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)?
  • $\begingroup$ Where did you find this list of assumptions? $\endgroup$ Commented Apr 15 at 13:04
  • $\begingroup$ @RichardHardy I've created this list based on information from several webpages combined with my own knowledge. Is anything wrong with this list? $\endgroup$ Commented Apr 16 at 13:18
  • 1
    $\begingroup$ E.g. point 3: I do not think zero correlation and lack of multicollinearity are the same thing. $\endgroup$ Commented Apr 16 at 13:29
  • $\begingroup$ @RichardHardy thank you, you're right, I removed zero correlation part $\endgroup$ Commented Apr 16 at 17:54
  • $\begingroup$ What about the zero autocorrelation of Xs? Is that a must? $\endgroup$ Commented Apr 16 at 19:32

1 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:


#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)

Which yields:

            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.


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