Detection of Structural Changes in Correctly Specified and Misspecified Conditional Quantile Polynomial Distributed Lag (QPDL) Model Using Change-point Analysis

Change-point analysis is a powerful tool for determining whether a change has taken place or not. In this paper we study the structural changes in the Conditional Quantile Polynomial Distributed Lag (QPDL) model using change-point analysis. We employ both the Binary Segmentation (BinSeg) and Cumulative Sum (Cusum) methods for this analysis. We studied the structural changes in both correctly specified and misspecified QPDL models. As an economic application we considered the production of rubber and its price returns. We observe that both Cusum and BinSeg methods correctly detected the structural changes for both the correctly specified and the misspecified QPDL model. The Cusum method gave the exact positions where the structural changes occurred and the BinSeg gave the approximated positions where the changes occurred. Both methods were able to detect the shift in time for both the mean and variance for the missspecified QPDL model, hence both methods were better for predicting structural stability in a QPDL models. The impact of this is that, when there are changes made to a data knowingly or unknowingly, they can be detected, as well as when these changes were effected. We further observed that both methods were powerful tools that better characterizes the changes, controls the overall error rate, robust to outliers, more flexible and simple to use.


Introduction
The question of structural stability of models is very important in predicting the future in various diverse areas of science such as economics, finance, physics, geology, medicine as well as in quality control and agriculture. Questions like did a change occur? Did more than one change occur? When did these changes occur? Can be answered by performing a change-point analysis, (Taylor, Wayne 2000a, 2000b [13,14]. In order to study the structural changes in models came the evolution of change-points which was introduced by the context of quality control by Csorgo and Horvath, (1997) [3].
Change-points has been employed in finding the possible changes in otherwise independent identically distributed random variables and has been extended to the stability tests of parameters of the regression functions, (Andrews, D. W. K. 1993) [1]. Killick, et al, (2012) [9], considered the common approach of detecting change-points through minimising a cost function over possible numbers and locations of change-points. They applied several established procedures for detecting changing points, such as penalised likelihood and minimum description length. In their study, they observed that Binary Segmentation is quicker as a search method, and believed this would be the case in almost all applications. Auger and Lawrence (1989) [2] propose an alternative, exact search method for changepoint detection, namely the Segment Neighbourhood (SegNeigh) method which searches the entire segmentation space using dynamic programming. The problem with this exhaustive search method is that it has significant computational cost of Ο( 2 ).
We use both the Binary Segmentation (BinSeg) and Cumulative Sum (cusum) methods for detecting the structural changes in the QPDL model because they are capable of detecting multiple changes (Mueller, 1992) [10]. The advantage of the BinSeg method is that it is computationally efficient, resulting in an O(n log n) calculation (Eckley et al, 2011) [4]. Both methods are powerful tools that better characterize the changes, control the overall error rate, robust to outliers, more flexible and simple to use, (Efron et al, 1993) [5].
The objectives of this study are, to detect the structural changes in correctly specified QPDL model, to detect the structural changes in misspecified QPDL model, and to study the mean-shift in time of the misspecified QPDL model. This study is important because as the part of our world moves into more sophisticated time periods, the accuracy of predicting the exact occurrences of future events as much as possible to reduce severe loses or impact is very important rather than using approximate estimates. It is also important in modeling to know when the exact change occurred and its impact.

Data Source
In order to answer the issues raised above, secondary annual data was collected from FAOSTAT, food balance sheet, price statistics, available with the Department of Census and Statistics Sri Lanka [6], and the World Bank (pink sheet) [15]. These data comprises of the production, imports, exports and prices of rubber. The rubber data ranges from 1961-2011.

Statistical Software
The R software, with the package 'Change-point' was used in analysing the structural changes in the conditional quantile polynomial distributed lag models.

Description of the test procedure for the detection of the change-point
Let us now consider a conditional 2 nd degree polynomial − model with a change after an unknown time point Where { } is some QPDL which differs distributionally from { } . The unknown parameter * is called the change-point.
We are now interested in the testing problem 0 : * = vs. 1 : * < Our testing procedures are based on various functionals of the partial sums of estimated residuals with respect to the model (1) where ̂ is the least-squares estimator of 0 (assuming the null hypothesis holds true). Precisely we minimize the nonlinear least squares (NLLS) with respect to .
Thus we consider the nonlinear least squares estimator for a suitable compact set K. The minimization is usually obtained by solving the nonlinear score function Which yields The behaviour of the estimator is investigated in a variety of situations including the correctly specified case without change as well as possibly misspecified cases with and without change. According to Horvath et al (2004) [8] and Wu, (2004) [16], under appropriate assumptions ̂ is eventually in the interior of the compact set K, so that we can assume for limit considerations.
In fact, if ̂ is not in the interior of K, (Hinkley, D. V. 1971) [7], we will reject the null hypothesis immediately since either a change occurred or the model is not capable of modeling the observed time series sufficiently well.
Test statistics are of the form where q(·) and r(·) are weight functions defined on (0,1) specified below and G < n. In this case, we obtain a consistent change-point estimator which is related to the test statistics.

Detection of structural Changes for the correctly specified QPDL model
Detection of structural changes in the QPDL model before misspecification are shown on tables 1 -4, using the methods Cumulative Sum (cusum) and Binary Segmentation (BinSeg) with Schwarz Information Criterion (SIC) penalty.          Table 3 shows an optimum of 5 changes for Cusum and 4 for BinSeg. The minimum average production 1362.          Similarly, from figure 8, we observe a shift of the change in the average production which occurred in 1980 for the correctly specified QPDL model to 1979 in the misspecified QPDL model. There was a drop in the mean production between 1979 and 1986 and then a decrease occurred for a short period in 1987. There was a high drop again between 1988 and 2005. There was an increase in average production in 2006. There was a further high increase in production from 2007 till 2011.

Conclusions
From the analysis, we observed that both methods detected the exact time change and the magnitude of the change. We also observed that both the Cusum and the BinSeg methods detected the structural changes for both the correctly specified and the misspecified QPDL model. The Cusum method gives the exact positions where the structural changes occurred and the BinSeg gives the approximated positions where the changes occurred. Both methods were able to detect the shift in time for both the mean and variance for the missspecified QPDL model, hence both methods were better for predicting structural stability in a QPDL models. The impact of this is that, when there are changes made to a data knowingly or unknowingly, they can be detected, as well as when these changes were effected. We further observed that both methods were powerful tools that better characterizes the changes, controls the overall error rate, robust to outliers, more flexible and simple to use.