*! version 1.0.1 16may2026

Title

ralsadf -- RALS-augmented Dickey-Fuller unit-root test of Im, Lee & Tieslau (2014)

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Syntax

ralsadf varname [if] [in] [, trend maxlags(#) ic(string) level(#) graph noheader]

varname must be a single time-series-operated variable; tsset must already be active.

trend include a linear trend (model 2); default is constant only (model 1).

maxlags(#) maximum number of dy lags to consider (default = 8).

ic(string) lag-selection rule: aic | bic | tstat (default).

level(#) confidence level used in the rejection message (default 95).

graph draw a time-series plot annotated with the RALS-ADF statistic and 5% CV.

noheader suppress the boxed header.

Description

ralsadf performs the two-stage residual-augmented Dickey-Fuller
unit-root test of Im, Lee & Tieslau (2014). Stage 1 fits

Dy_t = alpha + rho*y_ + sum gamma_i Dy_ + eps_t

(with an optional linear trend) and records the ADF statistic tau_ADF.
Stage 2 augments the regression with

w_t = (e_t^2 - m_2, e_t^3 - m_3 - 3*m_2*e_t)'

and refits by OLS; tau_RALS = b_rho / se(b_rho) is the resulting test
statistic. The asymptotic distribution of tau_RALS depends on the
variance-ratio nuisance parameter

rho^2 = sigma^2_RALS / sigma^2_ADF,

estimated as the ratio of the second-stage to the first-stage regression
variances. Critical values come from the Hansen (1995) CADF tables built
into the GAUSS source file rals_adf.src and are interpolated linearly
between adjacent rows of the rho^2 grid.

Decision rule: reject H0 (unit root) when the RALS-ADF statistic is less
than the (negative) critical value at the chosen level.

Options

trend selects the constant + trend specification (model 2 of the
GAUSS code). When omitted, only a constant is included.

maxlags(#) fixes the maximum number of Dy_ lags allowed
in stage 1. The default of 8 follows Im, Lee & Tieslau (2014).

ic(string) controls how the lag length is chosen:

tstat (default) starts at maxlags and removes the highest lag
whose |t| is below 1.645.

aic picks the lag minimising the Akaike information criterion.

bic picks the lag minimising the Schwarz information criterion.

graph draws a time-series plot of the analysed variable, with
the RALS-ADF statistic and the 5% critical value displayed in the subtitle.

noheader suppresses the boxed header.

Examples

. sysuse sp500, clear

. gen t = _n

. tsset t

. ralsadf close, trend

. ralsadf close, trend maxlags(12) ic(aic) graph

Stored results

ralsadf stores the following in r():

r(tauADF) stage-1 ADF statistic

r(tauRALS) RALS-ADF statistic

r(rho2) estimated rho^2

r(lag) lag length selected

r(T) sample size used

r(cv01_DF), r(cv05_DF), r(cv10_DF) stage-1 ADF critical values

r(cv01), r(cv05), r(cv10) stage-2 RALS critical values

References

Im, K.S., Lee, J., Tieslau, M.A. (2014). More powerful unit-root tests with non-normal errors. In Festschrift in Honor of Peter Schmidt, 315-342.

Hansen, B.E. (1995). Rethinking the univariate approach to unit-root testing. Econometric Theory 11: 1148-1171.

See also

Back to overview: rals

Unit-root tests: ralsadf, ralslm, ralslmb, ralsfadf, ralsfkss

Battery (run-all): ralsbattery

Cointegration tests: ralscoint, ralsfadl

Diagnostics: ralsdiag

Author

Dr Merwan Roudane -- merwanroudane920@gmail.com