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Random Level-Shift Time Series Models, ARIMA Approximations, and Level-Shift Detection Chung Chen; George C. Tiao Journal of Business & Economic Statistics, Vol. 8, No. 1. (Jan., 1990), pp. 83-97. Stable URL: http://links.jstor.org/sici?sici=0735-0015%28199001%298%3A1%3C83%3ARLTSMA%3E2.0.CO%3B2-%23 Journal of Business & Economic Statistics is currently published by American Statistical Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/astata.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Thu Jan 24 07:21:20 2008 O 1990 American Statistical Association Journal of Business & Economic Statistics, January 1990, Vol. 8, No. 1 Random Level-Shift Time Series Models, ARIMA Approximations, and Level-Shift Detection Chung Chen Department of Quantitative Methods, Syracuse University, Syracuse, NY 13244-2130 George C. Tiao Graduate School of Business, University of Chicago, Chicago, IL 60637 The main purpose of this article is to assess the performance of autoregressive integrated moving average (ARIMA) models when occasional level shifts occur in the time series under study. A random level-shift time series model that allows the level of the process to change occasionally is introduced. Between two consecutive changes, the process behaves like the usual autoregressive moving average (ARMA) process. In practice, a series generated from a random level-shift ARMA (RLARMA) model may be misspecified as an ARlMA process. The efficiency of this ARlMA approximation with respect to estimation of current level and forecasting is investigated. The results of examining a special case of an RLARMA model indicate that the ARlMA approximations are inadequate for estimating the current level, but they are robust for forecasting future observations except when there is a very low frequency of level shifts or when the series are highly negatively correlated. A level-shift detection procedure is presented to handle the low-frequency level-shift phenomena, and its usefulness in building models for fore- casting is demonstrated. KEY WORDS: ARMA models; Estimation of current level; Forecasting; RLARMA models. 1. INTRODUCTION be written as Consider the general Gaussian (autoregressive mov- (1 + .40B + .27B2)(1 - B)z, = a,, (1.2) ing average) ARMA(p, q ) model where Z, = (1 - .81B1')-'(1 - B12)y 17 (1.3) where {z,) is the observable time series, q(B) = (1 - q l B - ... - qpBP), 0(B) = (1 - BIB - ... - OqBq), {y,) is the observable logarithmic series, and {a,) is sup- B is the backshift operator, all the zeros of q(B) and posed to be a series of homogeneous white noise. For B(B) are on or outside the unit circle, and {a,) is a series simplicity, we shall work with the seasonally filtered of iid N(O, 02)variables. During the past 15 years, this series {z,) to motivate the problems considered in this class of models has been widely used to represent real article. time series in many areas, especially for forecasting in From (1.2), z, follows an AR(3) model in which the business and economic applications. Model (1.1) as- autoregressive polynomial contains a differencing factor sumes that observations are homogeneously generated (1 - B). On the other band, Figure 1 shows that a from the same probabilistic structure. In practice, how- linear trend appears to exist in the data with an obvious ever, time series data often contain structural changes level drop occurring in 1976. If this level drop is ignored, or discrepant observations. It is critical to investigate one may entertain a model containing a linear-trend to what extent such phenomena can make standard sta- term and an AR(3) noise term of the form tistical methods less efficient or even invalid. Z, = c + mot + N,, (1.4) 1.1 An Example where (1 - q l B - q2B2- q3B3)N,= a,. The estimation Figure 1 shows the seasonally filtered series of log- results of this model are given in Table 1. Standard arithmic monthly variety-store sales from January 1968 model checking based on residuals autocorrelations to September 1979. The original logarithmic variety- does not indicate evidence of model inadequacy. The store series was analyzed by Chang and Tiao (1982); estimated trend coefficient k0 = .24 x lo-' with a t see also Bell, Hillmer, and Tiao (1983). In their anal- value of .02 is exceedingly small, however, which seems ysis, the fitted model before adjustment for outliers can incongruent with the behavior of the series. 84 Journal of Business & Economic Statistics, January 1990 with the behavior of the series exhibited in Figure 1. am Now, examining the zeros of the fitted autoregressive polynomial (1 - q , B - q2B2 - q,B3), we consider an an alternative parameterization ( 1 - v ; B ) ( 1 - @ B - .3 00 @ B 2 )for the noise term N , , where 9; is the zero of B - ' a05- closest to the unit circle. The factorization of fitted \ I 413 N-J I I I I I I I' results gives for (1.4), (1 - .97B)(1 + .35B + .24B2); for (1.5), ( 1 - .93B)(1 + .48B + .31B2);and for (1.6), 6 7 8 8 n n n m 81 ( 1 - .89 B)(1 + .51 B + .34B2).In particular, the es- Figure 1. Seasonally Filtered Logarithmic Monthly Variety-Store timate of v ; decreases from 4; = ,97 for (1.4) to 4; Sales. = .93 for (1.5) and then to 4; = .89 for (1.6), and we This calculation shows the need for careful exami- see that prior to level-shift adjustments the fitted model nation of the homogeneity assumption underlying (1.4) is in reasonably close agreement with the factor Model (1.4). Indeed, applying the level-shift detection (1 + .40B + .27B2)(1 - B ) in (1.2). Although one cannot reject the hypothesis that there is a unit root in procedure considered by Chen (1984), which is sum- the A R polynomials for (1.4), (1.5), and (1.6) (Dickey marized later in (4.17) and (4.18), to the residuals of and Fuller 1979; Said and Dickey 1985), it is clear that (1.4),we find strong evidence of a level drop occurring the fitted A R part of Model (1.6) shows much less at the time point t(l) = April 1976. To adjust for the effect of this unusual event, we may entertain the fol- evidence of nonstationarity compared with that of (1.4). lowing modified model: This is implying that the differencing operator ( 1 - B ) simply attempts to model the occasional level changes z, = c + w,)t + w , L S , ( t ( ' ) )+ N,, (1.5) in this series. It should be pointed out that, as reported where by Chang and Tiao (1982), the level shift at t'l) = April 1976 was associated with the fact that a major chain store went out of business at that time, but the reasons for the other two indicated shifts are unknown. The key point of the preceding example is that oc- The estimates of Model (1.5) are