# [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational Intelligence - Mining multivariate time series for product migration analysis

Post on 11-Mar-2017

214 views

TRANSCRIPT

Third International Workshop on Advanced Computational Intelligence August 25-27,2010 - Suzhou, Jiangsu, China

Mining Multivariate Time Series for Product Migration Analysis

Tom S. Au, Rong Duan, and Wei Jiang

Abstract- As new technologies or products emerge, customer may migrate from a legacy product to a new product. One way to find out who migrate, how migrations look like, and the relationship between the legacy product and the new product is through mining the customer transaction history over time. For these purposes, we propose two customer segmentation procedures to quantify business impact of technology substitution. By assuming a general linear relationship between two substitutable products, we first develop a co-integration model to describe the dynamic relationship of two substitutable products. We then add structure breaks in the co-integration model to capture business changes along time. Structure breaks in either slope or intercept of the linear model are considered and the Least Angle Regression (LARS) algorithm is applied to estimate the structure break co-integration model. The estimated parameters are used to segment migration customers. The contribution of the proposed method is the mining of multivariate time series relationships of various customers, which is different from traditional time series mining research where univariate similarities among time series of different customers are explored. Another advantage is the inclusion of multiple break points with different types in a unified cointegration framework. To validate the accuracy and efficiency of the model, an industrial example from a telecommunication company is demonstrated.

I. INTRODUCTION

In this paper we consider N bivariate time series {Xit, Yit}, where i = (1,,,, ,N) represents the ith pair of time series and t = (1, ,T) represents the tth point of the time sequence. This type of time series is referred to as cross-sectional time series where the bivariate time series are taken for one unit. And the two time series are additive, which describe the overall spending/web-clicks for each customer, and our objective is to mine the time series to explain the changes of the overall time series Ll Xit and Ll Yit. In particular, we are interested in quantifying the association between the bivariate time series in order to identify important customers that contribute to the changes significantly.

Our motivating application is the problem of product migration, which is of great practical interest in telecommunications industry. The telecommunications industry is fast-paced and constantly innovating, where technology substitution is a common practice and a serious concern for all services providers. Recently, the industry is undergoing a series of changes as people are increasingly turning to new technology such as IP and wireless as their primary mode of communication. With the arrival of IP and MPLS, the old data networks

Tom S. Au and Rong Duan are with the AT&T Research Labs, Florham Park, NJ (email: {sau.rongduan}@research.att.com).

Wei Jiang is with the Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong (email: jiangwei@ust.hk).

978-1-4244-63374/10/$26.00 @2010 IEEE 350

are rapidly being replaced by next generation networks and IP migration is becoming a business imperative. According to an analysis by Yankee Group October 2007 [11], the global corporate data service packet revenue was declined from 24 to 19 billion from 2003 to 2007 and is forecasted to further decline to 11 billion in 2011. On the other hand, IP VPN revenue was increased from 8 to 25 billion during the same period and is forecasted to increase to 44 billion in 2011. Due to unavoidable trend of product migrations, it is crucial for companies to understand migration paths in order to retain customers and remain their competitiveness in rapidly changing markets.

Product migration is not only a bivariate time series problem. In reality, it might involve multiple legacy products and multiple new products and complicated migration paths. In this paper, for simplicity, we focus our research on two products (services, programs, locations etc) only - one legacy product which typically has overall decline spending y(t) and the other new product which typically has overall increasing spending x(t). There are a number of interesting questions being asked while studying the product migration problem:

1) How are the declines in the legacy product related to the growth in the new product? How much of the

declines are due to migration or losses?

2) Who are migrating from legacy product to new product?

3) How does the migration look like? Do we lose any revenue due to migration and gain any afterward?

4) Which customers are at risk and should be migrated? 5) What percentage of the legacy product has already

migrated?

6) When and at what level the two products will stabilize?

To address these questions, the typical approach is through surveys of a small number of customers with significant declines in legacy product and increases in new product, and find out the underlying reasons. Recently, Constantiou, etc [2] survey the Denmark IP telephony and identify the key economic factors and their impact on the diffusion process. Allenet, etc [1] study the substitution of branded drugs by generic drugs through survey of French market. There are also other methods, such as multiple regression and logistic regression are used on analysis self-service and clerk-based service substitution [6], Prins [10] study the adoption of mobile phone use split-harzard approach. All these literatures focus on finding the explanatory variables for adoption probability. But there is a few discussions on how to identify the migration customers except through surveys.

