# adf.test: Augmented Dickey-Fuller Test In aTSA: Alternative Time Series Analysis

## Description

Performs the Augmented Dickey-Fuller test for the null hypothesis of a unit root of a univarate time series `x` (equivalently, `x` is a non-stationary time series).

## Usage

 `1` ```adf.test(x, nlag = NULL, output = TRUE) ```

## Arguments

 `x` a numeric vector or univariate time series. `nlag` the lag order with default to calculate the test statistic. See details for the default. `output` a logical value indicating to print the test results in R console. The default is `TRUE`.

## Details

The Augmented Dickey-Fuller test incorporates three types of linear regression models. The first type (`type1`) is a linear model with no drift and linear trend with respect to time:

dx[t] = ρ*x[t-1] + β*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t],

where d is an operator of first order difference, i.e., dx[t] = x[t] - x[t-1], and e[t] is an error term.

The second type (`type2`) is a linear model with drift but no linear trend:

dx[t] = μ + ρ*x[t-1] + β*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t].

The third type (`type3`) is a linear model with both drift and linear trend:

dx[t] = μ + β*t + ρ*x[t-1] + β*dx[t-1] + ... + β[nlag - 1]*dx[t - nlag + 1] +e[t].

We use the default `nlag = floor(4*(length(x)/100)^(2/9))` to calcuate the test statistic. The Augmented Dickey-Fuller test statistic is defined as

ADF = ρ.hat/S.E(ρ.hat),

where ρ.hat is the coefficient estimation and S.E(ρ.hat) is its corresponding estimation of standard error for each type of linear model. The p.value is calculated by interpolating the test statistics from the corresponding critical values tables (see Table 10.A.2 in Fuller (1996)) for each type of linear models with given sample size n = length(`x`). The Dickey-Fuller test is a special case of Augmented Dickey-Fuller test when `nlag` = 2.

## Value

A list containing the following components:

 `type1` a matrix with three columns: `lag`, `ADF`, `p.value`, where `ADF` is the Augmented Dickey-Fuller test statistic. `type2` same as above for the second type of linear model. `type3` same as above for the third type of linear model.

## Note

Missing values are removed.

Debin Qiu

## References

Fuller, W. A. (1996). Introduction to Statistical Time Series, second ed., New York: John Wiley and Sons.

## See Also

`pp.test`, `kpss.test`, `stationary.test`

## Examples

 ```1 2 3 4 5``` ```# ADF test for AR(1) process x <- arima.sim(list(order = c(1,0,0),ar = 0.2),n = 100) adf.test(x) # ADF test for co2 data adf.test(co2) ```

### Example output

```Attaching package: 'aTSA'

The following object is masked from 'package:graphics':

identify

Augmented Dickey-Fuller Test
alternative: stationary

Type 1: no drift no trend
lag   ADF p.value
[1,]   0 -7.00    0.01
[2,]   1 -6.28    0.01
[3,]   2 -5.45    0.01
[4,]   3 -5.44    0.01
[5,]   4 -5.27    0.01
Type 2: with drift no trend
lag   ADF p.value
[1,]   0 -6.97    0.01
[2,]   1 -6.25    0.01
[3,]   2 -5.43    0.01
[4,]   3 -5.43    0.01
[5,]   4 -5.24    0.01
Type 3: with drift and trend
lag   ADF p.value
[1,]   0 -6.95    0.01
[2,]   1 -6.24    0.01
[3,]   2 -5.47    0.01
[4,]   3 -5.50    0.01
[5,]   4 -5.27    0.01
----
Note: in fact, p.value = 0.01 means p.value <= 0.01
Augmented Dickey-Fuller Test
alternative: stationary

Type 1: no drift no trend
lag   ADF p.value
[1,]   0 1.852   0.984
[2,]   1 0.715   0.850
[3,]   2 1.310   0.952
[4,]   3 1.706   0.978
[5,]   4 1.846   0.984
[6,]   5 2.176   0.990
Type 2: with drift no trend
lag    ADF p.value
[1,]   0 -0.515   0.872
[2,]   1 -1.780   0.416
[3,]   2 -0.929   0.726
[4,]   3 -0.453   0.894
[5,]   4 -0.372   0.909
[6,]   5 -0.222   0.929
Type 3: with drift and trend
lag    ADF p.value
[1,]   0  -5.19    0.01
[2,]   1 -15.09    0.01
[3,]   2  -9.47    0.01
[4,]   3  -7.68    0.01
[5,]   4  -7.58    0.01
[6,]   5  -6.84    0.01
----
Note: in fact, p.value = 0.01 means p.value <= 0.01
```

aTSA documentation built on May 1, 2019, 8:47 p.m.