When do you use the Dickey Fuller Test and the Ljung-Box Test? 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.
 A: *

*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.
A: In a simpler terms, we can say the Unit Root tests (including KPSS test or Augmented Dickey Fuller (ADF) test) are used to test if the the time series of interest is non-stationary or not. If it is, we keep differencing the time series until we have a stationary time-series (usually maximum number of differencing is 2).
The main idea of differencing is to help stabilize the mean of a time series by removing changes in the level of a time series, and therefore eliminating (or reducing) trend and seasonality. In that case, the transformed time series is ready for the next step which is regression.
On the other hand, Portmanteau tests (including Box-Pierce test or Ljung-Box test (more accurate)) are used to see if the model trained is ready for inferencing and forecasting meaning that the model has been able to capture all of the trend and seasonality within a reasonable threshold or not. In other words, they test if the residuals of the trained model is a white noise (i.e. normal distribution (0, 1)).
It is interesting to note that if there are a few significant spikes in the ACF, and the model fails Portmanteau tests but it can still be used for forecasting, but the prediction intervals may not be accurate due to the correlated residuals.
