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 (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 = contant-only, 3 = trend only, 4 = constant and trend, 5= const, trend and tredd 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), so it allows 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 exaimine 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.

## Examples

**Example 1: **

A | B | |
---|---|---|

1 | Date | Data |

2 | January 10, 2008 | -2.827 |

3 | January 11, 2008 | -0.947 |

4 | January 12, 2008 | -0.877 |

5 | January 14, 2008 | 1.209 |

6 | January 13, 2008 | -1.669 |

7 | January 15, 2008 | 0.835 |

8 | January 16, 2008 | -0.266 |

9 | January 17, 2008 | 1.361 |

10 | January 18, 2008 | -0.343 |

11 | January 19, 2008 | 0.475 |

12 | January 20, 2008 | -1.153 |

13 | January 21, 2008 | 1.144 |

14 | January 22, 2008 | -1.070 |

15 | January 23, 2008 | -1.491 |

16 | January 24, 2008 | 0.686 |

17 | January 25, 2008 | 0.975 |

18 | January 26, 2008 | -1.316 |

19 | January 27, 2008 | 0.125 |

20 | January 28, 2008 | 0.712 |

21 | January 29, 2008 | -1.530 |

22 | January 30, 2008 | 0.918 |

23 | January 31, 2008 | 0.365 |

24 | February 1, 2008 | -0.997 |

25 | February 2, 2008 | -0.360 |

26 | February 3, 2008 | 1.347 |

27 | February 4, 2008 | -1.339 |

28 | February 5, 2008 | 0.481 |

29 | February 6, 2008 | -1.270 |

30 | February 7, 2008 | 1.710 |

31 | February 8, 2008 | -0.125 |

32 | February 9, 2008 | -0.940 |

Formula | Description (Result) | |
---|---|---|

=ADFTest(\$B\$2:\$B\$32,1,,1,TRUE,1) | P-Value with no Constant (0.01) | |

=ADFTest (\$B\$2:\$B\$32,1,,2,TRUE,2) | Test Stats with Constant Only (-8.784) | |

ADFTest (\$B\$2:\$B\$32,1,,3,TRUE,3) | Critical Value with Constant and Trend (-3.966) | |

ADFTest (\$B\$2:\$B\$32,1,,4,TRUE,1) | P-Value with Constant, Trend, and Trend Squared (0.01) |

## 0 Comments