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Stata Technical Bulletin

the GLS detrended test are interpolated from Table 1 of ERS (page 825). Approximate 1% critical values for the GLS demeaned
test are identical to those applicable to the no-constant, no-trend Dickey-Fuller test and are computed using the dfuller code.
The ERS option specifies that the ERS (and Dickey-Fuller) values are to be used for all levels of significance (eschewing the
response surface estimates).

If the maximum lag order exceeds one, the optimal lag order is calculated by the Ng and Perron (1995) sequential t test
on the highest order lag coefficient, stopping when that coefficient’s р-value is less than 0.10. The lag minimizing the Schwarz
criterion (SC, or BIC) is printed with its minimized value.

kpss performs the Kwiatkowski-Phillips-Schmidt-Shin test introduced in Kwiatkowski et al. (1992) for stationarity of
a time series. This test differs from those in common use (such as dfuller and pperron) by having a null hypothesis of
stationarity. The test may be conducted under the null hypothesis of either trend stationarity (the default) or level stationarity.
Inference from this test is complementary to that derived from those based on the Dickey-Fuller distribution (such as dfgls,
dfuller and pperron). The KPSS test is often used in conjunction with those tests to investigate the possibility that a series is
fractionally integrated; that is, neither ʃ(ɪ) nor 7(0); see Lee and Schmidt (1996).

The series is detrended (demeaned) by regressing y on z_t = (l,t) (z_t = (l)ɔ , yielding residuals e_t. Let the partial sum
series of e_t be s_t. Then the zero-order KPSS statistic k_ll = T~² ɪɪɪ s)ⁱ∕T^-1 ∑t=ι ^et∙ For maxlag > 0, the denominator is
computed as the Newey-West estimate of the long run variance of the series; see [R] newey.

Approximate critical values for the KPSS test are taken from Kwiatkowski et al. (1992).

Examples

Data from Terence Mills’ Econometric Analysis of Financial Time Series on the UK FTA All Share Index of stock prices
(ftap) and stock returns (ftaret) are analyzed.

. use http ://fmwww.be.edu∕ec-p∕data∕Mills2d∕fta.dta
. tsset

time variable: month, 1965ml to 1995ml2
. dfgls ftap
Number of obs =   355

Maxlag = 16 chosen by Schwert criterion

Opt Lag (Ng-Perron sequential t) = 15 with RMSE 35.59803

Min SC = 7.275482 at lag 2 with RMSE 37.0745
. kpss ftap

KPSS test for ftap

Maxlag = 16 chosen by Schwert criterion

Critical values for HO: ftap is trend stationary

10%: 0.119 5% : 0.146 2.5%: 0.176 1% : 0.216

Test statistic

7.90141

4.18402

2.86036

2.18027

1.76579

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