| KPSS.test {VGAMextra} | R Documentation |
KPSS tests for stationarity
Description
The Kwiatkowski-Phillips-Schmidt-Shin (KPSS)
test for the null hypothesis that the series x is
level or trend stationary
Usage
KPSS.test(x, type.H0 = c("level", "trend")[1],
trunc.l = c("short", "large")[1],
show.output = TRUE)
Arguments
x |
Numeric. A univariate series. |
type.H0 |
The null hypothesis to be tested: either |
trunc.l |
The lag truncation parameter. See below for more details. |
show.output |
Logical. Should the results be displyed? Default is
|
Details
To test the null hypothesis that a univariate time series is level–stationary or stationary around a deterministic trend. The alternative states the existence of a unit root.
Under this methodology, the series, say
\{ y_t;~t = 1, \ldots, T\}
is assumed to be decomposed as
y_t = \rho t + \xi_t + \varepsilon_t,
that is, as the sum of a deterministic trend,
a random walk (\xi_t), and a stationary error
(\varepsilon_t \sim N(0, \sigma^2_z).
Hence, this test reduces to simply test the
hypothesis that \{ \xi_t \}
is stationary,
that is, H_0: \sigma^2_z = 0.
The test statistic combines the one–sided Lagrange
multiplier (LM) statistic
and the locally best invariant (LBI) test
statistic (Nabeya and Tanaka, (1988)).
Its asymptotic distribution is discussed
in Kwiatkowski et al. (1992), and depends on
the ‘long–run’ variance \sigma^2.
The test statistic is given by
\eta = T^{-2} \sum_i S^2_i / \widehat{\sigma}^2=
T^{-2} \sum_i S^2_i / s^2(l).
where s^2(l) is a consistent estimate of
\sigma^2, given by
s^2(l) = (1/T)\sum_{t = 1}^T\varepsilon^2_t +
(2 / T) \sum_{s = 1}^l w(s, l) \sum_{t = s + 1}^T \varepsilon_t
\varepsilon_{t - s}.
Here, w(s, l) = 1 - s/(l + 1), where
l is taken from trunc.l, the
lag–truncation parameter. The choice "short"
gives the smallest integer not less than
3 \sqrt{T} / 11, or else,
9 \sqrt{T} / 11, if
trunc.l = "large".
Note, here the errors, \varepsilon_t, are
estimated from the regression
x ~ 1 (level) or x ~ 1 + t (trend), depending upon
the argument type.H0.
Unlike other software using linear interpolates, here
the p–values for both, trend and level stationarity,
are interpolated by cubic spline interpolations
from the tail critical values given
in Table 1 in Kwiatkowski et al. (1992).
The interpolation takes place on \eta.
Value
A list with the following:
1) Test statistic and P-value,
2) Critical values,
3) Residuals, \varepsilon_t.
Note
There is no standard methodology to select an
appropriate value for trunc.l, however,
satisfactory results have been found for
trunc.l proportional to T^{1/2}.
See Andrews, D.W.K. (1991) for a discussion on this.
Empirically, this parameter may be suggested
by the problem in turn, and should be large enough to
approximate the true dynamic behaviour of the series.
Author(s)
Victor Miranda.
References
Andrews, D.W.K. (1991) Heteroskedasticiy and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–858.
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., and Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159–178.
Nabeya, S. and Tanaka, K. (1988) Asymptotic theory of a test for the constancy regression coefficients against the random walk alternative. Annals of Statistics, 16, 218–235.
Phillips, P.C.B. and Perron, P. (1988) Testing for a unit root in time series regression. Biometrika, 75, 335–346.
Phillips, P.C.B. (1987) Time series with unit roots. Econometrica, 55, 277–301
See Also
Examples
set.seed(2802)
test <- KPSS.test(rnorm(20), type.H0 = "trend")
class(test)
test$crit.value