Many corporations nowadays record and store customer level spending and usage (detail calls or datallP traffic) by

products overtime. There is so much behavior information contained in the monthly, daily, and even transactional information through thousands or millions multivariate time series, which provides an opportunity to explore these questions through data mining techniques. With thousands or millions bivariate time series, this paper casts Questions 1 and 2 into data mining problems with some preliminary solutions. For Question 1, we explore and group the bivariate time series data in order to find the time series patterns that drive the declines of the legacy product and/or the increases of the new product. For Question 2, we are interested in defining a measure or a set of features to quantify customer migrations and developing a classifier to identify the customers who are migrating during a certain period of time. In our study, we propose bivariate time series relations mining through 3 different models: linear regression, co-integration, and structure break co-integration models. For the structure break co-integration model, we propose LARS estimation to tailor the structure break co-integration model suitable for large dataset data mining method from computational view. With this time series data mining problem framed, we look forward better solutions from the research community that can help different industries to better analyze their migration problem.

The rest of the paper is organized as follows. Section 2 discusses the dataset properties and the distinct time series characteristics of the migration problem. Section 3 develop a bivariate time series data mining framework with different measurements and features to quantify associations between two time series. Section 4 illustrates the implementation of our framework using a case study of IP migrations. Section 5 concludes the paper with ongoing research.

II. PRODUCT MIGRATION CUSTO MER CHARACTERISTICS

Based on data exploration and business knowledge, we believe that customers can be classified into four meaningful groups for the purpose of migration analysis: new, disconnect, migration, and others. Figs. I and 2 show the different customer activity type regarding two product revenue time series. Due to proprietary reasons, the scale of time series have been removed and only the time series patterns are apparent from the graphs.

a. New customers Fig: I (a) shows an example of a new customer whose legacy product didn't change in the history but with increased revenue of the new product at some point within the time frame. The customer may have no legacy product with the company before, which indicates a totally new customer to the company, or have stable legacy product revenue, which indicates an existing customer just adopting the new technology in their service. New customers may be competitive wins from a competitors' legacy product and are the types of customers the corporation likes the most.

b. Disconnected customers Fig: I (b) shows an example of a disconnected customer whose legacy product revenue was totally lost at certain point within the time frame and has no change in the new product revenue. They may be

351

New Customer

.J'.

(a)

............ .............. '

Disconnect Customer

(b)

Fig. 1. New and Disconnect Customer

Migration Customer Migration Customer

2 Y--.. . .......... . . ..

N .NowPmducl

. ...... ......... u..

. .. U_ JIfOOood ,--

..... .s, ", . . " "' , .

(a) (b) Migration Customer Migration Customer wh Lag

(e) (d)

Fig. 2. Different Types of Migration Customer

competitive losses to a competitor' new product and are the types of customers the corporation should put more effort to retain. This type of customers is the churn customer in Question 4 that will be addressed in another paper.

c. Migrated customers Fig. 2 shows a few examples of the most interesting customers discussed in the paper - migration customers - whose legacy product revenue declined while the new product revenue increased at around the same time. It seems that the revenue loses and the revenue increases have a strong linear correlation in change time points. Fig. 2(a) shows an obvious migration point and Fig. 2(b) shows a gradual migration. Different from these two types of migration, Fig. 2(c) shows another type of migration. The two time series have different level of revenue change magnitude and one product has obvious level shift, the other is relatively smooth. Business knowledge reveals that it is either partial migration (new product revenue gain only recovers partial legacy product revenue loss) or full migration with new product development (new product revenue gain is more than legacy product revenue loss). Fig. 2(d) shows the legacy product revenue drop and new product revenue increase are at the different time. The lagged response of the new product revenue is due to the fact that most migration customers would like to secure the continuity of service before migrating to the new product.

d. Others The rest of customers besides the new, disconnected, and migrated corresponds to customers with no significant revenue changes in both products. They may be

customers before potential migrations (which is related to Question 4) or have already experienced migration from the legacy product to the new product. There could be many possible revenue patterns for these customers and depending on the variations, it is generally hard to distinguish them from the above three types of customers. Therefore a data mining strategy is needed to extract the time series patterns so that these basic customer patterns can be clearly differentiated.

In addition, not only the patterns of a time series are important, but also the scale and baseline are critical. Therefore normalization of the time series, which is popularly used for clustering, may not be appropriate in this study.

