ADFTest - Augmented Dickey-Fuller Stationary Test

1. The testing procedure for the ADF test is applied to the following model: $$\Delta y_t = \alpha + \beta_1 t + \beta_2 t^2 + \gamma y_{t-1} + \phi_1 \Delta y_{t-1} + \cdots + \phi_{p-1} \Delta y_{t-p+1} + \varepsilon_t$$ Where:
   
   • $\Delta $ is the first different operator
   • $ \alpha$ is a constant
   • $ \beta_1$ is the coefficient on a time trend
   • $ \beta_2$ is the coefficient on a squared time trend
2. This model can be estimated, and testing for a unit root is equivalent to testing that $\gamma = 0$.
3. In sum, the ADF test hypothesis is as follows: $$H_{o}: \gamma=0$$ $$H_{1}: \gamma \lt 0$$ Where:
   
   • $H_{o}$ is the null hypothesis (i.e. $y_t$ has a unit-root)
   • $H_{1}$ is the alternate hypothesis (i.e. ${y_t}$ does not have a unit-root)
4. The test statistics ($\tau$) value is calculated as follows: $$\tau = \frac{\hat{\gamma}}{\sigma_{\hat\gamma}}$$ where:
   • $\hat{\gamma}$ is the estimated coefficient
   • $\sigma_{\hat\gamma}$ is the standard error in the coefficient estimate
5. The test statistics value ($\tau$) is compared to the relevant critical value for the Dickey–Fuller Test. If the test statistic is less than the critical value, we reject the null hypothesis and conclude that no unit-root is present.
6. The number of non-missing values in the input time series must be at least 10.
7. The ADF test does not directly test for stationarity, but indirectly through the existence (or absence) of a unit-root. Furthermore, ADF incorporates a deterministic trend (and trend squared), so it allows a trend-stationary process to occur.
8. The main difference between the ADF test and a normal Dickey-Fuller test is that ADF allows for higher-order autoregressive processes.
9. For the test-down approach, we start with a given maximum lag length and test down by running several tests; in each, we exaimine the high-order coefficient's t-stat for significance.
10. It is not possible to use a standard t-distribution to provide critical values for this test. Therefore this test statistic (i.e. $\tau$) has a specific distribution simply known as the Dickey–Fuller table.
11. The ADF probability tables were simulated for different scenarios (e.g. no deterministic part, constant only, etc.) from 2,000,000 replications with gaussian errors.
12. The time series is homogeneous or equally spaced.
13. The time series may include missing values (e.g. #N/A) at either end.

Example 1:

Files Examples

Related Links

• Jarque, Carlos M.; Anil K. Bera (1980). "Efficient tests for normality, homoscedasticity and serial independence of regression residuals". Economics Letters 6 (3): 255-259.
• Ljung, G. M. and Box, G. E. P., "On a measure of lack of fit in time series models." Biometrika 65 (1978): 297-303
• Enders, W., "Applied econometric time series", John Wiley & Sons, 1995, p. 86-87
• Shapiro, S. S. and Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)", Biometrika, 52, 3 and 4, pages 591-611

External Links

• Wikipedia - Augmented Dickey-Fuller test
• Wikipedia - JDickey-Fuller test
• A Primer on Unit-Roots and Cointegration