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help for ^dfao^ (Statalist distribution, 28 March 2002)
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Augmented Dickey-Fuller unit root test with additive outliers
^dfao^ varname [^if^ exp] [^in^ range] [, ^M^axlag^(^#^)^ ^LLTC^ | ^NOLA^gtest
^NOAO^test ^NOCON^stant ^NOTR^end
^GEN^erate(newvar) ^REG^ress ^LEV^el(integer)]
^dfao^ is for use with time-series data. You must ^tsset^ your data before
using ^dfao^; see help @tsset@.
^varname^ may contain time-series operators; see help @varlist@.
Description
-----------
^dfao^ performs a Dickey-Fuller (DF 1979, 1981) test for a single unit root in a time series when
one or more additive outliers are present. An additive outlier is a temporary shock or aberration
in the data, as exemplified by the shock to U.S. grain prices in the mid-1990s. The approach used
here follows the method outlined in Vogelsang (1999) and Franses and Haldrup (1994). The presence
of additive outliers induce "in the errors a moving-average component with a negative coefficient"
(Vogelsang 1999, p. 237). Schwert (1989) showed that MA errors with a negative component result in
oversized unit root tests. Following Franses and Haldrup (1994), Vogelsang adds outlier dummy
variables to the augmented DF regression. The addition of these dummies does not affect the DF dis-
tribution if the number of dummies is not too large (see Franses and Haldrup). Outliers are de-
tected using an iterative method [see Vogelsang (pp. 240-2)]. Note that k+2 dummy variables are
added to the DF regression for each outlier found (the contemporaneous dummy and k+1 of its lags,
where k is the lag length) because an additional lagged dummy is needed "to remove the influence of
the outlier on the [kth lagged difference of the dependent variable].... If there is more than one
outlier and k is moderately large, the degrees of freedom lost ... from outlier removal can be non-
trivial" (Vogelsang 1999, p. 240).
The data are detrended unless the ^notrend^ option is specified, in which case the data are de-
meaned. If ^noconstant^ is specified, then the data are neither detrended nor demeaned, though this
option cannot be used when testing for additive outliers (see the explanation of the ^noconstant^
option below). The maximum lag order for the test is by default calculated from the sample size using
a rule provided by Schwert (1989, p. 151). The maximum lag order may also be provided with the ^maxlag^
option, and may be zero. If the maximum lag order exceeds one, a sequential lag-length test is auto-
matically performed for each lag using a user-specified criterion (choices are sequential-t test
[see Ng & Perron (1995)], Akaike, Schwarz, or Hannan-Quinn information criteria [see references]). If
the sequential-t test is chosen, then lag-length testing stops when that coefficient's p-value is less
than the test size specified by the user (see the explanation of the ^level^ option below). Note that
the sample size is held constant over lags at the maximum available sample. Automatic lag selection can
be suppressed using the ^nolagtest^ option.
Approximate 1%, 5%, and 10% critical values for the ADF unit root test are calculated using the
response surface equations in Cheung and Lai (1995).
^dfao^ prints the variable tested, the time period covered by the DF regression, the number of
observations used in the DF regression, the value of maxlag and how it was selected (by the user
or the Schwert criterion), deterministic terms used in the regression (none, constant, or constant +
trend), the DF test statistic and the 1%, 5%, and 10% critical values, the MacKinnon (1994)
approximate p-value for the test statistic, the optimal lag order, the RMSE (calculated from sample
size T), and the outliers together with their respective t-statistics, and critical values. If the
^regress^ option (see below) is selected, the coefficient table for the DF regression is also dis-
played. Note that t-statistics from the outlier test are printed as tau# and dummy variables are
printed as d#, where # is the observation number of the outlier. The outlier t-statistics and out-
lier dummy variables are printed in the order in which outliers were detected, from most to least
significant. The particular information printed depends of course on the options chosen.
^dfao^ places in the return array the number of outliers, the t-statistics for each outlier de-
tected in the outlier test, the value of the t-statistic or information criterion and RMSE calcu-
lated at each lag, the number of observations used in the DF regression, the value of maxlag, the
actual number of lags used in the DF regression, the coefficient of the lagged dependent variable
(rho), the DF test statistic (Zt), and the p-value corresponding to Zt. If outliers are detected,
^dfao^ also returns a macro listing the dates of the outliers in the order in which outliers were
detected (see the discussion of outlier t-statistics and dummy variables in the preceding para-
graph). The particular data returned depends on the options chosen.
Options
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^M^axlag(^#^) specifies the maximum lag order to be used in augmenting the Dickey-Fuller regression
and only accepts nonnegative integer values. If omitted, the maximum lag order is calculated as
described above.
^LLTC^(^t^|^aic^|^sc^|^hq^) stands for "lag length test criterion." Choices are "t" (sequential
t-test), "aic" (Akaike information criterion), "sc" (Bayesian Schwarz criterion), and "hq"
(Hannan-Quinn criterion). Only one option may be specified. If omitted, ^LLTC^ defaults to the
sequential t-test.
