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Unit Root investigation - KPSS test
Unit root investigation in case of structural brakes in trend - Perron test
The KPSS test
The KPSS was introduced by Kwiatkowski, Phillips, Schmidt and Shin in 1992 (see Kwiatkowski et al. 1992). This test is regard as innovative in comparison with earlier Dickey- Fuller test, or Perron type tests. The KPSS test has a null of stationarity of a series around either mean or a linear trend; and the alternative assumes that a series is non-stationary due to presence of a unit root.
In the KPSS test the start point is model, where series of observations is represented as a sum of three components: deterministic trend, a random walk, and a stationary error term.
The procedure of KPSS test is discussed by Kwiatkowski et al. (1992), Maddala and Kim (1998 p. 120-122), Syczewska (2010) among others, see also Nabeya and Tanaka (1988).
The model has the form as follows:
𝑦𝑡= 𝜉𝑡 + 𝑟𝑡+ 𝜀𝑡 (17a)
𝑟𝑡 = 𝑟𝑡−1 + 𝑢𝑡 (17b)
where:
𝑦𝑡 - observation of analysed variable;
𝑡 - deterministic trend;
𝑟𝑡 - random walk process;
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𝜀𝑡 - error term in the equation (17a) which is assumed to be stationary under the null hypothesis 𝑦𝑡 is trend stationary;
𝑢𝑡 - error term in the equation (17b), 𝑢𝑡~𝐼𝐼𝐷(0, 𝜎𝑢2);
The initial value is treated as fixed and serves the role of an intercept. It is also consider a special case of model (17a) in which is set 𝜉 = 0. In this case under the null hypothesis 𝑦𝑡 is stationary around a level (𝑟0) rather than around a trend.
In the KPSS test the set of hypotheses is as follows:
H0: 𝜎𝑢2 = 0 or 𝑟𝑡 is a constant
𝑦𝑡 is stationary and it represents process integrated of order 0, 𝑦𝑡 ~ I(0) vs.
H1: 𝜎𝑢2 ≠ 0
𝑦𝑡 is nonstationary and it represents process integrated of order 1 at least
The test statistics is as follows:
𝐿𝑀 =∑𝑇𝑡=1𝑆𝑡2 𝜎̂𝑒2
(18)
where:
𝑒𝑡 - residuals from the regression of 𝑦𝑡;
𝜎̂𝑒2 - the residual variance from this regression (residual sum of square divided by T);
T - size of sample;
𝑆𝑡2 - partial sum of 𝑒𝑡:
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𝑆𝑡 = ∑ 𝑒𝑖
𝑇
𝑡=1
(19)
The KPSS test is an upper tail test. The asymptotic distribution of the LM test statistic has been proposed by (Nabeya and Tanaka 1988). However this is valid only if the errors are IID (see Maddala and Kim (1998, p. 121).
Unit root tests (summary)
The test using stationarity as a null can be used for confirmatory analysis.
However, if both tests fail to reject the respective nulls or both reject the respective nulls, we do not have a confirmation.
Test 1 ("typical" test)
Test 2 („stationarity” test) (counterpart of the "typical" test”) H0: yt is nonstationary process (unit root
exists);
yt ~I(1)
H0: yt is stationary process yt ~(I(0)
H1: yt is stationary process yt ~I(0)
H1: yt is nonstationary process (unit root exists);
yt ~I(1)
"typical" test
stationarity” test) (counterpart of the "typical" test”) Augmented Dickey-Fuller test (1981) Leybourne-McCabe test (1994)
Phillips-Perron test (1988) Kwiatkowski-Phillips-Schmidt-Shin test (1992)
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Note: ADF and PP test are asymptotically equivalent. PP has better small sample properties than ADF.
Confirmation analysis
We receive a confirmatory result in two cases:
in the :typical" test we reject the null hypothesis (yt ~I(1)) and simultaneously in the "stationary" test we do not reject the null hypothesis (yt ~I(0)); as a result, we can say that the analyzed time series is stationary (yt ~I(0)).
in the :typical" test we do not reject the null hypothesis (yt ~I(1)) and
simultaneously in the "stationary" test we reject the null hypothesis (yt ~I(0));
as a result, we can say that the analyzed time series is non-stationary (yt ~I(1)).
The results are not confirmed in case if in both tests we reject the null hypothesis, or we do not reject the null hypothesis.
In the literature is also proposed confirmation analysis by using joint ADF and KPSS tests (see Syczewska and Charemza 1998 , Carrion-i-Silvestre, Sanso-i-Rossello and Ortuño 2001, Kębłowski and Welfe 2004, or Carrion-i-Silvestre and Sansó 2006).
