When do you use the Dickey Fuller Test and the Ljung-Box Test?

They seem to serve the same purpose?

I think that the Durbin-Watson statistics just checks for autocorrelation at lag 1.

  • The Dickey-Fuller test addresses the question of whether the time series of interest has a unit root.
  • The Ljung-Box and the Durbin-Watson tests help assess whether the time series of interest is autocorrelated.

These are two different questions. According to the tag description of a unit root,

A unit root is a property of a non-stationary time series which can lead to spurious regressions and wrong inference. A series $$ y_t = y_{t-1} + e_t $$ with $e_t \sim i.i.d.(0,\sigma^2)$* has a unit root if it can be expressed as $(1-L)y_t = e_t$ where $L$ is the lag operator. Then the characteristic equation of the above process has one unit root. A property of the unit root is that when an $I(1)$ autoregressive process is first differenced, it becomes an $I(0)$ process, i.e. it will be stationary.

According to the tag description of autocorrelation,

Autocorrelation (serial correlation) is the correlation of a series of data with itself at some lag. For example, if a given value were higher than average, it might be more likely than not that the next value would also be higher than average; if so, there is a positive autocorrelation at lag 1.

The autocorrelation for a time series, between times $s$ and $t$ is defined as:

$$R(s,t) = \frac{\mathbb{E}[(X_t - \mu)(X_s - \mu)]}{\sigma_t \sigma_s}$$

where $\mathbb{E}$ is the expected value operation.

Some differences between the Ljung-Box (LB) and the Durbin-Watson (DW) tests

  • LB tests a joint null hypothesis of autocorrelation at a set of lags being equal to zero, while DW tests a null hypothesis of autocorrelation at a single lag being equal to zero.
  • LB test can be applied to any selected group of lags (though usually the first $k$ for some natural number $k$), while DW tests autocorrelation at lag 1 specifically.

*It is enough that $e_t$ is a stationary process or an I(0) process which are almost the same.


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