III. A MATHEMATICAL FRAMEWORK FOR MIGRATION ANALY SIS

We now describe our mathematical model to extract important features for distinguishing migration customers from all other types. Assume N pairs of bivariate time series {Xit, Yit}, where Yit represents the time series for New Product, and Xit represents the time series for Legacy Product. Since the migration customers correspond to certain relationships between the two time series within certain periods, we use a sliding window approach to capture the complex relationships and compare linear regression models, co-integration models and structure break co-integration models in lagged windows.

A. Linear Regression Model

The first test evaluates the linear relationship between two vectors Xit and Yit. We will write Xi and Yi interchangeably with Xit and Yit respectively to denote vectors. Linear regression is the most popular statistical method in time series data analysis. It assumes variables Xi and Yi have linear relationships, a simple form through a general linear regression model is,

(1)

where Yi is the dependent variable, Xi is the single exogenous predictor, and c is a zero mean white noise.

In order to capture the dynamic relationship between Xi and Yi, a time lag 7 E [0, 3] is allowed in the linear model (1) within a sliding window of length (T -7). If two products are highly correlated beyond 3 lags, it will not be considered as driven by migration. The forward relationship model with time lag 7 is

tE(1,T-7) (2)

and the backward relationship model is

t E (7 + 1, T) (3)

To select the appropriate model /31 or (3)/, we can calculate R;T for each model (2) and (3) (7 = 0, 1,2, 3) as

R;T SSRiT/SSTiT 1 ",T-T( AT )2/ ",T-T( -T)2 - ut=1 Yit - Yit ut=1 Yit - Yit

(4)

352

and pick the time lag 7 that maximizes R;T that best fits the data. To test the linear relationship between XT and yT, the following hypothesis can be constructed,

Ho : the overall slope is zero (no linear relationship) HI : the overall slope is not zero (linearly related) Linear regression model is simple which can provide some

business insights. However, its assumption that both time series are stationary is too restrict in practice. Especially for economic time series, it is very common to have nonstationary customer behaviors with relatively large volatilities and more complicated patterns. The regression results may be spurious when the data are nonstationary as discussed below.

B. Co-integration Model Although the linear regression model in Eq. (1) is at

tractive, from Granger Representation Theorem [5], it may have multiple solutions of /3 if X and Y are not stationary. When and only when c(t) is J(O), a stationary time series, /3 can be uniquely determined and carry insights of business relationships. In this case, variables X and Y are called cointegrated. The co-integration concept, introduced in Granger [5], has turned out to be extremely important in the analysis of nonstationary economic time series. A generalization to J (d) variables is also possible, in which case the linear combination of co-integrated variables has to be J(d - do) where do> 0.

The Dickey-Fuller (DF) test is popularly used to determine if a time series is stationary. To overcome the problem of autocorrelation in the basic DF test, the DF test can be augmented by adding various lagged dependent variables (ADF)[3]. The correct value for number of lags can be determined by reference to a commonly produced information criteria such as the Akaike information criteria (AIC) or Schwarz-Bayesian information criteria (HIC).

To test the co-integration relationship between two or more non-stationary time series, Engle and Granger [4] suggest running an ordinary least square (OLS) estimation, obtaining the residuals and then running the augmented Dicky-Fuller test on the residuals to determine if the residual is stationary. The time series are co-integrated if the residual is itself stationary. In this paper, we will use Engle-Granger test for co-integration relationships between X and Y. Detailed procedures are as follows.

In order to accommodate lagged response in migration customers, same as the linear regression model, We extend the co-integration model by using sliding windows. The sliding window is used on each pair of Xit and Yit to estimate a co-integration coefficient vector /3iT" The residual matrix Ci(t-T) is tested using ADF based on an autoregressive regression with lag k as follows,

k

6ct' = (p - l)Ct'-1 + 'f/i 2: 6ct'_j + JLt' (5) j=1

where tf = t -7. In theory, lag k used in the autoregressive model can be determined by AIC or BIC, however, we simply

use k = (T -1) i which is the upper bound on the rate growth with the sample size T. Define TJ the root of the determination function of the autoregressive model in Eq. (5). We construct the following null and alternative hypotheses

{ Ho : TJ = HI : TJ <

1 1

(reject co - integration) (accept co - integration) (6)

to determine if the co-integration relationship exists for a particular delay response.