also reported in casional level shifts in real time series can lead to models Table 1. It shows that the adjustment of a single level of a very different nature, depending on whether these shift leads to appreciable changes in the estimates of changes are appropriately taken into account. It is also the parameters ( p i , y z , q 3 ,C , w,,)and a large reduction of interest to examine the potential influence of ad- in the estimated variance a?.Repeating the level-shift justing occasional level shifts on the forecasting per- detection procedure based on the residuals of Model formance. For illustration, let t, = September 1978 be (1.5),two additional possible level shifts have been in- the forecast origin and observations from October 1978 dicated at t(?) = September 1970 and t") = January to September 1979 be reserved for calculating forecast 1972. If all three level shifts are taken into account, the errors. A model of the form (1.2), which contains an estimation results of Model (1.6) are again in Table 1: exact differencing factor ( 1 - B ) , and Models (1.4) and 3 (1.6) are estimated, respectively, based on observations z, = c + w,,t + ~ , L s , ( t ( ~+ )N,. ) (1.6) up to to. Table 2 lists the 12-period forecast errors. As k= 1 expected, the forecast errors of (1.2) and (1.4) are in We see that the estimated linear-trend coefficient oj,, close agreement and are much worse than those of (1.6). changes markedly from .24 x for (1.4) to .17 X For this particular forecast period, we find that adjust- l o - ? for (1.6) after the level adjustments. The latter ing occasional level shifts helps improve forecast ac- estimate, with a t value of 3.84, is seen to be compatible curacy. Table 1. Time Series Modeling of Variety-Store Series Model C Wo PI P2 p3 103 x c2 uc NOTE: Esttmated standard errors are In parentheses Chen and Tiao: Random Level-Shift Models and a Detection Procedure 85 Table 2. Forecast Errors for Variety-Store Series (lead 1 = October 1978) Number of lead Model 1 2 3 4 5 6 7 8 9 10 11 12 (1.2) - ,023 - ,006 ,002 - ,052 - ,009 - ,031 - ,061 - ,046 - ,041 - ,063 - ,077 - ,086 (1.4) -.024 -.007 .004 -.053 -.011 -.033 -.064 -.050 -.045 -.068 -.083 -.092 (1.6) -.015 ,001 -.008 -.041 ,003 -.Ole -.047 -.030 -.024 -.044 -.057 -.064 1.2 The Random Level-Shift Problem (1984). The random-effect formulation is intuitively appealing because economic and environmental time The preceding example motivates the study of the series are affected by many unusual events, the oc- following problems: (a) What is an appropriate model currence and the impact of which may be described to describe the phenomena of occasional level shifts in by probability laws. Furthermore, it provides a con- time series? (b) What would be the potential loss of venient framework to assess the performance of stan- forecasting efficiency by ignoring the information of dard time series method on series with level shifts by such level shifts? (c) Is there an efficient way to detect evaluating the contributions of the level shifts to fore- such occasional level shifts? This article attempts to cast (or level estimation) mean squared error (MSE). address the issues in (a) and (b). A procedure for (c) In Section 2, a random level-shift ARMA model is will be given with illustrations of its usefulness in model introduced. The connection between this model and building, but its detailed properties will be discussed standard autoregressive integrated moving average elsewhere. (ARIMA) model-building procedures will be discussed General statistical approaches to problems of occa- in Section 3. In Section 4, the efficiency of an ARIMA sional stuctural changes so far consist of (a) developing approximation to the random level-shift ARMA model methods to detect changes or outliers at known or un- is investigated. Section 5 outlines some possible gen- known time point (e.g., Box and Tiao 1975; Chang and eralizations. Tiao 1982; Hinkley 1970, 1972; Page 1955, 1957), (b) devising robust prdcedures for data-analysis in the pos- 2. THE RANDOM LEVEL-SHIFT ARMA MODEL sible presence of structure changes or outliers (Martin 1980), and (c) modeling the nature of the changing A time series { y , } is said to follow a random level- behavior or outlying observations and deriving methods shift ARMA (RLARMA) model with order ( p , q), if according to the proposed models (Box and Tiao 1968; the stochastic structure of { y J is defined as Chernoff and Zacks 1964; Kander and Zacks 1966; Tay- lor 1980; Yao 1984). In particular, the models consid- ered by Chernoff and Zacks, Taylor, and Yao share the where p, = j,q, + pI- ( t = 2, 3, . . . , p,) is an unknown characteristic that the structure of the series may change initial level of the process; { x J follows a stationary occasionally and the time periods of structural changes Gaussian ARMA ( p , q) model such that 4(B)x, = are determined by a stochastic process. The models O(B)a,, 4(B) = (1 - 4 , B - ... - 4pBp), and O(B) = considered by Chernoff and Zacks and Yao, however, (1 - OIB - ... - 0,Bq); all the zeros of 4(B) and O(B) are perhaps too simple to describe usual time series are outside the unit circle; and {a,} are iid N(0, a ? ) . It data, and Taylor's model is so flexible that it needs an is also assumed that {q,} are iid N(0, ~ 0 ~{ j , } ; iid ) are identification procedure to specify a suitable form of . such that Pr( j, = 1) = 1 and Pr( j, = 0) = 1 - j; and . the model through the data to gain any insight from the {a,}, {q,}, and { j , } are mutually independent. model. A random level-shift ARMA process { y , } can be re- In this article, we focus on the phenomena of occa- garded as the sum of a level process b,} a stationary and sional level shifts in time series. It is known that sta- ARMA process {x,}. In the level process bl,},the ran- tionary ARMA models can represent series that dom variable j, is an indicator variable for the occur- homogeneously fluctuate around a constant level and rence or nonoccurrence of level shift, the frequency of nonstationary ARMA models can describe series in the level shift is determined by I., and the magnitude which the level changes from one time period to an- of the shifts are given by the random variables qr If 1 . other. When the level of the series changes only oc- = 0, the level of { y , ) stays unchanged and the structure casionally, as illustrated by the preceding example, the of { y , } reduces to that of a stationary A R M A ( p , q) usual ARMA model may not be appropriate. Now oc- process. If 3. = 1, the level changes from period to casional level shifts may be regarded as interventions period. In this case, { y , } is the sum of