Unsure about assumptions of linear model with time series variables, spurious regression and periodic patterns

Background

I'm learning about time series in context of linear regression. The goal of this question is to understand how seasonality of either X or Y can affect the model.

Linear model assumptions

1. $$Y \in R^t$$ is a linear combination of $$X_1, ... X_n \in R^t$$ with added noise term, i.e. $$Y = \beta_0 + \beta_1 X_1 + ... + \beta_n X_n + \epsilon.$$

2. $$\epsilon \sim N_t(0, I\sigma^2)$$ (homoscedasticity, multivariate normality and lack of autocorrelation of residuals)

3. lack of multicollinearity of observations

What I don't understand

1. What happens if X/Y/both are non-stationary? Are there any risks related to that other than "spurious regression"?
2. As far as I understand, spurious regression can happen when dependent variable and one of independent variables have the same non-stationary pattern (cointegration), for instance they both have positive trend or the same seasonal fluctiations pattern. If so, do we need to care about non-stationarity of $$X$$, in case we know that $$Y$$ is stationary?
3. What about periodic patterns which are technically stationary (cyclic patterns)?
• Where did you find this list of assumptions? Commented Apr 15 at 13:04
• @RichardHardy I've created this list based on information from several webpages combined with my own knowledge. Is anything wrong with this list? Commented Apr 16 at 13:18
• E.g. point 3: I do not think zero correlation and lack of multicollinearity are the same thing. Commented Apr 16 at 13:29
• @RichardHardy thank you, you're right, I removed zero correlation part Commented Apr 16 at 17:54
• What about the zero autocorrelation of Xs? Is that a must? Commented Apr 16 at 19:32

For your first and third question, why not try it and see? Here is a very crude shot at it with R; it would be easy to play with this:

set.seed(1234)

#Generate random varaibles
x <- rnorm(1000)
y <- rnorm(1000)

xcyl <- x + rep(c(1, 2, 3, 4), 250)
ycyl <- y + rep(c(1,-1,1, -1), 250) #different cycle

m1 <- lm(ycyl~xcyl)
summary(m1)


Which yields:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.47306    0.08507   5.561 3.45e-08 ***
xcyl        -0.18539    0.02944  -6.297 4.55e-10 ***


For your second question: No, they can have different patterns and still be spurious. For one thing, they can be opposite. You could, e.g., regress the Dow Jones Average on deaths from polio and find a strong negative relationship.