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I'm trying to understand the issues with using OLS regression when our data exhibits autocorrelation. Let's say you simulate a process where:

t = [1, 2, 3, ..., 100]

e = 100 random normal variables from N(0, 5)

y = 2 * t + e

Because of this set up, there is obvious autocorrelation of y. You nevertheless build a basic OLS model where y is the response and t is the predictor.

What are the issues autocorrelation like this causes? In many of the regression books, we talk about issues if variance is non-constant or distribution of Y | t is non-normal. If I recall correctly, such model violations affect the standard errors of our estimates and thus make it tougher to estimate true parameters.

But I don't recall any discussion on what happens when autocorrelation is present. Yet anywhere I look, it just says autocorrelatoin or non independence is bad. What exactly is the issue that prevents us from using OLS to model the process described above - and thus necessitates a time series technique.

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    $\begingroup$ This is an interesting question, but I don’t think you’re simulating any autocorrelation. $\endgroup$ – Dave Feb 16 '20 at 18:12
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    $\begingroup$ What we care about in regression is autocorrelation of the errors, not an "autocorrelation" such as you demonstrate above. $\endgroup$ – jbowman Feb 16 '20 at 22:29
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Assuming that you mean independent sampling when saying e = 100 random normal variables from N(0, 5), then there is no autocorrelation in your example. Assumptions such as independence in regression models apply to the error term, in your case e.

What you do have in the example is a (strong) trend, so what you should be preoccupied with is if this nonstationarity have consequences for the ols estimation. Here are some related questions with answers.

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Assuming there was autocorrelation you are dealing with the assumption that the error terms are not correlated with each other [they are independent of each other]. The impact of autocorrelation is to distort the statistical tests (by distorting the standard errors).

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