a Gaussian ran- with fixed effects at unknown time points; this approach dom-walk process and the stationary process {x,} and was taken by Hinkley (1970, 1972). An alternative way hence follows a nonstationary A R I M A ( p , 1, q * ) is to assume that the effect and the time point of the model, where q* = max(p, q + 1). When 0 < 3. < 1, level shift are both subject to some probability rules, the degree of nonstationarity of the { y , } process is be- as considered by Chernoff and Zacks (1964) and Yao tween that of a stationary A R M A ( p , q) process and a 86 Journal of Business & Economic Statistics, January 1990 nonstationary ARIMA(p, 1, q*) process. In this sense, the series is often signalled by the slowly decaying be- the RLARMA process can be used to represent series havior of the sample autocorrelation function (SACF). that exhibit nonstationarity between zero differencing The following theorem gives the asymptotic properties and first-order differencing. of the SACF for series generated from an RLARMA process. 3. RLARMA MODEL AND STANDARD ARIMA MODEL-BUILDING PROCEDURE Theorem I . Let { y,) ( t = 1, . . . , T ) be observations of an RLARMA(p, q) process, as defined in (2.1). If In this article, we shall not be concerned with direct . the probability of level shift i is strictly positive, then estimation of the parameters of the RLARMA model for any positive integer 1 (2.1). Our main focus will be on assessing the perform- ance of an ARIMA approximation to the RLARMA model for forecasting and estimation of current level. For the model in (1.1), widely used specification pro- cedure proposed by Box and Jenkins (1976) is (a) to difference y, as many times as is needed to produce The theorem follows from the results of lemma 2.6 and stationarity and then (b) to identify the resulting sta- corollary 2.6 of Tiao and Tsay (1983). tionary ARMA process. The necessity of differencing To illustrate the pattern of SACF for finite samples, Table 3. Mean and Standard Deviation (SD) of SACF of an RLARMA (0, 0) Process: 500 iterations i = .05 Mean SD i = .02 Mean SD i. = .01 Mean SD E. = .05 Mean SD i. = .02 Mean SD i = .01 Mean SD . = .05 Mean SD i = .02 Mean SD i. = .01 Mean SD i = ,05 Mean SD i .02 = Mean SD i = .01 Mean SD Chen and Tiao: Random Level-Shifl Models and a Detection Procedure 87 Table 3 gives the means and standard deviations of the SACF from a simulation experiment based on an RLARMA(0, 0) process with selected values of i., K, and T. The averaged SACF behaves as one would ex- pect from Theorem 1. The convergence rate of the sam- ple autocorrelations to 1 depends on the magnitude of ~ E.K. When i . is small, the sample autocorrelations may not be near 1, but the slowly decreasing pattern always exists. This pattern of SACF usually leads to differ- encing the series. Applying V = (1 - B) to both sides of Equation (2.1) yields Figure 2. Time Series Plot of Series A. WI = jrqr + VX,, (3.2) It could be interpreted that the ARMA model is able where w, = Vy,. The differenced series {w,) is the sum to explain the behavior of the first two moments of a of a stationary ARMA process {Vx,) and a stationary suitably differenced RLARMA process. It is then nat- process {j,q,): hence it is stationary but non-Gaussian. ural to ask (a) whether the RLARMA formulation pro- Let p,,(l) be the autocorrelation function (ACF) of se- vides any advantage over the ARIMA model for the ries {w,). It is easy to see that purpose of data analysis and (b) whether, if one starts with the ARMA modeling, it is possible to discover the random level-shift ARMA structure. where c = + i ~ a ~ / v a r ( V x , ) ) -This shows that the (1 '. 4. PERFORMANCE OF THE ACF of a differenced RLARMA series, {w,), will have ARIMA APPROXIMATION the same pattern as that of a stationary ARMA process, {Vx,), except for the initial dropoff from p(0) to p(1). Statistical models are built to explain phenomena that involve uncertainty. The choice of suitable models usu- Hence, following the Box-Jenkins iterative-modeling ally depends on the purpose of the practitioners. In procedure, an RLARMA process may easily be mis- Section 3, it was shown that a random level-shift ARMA specified as an ARIMA process. model may often be misspecified as an ARIMA model. As an illustration, series A with 150 observations is Inferential procedures-for instance, forecasting- generated by the model based on this misspecified ARIMA model may be re- garded as an approximation to those corresponding to the RLARMA process. In this section the efficiency of this approximation is investigated by comparing the a,, j,, and q, are as defined in (2.1); E. = .02; K = 9; a 2 MSE's of forecasts and those of estimating the current = 1; and the initial level p, is set to be p1 = 3. 'Figure level of the process. We assume first that all of the 2 shows the plot of the series, exhibiting two level shifts parameters in (2.1) are known, but this assumption will at t, = 48 and t2 = 113. The SACF of the original and be relaxed later. the differenced series are given in Table 4. The SACF 4.1 Minimum Mean Squared Error Estimates of of the original series dies out slowly, and that of the differenced series gives insignificant values after lag 2. YT+I and PT With the preceding patterns of SACF's, one would ten- Let yT = ( y , , . . . , yT)I be the T x 1 vector of tatively suggest an ARIMA(0, 1, 2) model for the observations of an RLARMA model (2.1) and J = ( j, , original series. The estimated model for series A is . . . ,jT)' be the T x 1 vector of indicator variables for level shifts. In practice, J is usually unknown. For es- timating (forecasting) the future observations yT+/ =(1 1, 2, . . .) and the current level p T based on yT, the minimum MSE (MMSE) estimates are the conditional expectations E ( y T + // yT) and E(pT I yT). We shall de- where the values in parentheses are the associated stan- note the errors as dard errors. Portmanteau tests on the SACF of the residuals does not show evidence of inadequacy of ARIMA(0, 1, 2). Table 4. Sample ACF' of Series A and {Cy,} 88 Journal of Business & Economic Statistics, January 1990 and the corresponding MSE of e(1) as where {q:) are iid with zero means, common variance Kl.o2,and Gaussian, then y, will follow a standard Gauss- ian ARIMA model. The covariance structure of the where the expectation is taken over the joint distribu- vector series { y,, p,) corresponding to the model in (2.1) tion of (p,, y,,,, yT). When 1 > 0, e(1) is the lead-1 and that corresponding to the replacement (4.5) are forecast error and e(0) is the error of estimating the exactly the same, however. Thus the misspecification current level. Now