ARIMA Procedure

Stationarity Tests

When a time series has a unit root, the series is nonstationary and the ordinary least squares (OLS) estimator is not normally distributed. Dickey and Fuller studied the limiting distribution of the OLS estimator of autoregressive models for time series that have a simple unit root (Dickey 1976; Dickey and Fuller 1979). Dickey, Hasza, and Fuller (1984) obtained the limiting distribution for time series that have seasonal unit roots. Hamilton (1994) discusses the various types of unit root testing.

The augmented Dickey-Fuller (ADF) test (Dickey and Fuller 1979) and the Phillips-Perron (PP) test (Phillips and Perron 1988) are usually used to test stationarity. Both tests can be used to test three types of data generation models: when the series is zero-mean stationary (zero mean), nonzero-mean stationary (single mean), and linear time trend stationary (trend). The following sections discuss these three models of order p plus 1.

Zero Mean

A zero-mean, stationary autoregressive process of order p plus 1, AR(p plus 1), can be described as follows:

y Subscript t Baseline equals alpha 1 y Subscript t minus 1 Baseline plus alpha 2 y Subscript t minus 2 Baseline plus midline-horizontal-ellipsis plus alpha Subscript p plus 1 Baseline y Subscript t minus p minus 1 Baseline plus e Subscript t

It could also be written as

nabla y Subscript t Baseline equals delta y Subscript t minus 1 Baseline plus theta 1 nabla y Subscript t minus 1 Baseline plus midline-horizontal-ellipsis plus theta Subscript p Baseline nabla y Subscript t minus p Baseline plus e Subscript t

where nabla y Subscript t Baseline equals y Subscript t Baseline minus y Subscript t minus 1, delta equals alpha 1 plus midline-horizontal-ellipsis plus alpha Subscript p plus 1 Baseline minus 1, and theta Subscript k Baseline equals minus alpha Subscript k plus 1 Baseline minus midline-horizontal-ellipsis minus alpha Subscript p plus 1.

In this alternate form, y Subscript t is difference stationary if delta equals 0. The zero-mean form of the ADF and PP tests is useful for testing whether y Subscript t is difference stationary (null) or zero-mean stationary (alternative).

Single Mean

A stationary autoregressive process of order p plus 1, AR(p plus 1), with mean mu can be described as follows:

y Subscript t Baseline minus mu equals alpha 1 left-parenthesis y Subscript t minus 1 Baseline minus mu right-parenthesis plus alpha 2 left-parenthesis y Subscript t minus 2 Baseline minus mu right-parenthesis plus midline-horizontal-ellipsis plus alpha Subscript p plus 1 Baseline left-parenthesis y Subscript t minus p minus 1 Baseline minus mu right-parenthesis plus e Subscript t

It could also be written as

nabla y Subscript t Baseline equals minus delta mu plus delta y Subscript t minus 1 Baseline plus theta 1 nabla y Subscript t minus 1 Baseline plus midline-horizontal-ellipsis plus theta Subscript p Baseline nabla y Subscript t minus p Baseline plus e Subscript t

In this alternate form, y Subscript t is difference stationary if delta equals 0. The single-mean form of the ADF and PP tests is useful for testing whether y Subscript t is difference stationary (null) or nonzero-mean stationary (alternative).

Trend

A stationary autoregressive process of order p plus 1, AR(p plus 1), with linear time trend mu plus beta t can be described as follows:

y Subscript t Baseline minus mu minus beta t equals alpha 1 left-parenthesis y Subscript t minus 1 Baseline minus mu minus beta left-parenthesis t minus 1 right-parenthesis right-parenthesis plus alpha 2 left-parenthesis y Subscript t minus 2 Baseline minus mu minus beta left-parenthesis t minus 2 right-parenthesis right-parenthesis plus midline-horizontal-ellipsis plus alpha Subscript p plus 1 Baseline left-parenthesis y Subscript t minus p minus 1 Baseline minus mu minus beta left-parenthesis t minus p minus 1 right-parenthesis right-parenthesis plus e Subscript t

It could also be written as

nabla y Subscript t Baseline equals minus delta mu minus delta beta t plus nu beta plus delta y Subscript t minus 1 Baseline plus theta 1 nabla y Subscript t minus 1 Baseline plus midline-horizontal-ellipsis plus theta Subscript p Baseline nabla y Subscript t minus p Baseline plus e Subscript t

where nu equals alpha 1 plus 2 alpha 2 plus midline-horizontal-ellipsis plus left-parenthesis p plus 1 right-parenthesis alpha Subscript p plus 1.

In this alternate form, y Subscript t is difference stationary (with nonzero mean) if delta equals 0. The trend form of the ADF and PP tests is useful for testing whether y Subscript t is difference stationary with nonzero mean (null) or trend stationary (alternative).

When there is a unit root (that is, the series is nonstationary), the sum of the autoregressive parameters is 1 and hence delta equals 0. The ADF tests and the PP tests both build on delta. There are three kinds of tests under the ADF tests: rho (rho) test, tau (tau) test, and F test. The rho test is the regression coefficient test, which is also called the normalized bias test. The tau test is the studentized test. The F test is a joint test for unit root. For more information about test statistics under the ADF tests, see Dickey (2005), the section Stationarity Tests, and Hamilton (1994). There are two kinds of test statistics under the PP tests: rho test and tau test statistics. For more information about test statistics under the PP tests, see Chapter 8, AUTOREG Procedure. The following table presents null hypotheses and decision rules for the three test statistics under the three different model types:

Rho Test Tau Test bold upper F Test
Types upper H 0 p-value upper H 0 p-value upper H 0 p-value
Zero mean delta equals 0 Stationary if low delta equals 0 Stationary if low N/A N/A
Single mean delta equals 0 Stationary if low delta equals 0 Stationary if low delta equals alpha 0 equals 0 Stationary if low
Trend delta equals 0 Stationary if low delta equals 0 Stationary if low delta equals gamma equals 0 Stationary if low

In this table, alpha 0 is the intercept in the test regression for the single-mean model,

nabla y Subscript t Baseline equals alpha 0 plus delta y Subscript t minus 1 Baseline plus theta 1 nabla y Subscript t minus 1 Baseline plus midline-horizontal-ellipsis plus theta Subscript p Baseline nabla y Subscript t minus p Baseline plus e Subscript t

and gamma is the parameter of t in the test regression for the trend model,

nabla y Subscript t Baseline equals alpha 0 plus gamma t plus delta y Subscript t minus 1 Baseline plus theta 1 nabla y Subscript t minus 1 Baseline plus midline-horizontal-ellipsis plus theta Subscript p Baseline nabla y Subscript t minus p Baseline plus e Subscript t

As shown in the regression for single-mean model, when delta equals 0 (that is, there is a unit root), the intercept alpha 0 equals minus delta mu equals 0. The F test with a joint hypothesis of zero intercept and zero slope can therefore be used as a unit root test for the single-mean model. The F test for the trend model follows a similar logic. When delta equals 0, gamma equals minus delta beta equals 0. The F statistic for the trend model therefore tests the joint null hypothesis: delta equals gamma equals 0. When you are testing for unit root, the tau (tau) test is more powerful than the F test.

For a more detailed description of the Dickey-Fuller tests, see the section PROBDF Function for Dickey-Fuller Tests. For a description of Phillips-Perron tests, see Chapter 8, AUTOREG Procedure. The random-walk-with-drift test suggests whether or not an integrated times series has a drift term. Hamilton (1994) discusses this test.

Last updated: June 19, 2025