# adftest

Augmented Dickey-Fuller test

## Syntax

## Description

returns
the rejection decision `h`

= adftest(`y`

)`h`

from conducting an augmented Dickey-Fuller test for a
unit root in a univariate time series `y`

.

returns the table `StatTbl`

= adftest(`Tbl`

)`StatTbl`

containing variables for the test results,
statistics, and settings from conducting an augmented Dickey-Fuller test for a unit root in
the last variable of the input table or timetable `Tbl`

. To select a
different variable in `Tbl`

to test, use the
`DataVariable`

name-value argument.

`[___] = adftest(___,`

specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
`Name=Value`

)`adftest`

returns the output argument combination for the
corresponding input arguments.

Some options control the number of tests to conduct. The following conditions apply when
`adftest`

conducts multiple tests:

For example, `adftest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1])`

conducts two tests, at a level of significance of 0.025, for the presence of a unit root in
the variable `GDP`

of the table `Tbl`

. The first test
includes `0`

lagged difference terms in the AR model, and the second test
includes `1`

lagged difference term in the AR model.

## Examples

## Input Arguments

## Output Arguments

## More About

## Tips

To draw valid inferences from the test, determine a suitable value for

`Lags`

.One method is to begin with a maximum lag, such as the one recommended in [7], and then test down by assessing the significance of $${\widehat{\beta}}_{p}$$, the coefficient of the largest lagged change in

*y*. The usual_{t}*t*statistic is appropriate, as returned in the`reg`

output structure.Another method is to combine a measure of fit, such as the SSR, with information criteria, such as AIC, BIC, and HQC. These statistics are also returned in the

`reg`

output structure. For more details, see [6].With a specific testing strategy in mind, determine the value of

`Model`

by the growth characteristics of*y*. If you include too many regressors (see_{t}`Lags`

), the test loses power; if you include too few regressors, the test is biased towards favoring the null model [4]. In general, if a series grows, the`"TS"`

model (see`Model`

) provides a reasonable trend-stationary alternative to a unit-root process with drift. If a series is does not grow, the`"AR"`

and`"ARD"`

models provide reasonable stationary alternatives to a unit-root process without drift. The`"ARD"`

alternative model has a mean of*c*/(1 –*a*); the`"AR"`

alternative model has mean 0.

## Algorithms

Dickey-Fuller statistics follow nonstandard distributions under the null hypothesis (even
asymptotically). `adftest`

uses tabulated critical values, generated by
Monte Carlo simulations, for a range of sample sizes and significance levels of the null model
with Gaussian innovations and five million replications per sample size.
`adftest`

interpolates critical values `cValue`

and *p*-values `pValue`

from the tables. Tables for tests
of `Test`

types `"t1"`

and `"t2"`

are
identical to those for `pptest`

. For small samples, tabulated values are
valid only for Gaussian innovations. For large samples, values are also valid for non-Gaussian
innovations.

## References

[1] Davidson, R., and J. G. MacKinnon. *Econometric Theory and Methods*. Oxford, UK: Oxford University Press, 2004.

[2] Dickey, D. A., and W. A. Fuller.
"Distribution of the Estimators for Autoregressive Time Series with a Unit Root."
*Journal of the American Statistical Association*. Vol. 74, 1979, pp.
427–431.

[3] Dickey, D. A., and W. A. Fuller.
"Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root."
*Econometrica*. Vol. 49, 1981, pp. 1057–1072.

[5] Hamilton, James D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

[6] Ng, S., and P. Perron. "Unit Root
Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag."
*Journal of the American Statistical Association*. Vol. 90, 1995, pp.
268–281.

## Version History

**Introduced in R2009b**

## See Also

`kpsstest`

| `lmctest`

| `pptest`

| `vratiotest`

| `i10test`