it will be useful to decompose e(1) of an RLARMA model by an ARIMA model concerns as the sum only the distributional differences between {q:) in (4.5) and {j,q,) in (2.1). Now with the replacement (4.5), e(1) = el(/) + en(/), (4.3) minimum MSE estimates of y,+, and p, based on yT where will be linear in y,. We can alternatively regard our comparison of the efficiency of forecasting y,,, and estimating p T between the RLARMA model and the - ARIMA approximation as a study of the loss in the I, - Yr+/ - E ( ~ T + yr, J ) , 1 > 0, efficiency of linear estimation. Specifically, for estimating (forecasting) y,+, based on y,, note first that from (2.1) and (4.5), the overall ARIMA model for y, takes the form and E(. / y,, J ) are expectations conditional on (y,, J). Using the fact that where a(B) = (1 - a l B - ... - a,*B47, q* = max(p, q + I), {c,) are iid N(0, o f ) , and o f and a(B) can be obtained from the relation we can then write MSE(1) as cy(B)a(F)af @(B)@(F) , = Kl. + (1 - B)(l - F) 4(B)4(F)a- 4(B)4(F)' where V1(l) = C, P(J) ~ [ e : ( l )I J ] and V"(1) = E[ei(l)]. Note that in (4.4) the expectation E[e:(l) J ] is taken where F = B-' (e.g., see Cleveland and Tiao 1976; over the joint distribution of (p,, y7+,, yT), conditional Hillmer and Tiao 1982). In particular, for 1. > 0, a(B) on J , and p(J) is the distribution of J . On the other # 0 for 1 BI = 1. For large T, the results of Box and hand, the expectation E[ei(l)] is taken with respect to Jenkins (1976) can be used to obtain the optimal fore- the joint distribution of (y,, J ) . As will be shown in cast for y,+,, );,(/), whose MSE is Appendix A, where explicit expressions of V,(l) and (~E M S E A R M A = ) ( Y T +-~ V,,(l) are derived, the main reason for the decomposi- tion in (4.3) and (4.4) is that E[e:(l) I J] can be readily 1- 1 calculated, but determination of E[e:(l)] is considera- = of C v/5, s=n v/,, = 1, 1 > 0, bly more burdensome. Note that, since V,,(l) 2 0, we can regard V1(l) as a lower bound for MSE(1). There- (4.8) fore, in our comparison of MSE(1) with the MSE from where the W,'s satisfy the relation 4(B)(l - B)(1 + - ARIMA approximations to be presented later in Sec- tylB + F 2 B 2 + ...) = ff(B). tion 4.3, we shall mainly use simulation estimates of the Next, for estimating the current level p, from (2.1) lower bound V,(l). In a few cases, simulation estimates and ( 4 . 9 , we may employ the signal-extraction theory of V,(l) will also be given to illustrate the magnitude for a nonstationary ARIMA model given, for example, of V"(1). by Box, Hillmer, and Tiao (1976). Substituting the fu- ture y's with the forecast values based on (4.6) and { y , , 4.2 The MSE of the ARIMA Approximation . . . , yT), it can be shown that the best linear estimate From the discussion in Section 3, we see that the of current level is RLARMA model (2.1) may be misspecified as a Gauss- ian ARIMA model following standard model-building procedures. The reason is that these procedures assume normality of the observations and consider mainly the As shown by Beveridge and Nelson (1981), the right covariance structure of {y,) and that of the differenced side is, in fact, the limit lim,,, jT(1), and they have series {w,). o see this misspecification clearly, note that T regarded this as an estimate of the permanent level of in (2.1) the level component p, is such that for t 2 2, y7-. It is shown in Appendix B that the MSE of fi,, p, = p,-, + j,q,, where the j,q,'s are independent and denoted as MSEARMA(0), is have zero means and common variance k-l.02. If we MSEARMA(O) E(PT - lll^~)? = assume instead that Chen and Tiao: Random Level-Shift Models and a Detection Procedure 89 where 4'(1) and a'(1) are, respectively, the first deriv- The MSE of the ARIMA approximation MSEARMA(l) atives of 4(B) and a(B) with respect to B evaluated at can be easily obtained from (4.8) and (4.10). From (4.4) B = 1. The results in (4.8) and (4.10) can now be used and applying the weak law of large numbers, a con- to compare with the MSE(1) in (4.4). sistent estimate of V1(l) can be obtained by calculating Note that in the Gaussian ARIMA framework, de- 2 E[ei(l) I J]/N,, where the summation is taken over composition of an overall model for y, into a nonsta- a random sample of J'S from the distribution ( A . l ) and tionary component model for p, and a stationary E[e:(l) / J] is obtained for each J from (A.8). The length component model for x, is, in general, not unique (e.g., of the observed series is set equal to 100; that is, T = see Tiao and Hillmer 1978); also the estimation of p, 100. After some preliminary experimentation over var- will depend on the component model assumed. For the ious values of i.'s, we choose the number of samples N, present problem, we are investigating the efficiency of to be 1,000. An upper bound for the loss of efficiency the ARIMA approximation to the RLARMA model by using the ARIMA forecasts is obtained by calculat- defined in (2.1); here the random-walk formulation ing the ratio (4.5) for p, seems to be the natural one to use, since, as mentioned earlier, it leads to an identical covariance structure of (y,, p,) for both the RLARMA and the ARIMA models. One may raise the broader issue of the uniqueness of the RLARMA component structure, For forecasting comparisons, the R(1)'s have been but this topic is beyond the scope of this article. The calculated for E. = (.01, .02, .05, .08, . l o , .15), 4 = uniqueness issue discussed here only affects estimation (-.9, -.5, .O, .5, .9), K = (9.0,25.0),and1 = 1 , . . . , of current level and has no impact on forecasting. 10. Only partial results are reported here to show the pattern of relative forecasting performances. Specifi- 4.3 Efficiency of the ARIMA Approximation to cally, Table 5 gives R(1)'s when 4 = .O and K = 25.0 the RLARMA Model for various values of i. and 1, and Figure 3 shows plots of R(1) (1 = 1, . . . , 10) for i. = (.01, .05, .15), = To illustrate the loss in efficiency of the ARIMA ( - .9, - .5, .O, .5, .9), and K = 25. The loss in MSE's approximation to the RLARMA model, we consider by using the ARIMA approximation varies with the the following simple model: frequency of level shift I., the autoregressive coefficient 4 of the noise term, and the lead time I. In general, the loss in efficiency decreases as E. is increased, and it becomes very small for large 1. For short-term forecasts (low values of I), the potential maximum loss by using where {a,), {j,), and {q,) are defined in (2.1). Chernoff the ARIMA approximation is about 25% when 4 2 0. and Zacks (1964) considered methods of estimating the Although this loss can be substantially higher for 1 = current level for a special case of this model ( 4 = 0). 