Copula-Based Univariate Time Series Structural Shift Identification Test

The issue of structural breaks in time series was originally investigated by Wald (1947). The problem has attracted a lot of attention since; see Perron (2005) for a comprehensive review. Recently the area of joint distributions has attracted significant importance given applications in financial risk management and actuarial science. Copula models are often used in this context when it is useful to decompose joint distribution modeling into two steps: modeling marginal distributions and dependence (i.e. copulas themselves). As a matter of fact structural tests for copulas started being developed to research the stability of copulas during the time. Among the recent papers one might find Harvey The objective of the current paper is to research univariate, not multivariate time series using same copula structural shift identification approach. The paper contributes in several ways:  The unique feature of time series in terms of copula is presented. Particularly, the stability of marginals for various lagged components of the series;  Conventional (in copula theory) independence test is interpreted as a non-linear correlogram test;

 Copula structural break test procedure application enables to reveal shifts, missed by conventional (linear) tests on structural break identification (empirical example of US GDP is considered).

As a result the paper is organized as follows. First Section 2 is devoted to brief literature review.

Second theoretical framework is given in Section 3. Then Section 4 presents the data used. Section 5 provides the results of test procedure application. Section 6 concludes.

The most common structural break tests are that of Andrews-Zivot (e.g. Andrews (1993)) and Philips-Perron (cf. Perron (2005)). The idea is to consider the change in intercept and (or) trend for the linear time series model. A dummy variable approach is used to detect the moment when the change is significant to be considered as the break point.

Previous works dealing with copula structural break identification of similarly copulas comparison included Genest, Remillard (2004) Before discussing copula models application to time series analysis it is necessary to be fair enough and mention the works of Darsow et al. (1992) and Ibragimov (2009) who already researched the properties of copulas when applied to Markov processes. Particularly, Ibragimov (2009) defines rand m-dependence properties for copulas to be suitable for time series modeling.

Copulas represent a way of joint probability distribution function decomposition as it is given below in (1). Extensive overview of copulas and their properties as the linkage to triangular norms might be found in Nelsen (2006) and Alsina at al. (2006), respectively.

Stationarity Hypothesis 1. Marginal distributions when decomposing time series into copula and marginal are the same.

, the following representation (2) holds given (3) that is true for large rows. In case of few observations test restrictions should be studied in greater detail.

The property (3) is of great importance for the testing procedure as it clearly states that having once modeled the marginals their relationship is fully captured by copulas that do not limit the dependence nature to linearity.

Briefly to remind the testing procedure taken from Brodsky et al. (2009).

Two empirical copulas (4) before and after potential break point l are estimated.

,,

,

)

and for every [1, , ]

N is fixed and the following modification of the Kolmogorov-Smirnov statistics ( 6) is applied:

Then the statistics (7) takes its maximum value in the break point (8

Further properties of the test statistics might be found in Brodsky et al. (2009).

As opposed to Brodsky et al. (2009) who considered multivariate time series, the present paper focused on copulas applied not to a time series vector, but to a univariate time series with special attention to the dependence structure of lagged components in it. US GDP official quarterly data is taken as an example. Data description and test results follow below.

To apply non-linear structural break test a very common data row was chosen, i. It is important to note is that GDP time series is rather a low frequency and relatively nonvolatile time series compared to minute-or transaction based financial time series. The latter are prime candidates to assume nonlinear dependence. Nevertheless, it was desirable to start from ordinary low frequency macroeconomic time series for test validation.

Financial time series research might well be subject of another paper where conditional heteroscedasticity might also need revision with respect to non-linearity of variance dependence on its previous values and previous squared residuals’ values.

GDP level-data is non-stationary is it follows solely from the visual analysis of data. That is why further analysis was proceeded with the data transformed to growth rates.

Table 1 and Figure 2 below present marginal descriptives proving ma

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