Perron test
One of the assumption for almost all econometric models is constant coefficients over time (over the sample). If it is not fulfilled this usually denotes that the form of the model is misspecified. In other words, the model is not appropriate representation of the data. One of the reason could be structural breaks in the DGP (data generating process). Structural changes are usually interpreted as changes in regression parameters. Such changes have impact on the unit root test results.
Most popular development in unit root test are concerned with the effect of changes in the coefficients of the deterministic variable such as deterministic trend (see Perron test). We can apply Perron test as long as simple known break. Perron test is an modification of ADF test.
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Perron test recognizes three types of deterministic trend break:
- moving the trend line without changing the slope (fairly stable slope) (case 1, see Figure 4)
- change of slope (case 2, see Figure 5)
- moving the trend line with changing the slope (case 3, see Figure 6)
Figure 4. Moving the trend line without changing the slope
0 20 40 60 80 100 120
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
t
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Figure 5. change of slope
Figure 6. Moving the trend line with changing the slope
0 20 40 60 80 100 120 140
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
t
0 20 40 60 80 100 120 140 160
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
t
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CASE 1. Moving the trend line without changing the slope (Figure 4) In the first step we estimate parameters of following regression:
𝑦𝑡 = 𝛼0+ 𝛽 ∙ 𝑡 + 𝜃 ∙ 𝐷(𝑈)𝑡+ 𝛿 ∙ 𝐷(𝑇𝐵)𝑡+ 𝛼1 ∙ 𝑦𝑡−1+ ∑ 𝑐𝑖 ∙ ∆𝑦𝑡−𝑖
𝑘
𝑖=1
+ 𝜀𝑡
(20a)
where:
𝑦𝑡 - analysing time series (process),
t, 𝛼0, 𝛽, 𝜃, 𝛿, 𝛼1, 𝑐𝑖 - parameters of the regression, 𝑡 - linear trend, t = 1,2, ..., T,
𝐷(𝑇𝐵)𝑡 = {1, for 𝑡 = 𝑇𝐵+ 1 0, for 𝑡 ≠ 𝑇𝐵+ 1
𝑇𝐵 - the moment of the change in the structure
(20b)
𝐷(𝑈)𝑡 = {1, for 𝑡 > 𝑇𝐵
0, for 𝑡 ≤ 𝑇𝐵 (20c)
The set of hypotheses is as follows:
𝐻0: 𝜃 = 0; 𝛽 = 0; 𝛿 ≠ 0; 𝛼1 = 1 𝐻1: 𝜃 ≠ 0; 𝛽 ≠ 0; 𝛿 = 0; 𝛼1< 1
(21)
We assume the existence of a unit root (𝛼1 = 1) and the occurrence of a single change in the deterministic trend (so-called shock, the trend is returning to its original path) in the null hypothesis. The alternative hypothesis assumes that there is no unit root (process is stationary - 𝛼1 < 1), and the change is not shock-related but concerns the constant change in the trend.
CASE 2. Change of slope in the trend function (Figure 5)
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In the first step we estimate parameters of following regression:
𝑦𝑡 = 𝛼0+ 𝛽 ∙ 𝑡 + 𝜃 ∙ 𝐷(𝑈)𝑡+ 𝛾 ∙ 𝐷(𝑇∗)𝑡+ 𝛼1∙ 𝑦𝑡−1 + ∑ 𝑐𝑖∙ ∆𝑦𝑡−𝑖
𝑘
𝑖=1
+ 𝜀𝑡
(22a)
where:
𝑦𝑡 - analysing time series (process),
t, 𝛼0, 𝛽, 𝜃, 𝛾, 𝛼1, 𝑐𝑖 - parameters of the regression, 𝐷(𝑇∗)𝑡= {𝑡 − 𝑇𝐵, 𝑑𝑙𝑎 𝑡 > 𝑇𝐵
0, 𝑑𝑙𝑎 𝑡 ≤ 𝑇𝐵 (22b)
𝑇𝐵 - the moment of the change in the structure
The set of hypotheses is as follows:
𝐻0: 𝜃 ≠ 0; 𝛽 = 0; 𝛾 = 0; 𝛼1= 1 𝐻1: 𝜃 = 0; 𝛽 ≠ 0; 𝛾 ≠ 0; 𝛼1 < 1
(23)
We assume the existence of a unit root (𝛼1 = 1) in the null hypothesis. The alternative hypothesis assumes that there is no unit root (process is stationary - 𝛼1 < 1), and the change of slope in trend is significant and permanent.
