) (58.951) ( ) ( ) (94.892) (99.839)

12 Figure 4: The Power of the RE test in Different Zones of the Stability Region for φ and θ. we find that both RE and ME tests perform very well when the parameters are in Zone 1, perform poorly in Zone 2 and fail in Zone 3 where the mean of residuals is decaying oscillations. Proposing new tests to detect decaying osillations deserves further study. This topic is currently under investigation. Acknowledgements The work of F. He was supported by the National Natural Science Foundation of China (no ) and the SRF for ROCS, SEM. References [1] Adenstedt, R. K. (1974) On Large Sample Estimation for the Mean of a Stationary Random Sequence, Annals of Mathematical Statistics, 2, [2] Alwan, L. C. and Roberts, H. V. (1988) Time Series Modeling for Statistical Process Control, Journal of Business & Economic Statistics, 6, [3] Baillie, R.T. and Chung, C.F. (1996) Analysing Inflation By The Fractionally Integrated ARFIMA-GARCH Model, Journal of Applied Econometrics, 11,

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