In regular regression, the expected value of Y | X is allowed to change. In fact we generally use regression when we want to model this change in conditional mean.
I am not understanding why in time series, we want our series to be mean stationary. I get the stationary variance assumption as that's similar to the identically distributed assumption in regular regression. But why is mean stationarity so important?
In the case of time series forecasting, first of all, you need to understand that stationarity is important mostly in the context of ARMA and related models (AR: Auto-Regressive, MA: Moving Average). There are other types of time series forecasting models where stationarity is not a requirement, such as Holt-Winters or Facebook Prophet.
Here are two intuitive, if not entirely mathematically rigorous, explanations of why mean stationarity is important in the ARMA case:
The AR component of ARMA models, treats time series modeling as a supervised learning problem, $Y_t = a_1Y_{t-1}+...a_nY_{t-n}+c+\sigma(t)$. A common rule of thumb in supervised learning is that the distribution of the training data and the distribution of the test data should be the same, otherwise your model will perform poorly on out-of-sample tests and on production data. Since for time series data, you train set is the past, and your test set is the future, the stationarity requirement is simply ensuring that the distribution stays the same over time. This way you avoid the problems that come with training your model on data that has a different distribution than the test/production distribution. And mean stationarity in particular is just saying that the mean of the train set and the mean of the test should stay the same.
An even simpler consideration: take the most basic ARMA model possible, an $AR(1)$ model: $$Y_t = aY_{t-1}+c+ \sigma$$ so the recursive relationship for estimating on step based on the previous one is: $$\hat{Y}_t = a\hat{Y}_{t-1}+c$$, $$\hat{Y}_t - c = a\hat{Y}_{t-1}$$ taking the expected value: $$E(\hat{Y}_t) - c = aE(\hat{Y}_{t-1})$$ meaning that: $$a = \frac{E(\hat{Y}_t) - c}{E(\hat{Y}_{t-1})}$$ so if we want $a$ to stay constant over time, which is the starting assumption of an $AR(1)$ model since we want it to be similar to a linear regression, then $E(\hat{Y}_t)$ has to stay the same for all $t$, i.e. you series has to be mean stationary.
The above considerations are applicable as well to the general ARMA case, with $AR(p)$ and $MA(q)$ terms, although the math is somewhat more complicated than what I describe, but intuitively, the idea is still the same. The 'I' in ARIMA stands for "Integrated" which refers to the differencing process that allows one to transform a more general time series into one that is stationary and can be modeled using ARMA processes.
I disagree with @Alexis characterization that "that time series are stationary is more or less embodying the worldview that the past does not matter" - if anything it is the other way around: Transforming a time series into a stationary one for modeling purposes is exactly about seeing whether there are any causal/deterministic structures in the time series beyond just trend and seasonality. I.e. does the past impact the present or the future in ways more subtle ways than just the large scale variations? (But I might simply misinterpreting what she is trying to say).
Stationarity is important because it is a mathematically strong assumption that's still much weaker than independence or finite-range dependence.
In some settings, it's important primarily for the mathematical tractability: it's easier to first find out what is true for stationary time series, then you can work on how to relax the assumptions. Perhaps you only need weak-sense stationarity, or mean stationarity plus some tail condition, or whatever. Or perhaps you need stationarity for a result to hold exactly, but it holds approximately under weaker assumptions.
In other settings stationarity is important because there are so many ways to be non-stationary that it would be hard to handle every one of them. If a problem can be approximated by a stationary series that's a big practical advantage. Here it's important to remember that the stationary series $X(t)$ that appears in the maths may not be your raw data. For example, traditional ARMA models are stationary, but you would typically want to remove season and trend relationships before fitting one. You might want to log-transform a series that has increasing mean and variance. And so on.
First, your mean estimates and your standard errors will be badly biased if you are using any of the inferential tools which assume i.i.d, meaning your results risk being spurious. This can even be true if your data are weakly stationary, but your study period is shorter than the time it takes your series to reach equilibrium after a disturbance.
Second, assuming that time series are stationary is more or less embodying the worldview that the past does not matter (e.g., the prevalence of COVID-19 today is completely independent of COVID-19 prevalence yesterday; the \$ per capita spent on addictive goods such as cigarettes this year is completely independent of the \$ per capita spent on them last year)… kinda unrealistic.
Stationary means that the statistics that describe the random process are constant. ‘A memoryless Markov process’ is another way to say stationary as is saying that the probability generating function has no “feedback” terms, but if you recognized those words you might not be asking this question. FWIW “weakly stationary” isn't quite the same, a constant or knowable rate of change of the stats would be weakly stationary, as would something that averages out, but it’s a little more involved so consider this fair warning that there’s more to know in case that’s part of the puzzle, but describing everything that isn’t stationary in detail would turn a simple answer a complex answer.
Why is stationary important? The commonly used statistical formulae are crafted to use a data set to extract an imprecise description with an estimable accuracy of an otherwise unknown random process. The formulae assume that adding more samples increases the accuracy of the description by reducing the uncertainty. For that the Mean Central tendency, i.e. ergodic in the mean, has to be true. If the random process itself is changing, e.g. the average value or the variance is changing, then an essential underlying assumption is invalid, you can’t make a better estimate.
As a general “what happens” if the mean is moving as a linear function of time, the computed mean will represent the mean at a weighted mean time, and the computed variance will be inflated. Is possible to compute an ‘optimal a posteriori” (after the fact) estimate of a non stationary process and then use that to extract meaningful stats because the best estimate of the time function minimizes the variance. It’s also easy to hypothesize some high order time function and create a complex model that appears to be valid and predictive that in fact has no predictive power because it modeled a snapshot of randomness, not an underlying time trend.
Short and sweet:
The parameters need to be constant. If the series is not stationary, then the parameters that you estimate are going to be functions of time themselves. But the model assumes that they are constants, as such, you will estimate the average parameter value over the time-period. See Skander's answer for why, I won't dive into the math since he already did.
This presents at least 2 problems:
Getting to stationarity is actually pretty easy. We just need to difference until we have a stationary series. So just do that.