1 when 4 < 0, it declines rather precipitously for 1 2 The purpose of our study is to investigate the perform- 1. We remark that for K = 9, the same pattern of R(1) ance of the ARIMA approximation to this RLARMA is found and, as would be expected, the loss is uniformly model in terms of both forecasting and current-level smaller than the cases reported here. For instance, for estimation. Let w, = Vy,; it is straightforward to see ( 4 = .O, I. = .01), R(1) is 1.173 for K = 9 versus 1.267 that for K = 25; for ( 4 = - .5, E. = .01), it is 1.341 for h- = 9 versus 1.487 for K = 25; for ( 4 = .5, 1. = .01), it is 1.087 for K = 9 versus 1.105 for K = 25. In sum, our results suggest that ARIMA approximation performs quite well for forecasting except when the frequency of shift 1. is very small andlor the autoregressive coefficient 4 has a large negative value. and p,,(l) = $p,(l - 1) for 1 2 2. From the ACF of Accuracy of the Upper Bound R (I) and Current Level w,, the ARIMA approximation to {y,) would be the Estimation Loss. From (4.4), the exact loss of effi- model ciency is where a is a constant parameter and {c,) would be re- garded as iid N(0, o f ) .The quantities a and afcan be determined in terms of the parameters in (4.11) by To assess the sharpness of the upper bound R(1) of the exact loss, evaluation of V,(l) is required. From (A.13), (4.14) VO(l)may be estimated by the average of eZ(1) over, where rn = ((1 + 4 2 ) i + 2)1(1 ~ + $1.~)and 0 < a say, N1 random drawings of ( J , yT) from the joint dis- 5 1. tribution P(J)P(yTI J ) given in ( A . l ) and ( A . l l ) . Now, Chen and Tiao: Random Level-Shift Models and a Detection Procedure 91 Table 6. V,(I)I(V,(I) + V,(I)): = 25, T = 100 = 25 and selective values of ( 4 , I., 1). It shows that for given in (4.5) will lead to very inefficient results if in the comparisons of forecasting performances the upper fact p, only changes occasionally. bound R(1) (1 2 1) will be close to the exact ratio Eff(1) Now nonstationary models of nearly the forms (1 - for measuring the loss by using the ARIMA approxi- B)dp, = yl: for d = 1, 2 or (1 - BS)p, = yl: for s = 4, mation. The entries for 1 = 0 in Table 6 imply that the 12 have been employed to represent the underlying upper bound R(0) will substantially overestimate the stochastic structures for the trend and seasonal com- loss in efficiency for estimating the current level p. It ponents of commonly used seasonal-adjustment pro- is for this reason that the R(0)'s have not been included cedures such as the Census X-11 method (e.g., see in Table 5. The estimated loss of efficiency Eff(0) is Burridge and Wallis 1984; Cleveland and Tiao 1976), given in Table 7 for K = 25 and selected values of ( 4 , model-based approach (Hillmer and Tiao 1982), and i.). Although because of computational complexity we Kalman-filtering method (Engle 1976). Since the main have only estimated Eff(0) for a much smaller number interest in these adjustment procedures is in estimating of combinations of values of ( 4 , j) it is clear from ., the components rather than forecasting, our findings Table 7 that, in contrast to the forecasting comparison suggest that the procedures could be very sensitive to shown in Table 5 and Figure 3, a very substantial gain occasional level shifts in the series, and appropriate in efficiency will be obtained by knowing the true model statistical analysis should be performed on the data to when estimating the current level. Moreover, this gain ascertain if such shifts indeed occur. is higher when the noise term is negatively correlated and there is a general tendency for Eff(0) to decrease as E. is increased for fixed 4 . Notice that the results are 4.4 A Level-Shift Detection Procedure obtained from approximation and the accuracy is not high enough to establish the property that Eff(0) is A full analysis of the RLARMA process (2.1) would uniformly decreasing in E. for fixed 4 . The overall mag- involve specification of the model for x, and joint es- nitude of Eff(0) is our major concern, however. timation of the parameters (I., K) for the level process p,, as well as those for the x, process. For series exhib- Discussion. When 3. = 1.0, the model in (4.11) be- iting occasional level shifts, accurate estimates of (I., K) comes exactly a nonstationary ARIMA(1, 1, 1) model. would require a very large sample. The simulation re- The findings reported in Table 3 and Figure 3 show the sults in Section 4.3 suggest that, at least for forecasting remarkable degree of robustness of the ARIMA(1, 1, purposes, one needs only be concerned with the situ- 1) model to departure from i. = 1.0 for forecasting ation in which i. is very small or x, is highly negatively future observations. The ARIMA approximation, how- autocorrelated, but in these cases the sample would ever, becomes inadequate when the probability . of ; contain much information concerning the time points level change is very small or the series are negatively and the magnitudes of possible level changes. The rea- correlated. sons are: (a) when i. is very small, the changes are On the other hand, Table 7 shows that the ARIMA infrequent so that there will tend to be a large number approximation is clearly inadequate for estimating the of observations available between changes, and (b) it current level of an RLARMA model. This finding sug- is well known that negative autocorrelations increase gests that for a component model of the form y, = p, the efficiency in estimating the mean of observations. + x,, where x, follows a stationary stochastic model, Now for series of moderate length with very infrequent estimation of p, depends rather critically on the true level shifts of unknown magnitude at unknown time distributional structure of p,. In particular, a Gaussian points, there is little to choose between a random-effect random-walk type of approximation for p, such as that formulation such as that given by the RLARMA model in (2.1) and a fixed-effect formulation for practical pur- Table 7. Eff(0) for ti = 25 and T = 100 poses. A potentially useful modeling procedure to im- prove forecast performance might consist of (a) con- structing an ARIMA model for the observed series {y,}, (b) detecting the possible level shifts (i.e., estimat- ing J by appropriate analysis of the residuals, and (c) re- modeling the data and making forecasts based on the es- timated information of J. 92 Journal of Business & Economic Statistics, January 1990 Specifically, let y, = ( y , , . . . , y,)' be a set of ob- Examining the estimated ARMA coefficients, there is servations and this series be empirically