At last, the appropriate lag T of the delayed relationship is selected with the best fit of R2 among those rejecting unit root tests in Eq. (6) for CiT> T* = argmaxrE(0,3)R2 , Consequently, the best fit co-integration coefficient is then (3i = (3ir* ' Otherwise, if no T in the delayed relationships rejects hypothesis in Eq. (6), we set (3i = O.

C. Co-integration Model with Structure Breaks

As seen in Fig. 2(c), sometimes one product may be linearly related with the other product only when level shifts are considered. Ignoring these structure breaks can lead to incorrect inference in particular in unit-root and co-integration tests. Structural breaks have been discussed extensively in the context of univariate autoregressive time series with unit root. Perron [8] [9] suggested three models that allow breaks in the deterministic terms: (A) changes in the intercept but slope of the linear trend is unaffected, (B) no change in intercept but changes in slope of trend function, and (C) both intercept and slope are changed at the time of the break. Liitkepohl [7] propose a structure shift at unknown time for VAR models. It is proposed to estimate the break time first on the basis of a full unrestricted VAR model. Two alternative estimators are considered and their asymptotic properties are derived. The test statistic is shown to have a well-known asymptotic null distribution that does not depend on the break time. However, the model is too complicated to be implemented when testing a large volume of time series.

In this paper, we propose a co-integrating model for linear regression model with breaks in the deterministic trend or intercept components at unknown time points. Assume two basic time series functions: step fs(t) = It>o and ramp fr(t) = t,It?:.o, where t represents time along which a total of T consecutive data points of time series are recorded. These primitives can be extended to include time delay w so that a general representation follows fk(t - w) (w = 1,2"" ,T, k E K = {r, s }) represents two types of primitives.

The breaks we concerned here represent changes of the time series structure which may be a sudden change in the mean level of the sequence that we are interested and/or a sudden change in the slope of the relationship. Suppose the change time is te, the former can be represented by a step component with delay w = te, i.e., Cs = fs(t - te), while the later can be represented by a ramp component, i.e"Cr = fr(t - te). If both mean and slope of the time series change at the same time, a linear combination of step and ramp is sufficient for characterization. Based on

353

these representations, it is easy to see that the general linear regression model can be expressed as,

Y(t) (3xX(t) + (3dr(t) + L:: (32wfs(t - w) + L:: (33wfr(t - w) + E(t).

Rewriting it in a compact form as

Y = (3X+ E,

(7)

(8)

where XTxp is predictors' matrix combining L X(t) and {Ctr,Cs,Cr}, and (3 = ((31,'" ,(3p)' is the coefficients vector and p = 2T -2.

It is obvious that model (7) is over-specified. In order to efficiently solve the linear regression problem in Eq. (7) and automatically pick the most significant features that best fit the data without over-fit, we here propose to utilize the LASSO regression method for feature selection. The LASSO method can be formulated as a penalized LS solution as follows,

P

(3LASSO = argmin(IIY - X(3112 + A L l(3i l), (9) f3 i=1 where the (3LASSO is the LASSO coefficient and A is the tuning parameter. LASSO regression can be formulated as the following quadratic programming problem:

(3LASSO = argminf3(IIY - X(3112) s.t. L:f=1 l(3i I ::::: (

(10)

where ( :::: 0 the tuning parameter. Problems (9) and (10) are equivalent and there is a one-to-one relationship between the A and (. Once important features are extracted from the penalized LS solution, linear regression model (8) can be solved perfectly using OLS method and general linear regression and co-integration tests can be applied to check the model fit.

D. Migration Identification and Risk Index

It is important to note that the above statistical tests only consider the statistical significance of the time series patterns while the actual customer revenue is ignored from the tests. In practice, as enterprizes care more about profits, customers should not be treated equally even though they have similar statistical tests. Business values have to be extracted from the customer information in order to prioritize the migration effects. In order to enhance the power of different statistical tests and, more importantly, to consider the business impacts of different types of customers, we define a risk index for each customer so that customers can be ranked based on their business importance in terms of product migrations.

First, a migration revenue impact is defined as the minimum of the new product revenue increase and the legacy product revenue decrease for each customer, i.e.,

OOi = min(( -1 * 6Xi), 6}i).

