Methods and formulas for Augmented Dickey-Fuller Test

Select the method or formula of your choice.

Regression models

The calculations use the following definitions:
TermDescription
the observed time series values at time = 1, …, T
the difference of two consecutive observations at time t, , where t = 2, …, T
the constant term in a regression model
the coefficient of a linear time trend in a regression model
the coefficient of a quadratic time trend in a regression model
the lag order of the autoregressive process
the serially independent error term at time t for t = 2, ..., T
The Augmented Dickey-Fuller Unit Root Test (ADF) uses ordinary least squares regression estimates. Specifications for the analysis in Minitab Statistical Software set the constant, linear, and quadratic coefficients to 0.
A model with only a constant coefficient
A model with a constant coefficient and a linear coefficient
A model with a constant coefficient, linear coefficient, and a quadratic coefficient
A model with no regression coefficients

Hypotheses

Each Augmented Dickey-Fuller test uses the following hypotheses:

Null hypothesis, H0:

Alternative hypothesis, H1:

The null hypothesis says that a unit root is in the time series sample, which means that the mean of the data is not stationary. Rejecting the null hypothesis indicates that the mean of the data is stationary or trend stationary, depending on the model for the test.

Test statistic

The test statistic for the ADF has the following form:

where

TermDescription
the least square coefficient estimate of the coefficient
the standard error of the least squares estimate of the coefficient from the regression model

MacKinnon's approximate p-values

Under the null hypothesis, the asymptotic distribution of the test statistic does not follow a standard distribution. Fuller (1976)1 provides a table with common percentiles of the asymptotic distribution. MacKinnon (19942, 20103) applies response surface approximations to simulated data to provide an approximate p-value for any value of the ADF test statistic.

If the specifications for the analysis use 0.01, 0.05, or 0.1 as the significance level, then the evaluation of the null hypothesis compares the test statistic to the critical value for that significance level. If the test statistic is less than or equal to the critical value, reject the null hypothesis.

If the specifications for the analysis give a different significance level, then the evaluation of the null hypothesis compares the approximate p-value to the significance level. If the p-value is less than the significance level, reject the null hypothesis.

Critical values for the significance levels 0.01, 0.05, and 0.1

Mackinnon (2010) provides the following general formula for the calculation of the critical value for three significance levels: 0.01, 0.05, and 0.1:

where n is the number of observations that the analysis uses to fit the regression model. The values for and come from tables in MacKinnon (2010). If the test statistic is less than or equal to the critical value, reject the null hypothesis.

Approximate p-values

The calculation of the approximate p-value comes from Mackinnon (1994). Compare the p-value to the significance level to make a decision. If the p-value is less than or equal to the significance level, reject the null hypothesis.

Determination of lag order

The conduction of an ADF requires specification of the lag order for the regression model. Specifications for the analysis provide the lag orders to evaluate. The default maximum order to evaluate has the following form:

The selection of the lag order depends on the criterion in the specifications of the analysis. If the specifications for the analysis do not include a criterion, then the regression model for the test is the maximum order of p.

In the calculations to determine the lag order, the number of observations depends on the maximum lag order such that m = np – 1.

where
TermDescription
nthe total number of observations
pthe maximum lag order of the differenced terms that are in the model

The calculation of each criteria follows:

Akaike Information Criterion (AIC)

The analysis evaluates a regression model for each lag order in the specifications of the analysis. The lag order for the test is the regression model with the minimum value of the AIC.

where

TermDescription
mthe number of observations that depends on the maximum lag order
kthe number of coefficients in the model, including the constant if the regression model has a non-zero constant
RSSthe residual sum of squares of the regression model

Bayesian Information Criterion (BIC)

The analysis evaluates a regression model for each lag order in the specifications of the analysis. The lag order for the test is the regression model with the minimum value of the BIC.

where

TermDescription
mthe number of observations that depends on the maximum lag order
kthe number of coefficients in the model, including the constant if the regression model has a non-zero constant
RSSthe residual sum of squares of the regression model

t-statistic

When the criterion is the t-statistic, the analysis begins with the regression model with the maximum lag order for the analysis. The analysis begins with the regression model where the lag order is p and reduces the order sequentially. The lag order for the test is the first regression model where the highest-order lag term is significant at the 0.05 level. The t-statistic has the following form:

where i = 1, …, p

TermDescription
the least squares estimate of the coefficient in the regression model
the standard error of the least squares estimate of the coefficient in the regression model
1 Fuller, W. A. (1976). Introduction to statistical time series. New York, Wiley.
2 MacKinnon, J. G. (1994). Approximate asymptotic distribution functions for unit-root and cointegration tests. Journal of Business and Economic Statistics, 12, 167-176.
3 MacKinnon, J. G. (2010). Critical values for cointegration tests: Working paper 1227. Queen's University, Department of Economics.