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CASE 3. Moving the trend line with changing the slope (Figure 6) In the first step we estimate parameters of following regression:
𝑦𝑡 = 𝛼0+ 𝛽 ∙ 𝑡 + 𝜃 ∙ 𝐷(𝑈)𝑡+ 𝛾 ∙ 𝐷(𝑇)𝑡+ 𝛿 ∙ 𝐷(𝑇𝐵)𝑡+ 𝛼1 ∙ 𝑦𝑡−1+ ∑ 𝑐𝑖 ∙ ∆𝑦𝑡−𝑖
𝑘
𝑖=1
+ 𝜀𝑡
(24a) where:
𝑦𝑡 - analysing time series (process),
t, 𝛼0, 𝛽, 𝜃, 𝛾, 𝛿, 𝛼1, 𝑐𝑖 - parameters of the regression, 𝐷(𝑇)𝑡 = {𝑡, 𝑑𝑙𝑎 𝑡 > 𝑇𝐵
0, 𝑑𝑙𝑎 𝑡 ≤ 𝑇𝐵 (24b)
𝑇𝐵 - the moment of the change in the structure
The set of hypotheses is as follows:
𝐻0: 𝜃 = 0; 𝛽 = 0; 𝛾 = 0; 𝛿 ≠ 0; 𝛼1 = 1 𝐻1: 𝜃 ≠ 0; 𝛽 ≠ 0; 𝛾 ≠ 0; 𝛿 = 0; 𝛼1 < 1
(25)
We assume the existence of a unit root (𝛼1 = 1) in the null hypothesis. The alternative hypothesis assumes that there is no unit root (process is stationary - 𝛼1 < 1), and the change of slope and constant term in trend function is significant and permanent.
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The test statistic in the Perron test
𝑡∗ =𝛼̂1− 1 𝑠𝑒(𝛼̂1)
(26)
where:
𝛼̂1 - estimator of parameter 1,
𝑠𝑒(𝛼̂1) - standard error for estimated 1 parameter.
Critical values for the Perron test differ from those used in the DF test. The author of the simulation-based test set a new one and presented it at work [Perron P.: The great crash, the oil price shock, and the unit root hypothesis, Econometrica, Vol. 57, No. 6, 1989, s. 1361- 1401].
Critical values for the Perron test are given for the significance and for the parameter , where = TB/T. For other parameters and its significance verification, the Student's t test is used. The choice of the moment the structure change occurs can be determined on the basis of the Student's t-statistic calculated for the variable showing the moment of change. Then the point for which the value of this statistic is the largest (in absolute value) is selected.
References:
1. Carrion-i-Silvestre, J. L., & Sansó, A. (2006). A guide to the computation of stationarity tests. Empirical Economics, 31(2), 433.
2. Carrion-i-Silvestre, J. L., Sanso-i-Rossello, A., & Ortuño, M. A. (2001). Unit root and stationarity tests’ wedding. Economics Letters, 70(1), 1-8.
3. Charemza W.W., & Syczewska E.M. (1998). Joint Application of the Dickey-Fuller and KPSS Tests, Economic Letters, 61, 17-21.
4. Enders W., (2008), Applied econometric time series. John Wiley & Sons.
5. Kębłowski, P., & Welfe, A. (2004). The ADF–KPSS test of the joint confirmation hypothesis of unit autoregressive root. Economics Letters, 85(2), 257-263.
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6. Kwiatkowski, D., Phillips, P. C., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?. Journal of econometrics, 54(1-3), 159-178.
7. Leybourne, S. J., & Newbold, P. (1999). The behaviour of Dickey–Fuller and Phillips–
Perron tests under the alternative hypothesis. The Econometrics Journal, 2(1), 92-106.
8. Leybourne, S., & Newbold, P. (1999). On the size properties of Phillips–Perron tests.
Journal of Time Series Analysis, 20(1), 51-61.
9. Libanio, G. A. (2005). Unit roots in macroeconomic time series: theory, implications, and evidence. Nova Economia, 15(3), 145-176.
10. Maddala G.S., Kim I.M., (1998), Unit roots, cointegration, and structural change. Series:
Themes in modern econometrics, Cambridge University Press.
11. Nabeya, S., & Tanaka, K. (1988). Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative. The Annals of Statistics, 16(1), 218-235.
12. Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis.
Econometrica: Journal of the Econometric Society, 1361-1401.
13. Syczewska, E. M. (2010). Empirical power of the Kwiatkowski-Phillips-Schmidt-Shin test (No. 45).
14. Zivot, E., & Andrews, D. W. K. (2002). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of business & economic statistics, 20(1), 25- 44.