In fact, the coefficient (3 also measures the likelihood of the migration. Therefore a migration index for customer i is

defined as follows to quantify the risks of product migration,

(11)

It combines the likelihood and impact of the migration. To screen the most important migration customers, a threshold TM is set by business experts and the migration customer is identified when

{ i E migration if i E nomigraion if (12)

In the next section, we shall compare the above three linear models for describing the relationship between legacy and new products and demonstrate their identification power of migration customers in an industrial case study.

IV. P ERFORMANCE ANALY SIS

As discussed in the introduction, in the telecommunication industry, product migration is popular due to technology advances. It is important for a company to understand patterns of product migrations, identify customers who have migration features, and quantify the business impact of product migrations. We now apply the above 3 statistical models and associated hypothesis testing methods to identify migration customers in a telecommunication company to assist marketing decisions when introducing new product. Here the legacy product may refer to packet and related services and the new product may refer to IP or other related products.

A. Data Description and Preprocessing

The dataset that involves in this migration analysis has millions of customers with consecutive 36 months of billing revenue for both legacy and new products. As illustrated in Fig. 3, customer billing time series data is always noisy but includes a large amount of customer behavior information, e.g., spending trends, seasonal effects, account adjustments, contract renews, business development, and of course billing errors. These time series patterns are so complex and noisy that the standard statistical hypothesis testing may fail to capture the true relationship between the two products without cleaning and pre-processing the data. It is therefore important to filter out attributes that are not related to the migration study, e.g., outliers which are not of our interests since they are not a feature for migration analysis but could bias the estimation results seriously. In billing revenue, the outliers usually driven by the adjustment one month later. In order to alleviate the effects caused by outliers, we applied 3-month median filter for pre-processing and smoothing the data while retaining trend and change point as key migration features in the analysis.

B. Experiment Design As discussed in Section 4, the lag we used in sliding

window linear regression and extended co-integration model is T = 3. The goodness of fit statistic RTT is used to pick the best-fit lag T. For each sliding window T, the original

354

---.L. A..-. '-

c= ......,,::: .Jt.F= --- :::::::::-,.. - L----- ------ L..t:

"J,.- ::I::=1

::::::::,.. ==If-

=:::! ---+- \------

Fig. 3. Original Time Series

data are denoted as Xi and T. For the structure break cointegration test, the design matrix X consists of 4 columns of X (t - T) , where T is from 0 to 3 to represent the lagged response between Xi(t) and Yi(t). Since there are T - 1 columns of step components and T - 1 columns of ramp components, the total dimension of the design matrix X is 36 x 70. The major components of Yi (t) is calculated by LARS estimation. If any of the 4 columns of Xi (t - T) is picked and the corresponding co-integration test for the residual rejects the null co-integration hypothesis, Xi(t) and Yi(t) is identified as co-integrated for customer i.

Once co-integrated relationships are identified for customers, their migration index can be evaluated and ranked based on business knowledge. The threshold TM can be applied to the ranked index to discriminate Migration customers from all others. The cut-points are usually set by marketing experts according to business rules such as revenue impacts, customer targetability, etc. To make our analysis comprehensive, we tested the three models using different statistical threshold values and evaluated their classification probability and associated revenue scores.

C. Peiformance Comparison

Fig. 4 shows scatter plots of the revenue changes of the new and legacy products for the migration customers identified at 5% significance level by the three methods. "0" represents sliding window linear regression (SWLR), "6" represents Extended Co-integration model (ECIT) and "+" represents structure break co-integration model (SBCIT). To validate the result, we randomly picked 3000 customers and asked market experts to label the customers as the ground truth for verification. A close look at the figure shows that the SWLR method has high true positive and false positive rate, identified 478 migration customers among which 353 are false positives. On the other hand, the ECIT method identified 199 migration customers, which is far less than SWLR. This is because that the SWLR method makes stationary assumptions of the bivariate revenue time series, which failed to account for conspicuous variations hidden in the customer level revenue data. The ECIT method has low true positive and low false positive rates since it is more strict to consider the order of the bivariate time series. However, it may misclassify customers that contain structure break in their regression model. On the other hand, the SBCIT method identified 237 migrations customers and showed high true positive and low false positive.

