Augmented Dickey–Fuller test

In statistics and econometrics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. In statistics and econometrics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model where α {displaystyle alpha } is a constant, β {displaystyle eta } the coefficient on a time trend and p {displaystyle p} the lag order of the autoregressive process. Imposing the constraints α = 0 {displaystyle alpha =0} and β = 0 {displaystyle eta =0} corresponds to modelling a random walk and using the constraint β = 0 {displaystyle eta =0} corresponds to modeling a random walk with a drift. Consequently, there are three main versions of the test, analogous to the ones discussed on Dickey–Fuller test (see that page for a discussion on dealing with uncertainty about including the intercept and deterministic time trend terms in the test equation.) By including lags of the order p the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p has to be determined when applying the test. One possible approach is to test down from high orders and examine the t-values on coefficients. An alternative approach is to examine information criteria such as the Akaike information criterion, Bayesian information criterion or the Hannan–Quinn information criterion. The unit root test is then carried out under the null hypothesis γ = 0 {displaystyle gamma =0} against the alternative hypothesis of γ < 0. {displaystyle gamma <0.} Once a value for the test statistic is computed it can be compared to the relevant critical value for the Dickey–Fuller Test. If the test statistic is less (this test is non symmetrical so we do not consider an absolute value) than the (larger negative) critical value, then the null hypothesis of γ = 0 {displaystyle gamma =0} is rejected and no unit root is present. The intuition behind the test is that if the series is characterised by a unit root process then the lagged level of the series ( y t − 1 {displaystyle y_{t-1}} ) will provide no relevant information in predicting the change in y t {displaystyle y_{t}} besides the one obtained in the lagged changes ( Δ y t − k {displaystyle Delta y_{t-k}} ). In this case the γ = 0 {displaystyle gamma =0} and null hypothesis is not rejected. In contrast, when the process has no unit root, it is stationary and hence exhibits reversion to the mean - so the lagged level will provide relevant information in predicting the change of the series and the null of a unit root will be rejected. A model that includes a constant and a time trend is estimated using sample of 50 observations and yields the D F τ {displaystyle DF_{ au }} statistic of −4.57. This is more negative than the tabulated critical value of −3.50, so at the 95 percent level the null hypothesis of a unit root will be rejected.