Does Stationarity for Time Series extend to Independent Variables? There have been many questions about the importance of stationarity and also its means of calculation here on CV, but one question that I have not seen an answer to is whether or not stationarity (in the context of a time series regression) is required for both the independent (X) variables, or is just required for the dependent (Y) variables. 
Many thanks for someone who can elucidate this question! 
 A: Stationarity should be sought for both to avoid any spurious correlations (that some or all variables actually just increase/decrease over time, independent of other factors), and there are methods of correction that can be applied to the entire model (i.e., including a trend as a DV) or just to specific variables (i.e., taking first differences of non-stationary variables) to address this if it is a problem.
Level variables are frequently violated by non-stationarity; for example, the number of Internet users in the world or the amount of pollution generally continually increase. So, if you have one of those as either independent or dependent variable, it will cause an issue. Non-stationarity both inflates t-statistics and biases betas. Including a trend often helps to control for that (so that you can say "the effect of the number of Internet users on Y, controlling for time, is predicted to be..."). And again, first differences of the specific variable at fault is also an option.
A: Assume you have time series on a variable $Y$ and on a variable $X$, say both non-negative, and based on some idea of yours you believe that there is a reasonable argument that they can be linked by a linear relationship
$$y_t =  \beta x_t + u_t \tag{1}$$
Now say that your theoretical argument is basically sound, but it captures how $Y$ relates to $X$ around a positive deterministic time trend, denote it $d_t =1,2,...$. In reality therefore, the true relationship is
$$y_t =  \beta x_t + \gamma d_t + v_t \tag{2}$$
where $v_t$ has all the nice properties.
But you are impatient, you don't even graph the series to see how they evolve over time, and you go on and estimate model $(1)$ by OLS. You will obtain
$$\hat \beta = \frac {\sum_{t=1}^Tx_ty_t}{\sum_{t=1}^Tx_t^2}$$
and inserting the true equation $(2)$ for $y_t$ into this, you will get
$$\hat \beta  = \beta + \frac {\gamma\sum_{t=1}^Tx_id_t +\sum_{t=1}^Tx_tu_t}{\sum_{t=1}^Tx_i^2}$$
Consider the expected value of the estimator conditional on the regressor series
$$E(\hat \beta \mid X) = \beta + \frac {\gamma\sum_{t=1}^Tx_id_t }{\sum_{t=1}^Tx_i^2}$$
which explodes as $t$ increases since $d_t$ increases, going away from the true value of the parameter. This happens because the estimator is forced to incorporate the existence of the time trend into the estimate of $\beta$. In practice if you start with a sample of length $T$ and then gradually increase the sample length, you will see the OLS estimate of $\beta$ to increase also, making the whole estimation useless.  
This is the most basic example to show that stationary and non-stationary data "don't mix".  

It would be instructive to work the reverse case. Assume that $y_t$ is
  stationary, and you try to regress it on a time trend. Say, the true
  model is
$$y_t = \alpha + v_t$$ but you specify and estimate
$$y_t = \alpha + \gamma d_t + u_t$$
What will happen here?

ADDENDUM
Responding to conversation in the comments, the essence of the answer above is that if some of our data series are stationary, and some non-stationary, putting them together through an estimation algorithm will not provide meaningful results.
But, there exist techniques to transform the data series so that they become stationary, and then we can execute estimation (since now all data series involved in the estimation have become stationary), and obtain meaningful results. Detrending or Differencing a non-stationary series is the most usual such technique.  
Moreover, there exists the phenomenon of co-integration, where, say, two stochastic processes are both non-stationary, but there exists a vector of constants that makes their linear combination weighted by this vector a stationary process. In such cases, regressing the one non-stationary series on the other non-stationary series provides meaningful results, and in some cases it is even to be preferred from the alternative approach, i.e. to, say, differencing both series to make them both stationary.  
A: Stationarity means, among other things, that a


*

*All the random variables $X_t$ have the same distribution

*All pairs of random variables $(X_t, X_s)$ have a joint distribution that depends only on the difference $t-s$ and not on the
individual values of $t$ and $s$: the joint distribution of $(X_1,X_3)$ is the
same as the joint distribution of $(X_2,X_4)$.  

*More generally, the $n$-th order
distributions of $(X_t, X_s, X_u, \ldots, )$ are the same as the $n$-th order
distributions of the random variables 
$(X_{t+\tau}, X_{s+\tau}, X_{u+\tau}, \ldots, )$
If the $X_t$ are all independent random variables, then the first of
the above conditions is not satisfied unless we also say, claim, assert,
or have reasons for believing, that all the $X_t$ have the same distribution.
This is often referred to in abbreviated fashion as iid or i.i.d. which
stands for independent and identically distributed random variables.
In this case, all the other conditions are automatically satisfied.
In other words,

If the random variables constituting the process or time series are
  iid random variables, then the process is stationary. If the random
  variables are independent but not necessarily identically distributed,
  then the process is non-stationary.