It is important to note that, for a given level of statisti-

:&",,,:m::r..""'T*r.:::-

I . Fig. 4. ROC Curve

Fig. 5. ROC Curve - Migration Probability

cal significance, the three methods may generate different classification results. To show the power of three different methods with the same level of true positive rate, true and false positive rates for different p-values are calculated for the SWLR, ECIT and SBCIT methods and are plotted in Fig. 5. This curve corresponds to Relative Operating Characteristic (ROC) curve of the relationship between true positive rate vs. false positive rate. In general, the true positive rate (TPR) and false positive rate (FPR) can be assessed by the instance count and plotted in Relative Operating Characteristic (ROC) curve. If the instance is positive and it is classified as positive, it is counted as a true positive; if the instance is negative and it is classified as positive, it is counted as a false positive. It is easy to see from the ROC curve that, the SBCIT method is the best among the three models and tests, which has the highest true positive and lowest false positive rates. The SWLR method is the worst which has a very high false positive rate.

On the other hand, as discussed before, it is not only the number of migration customers is of interest to practitioners, but also the revenue that these customers represent. To assess the revenue impact of migration customers using the ROC concept, migration scores in terms of revenue can be defined. Therefore, false positive and true positive scores weighted by the absolute value of legacy product revenue change can be calculated after identifying the migration customers. These scores carry more business meanings than the probability rate in the migration study. Fig. 6 shows the migration revenue scores for the three models when using different threshold/false positive errors. It is clear that the SBCIT method outperforms the other 2 models significantly. This manifests that there are tremendous business changes in the customer database, which considerably reflects the migration activities.

V. DISCUSSION AND FUTURE RESEARCH In this paper, we elaborate the product migration prob

lem in competitive market and propose a framework for migration customer identification and quantification using

355

Fig. 6. ROC Curve - Migration Revenue Score

statistical data mmmg techniques. Three methods, Sliding Window Linear regression, Extended co-integration model and Structure Break Co-integration model, are developed to explore relationships between two time series of product revenue. An industrial case study in telecommunication to illustrate the real world data mining process for migration analysis. The numerical results show that the Structure Break Co-integration model is the most effective among the three methods in both statistical and business sense.

The methodology reported in this paper is only a preliminary study of the bi-products migration analysis. Moreover, when multiple products are involved in the business applications, multivariate time series models are necessary to address the co-integration issue of among different product revenues. It is important to identify these multiple migration paths when multiple new products are launched in market.

REFERENCES [1] B. Allenet and H. Barry, "Opinion and behaviour of pharmacists towards

the substitution of branded drugs by generic drugs: survey of 1,000 French community pharmacists", Pharmacy World & Science, vol. 25(5), pp. 197-202,2003

[2] I. D. Constantiou and K. Kautz, "Economic factors and diffusion of IP telephony: Empirical evidence from an advanced market", Telecommun. Policy,Vol. 32, no. 3-4, pp. 197-211, Pergamon Press, Inc. Tarrytown, NY, 2008.

[3] G. Elliott , T.J. Rothenberg and J.H. Stock, "Efficient Tests for an Autoregressive Unit Root", Econometrica,vol. 64(4), pp. 813-836,1996.

[4] R.E Engle and C.W.J. Granger, "Co-Integration and Error Correction: Representation, Estimation, and Testing", Econometrica, vol. 55(2), pp. 251-276, Mar. 1987.

[5] C. W. J. Granger, "Some properties of time series data and their use in econometric model specification", Journal of Econometrics, vol. 16, pp. 121-130, 1981.

[6] M.L. Meuter , M.J. Bitner , A.L.Ostrum and S.W.Brown, "Choosing Among Alternative Service Delivery Modes: An Investigation of Customer Trial of Self-Service Technologies", Journal of Marketing, vol. 69(2), pp. 61-83, April, 2005.

[7] H. Lutkepohi, P. Saikkonen and C. Trenkler , "Testing for the Cointegrating Rank of a VAR Process with Level Shift at Unknown Time ", Econometrica, vol. 72, pp. 647-662, 2004.

[8] P. Perron, "The Great Crash, the oil price shock, and the unit root hypothesis", Econometrica, vol. 57, pp. 13611401, 1989.

[9] P. Perron, "Testing for a unit root in a time series with a changing mean", Journal of Business and Economic Statistics, vol. 8, pp. 301320, 1990.

[10] R. Prins, "Modeling Consumer Adoption and Usage of Value-Added Mobile Services", Erasmus Research Institute of Management Ph.D. Series Research in Management, vol. 128, 2008.

[11] Yankeegroup, http://www.yankeegroup.com/link.do? method=getByTaxonomy, 2007.

Recommended