Bayesian Unit Root Test for AR(1) Model with Trend Approximated by Linear Spline Function

Bayesian Unit Root Test for AR(1) Model with Trend Approximated by Linear Spline Function Jitendra Kumar1,∗ , Varun Agiwal1 , Dhirendra Kumar1 , and Anoop Chaturvedi2 1Department of Statistics, Central University of Rajasthan, Bandarsindri, Ajmer, India 2Department of Statistics, University of Allahabad, Allahabad, India Abstract The objective of present study is to develop a time series model for handling the non-linear trend process using a spline function. Spline function is a piecewise polynomial segment concerning the time component. The main advantage of spline function is the approximation, non linear time trend, but linear time trend between the consecutive join points. A unit root hypothesis is projected to test the non stationarity due to presence of unit root in the proposed model. In the autoregressive model with linear trend, the time trend vanishes under the unit root case. However, when non-linear trend is present and approximated by the linear spline function, through the trend component is absent under the unit root case, but the intercept term makes a shift with r knots. For decision making under the Bayesian perspective, the posterior odds ratio is used for hypothesis testing problems. We have derived the posterior probability for the assumed hypotheses under appropriate prior information. A simulation study and an empirical application are presented to examine the performance of theoretical outcomes. Keywords AR Model, Linear Spline Function, Posterior Odds Ratio, Unit Root AMS 2010 subject classifications 62M10, 62F15, 41A15 DOI: 10.19139/soic- • Introduction The Box-Jenkins approach for analyzing and modelling time series emerged as one of the most acceptable field of data analytic in which anyone may integrate various information from the recorded data through a modelling or methodology approach. It assists in recognizing the order dependence within observations, linear dependence among a set of variables, and prediction for a better perspective. In order to achieve these features, the first and foremost objective of the researchers is to expose the key elements of time series such as linearity, stationarity, trending, co-integration, etc. in the process. Then, one can draw significant inferences about the data generation process through the best-fitted model under some estimation, testing, and model selection procedures. In a time series model, linear dependence occurs when series is nearly symmetric to a lagged version of itself, and disturbance coefficients are constant over time, i.e., series is stationary with its mean, variance and autocorrelation. Even in various applications, series contain non-linear dynamics in case of non-stationarity, strong asymmetric nature, irregular time, seasonal variation, etc. In time series, non-stationarity occurs due to various components such as time trend, presence of unit root, structural break, outliers, etc. If the time series has a unit root and becomes stationary after taking the first difference, it is termed as difference stationary or integrated of order one. In time series, the classical tests of unit root are discussed by [1], [9], and [8]. [2] considered a time series model ∗Correspondence to: Jitendra Kumar (Email: ). Department of Statistics, Central University of Rajasthan, Bandarsindri, India. ISSN (online) ISSN 2311-004X (print) Copyright ⃝c 2020 International Academic Press with a polynomial trend where trend component did not disappear under the maintained hypothesis of unit root and developed a classical statistic for the presence of unit root. The classical tests are largely based on asymptotic justification and often lead to the low power of the test, particularly in finite samples. [3] and [4] demonstrated that Bayesian unit root tests based on flat prior assumptions perform better than classical methods. [5] used a Bayesian approach for testing the presence of unit root in various real exchange rate time series. [6] derived the posterior odds ratio for testing the unit root hypothesis under a vague prior assumption. [7] studied the unit root hypothesis for an autoregressive model with a polynomial trend under a Bayesian framework. To model the non-linear trend components, one may require a polynomial of high order in which estimates of coefficients are usually unstable. An alternative to fit a polynomial of higher order is to approximate it by a spline function. Spline function is a piecewise polynomial segment that has been joined together at the knots in a fashion that ensures certain continuity properties. In other words, spline function fits a curve of low order between different join points, known as knots. A knot is a common point that occurs because there are changes in pattern behavior at different intervals. [10] expressed that splines are the smoothest possible piecewise polynomial which retain a segmented nature, whereas [15] called splines as lines or curves function which are usually required to be continuous and smooth. [11] used the model based on a spline function for predicting the number of deaths due to cancer in the USA. [12] determined the number and positions of knots in the regression splines model using the new Gibbs sampler algorithm, where the model expressed as linear mixed with random effect term. [14] studied the Bayesian approach for modeling the partial linear model with AR(1) error belongs to the scale mixture of normal (SMN) distributions family. [13] used the spline, Bayesian spline, and penalized spline regression methods to model the distribution graph of ratios of export to import for Turkey. [17] considered Buys-Ballot and classical methods of decomposing to estimate the cubic trend as well as other components of the times series and obtained the chain base and fixed base estimators with their statistical properties. [19] proposed a particle swarm optimization B-spline network to improve the prediction accuracy of non-linear time series. They adopted a forecasting error square sum to evaluate the training effect of the B-spline network. [18] presented a new cubic B-spline approximation method for solving second order singular boundary value problems with application in physiological sciences. [16] considered smoothing spline (SS) and penalized spline (PS) methods for estimating the unknown functions in a conditional heteroscedastic non-linear autoregressive (CHNLAR) model and concluded that SS method performed superior to the PS method. In the present paper, the main emphasis is to build up a Bayesian approach for testing the unit root in an autoregressive time series (AR) model with a non-linear time trend. The non-linear time trend has been approximated by the linear spline function. For unit root testing, we obtain the posterior odds ratio for the considered hypotheses of the model under the appropriate prior assumptions. Due to the complex form of posterior probability, the Monte Carlo integration technique is used to achieve the results from the posterior odds ratio. The precision of linear spline function for the approximation of time trend AR model is justified in both simulation and empirical studies. In an empirical application, dataset on import series of Asian Regional Forum (ARF) countries are taken to demonstrate the applicability of linear spline function in the time series model. • Model specification Let us consider a time series model with a non-linear trend component, which is approximated linearly by a spline function with r join points 1 t1 t2 · · · tr T . Then, time series model with a spline function is given by, r yt = δ0 + δt + ψisi(t) + ut, (1) i=1 where, {yt; t = 1, 2, . . . , T } is an observed series, δ0 isthe intercept term, δ isthe trend coefficient,r isthenumber of knots containslocation of knots t1, t2, · · · , tr, ψi is the coefficient of i th knot and si(t) is a spline function describe as a linear polynomial form defined as follow, s (t) = (t −t ) + = t − ti t ti , 0 t ≤ ti. Let ut is a stochastic error term follows the AR(1) process, ut = ρut−1 + εt, t = 1, 2, . . . , T (2) where, ρ is the auwith zero mean and unknown variance τ −1 . Combining (1) with (2), we can write the model as, r yt = (1 −ρ)φ+ ρyt−1 + βt + ψi [si(t) −ρsi(t −1)] + ǫt, (3) i=1 where, φ = δ0 + ρ δ and β = (1 − ρ)δ. Notice that r r r Σψi[si(t) −ρsi(t −1)] = Σψi [si(t) −si(t −1)] + (1 −ρ)Σψisi(t −1), and i=1 i=1 i=1 ψ1 t1 + 1 t t2,