Stationarity and Unit Root Testing Dr. Thomas Kigabo RUSUHUZWA

Non Stationarity Testing  Various definitions of non-stationarity exist  There are two models which have been frequently used to characterize non- stationarity:  the random walk process with drift: Y t =  + Y t-1 + u t where ut is iid;  and the deterministic trend process: Y t =  +  t + u t where ut is iid;

Trend stationarity  yt = α + βt + εt; εt ~ (0,σ2) where βt is a linear trend  The mean and variance are: E(y) = α + βt (a function of time); var(y) = σ 2 ( not a function of time)  the process is stationary around the time trend : E(y - βt) = α

Trend stationarity ( CTD)  removing the time trend leads to (trend) stationarity in the series;  To do that we can: include a deterministic time trend as one of the regressors in the model; this defines another variable with its time trend removed (i.e., save the residuals from a regression of the variable on a time trend) ; The regression model will then operate with stationary series with constant means and variances

Trend stationary data and differencing  One may be tempted to try first-differencing all non stationary series, since it may be hard to tell if they are unit root processes or just trend stationary  For instance, a first difference of the trend stationary process gives: yt – yt-1 = β + εt – εt-1  Is this an improvement? The time trend is gone, but the errors are now an MA(1) process

Testing for unit roots The Dickey-Fuller unit root test  The most common unit root test is the Dickey-Fuller test  Write the zero-mean AR(1) process: yt = ρyt-1 + εt (1)  subtract yt-1 from both sides: yt – yt-1 = (1 – L)yt = Δyt = (ρ – 1)yt-1 + εt (2) The (one-sided) test corresponds to whether the coefficient on the lagged value on the right-hand side, (ρ – 1), is zero versus less than zero

CTD  When the null hypothesis, (ρ – 1) = 0 or, equivalently, ρ = 1, is true, (1) reduces to: Δyt = εt  such that Δyt is stationary, meaning that yt in levels is a random walk and thus non- stationary  H0 is rejected if the t-statistic is smaller than the relevant critical value –  this statistic does not have the conventional t- distribution

CTD  The DF test is based on the assumption that εt is white noise, i.e., serially uncorrelated  The augmented DF (ADF(p)) test: Δyt = μ + ρ0yt-1 + ρ1Δyt-1 + … + ρk-1Δyt-(k- 1) + εt (3)  uses additional lags of the dependent variable to get rid of serial correlation

Selecting the proper lag length  The choice of additional lags may be based on information criteria or a sequential testing procedure:  Hall (1994) showed than when the order p in the AR(p) model for yt is selected through t- tests on the ρ1 to ρk-1 parameters in (3) (or via an application of the information criteria), the relevant DF statistics still apply

Dealing with serial correlation  The Phillips-Perron unit root test  Rather than using (long) autoregressions to approximate general (serially) dependent processes, the unit root tests due to Phillips and Perron (1988) adjust for serial correlation non-parametrically

CTD  Phillips-Perron tests have the same limiting null distribution as the DF distribution and therefore the same critical values  But this test relies on asymptotic theory, which means that large samples (i.e., long data series) are required for it work well

Testing for unit roots The Dickey-Fuller unit root test  Under H0, the model for yt is: Δyt = μ + ρ1Δyt-1 + … + ρk-1Δyt-(k-1) + εt (4) which is an AR(k-1) in the first difference Δyt  Thus, if yt has a (single) unit root, then Δyt is a stationary  Because of this property, we say that if yt is nonstationary, but Δdyt is stationary, then yt is integrated of order d (denoted yt ~ I(d)) or has d unit roots

CTD  A (non-stationary) variable is said to be integrated of order d, denoted I(d), if it needs to be differenced d times to achieve stationarity;  The order of differencing depends on the number of unit roots - this means that, for example, an I(1) variable needs to be differenced once to achieve stationarity and that it has only one unit root

CTD Some observations: deterministic regressors  Potential models are:  Δyt = μ + δt + (ρ - 1)yt-1 + … + εt (6) where:  μ = δ = 0 is referred to as a model without drift;  δ = 0 is referred to as a model with drift; and otherwise as a model with trend  The test is not accurate if the original data series contains a constant and/or a trend

CTD  The key problem in practice is that: tests for unit roots are conditional on the presence of deterministic regressors; and tests for the presence of the deterministic regressors are conditional on the presence of a unit root

CTD  Campbell and Perron (1991) illustrated that researchers may fail to reject the null hypothesis of a unit root because of a misspecification concerning the deterministic part of the regression

Unit Root Testing Summary of Dickey-Fuller Tests

Unit Root Testing: Using E-views '-CONSTRUCTION DES DIFFERENCES PREMIERES SMPL 1999:1 2012:3 GENR DLGDP = LGDP-LGDP(-1) '-ESTIMATION DU MODELE LIBRE EQUATION MOD3.LS DLGDP C LGDP(-1) DLGDP(- SCALAR SCALAR '-ESTIMATION DU MODELE CONTRAINT EQUATION MOD3C.LS DLGDP C LGDP(-1) DLGDP(-1) SCALAR '-CONSTRUCTION DE LA STATISTIQUE F3 SCALAR F3=((SCR3C-SCR3)/2)/(SCR3/NDL)

Unit Root Testing -CONSTRUCTION DES DIFFERENCES PREMIERES SMPL 1999:1 2012:3 GENR DLGDP = LGDP-LGDP(-1) '-ESTIMATION DU MODELE LIBRE EQUATION MOD2.LS DLGDP C LGDP(-1) DLGDP(-1) SCALAR SCALAR '-ESTIMATION DU MODELE CONTRAINT EQUATION MOD2C.LS DLGDP LGDP(-1) DLGDP(-1) SCALAR '-CONSTRUCTION DE LA STATISTIQUE F2 SCALAR F2=((SCR2C-SCR2)/2)/(SCR2/NDL)

Power of unit root tests  power of a test is equal to the probability of rejecting a false null hypothesis given a sample of finite size  It may be difficult to reject the null hypothesis of unit root using ADF test when is close to one

The KPSS unit root test  Kwiatkowski et al. (1992) proposed a test (generally referred to as the KPSS test) that instead of testing for the presence of a unit roots tests for the absence of one  the null hypothesis is that the series under investigation is stationary

Unit root tests with more power  The test due to Elliott et al. (1996) ERS showed that it is possible to enhance the power of a unit root test by estimating the model using something close to first differences, which is called local GLS detrending  Instead of creating the first difference of yt, ERS preselect a constant close to unity, α, and subtract αyt-1 from yt

CTD  The value of α that seems to provide the best power is α = (1 – 7/T) for the case of an intercept and α = (1 – 13.5/T) if is an intercept and trend;  ERS then estimate the basic ADF regression using the local GLS detrended data, denoted ytd:

CTD

 The lag length m is selected using the information criteria ( BIC) and the null hypothesis of a unit root can be rejected of