Returns the p-value of the Augmented Dickey-Fuller (ADF) test, which tests for a unit root in the time series sample.

## Syntax

**ADFTest**(**X**, Order, Length, **Options**, test-down, Return_type, $\alpha$)

**X**- is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
**Order**- is the time order in the data series (i.e., the first data point's corresponding date (earliest date = 1 (default), latest date = 0)).
Order Description 1 Ascending (the first data point corresponds to the earliest date) (default). 0 Descending (the first data point corresponds to the latest date). **Length**- is the lag length of the autoregressive process. If missing, an initial value equal to the cubic root of the input data size is used.
**Options**- are the model description flags for the Augmented Dickey-Fuller test scenario/condition (1 = no constant, 2 = constant-only, 3 = trend only, 4 = constant and trend, 5= const, trend and trend squared).
Method Description 1 No deterministic component. 2 Constant only. 3 Trend only. 4 Constant and trend. 5 Constant, trend, and trend squared. **test-down**- is the mode of testing. If set to TRUE (default), ADFTest performs a series of tests; it starts with the input length lag, but the actual length lag order used is obtained by testing down.
**Return_type**- is a switch to select the return output (1 = P-Value (default), 2 = Test Stats, 3 = Critical Value.
Method Description 1 P-Value. 2 Test Statistics (e.g., Z-score). 3 Critical Value. - $\alpha$
- is the statistical significance of the test (i.e., alpha). If missing or omitted, an alpha value of 5% is assumed.

## Remarks

- 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.

- This model can be estimated, and testing for a unit root is equivalent to testing that $\gamma = 0$.
- 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).

- 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.

- 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.
- The number of non-missing values in the input time series must be at least 10.
- 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), allowing a trend-stationary process to occur.
- The main difference between the ADF test and a normal Dickey-Fuller test is that ADF allows for higher-order autoregressive processes.
- For the test-down approach, we start with a given maximum lag length and test down by running several tests; in each, we examine the high-order coefficient's t-stat for significance.
- 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.
- 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.
- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g., #N/A) at either end.

## Files Examples

## Related Links

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

## References

- 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.

## Comments

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