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14

Let's start from the basics. Variance tells us about the variability around the mean $$\operatorname{Var}(X) = E[(X - E[X])^2]$$ You can generalize this concept to two variables, the covariance $$\operatorname{Cov}(X, Y) = E[(X - E[X]) (Y - E[Y])]$$ where variance is a special case of it $$\operatorname{Cov}(X, X) = E[(X - E[X])^2]$$ Correlation is ...

11

Similar to simple covariance and correlation, the mean is subtracted while estimating the autocorrelation (autocovariance or cross-correlation and cross-covariance). The following is the covariance of $X$ and $Y$ for example, where the means of the random variables are subtracted from the random variable itself: \operatorname{cov}(X,Y)=\mathbb E[(X-\mathbb ...

9

One key downside is that ARIMA models tend not to forecast very well. (I'm sure I will get my share of pushback for that statement. And yes, it is too broad in a sense, but it serves as - I believe - a useful first-order approximation.) This came as something of a surprise at the earlier forecasting competition, at least to the statisticians who had gone ...

8

The whole notion of "spurious" correlation is easy to misinterpret. Correlation is correlation --- if it is estimated well (i.e., via a good estimator and with a reasonable amount of data) then we can confidently say that the correlation is such-and-such. Correlation is a statistical measure with an extremely weak interpretation --- it just ...

4

In my answer, I respectfully disagree with the accepted answer. First of all, the fact that ARIMA models do not forecast well in forecasting competitions is not a weakness of ARIMA but is evidence that the stochastic process that produced the time series in question was one other than ARIMA and ARIMA should not have been used in the first place. A time ...

3

Your reasoning makes sense. For a wide-sense stationary process, a correlogram can be interpreted in a straightforward way, since the population counterparts of the sample autocorrelations you see in it do actually exist. For a process that is not wide-sense stationary, it is still technically possible to plot a correlogram. However, it loses its ...

2

If you make your harmonic signal to have a zero mean you'd see the auto correlation have more negative values. Since your signal is almost always positive and the correlation is a sum of the values it makes sense ti will be almost always positive.

1

The problem with spurious relationships - in the narrow context of pair trading - is not even with causality. The problem is that the relationship doesn't hold out of sample. This means that when you actually start trading on the developed algorithm, you won't make any money. And that can be a little bit of an issue, right?

1

There are two concerns with correlations of time series Correlation when causal relationship is absent. Correlation does not imply causation. An example is the correlation between ice cream sales and the death rate due to drawing. These two are both high in summer and low in winter and they correlate in time, but this is not due to a direct causal ...

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