Augmented Dickey-Fuller Unit Root Test for Detecting Unit Roots in Time Series

Performs the augmented Dickey-Fuller test for a unit root in a time series, testing the null hypothesis of nonstationarity against the alternative of (trend-)stationarity. It extends the simple Dickey-Fuller test by including lagged difference terms to account for autocorrelation.

Usage

adfTest(
  y,
  type = c("none", "drift", "trend"),
  lags = 1,
  selectLags = c("fixed", "aic", "bic")
)

Arguments

y: numeric vector or univariate time series to be tested for a unit root.
type: the deterministic part of the test regression, one of "none" (default), "drift" or "trend".
lags: the number of lagged difference terms to be included. If selectLags is not "fixed", this is the maximum number of lags considered in the lag selection.
selectLags: the lag selection method, one of "fixed" (default, use lags as given), "aic" or "bic" (choose the lag order up to lags that minimizes the respective information criterion). Case-insensitive, so "AIC" and "BIC" are accepted as well.

Value

An object of class "htest" containing the following components:

statistic: the tau statistic, followed by the phi statistic(s) if type is "drift" or "trend". The p-value refers to the tau statistic.
parameter: the number of lagged differences included in the test regression (after lag selection, if requested).
p.value: the interpolated p-value of the tau statistic.
critical.values: a matrix with the 1%, 5% and 10% critical values of all reported statistics, interpolated for the effective sample size (not shown on screen).
alternative: a character string describing the alternative hypothesis.
method: a character string indicating the test performed.
data.name: a character string giving the name of the data.

Details

If type is set to "none" neither an intercept nor a trend is included in the test regression, "drift" adds an intercept and "trend" adds both an intercept and a linear trend.

The reported test statistic is the t statistic of the lagged level (tau1, tau2 or tau3, depending on type). For type = "drift" and type = "trend" the F statistics phi1 resp. phi2 and phi3 of Dickey and Fuller (1981), testing joint hypotheses on the deterministic terms, are reported as additional statistics; their critical values are contained in the critical.values component of the result.

The p-value refers to the tau statistic and is obtained by linear interpolation in the finite sample quantiles given by Fuller (1976), following the approach of tseries::adf.test(). If the statistic falls outside the range of the table, the p-value is reported as the respective boundary (0.01 or 0.99) and a warning is issued. The critical values are taken from Hamilton (1994) and Dickey and Fuller (1981).

Missing values are not allowed.

Note

Based on code by Bernhard Pfaff previously published in the urca package, adapted to conform to package standards.

References

Dickey, D. A. and Fuller, W. A. (1979) Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, 74, 427–431.

Dickey, D. A. and Fuller, W. A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057–1072.

Fuller, W. A. (1976) Introduction to Statistical Time Series, New York: Wiley.

Hamilton, J. D. (1994) Time Series Analysis, Princeton: Princeton University Press.

Examples

adfTest(AirPassengers, lags = 3, type = "trend")
#> Warning: p-value smaller than reported p-value
#> 
#> 	Augmented Dickey-Fuller Test
#> 
#> data:  AirPassengers
#> tau3 = -6.9358, phi2 = 16.4042, phi3 = 24.0601, lags = 3, p-value =
#> 0.01
#> alternative hypothesis: trend-stationary
#> 

# a random walk should not be rejected
set.seed(5)
rw <- cumsum(rnorm(200))
adfTest(rw, type = "drift")
#> 
#> 	Augmented Dickey-Fuller Test
#> 
#> data:  rw
#> tau2 = -2.8598, phi1 = 4.1428, lags = 1, p-value = 0.05381
#> alternative hypothesis: stationary
#>