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Time series are generally autocorrelated. I've learned this means that the values of a time series are correlated with prior values of itself. I'm struggling to understand why this is. I would appreciate any insights from the folks here.

One explanation I've seen is that in a time series, the value of the variable at one point in time is related to the value at a previous point in time. I think this makes sense -- snowfall in December is related to snowfall in January because its wintertime. Clearly snowfall would depend on the time of year. Or if there is a trend in more people ordering pizza, then pizza ordered at $t$ will be greater than pizza-ordered at $t-1$ in some sort of relationship.

But why can't we just handle dependencies like this with a regular linear model? Why is a timeseries correlated with itself (necessitating using things like differencing and ARIMA), rather than just with time (in which case, we could run plain-ole linear models)?

Is the idea that there are just so many reasons that a value at time $t$ is related to values at $t-1$ that we can't possibly control for them all with explicit parameters in a linear regression? Or is there something else going on?

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  • $\begingroup$ At stats.stackexchange.com/a/35524/919 I answered a generalization of this question to spatial models. In summary, you can handle many time relationships with a "regular linear model," but often that is neither a parsimonious nor realistic way to do it. $\endgroup$
    – whuber ♦
    Commented Mar 2, 2021 at 20:47
  • $\begingroup$ You're absolutely right that regression is an extremely flexible way to estimate a probability model that would express several very complex types of autocorrelation. However, it's not a complete framework, and there are exploratory analyses specific to time series to try to suss out the proper way of estimating dependency structures, like variograms or spectral analyses. $\endgroup$
    – AdamO
    Commented Mar 2, 2021 at 22:27

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Is the idea that there are just so many reasons that a value at time t is related to values at t−1 that we can't possibly control for them all with explicit parameters in a linear regression? Or is there something else going on?

Yes. If you regress the response on time (and possibly other variables) in OLS, the assumption is that residuals are independent. This might be the case for some data, and/or if you have included all relevant predictors, and/or the effects are linear. If the residuals are correlated over time, you need an autocorrelation structure.

stats.stackexchange.com/a/35524/919 indeed provides a more general and precise explanation.

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You're confounding a few ideas here.

Basically, let's try to estimate the temperature today. It is likely near what it was yesterday. Indeed this is due to a third variable (the season), as the temperature is more likely to be different that it would be in another season.

If data values evolve day-to-day (like stock prices, or something similar), a great deal of what determines the price tomorrow is the price today.

If you do statistics on such variables you confuse yourself. The R**2 - the measure of how well your regression fits to true data - will be something like 95%. But you can't predict tomorrow's stock price with 95% certainty (though you can say it will be near today's value, most likely).

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