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

Question

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)?

Answer 1

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)

#Add cyclic noise
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.