There are a number of ways to conduct a unit roo t test (e.g., Dickey and Fuller, 1979; Phillips and Perron, 1988; Kwiatkowski et al, 1992; Leybourne and McCabe, 1994;
Elliott et al, 1996; Perron and Ng, 1996; Ng and Perron, 2001), however, I will only
discuss two of the most popular used techniques which will be applied in the
following study: Augmented Dickey-Fuller test and Phillips-Perron test.
Augmented Dickey-Fuller (ADF) Test
The first unit root test has its roots in the paper by David Dickey and Wayne Fuller
(1979) and is named after them. The intuition and process of it can be described as
following:
Consider an AR (1) model:
1
t t t
where εt N(0,σε2). ythas a unit root if β=1. If one series has a unit root then its
autocorrelations will be near one and will not drop much as lag lengt h increases; it will have a long memory and exhibit trend behaviour. With the stochastic trend
contains in yt, standard regression inference measures can be very misleading in that t-
values and R-square can both be overstated. Thus it is essential to identify the
integration of the series when dealing with financial data.
By subtracting yt−1from both sides, we can get a modified version of the AR (1)
mod el: 1 t t t y c φy− ε ∆ = + + (5.2)
where φ β= −1. The Dickey-Fuller test now tests the null hypothesis of a unit root
0: 0
H φ = against the alternative H1:φ <0. Since the test is done over the difference term, it is not possible to use standard t-distribution to as critical values. These critical values are derived from a limiting distribution that can be represented as a functional
of Brow nian motion and, to this day, they are derived in tables through simulation
(see, e.g., MacK innon, 1996) of this distribution.
There are three main versions of the DF test when conside ring whether to include a
constant and/or deterministic trend:
Test for a unit root: ∆ =yt φyt−1+εt (5.3)
Test for a unit root with drift: ∆ = +yt c φyt−1+εt (5.4)
1
t t t
y c γt φy− ε
∆ = + + + (5.5) Each version of the test has its own distribution and critical values which depend on
the size of the sample. A sample value less negative than the critical values suggests
that we cannot reject the null that yt has a unit root.
While the DF test only applies to AR (1) model, the Augmented Dickey-Fuller test is
an extension version of the
series mode ls. The testing procedure for the ADF test is the same as for the DF test
but it is applied to AR (p) model:
1 1 1 ...
t t t p t p t
y c γt φy− δ y− δ y− ε
∆ = + + + ∆ + + ∆ +
(5.6)
where p is the lag order of the autoregressive process.
Choosing the order p of an autoregression is a very impor tant issue as it requires
balancing the benefit of including more lags against the cost of add itional estimation uncertaint y. O n the one hand, if the order of an estimated autoregression is too low,
potentially valuable information contained in the more distant lagged values will be
omitted. On the other hand, if it is too high, more coefficients than necessary will be
estimated, which in turn introd uces add itional estimation error. A widely used
approach in practice, Akaike Information Criterion (AIC) is app lied in the study to choose the optimal p. The calculation of AIC is as following:
( ) 2 ( ) ln(SSR p ) ( 1) AIC p p T T = + + (5.7)
where, SSR (p) is the sum of squared residuals of the estimated AR (p). The AIC
estimator of p, pˆ, is the value that minimizes AIC (p) among the possible choices p =
0,1,…, pmax , where pm ax is the largest value of p considered. The AIC trades off the two forces in the above equation: SSR (p) necessarily decreases when an add itional
lag added, while in contrast, the second term in (5.7) increases, so that the number of
lags that minimizes the AIC is a consistent estimator of the true lag length.
Again, the ADF statistic doe s not have a normal distribut ion, and the critical values
can be found in ADF simulation tables (see, e.g., MacK innon, 1996).
Phillips-Perron (PP) Test
An alternative way to test for the stationarity of series is the Phillips-Perron test,
which is also widely used in econometric software packages. Phillips and Perron
(1988) developed a more comprehensive theory of unit root nonstationarity. Basically
the test is similar to the standard DF or ADF test, but it is more powerful in term of it incorporates an automatic correction to the DF or ADF procedure to allow for
autocorrelated residuals.
Consider the AR (1) model (5.1), the PP method modifies the t-ratio of the φ coefficient so that serial correlation does not affect the asymptotic distribution of the
test statistic. The statistic PP test based on is:
1/ 2 0 0 0 1/ 2 0 0 ˆ ( )( ( )) ( ) 2 T f se t t f f s φ φ γ −γ φ = − (5.8)
whereφˆ is the estimate, and tφis the t-ratio of φ, se( )φˆ is coefficient standard error,
and s is the standard error of the test regression. In add ition, γ0is a consistent estimate of the error variance in (5.3) calculated as(T −k s) 2/T, where k is the number of regressors. The remaining term, f0, is an estimator of the residual spectrum at
frequency zero, and is based on kernel-based sum-of-covariances in our tests.
Besides the advantage that the PP test corrects for any serial correlation and
heteroskedasticity in the error term, it also has the advantage over the ADF test that
the user does not have to specify a lag length for the test regression. Its disadvantage
is that it is an asymptot ic procedure and may not be fully appropr iate with finite
sample. In order to seek a more rob ust result, I will apply bot h the ADF and PP tests
with the sample.