described by a a near cancellation between the MA and the AR parts. general ARMA model q(B)y, = O(B)a,. Let a, = (dl, This leads us to consider a more parsimonious model, . . . , 6,)' be the estimated residuals of the ARMA fitting to y,. Based on the fitted model and the resid- uals, we can define the following statistics: f, =~;IIL(~)/~~~(IIL(~))'IIL(~), (4.17) where is the residual variance; L(i) = (LS,(i), . . . , LST(i))' is a T x 1 vector to indicate the occurrence of a level shift at t = i; and II is a T x T lower triangular matrix with elements rjj such that z,, = 0 if j > i, r,, = Note that the fitted noise structure (1 + .49B)a, is close 1 if j = i, and otherwise r,, = - 7r,-,, where (1 - 7r,B to the structure of x, in (3.4). This example shows that - 7r2B2- ...) = q(B)O(B)-I. Notice that 2, is the like- the procedure not only detects the level shifts but also lihood ratio statistic for testing the hypothesis of no helps identify partially the true underlying structure of level shift versus that of a fixed level shift at t = i. Chen the process. (1984) studied the properties and the performance of a level-shift detection procedure based on 4.5 A Study of ARMA Approximation to RLARMA When the Parameters Are Estimated where rm,, = maxi t,l and C is a positive constant. Typ- The results of Section 4.3 are obtained assuming that ically, we choose C = 2.8 for high sensitivity level-shift the parameters of the RLARMA model (4.11) and detection, C = 3.0 for moderate sensitivity, and C = those of its ARIMA approximation (4.13) are known. 3.3 for low sensitivity. It can be shown that the statistic In reality, the model and the parameters are determined ?, is measuring difference of levels before and after time from the nature of the data. The following Monte Carlo period i. Consequently, the power of this statistic will experiment is designed to investigate, in finite-sample be less when i is near to either the beginning or the situations, the relative performance of ARIMA model ending of the sample. Sampling properties of this sta- building with and without incorporating the level-shift tistic and its relationship with the RLARMA model will detection procedure (4.17) and (4.18) when the data be discussed in another article. In practice, a more ef- are generated from RLARMA models. ficient approach is to apply the method in an iterative fashion. Once a level shift is identified, the residuals Step 1. For given values of E., K , and 4 , 200 obser- are adjusted by removing the effect of this level shift. vations of 2, are generated from an RLARMA (1, 0) Then a new set of r, statistics as defined in (4.17) may model in (4.11). be computed based on the adjusted residuals. The it- Step 2. Standard Box-Jenkins techniques are ap- eration stops when r,,, is smaller than or equal to C. plied to the first 190 observations of the simulated series In Section 1.1, the statistic (4.18) was applied to the to build an adequate ARIMA (or ARMA) model. To filtered variety-store sales data. To further illustrate the reduce the judgmental effect in the process of model potential usefulness of the method, we apply the iter- building, however, three models are sequentially con- ative procedure as described with C = 2.8 to series A sidered to fit the first 190 observations of the simu- generated by the RLARMA model in (3.4). The pro- lated series. They are AR(1), ARIMA(1, 1, O), and cedure successfully identifies two level shifts at t, = 48 ARIMA(1, 1, 1). We take the position that a simpler and = 113. To incorporate this information in the model is maintained unless there is strong evidence model-building process, we consider the fixed-effect against it. Model adequacy is tested by the Box-Ljung model with level shifts at known time points t, = 48 Q statistics with 12 lags, Q12,at the 1% level of signif- and t2 = 113 in (4.19) as an alternative to Model (3.4) icance. If both AR(1) and ARIMA(1, 1,O) models are and obtain the following estimates (numbers in paren- rejected, then the ARIMA(1, 1, 1) model will be fitted theses are standard errors of the estimates): to the data. Step 3. Lead-1 to lead-10 out of sample forecasts and the corresponding forecast errors are calculated at or- igin 190 based on the model specified in Step 2. Step 4. The level-shift detection procedure (4.18) is applied in an iterative fashion to the residuals of the fitted results in Step 2. In this study, the critical value C is chosen as 2.8. When level shifts are identified, an intervention model incorporating level shifts with AR(1) noise will be employed to estimate the magni- tudes of the shifts as well as the time series parameter Chen and Tiao: Random Level-Shift Models and a Detection Procedure 93 in a manner similar to (4.20). Lead-1 to lead-10 out- adjusting for the occasional level shifts, the estimated sample forecasts are computed based on this modified n weights are very close to the true structure of the model, assuming that the effect of level shift will persist underlying process. Figure 4 presents the ratios of mean to the future. squared forecast errors from ARIMA (or ARMA) Step 5. Repeat Steps 1-4 500 times for each set of models over those from models incorporating level i K , and 4 . Mean squared forecast errors based on . , shifts. The overall patterns are largely consistent with ARIMA (or ARMA) models and AR(1) models in- those obtained in Figure 3, corresponding to the cases corporated with the level-shift information in Step 4 are of known parameters. In most cases, the magnitudes of computed, respectively. R(1)'s are about the same except for the case of (1. = .01, 4 = .5), in which the improvement of forecast Results. The averages of the Q12'sof the residuals- accuracy is even higher when the parameters are esti- sample ACF over the 500 iterations for various com- mated. With critical value set to C = 2.8, about 75.4% binations of /. and 4 range from 7.52 to 13.83 for the of level shifts are correctly identified, and the average final model in Step 2 and from 9.06 to 12.75 for model number of false level shifts found is 267 for each com- fittings in Step 4. On the average, the results do not bination of i. and 4 over the 500 iterations. To further indicate any serious inadequacy in model specification. illustrate the comparison, we compute the ratios of Table 8 reports the average n weights of the fitted mean squared forecast errors when parameters are es- ARIMA (or ARMA) models for Step 2, as well as those timated to that when parameters are known. The results from the intervention models for Step 4. The average of ARIMA approximation and intervention model n weights without adjusting the level shifts match fairly (RLARMA consideration) are reported in Figures 5 well with the theoretical ones calculated from (4.13) and 6, respectively. For the same 6 , the magnitudes of and (4.14), which also indicates that model specification MSE ratios in both approaches are about the same; this has been conducted properly. We also found that, after suggests that the loss of accuracy when the parameters Table 8. Theoretical and Estimated n Weights: k- = 25,500 Iterations ARIMA models Intervention models .5 ,042 .O (.132) .5 - ,012 .o (.072) .5 - ,036 .O (.079) NOTE n, IS the theoretical value and R, IS the mean estlmate of 11.Numbers In parentheses are the standard devlat~ons the estimates of 94 Journal of Business & Economic Statistics, January 1990 I 0 0 ( 2 0 . .. 