^NOLA^gtest suppresses automatic lag-length testing. If ^nolagtest^ is specified and ^maxlag^ is
omitted, then the DF regression is run with the number of lags determined by the Schwert criterion
as described above. ^NOLA^gtest generates an error if ^LLTC^ is also specified.
^NOAO^test suppresses the test for additive outliers. If this option is used then ^dfao^ works like
^dfuller^, yet and unlike the latter routine, still provides for automatic lag-length determination
and response surface critical values.
^NOCON^stant suppresses the constant in the DF regression. The data are neither demeaned nor de-
trended. Because this option precludes both the inclusion of a trend in the DF regression and the
outlier test, specifying ^NOCON^stant on the command line automatically sets the ^NOTR^end and
^NOAO^test options.
^NOTR^end means that the data are demeaned rather than detrended. Since many time series exhibit
trending behavior, the default is to include a trend in the auxiliary and DF regressions.
^GEN^erate(^newvar^) specifies that the residuals from the DF regression should be saved as a new
variable for subsequent analysis. The residuals correspond to the DF regression using the lag-length
that is specified as an option or computed internally.
^REG^ress displays the coefficient table for the DF regression.
^LEV^el(^#^) controls both the width of the confidence intervals for displayed statistics and the test
size used in both outlier testing and lag-length determination. The test size is calculated as
100 - ^level^(^#^). If omitted, ^level^ equals the value contained in the global macro ^$S_Level^.
Examples
--------
. ^use http://fmwww.bc.edu/ec-p/data/macro/pervog92^
. ^dfao lfiuscpi^
. ^dfao lfiuscpi, notr lev(90)^
. ^dfao lfiuscpi, notr lev(90) lltc(aic)^
. ^dfao lfiuscpi, notr lev(90) reg^
. ^dfao lfiuscpi, notr lev(90) m(11) nola^
. ^dfao lfiuscpi, noao^
. ^dfao lfiuscpi, nocon^
. ^dfao lfiuscpi, gen(dfaores)^
. ^wntestq dfaores^
References
----------
Akaike, H. 1973. Information Theory and the Extension of the Maximum Likelihood Principle.
In 2nd International Symposium on Information Theory. eds. B.N. Petrov, and F. Csaki, 267-81.
Budapest: Akailseoniai-Kiudo.
Akaike, H. 1974. A New Look at the Statistical Identification Model. IEEE: Trans. Auto
Control 19: 716-23
Baum, C. F. 2000. sts15: Test for stationarity of a time series. Stata Technical Bulletin 57: 36-39.
Baum, C. F., and R. Sperling. 2000. sts15.1: Tests for stationarity of a time series: update.
Stata Technical Bulletin 58: 35-36.
Cheung, Y.-W., and K.S. Lai. 1995. Lag Order and Critical Values of the Augmented Dickey-Fuller
Test. Journal of Business & Economic Statistics 13: 277-80.
Dickey, D.A., and W.A. Fuller. 1979. Distribution of the Estimators for Autoregressive Time Series
with a Unit Root. Journal of the American Statistical Association 74: 427-31.
Dickey, D.A., and W.A. Fuller. 1981. Likelihood Ratio Tests for Autoregressive Time Series with a
Unit Root. Econometrica 49: 1057-72.
Franses, P.H., and N. Haldrup. 1994. The Effects of Additive Outliers on Tests for Unit Roots and
Cointegration. Journal of Business & Economic Statistics 12: 471-78.
Hannan, E.J., and B.G. Quinn. 1979. The Determination of the Order of an Autoregression. Journal of
the Royal Statistical Society, Series B 41: 190-95.
MacKinnon, J.G. 1994. Approximate Asymptotic Distribution Functions for Unit Root and Cointegration
Tests. Journal of Business & Economic Statistics 12: 167-76.
Ng, S., and P. Perron. 1995. Unit Root Tests in ARMA Models with Data-Dependent Methods for the
Selection of the Truncation Lag. Journal of the American Statistical Association 90: 268-81.
Schwarz, G. 1978. Estimating the Dimension of a Model. Annals of Statistics 6: 461-64.
Schwert, G.W. 1989. Test for Unit Roots: A Monte Carlo Investigation. Journal of Business &
Economic Statistics 7: 147-59.
Vogelsang, T.J. 1999. Two Simple Procedures for Testing for a Unit Root When There are Additive
Outliers. Journal of Time Series Analysis 20: 237-52.
Acknowledgements
----------------
Portions of the code were taken from dfuller.ado written by Stata Corp. and ^dfgls^ by Baum and
Sperling. Nick Cox provided valuable programming assistance. Thanks also to Tim Vogelsang for
clarifying comments. Finally, Kit Baum made several suggestions that significantly improved the
functionality of the program in addition to writing portions of this help file. Remaining errors
are mine.
Author
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Richard Sperling, The Ohio State University, USA
rsperling@@boo.net
Also see
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Manual: ^[R] dfuller^
On-line: help for @dfuller@, @time@, @tsset@, @dfgls@ (if installed)