Whether it is the traditional Cointegration or the ARDL approach, the first step in time series analysis consist of testing for stationarity. In this study, the data series is tested for stationarity using the Augmented Dickey fuller test (ADF) and the Kwiatwowski- Phillips-Schemidt-Shin test (KPSS). The reasoning behind the use of the former is based on the observation that it is one of the most commonly used technique to identify the presence of unit root in single time series. Regarding the use of KPSS, it is viewed as an appropriate complement to ADF unit root tests not only because it directly tests the stationarity but also because it provides reliable results for shorter time series such as the one that is being used in this study (Arltová and Fedorová, 2016). Given these reasons, ADF and KPSS are therefore suitable for unit root testing.
3.2.1 Arguemented Dicki-fuller (ADF) test
The testing procedure of the Arguemented Dicki-fuller test is based on the model of stochastic autoregressive AR(P) series that is presented as:
𝑦𝑡 = 𝜌𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 𝑝
𝑖=1
𝑡 = 1,2 … , 𝑇 (1)
Where 𝜌 is the auto regression parameter, 𝜀𝑡 is the non-systematic component of the model that meets the characteristics of the white noise process and p denote the number of augmenting lags which can be determined by minimizing the Schwartz Bayesian information criterion.
The objective is to test whether 𝜌=1. If this holds, it confirms the presence of unit root which means that the series is non Stationary. If 𝑦𝑡−1 is subtracted from each side of
model (1), the following model is obtained:
∆𝑦𝑡= (𝜌 − 1)𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (2) 𝑝
𝑖=1
21 ∆𝑦𝑡= 𝜃𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (3)
𝑝−1
𝑖=1
A unit root test can now be performed in terms of model (3) by testing the null hypothesis 𝐻0:𝜃=0 against the alternative hypothesis 𝐻𝑎:𝜃<0. If the null hypothesis
𝐻0:𝜃=0 holds, then 𝜌-1=0 𝜌=1, which implies that the series is a random walk process hence the series is non-stationary. However, if the alternative hypothesis
𝐻𝑎:𝜃<0 holds, then 𝜌<1=0 𝜌<1, which means that the series is stationary. Model (1) can also be expanded by a constant or a linear trend. When model (1) is extended by the intercept constant, the following equation is obtained:
𝑦𝑡 = 𝛼0+ 𝜌𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 𝑝
𝑖=1
(4)
Where 𝛼0 denote an intercept constant called a drift. The objective here is to test for a unit root (𝜌=1) in terms of an AR(P) process with a constant. Once again, if 𝑦𝑡−1 is subtracted from each side of model (4), the following model is obtained:
∆𝑦𝑡 = 𝛼0+ (𝜌 − 1)𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (5) 𝑝
𝑖=1
If we set: 𝜃 = 𝜌 − 1 then we obtain the following equation:
∆𝑦𝑡 = 𝛼0+ 𝜃𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (6) 𝑝−1
𝑖=1
A unit root test can now be performed in terms of model (6) by testing the null hypothesis { 𝐻0:𝜃=0, 𝛼0 ≠ 0} against the alternative hypothesis { 𝐻𝑎:𝜃<0, 𝛼0 ≠ 0}. If the null hypothesis { 𝐻0:𝜃=0, 𝛼0 ≠ 0} is valid, then 𝜌-1=0 𝜌=1, which implies that the series is random walk around a drift. However, if null hypothesis { 𝐻0:𝜃=0, 𝛼0 ≠ 0} is not valid, then 𝜌<1=0 𝜌<1, which means that the series is level stationary process. Finally, model (4) can further be extended by the linear trend which result to the following equation:
22 𝑦𝑡 = 𝛼0+ 𝛼1𝑡 + 𝜌𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡
𝑝
𝑖=1
(7)
Where 𝛼1 is the coefficient of a linear time trend. The objective here is to test for a unit
root (𝜌=1) in terms of an AR(P) process with a trend. If 𝑦𝑡−1 is subtracted from each side of model (4), the following model is obtained:
∆𝑦𝑡 = 𝛼0+ 𝛼1𝑡 + (𝜌 − 1)𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (8) 𝑝
𝑖=1
If we set: 𝜃 = 𝜌 − 1 then we obtain the following equation:
∆𝑦𝑡= 𝛼0 + 𝛼1𝑡 + 𝜃𝑦𝑡−1+ ∑ 𝛾𝑖∆𝑦𝑡−𝑖+ 𝜀𝑡 (9) 𝑝
𝑖=1
A unit root test can now be performed in terms of model (9) by testing the null hypothesis { 𝐻0:𝜃=0, 𝛼1 ≠ 0} against the alternative hypothesis { 𝐻𝑎:𝜃<0, 𝛼1 ≠ 0}. If the null hypothesis { 𝐻0:𝜃=0, 𝛼1 ≠ 0} is valid, then 𝜌-1=0 𝜌=1, which implies that the series is random walk around trend. However, if null hypothesis { 𝐻0:𝜃=0, 𝛼1 ≠ 0} is not valid, then 𝜌<1=0 𝜌<1, which means that the series is trend stationary process. 3.2.2 Kwiatwowski-Phillips-Schemidt-Shin (KPSS) test
To explain the Kwiatwowski-Phillips-Schemidt-Shin (hereafter, KPSS) test, this discussion closely follows Syczewska (2010). KPSS examines a series of observations of variable of interest, 𝑦𝑡 with t=1,2,…., T, and assumes it can be
decomposed into the sum of the deterministic trend, random walk and stationary error term:
𝑦𝑡= 𝜉𝑡 + 𝑟𝑡+ 𝜀𝑡 (10)
𝑟𝑡= 𝑟𝑡−1+ 𝜇𝑡 (11)
Where t denotes determestic trend, 𝑟𝑡 is random walk process with variance 𝜎𝜇2, 𝜀𝑡 is
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an error term of model (11) and by assumption is iid N(0, 𝜎𝜇2).Furthermore, an assumption is being made that the initial value of 𝑟0 of model (10) is constant and play the role of an intercept. The objective here is to test a null hypothesis of stationarity (𝜎𝜇2=0) around mean or around time trend, against the alternative of presence of unit
root.
To test a null hypothesis of stationarity around the mean {𝐻0:𝜎𝜇2=0, 𝜉=0} against the
alternative hypothesis {𝐻0:𝜎𝜇2>0, 𝜉=0}, we assume that time trend (t) from model (10)
is zero and let 𝑒𝑡, t=1,2,,……,T denote the estimated errors computed as residuals
from a regression of 𝑦𝑡 on a constant (i.e. 𝑒𝑡= 𝑦𝑡− 𝑦̅) which will result into the following model:
𝑦𝑡= 𝑟𝑡+ 𝜀𝑡 (12)
The test statistic can be given by:
𝐾𝑃𝑆𝑆 = 𝑇−2∑𝑆𝑡 2 𝜎̂𝜀2 𝑇 𝑖=1 (13)
Where 𝑆𝑡= ∑𝑇𝑖=1𝜀̂𝑡, t=1,2,…,T and 𝜎̂𝜀2 is the consistent estimator of the long-run
variance 𝜎𝜀2 of process 𝜀𝑡 from model (12) which is given by the formula:
𝜎̂𝜀2 = 𝑇−1∑ 𝑒 𝑡2 𝑇 𝑖=1 + 2 𝑇−1∑ 𝑤( 𝑘 𝑗=1 𝑗, 𝑘) + ∑ 𝑒𝑡𝑒𝑡−𝑠 𝑇 𝑡=𝑠+1 (14)
Where w(j,k) is a kernel function, for example, the Bartlett window:
𝑤(𝑗, 𝑘) = 1 − 𝑗 𝑘 + 1⁄ .
If the computed KPSS value is greater than critical values which were derived by simulation and listed on in Kwiatkowski, Phillips, Schmidt and Shin (1992), the null hypothesis of stationarity around constant is rejected at given level of significance.
To test a null hypothesis of stationarity around the linear time trend {𝐻0:𝜎𝜇2=0, 𝜉≠0}
against the alternative hypothesis {𝐻0:𝜎𝜇2>0, 𝜉≠0}, the estimated errors 𝑒𝑡 are
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definitions are unchanged. The test statistic of this null hypothesis is based on LM test which is represented by the following model:
𝐿𝑀 = ∑𝑆𝑡2 𝜎̂𝜀2
⁄ (15) 𝑇
𝑡=1
If the computed LM test statistic is greater than critical value, the null hypothesis of stationarity around linear time trend is rejected at given level of significance.