4 0 + .. + ... 6 0 ~~ . . * . ~ . . . ~ ~ + ~- .. 8 0 loo - - ...................... .................. . . O D 2 0 4 0 . 6 0 (10 LOO . D O . - -- . 2 0 6 0 6 0 - ~ ~ ~ * . ~ . .~. 8 0 ~, l o o Figure 4. Plots of R(1) When Parameters Are Estimated: K = 25; A: i = .01; B: i= .02; C: i . = .05. are estimated is about the same in both approaches. follows an RLARMA process. The first generalization There is no obvious pattern in Figures 5 and 6 with can be defined explicitly as respect to A. In both cases there is a slight increasing yr=pr+N,, t = 1 , . . . , T, (5.1) trend of MSE's with respect to the number of forecast leads, which seems to indicate that the short-run fore- ) where U d ( B N , = x,, {p,)and {x,)are as defined in (2.I ) , casts suffer less due to the uncertainty of estimated and U d ( B )is a polynomial of degree d, having all of its parameters. Further studies are called for to understand zeros lying on the unit circle. In particular, if {N,) is a the performance of the proposed detection procedure random-walk process, the model becomes a discretized and the sampling properties of the estimates in the in- version of the compound events models considered by tervention model. Press (1967) and Clark (1973).That type of model has been a candidate for describing the behavior of security 5. GENERALIZATIONS prices. The second possible generalization may be for- The idea of occasionally changing level in stationary mulated as Ud(B)y, = P, + & ( t = 1, . . . , T ) or process can be extended to the situations that the ho- mogeneous component {x,} in (2.1) is nonstationary. -r y , = - +P Xr (5.2) Two potentially useful models are (a) the observed se- Ud(B) I/d(B)' ries is the sum of a level process {p,)and a nonstationary where {x,)and {p,)are as defined in (2.1) and U d ( B )is ARMA process N, and (b) a suitably differenced series that given in (5.1). This class of model may be used to Figure 5. Ratios of MSE's: Estimated Parameters Versus Known Parameters, ARlMA Approximation: K = 25; A: A = .01; B: A = .02; C: i = .05. Chen and Tiao: Random Level-Shift Models and a Detection Procedure 95 Figure 6. Ratios of MSE's: Estimated Parameters Versus Known Parameters, RLARMA Consideration: h- = 25; A: i = .01; B: i = .02; C: i = .05. represent series with various types of abrupt structural suppose that, for t = 1, 2 , . . . . changes. For instance, if U d ( B ) = ( 1 - B ) in (5.2), then { y,) may reveal occasional changes in trend. Notice that ( 5 . 1 ) and ( 5 . 2 ) are very different in terms of the effect of the changing p,. For example, if U d ( B ) = 1 - B , then (5.1) implies y, has random level shifts and where the y,'s can be obtained from the relation nonstationary error, whereas (5.2) implies that y, has $ ( B ) y ( B ) = O(B) with y ( B ) = ( 1 + y I B + y z B 2 + random linear-trend shifts and nonstationary error. ...). From (2.1) and ( A . 2 ) , we can write Moreover, ( 1 - B ) y , has random point outliers and stationary error under (5.1) and random level shifts and stationary error under (5.2). Identifying these two models requires careful analysis of both the original series and suitably differenced data. In general, the where 0 is a T x 1 null vector, T and U are T x T preceding models can be employed to represent series lower triangular matrices such that with behavior of nonstationarity between d differencing and ( d + 1 ) differencing. It would be of interest to investigate the efficiency of A R I M A approximations to such models and for low frequency shift in p, (small values of /.) and to extend the level-shift detection pro- cedure in (4.17) and (4.18) to cover these cases. D is a T x T diagonal matrix with diagonal elements I ( j l r . . . . jr), +(I' = (yT+1-1, . . . . V I ) ' , rl = C5=l ACKNOWLEDGMENT h ~ T + ~ Y I T and? rz = 2:: W 5 a ~ + ~ - s + ~ Since { j,), { ~ r ) , and {a,) We thank the referees and an associate editor for are independent, it follows that, conditional on J , ( Y ~ , their helpful comments. pr, y T + / )are jointly normal and their covariance matrix is APPENDIX A: EXPLICIT EXPRESSIONS OF V,(I) AND V,(I) where We here give explicit expressions for V , ( l ) and V,,(l) in (4.4). From (2.1), the distribution p ( J ) is A = K ~ "J"t" ' ' U " m m For notational convenience, we now assume that the processes {p,) and {x,) start at t = 1. Specifically, we 96 Journal of Business & Economic Statistics, January 1990 m = J ' J is the number of level shifts, and q = where p ( J ( yT)is the conditional distribution of J given CJ=:/-l tyf. It follows that, for the Vl(l) component in yT. Moreover, by Bayes's theorem, we find that (4.41, I P(J I YT)= P ( J ) P ( Y ~J)/C I P ( J ) P ( Y ~J). (~.ll) J where p(yT 1 J ) = (2n)-T'21MJ11'2a-Texp{- (l/2a2)(r'r - r'wJ$m)}. It follows that where R = (KUDU' + q Y 1 ) - ' . For actual computation of ( A S ) , the main burden 0 1 = 1 - J L EJ 1= 0 lies in evaluating the T x T matrix R. Note that the = (I,, - W;+(l))'$,,, rank of UDU' is m, the number of nonzero values in J or the number of level shifts that occurred in the - EJJ 1 T - ) 1 > 0, observational period. For small m , computation of R (A.12) can be drastically simplified by employing the identity (I, + PQ)-I = I, - P(I,, + QP)-lQ, where P is an and hence n x n' and Q an n' x n matrix. Specifically, letting H be the T x T lower triangular matrix From (4.4), (A. I), (A.8), (A.12), and (A. 13), it is clear that exact evaluation of either V1(l) or V,,(l) will be extremely burdensome for large T, because there are 2Telements in the sample space of J . Since E[e:(l) I J ] is available from (A.8), however, estimates of VI(l)can where the elements h,'s can be obtained from the re- be readily obtained by performing simulation experi- lation B(B)(l - B)H(B) = 4(B) with H(B) = (1 + ments with respect to J only. Moreover, we see from hlB + h2B2+ If t,, . . . , t,,,denote the positions .a. .) (A.12) and (A. 13) that considerably more complex sim- where a level shift occurs, let WJ = (h,,, . . . , h,) be ulation experiments will be required to estimate the a T x m matrix such that j,, + ... + jlm = m. An component V,,(l). alternative expression for R is APPENDIX B: PROOF OF THE IN MSEARMA(0) (4.10) From (2.1) and (4.9), we have that Then Equation (AS) may be simplified as where w = a ( l ) / 4 ( l ) . Since {p,}and {a,} are indepen- dent, the covariance generating function of (pT - ,iT) is G(B, F) = Kia2(l - B)-'(l - F ) - ' where 1 is an m x 1 vector of ones, Mj = (I + , KW;WJ)-I is an m x m matrix, and KM,W;WJ = I - Mj. In (A.8), t ,, . . . , t, are.the positions at which a level shift occurs. Expressions ( A . l ) and (A.8) can now From (4.7), by setting B = F = 1, we obtain xE.a' = be employed to evaluate the component V,(l) in (4.4). azw2, and hence in (B.2) Turning now to the component V,,(l)in (4.4), we first consider the terms of e,,(l) in (4.3). From (A.3) and (A.4), it is readily shown that E(pT1 y,, J ) = lk4, and , where $ = h-MJW;r, r = IIy,, and II = Y-l is a T X T lower triangular matrix. Note that It follows that (B.2) can be written as E ( Y T +I IYT)= 2 p(J I YT)E(.YT+/ J), J / YT, (A.10) Chen and Tiao: Random Level-Shift Models and a Detection Procedure 97 Let -(1975). "Intervention Analysis With Applications to Eco- nomic and Environmental Problems," Journal of the American Statistical Association, 20. 70-79. Burridge. P., and Wallis, K. F. (1984). "Unobserved-Components Models for Seasonal Adjustment Filters," Journal of Business and and note that C:=,, 8, = w4(l)la(l) = 1. 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If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. References A Bayesian Approach to Some Outlier Problems G. E. P. Box; G. C. Tiao Biometrika, Vol. 55, No. 1. (Mar., 1968), pp. 119-129. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28196803%2955%3A1%3C119%3AABATSO%3E2.0.CO%3B2-B Unobserved-Components Models for Seasonal Adjustment Filters Peter Burridge; Kenneth F. Wallis Journal of Business & Economic Statistics, Vol. 2, No. 4. (Oct., 1984), pp. 350-359. Stable URL: http://links.jstor.org/sici?sici=0735-0015%28198410%292%3A4%3C350%3AUMFSAF%3E2.0.CO%3B2-A Estimating the Current Mean of a Normal Distribution which is Subjected to Changes in Time H. Chernoff; S. Zacks The Annals of Mathematical Statistics, Vol. 35, No. 3. (Sep., 1964), pp. 999-1018. Stable URL: http://links.jstor.org/sici?sici=0003-4851%28196409%2935%3A3%3C999%3AETCMOA%3E2.0.CO%3B2-N A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices Peter K. Clark Econometrica, Vol. 41, No. 1. (Jan., 1973), pp. 135-155. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28197301%2941%3A1%3C135%3AASSPMW%3E2.0.CO%3B2-G http://www.jstor.org LINKED CITATIONS - Page 2 of 4 - Decomposition of Seasonal Time Series: A Model for the Census X-11 Program W. P. Cleveland; G. C. Tiao Journal of the American Statistical Association, Vol. 71, No. 355. (Sep., 1976), pp. 581-587. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197609%2971%3A355%3C581%3ADOSTSA%3E2.0.CO%3B2-E Distribution of the Estimators for Autoregressive Time Series With a Unit Root David A. Dickey; Wayne A. Fuller Journal of the American Statistical Association, Vol. 74, No. 366. (Jun., 1979), pp. 427-431. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28197906%2974%3A366%3C427%3ADOTEFA%3E2.0.CO%3B2-3 An ARIMA-Model-Based Approach to Seasonal Adjustment S. C. Hillmer; G. C. Tiao Journal of the American Statistical Association, Vol. 77, No. 377. (Mar., 1982), pp. 63-70. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198203%2977%3A377%3C63%3AAAATSA%3E2.0.CO%3B2-9 Inference About the Change-Point in a Sequence of Random Variables David V. Hinkley Biometrika, Vol. 57, No. 1. (Apr., 1970), pp. 1-17. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28197004%2957%3A1%3C1%3AIATCIA%3E2.0.CO%3B2-9 Time-Ordered Classification D. V. Hinkley Biometrika, Vol. 59, No. 3. (Dec., 1972), pp. 509-523. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28197212%2959%3A3%3C509%3ATC%3E2.0.CO%3B2-D Test Procedures for Possible Changes in Parameters of Statistical Distributions Occurring at Unknown Time Points Z. Kander; S. Zacks The Annals of Mathematical Statistics, Vol. 37, No. 5. (Oct., 1966), pp. 1196-1210. Stable URL: http://links.jstor.org/sici?sici=0003-4851%28196610%2937%3A5%3C1196%3ATPFPCI%3E2.0.CO%3B2-K http://www.jstor.org LINKED CITATIONS - Page 3 of 4 - A Test for a Change in a Parameter Occurring at an Unknown Point E. S. Page Biometrika, Vol. 42, No. 3/4. (Dec., 1955), pp. 523-527. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28195512%2942%3A3%2F4%3C523%3AATFACI%3E2.0.CO%3B2-K On Problems in which a Change in a Parameter Occurs at an Unknown Point E. S. Page Biometrika, Vol. 44, No. 1/2. (Jun., 1957), pp. 248-252. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28195706%2944%3A1%2F2%3C248%3AOPIWAC%3E2.0.CO%3B2-K A Compound Events Model for Security Prices S. James Press The Journal of Business, Vol. 40, No. 3. (Jul., 1967), pp. 317-335. Stable URL: http://links.jstor.org/sici?sici=0021-9398%28196707%2940%3A3%3C317%3AACEMFS%3E2.0.CO%3B2-0 Hypothesis Testing in ARIMA(p, 1, q) Models Said E. Said; David A. Dickey Journal of the American Statistical Association, Vol. 80, No. 390. (Jun., 1985), pp. 369-374. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28198506%2980%3A390%3C369%3AHTIAM%3E2.0.CO%3B2-4 Conjectured Models for Trends in Financial Prices, Tests and Forecasts Stephen J. Taylor Journal of the Royal Statistical Society. Series A (General), Vol. 143, No. 3. (1980), pp. 338-362. Stable URL: http://links.jstor.org/sici?sici=0035-9238%281980%29143%3A3%3C338%3ACMFTIF%3E2.0.CO%3B2-R Some Consideration of Decomposition of a Time Series G. C. Tiao; S. C. Hillmer Biometrika, Vol. 65, No. 3. (Dec., 1978), pp. 497-502. Stable URL: http://links.jstor.org/sici?sici=0006-3444%28197812%2965%3A3%3C497%3ASCODOA%3E2.0.CO%3B2-4 http://www.jstor.org LINKED CITATIONS - Page 4 of 4 - Consistency Properties of Least Squares Estimates of Autoregressive Parameters in ARMA Models George C. Tiao; Ruey S. Tsay The Annals of Statistics, Vol. 11, No. 3. (Sep., 1983), pp. 856-871. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28198309%2911%3A3%3C856%3ACPOLSE%3E2.0.CO%3B2-Z Estimation of a Noisy Discrete-Time Step Function: Bayes and Empirical Bayes Approaches Yi-Ching Yao The Annals of Statistics, Vol. 12, No. 4. (Dec., 1984), pp. 1434-1447. Stable URL: http://links.jstor.org/sici?sici=0090-5364%28198412%2912%3A4%3C1434%3AEOANDS%3E2.0.CO%3B2-T

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