Paul I. Louangrath
2.2 Time Series Modeling
There are two main lines of methodology used for time dependent event analysis. The first approach is time series modeling. This method is commonly employed in econometrics (Brockwell & Davis, 2002). The second method, under which this research adopts, is the time-to-event modeling which uses system failure and survival functions analyses (Richards, 2012).
Conventional practice in econometrics relies on autoregressive family of models for sequential time-based event modeling. These time series modeling may be classified into univariate and multivariate models (NIST: Engineering Handbook, 2013). Univariate time series is common; it consists of a family of autoregressive (AR) modeling in the generally form:
1 1 2 2 ...
t t t p t p t
X X X X A
(1)
where Xt is the observed value for the time series and At is the white noise. The term is defined as:
1 1
p i i
(2)
The term is the process mean. In effect, AR model is a linear regression where the current value is regressed against the value of the prior period, i.e. Xt Y
and Xt1 X in the Cartesian space. The term p in the equation stands for the order of the AR model. The AR model is the first type among the three common autoregressive models.
The second type of autoregressive model for univariate time series is called the moving average (MA) model which is given by:
1 1 2 2 ...
t t t t q t q
X A A A A
(3)
where Xt = time series; = process mean; At1 = white noise; 1,...,q
= parameters; and q= order of the MA model (Shumway & Stoffer, 2011).
The AR model regresses the current time series Xt
against its prior period Xt1
; however, MA model regresses Xt
against its own white noise in prior period. Since the error
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term is not readily observable, forecasting under MA model is more complicated than in AR where the standard least square technique is used. In order to obtain the white noise term for the equation, an initial sample with adequate size must be used to generate the first MA model. Thereafter, more data points are added, and the changes is the error term is recorded for analysis of its distribution. This process requires larger sample size. In NPL case, this is not practicable since the problem requires quick intervention to prevent further losses (Masood et al., 2010).
A third approach to time series modeling is called the Box-Jenkins method which combined AR and MA; thus, it is called ARMA (autoregressive moving average) (Box &
Jenkins, 1970). ARMA is given by:
1 1 2 2 ... 1 1 2 2 ...
t t t p t p t t t q t q
X X X X A A A A
(4) ARMA assumes that the time series are stationary. Stationary means that the data series are reverting to its long-run equilibrium, i.e. mean reverting (Makridakis & Hibon, 1995). If there is an external shock, and the effect of the shock is integrated, the shock destroys the original long-run equilibrium. As the result, the time series lose the memory. The series are no longer mean reverting. If the effect of the shock establishes a new mean, it is said that the time series are integrated; thus, ARMA becomes ARIMA (autoregressive integrated moving average) which is given by:
1 1 1 1
ˆt t ... p t p t ... q t q Y Y Y A A
(5)
The second category of time series forecasting model is the multivariate type. The Box-Jenkins form for multivariate time series is called autoregressive moving average vector (ARIMAV):
1, 2 ,...,
Tt t t nt
X x x x
where t (6)
The term Xt is further defined as:
1 1 2 2 ... 1 1 2 2 ...
t t t p t p t t t q t q
X X X X A A A A (7)
where Xt and Atare n1 column vector, and At is the multivariate white noise. For the parameter term, k
k jj, ,k 1, 2,...,p and k
k jj, ,k 1, 2,...,p both are nn matrices for autoregressive and moving average parameters. Note that[ t] 0; [ t t k' ] 0, 0 E A E A A k
and
[ t t k' ] , 0
E A A
A k where
A is the dispersion or covariance matrix of At.
The weakness of ARIMA is that the estimate of parameters and covariance matrix is not easy. This complexities may be illustrated by ARIMA(p,q) where the parameter p2 and q0, the ARIMAV(2,0) model may be written as:
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1,11 1,12 1 2,11 2,12 2 1
1,21 1,22 1 2,21 2,22 2 2
t t t t
t t t t
X X X A
Y Y Y A
(8)
where X = input and Y = output; both with the mean vector of (0,0). The scalar form of ARIMA(2,0) is written separately for X and Y as:
1,11 1 2,11 2 2,12 1 2,12 2 1
t t t t t t
X X X Y Y A
(9)
1,22 1 2,22 2 1,21 1 2,21 2 2
t t t t t t
Y Y Y X X A
(10)
These complications (5–10) are not practicable and may be inaccessible for practitioners in NPL assessment. Both univariate and multivariate time series models could not effectively be used as tools for short-term risk assessment in NPL study; in NPL study the event-to-date is provided on a quarterly basis and the needed forecast is also quarter-to- quarter. Moreover, time series modeling requires larger sample size. This requirement is not suitable for NPL problem where immediate and short-term interventions are needed.
Secondly, time series modeling could not provide any information on the internal system that generates the NPL rates. The limitation of time series autoregressive models is due to the fact that the series regresses against its own past series. Neither AR, ARMA, ARIMA nor ARIMAV could tell us about “how reliable does the system behave?” The answer to this question could help us better model by assessing the current situation and forecast future event. Specifically, autoregressive in time series could not provide information on the current circumstance of the system. Autoregressive time series could not answer the question: “what is the reliability and risk of the failure of NPL as a system?”
In NPL analysis, the NPL level is considered as time-to-event data, not just a mere event occurring with corresponding time, i.e. time series data. The issue of system failure and system survival has practical importance on management decision making because time-to- event modeling allows us to capture the characteristics of failure with respect to time and assess the current condition of the system.
Time series data modeling does not differentiate between success or failure; the observed values in time series are not classified. Each value is connected to time: Xi:ti. This generalization of data makes time series modeling a tool for crude estimate. The weakness of time series comes from the fact that all time series depend on autoregression, i.e. the Y-array is obtained through a lag of the time series or that each Xt is lagged to Xt1; thus, the modeling is a regression of the data series against itself by using the previous period as the basis for the next event (Box et al., 1994, pp. 9-10). This type of autoregressive modeling could allow only simple forecasting without assessing the underlying condition of the system that produces the event. The weakness of time series modeling underscores the inadequacy of the current tools in econometrics for purposes of risk assessment or forecasting crisis. This inadequacy is best exemplified by these words of the former European Central Bank President, Jean-Claude Trichet who remarked:
“When the crisis came, the serious limitations of existing economic and financial models immediately became apparent. Arbitrage broke down in many market segments, as markets froze and market participants were gripped by panic. Macro models failed to predict
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the crisis and seemed incapable of explaining what was happening to the economy in a convincing manner. As a policy-maker during the crisis, I found the available models of limited help. In fact, I would go further: in the face of the crisis, we felt abandoned by conventional tools. In the absence of clear guidance from existing analytical frameworks, policy-makers had to place particular reliance on our experience. Judgement and experience inevitably played a key role.” (Trichet, 2010).
Times series modeling is a common method in econometrics (Lin et al., 2003). Time series modeling uses the prior period’s sequential values as the basis for predicting the subsequent period (Green, 2011). This approach tends to require large sample size (Brockwell & Davis, 2002). In NPL studies, managers and policy makers do not have the luxury of lengthy period for data collection prior to making the forecast. NPL is a practical problem requiring an immediate intervention. Therefore, time series may not be an appropriate tool. This paper proposes the use of non-parametric method in system analysis as a tool for short-term risk assessment in NPL studies under Extreme Value Theory (EVT).
This research employs data distribution verification through tail index calculation as the means for model selection and testing.