Does it really matter if a correlation is spurious?

Question

Let’s say you are trying to find if there is a correlation between two stock prices, where both are likely non stationary series. You have no concern as it relates to a potentially causal relationship...

You run a simple correlation analysis against all the rules. Both our series are autocorrelated and non stationary. You find there is a 98% correlation so you conclude they depend on each other.

This is the conversation I just had with a colleague... but I think they are 100% wrong and I’d like some confirmation.

If you find two autocorrelated and non stationary series to be 98% correlated, then the correlation is likely spurious. What this means to me is that the correlation we observe is likely due to complete chance (and their correlation is likely a result of their mutual dependence on something else outside of the two series themselves). So if our goal is to identify the extent to which these two series “depend” on each other, finding a valid correlation coefficient is necessary. Is this correct?

Answer 1

Here's a simulated example of two prices that are very highly correlated ($\rho = 0.9875$). When you attempt to predict the price change in one using the lagged value of the other, very little of the variation in the price change is explainable:

. clear

. set seed 12092021

. set obs 102
Number of observations (_N) was 0, now 102.

. gen t = _n

. tsset t

Time variable: t, 1 to 102
        Delta: 1 unit

. gen p1 = 1 + 3*t + rnormal(0,5) 

. gen p2 = 3 + 2*t + rnormal(0,10)

. corr p1 p2
(obs=102)

             |       p1       p2
-------------+------------------
          p1 |   1.0000
          p2 |   0.9875   1.0000


. reg FD.p2 p1

      Source |       SS           df       MS      Number of obs   =       101
-------------+----------------------------------   F(1, 99)        =      0.01
       Model |  .727541841         1  .727541841   Prob > F        =    0.9436
    Residual |  14322.4337        99  144.671048   R-squared       =    0.0001
-------------+----------------------------------   Adj R-squared   =   -0.0100
       Total |  14323.1613       100  143.231613   Root MSE        =    12.028

------------------------------------------------------------------------------
       FD.p2 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
          p1 |   .0009672   .0136392     0.07   0.944    -.0260959    .0280303
       _cons |   1.665843   2.420693     0.69   0.493    -3.137338    6.469024
------------------------------------------------------------------------------

. reg FD.p1 p2

      Source |       SS           df       MS      Number of obs   =       101
-------------+----------------------------------   F(1, 99)        =      0.01
       Model |  .683934381         1  .683934381   Prob > F        =    0.9171
    Residual |  6210.52068        99  62.7325321   R-squared       =    0.0001
-------------+----------------------------------   Adj R-squared   =   -0.0100
       Total |  6211.20461       100  62.1120461   Root MSE        =    7.9204

------------------------------------------------------------------------------
       FD.p1 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
          p2 |  -.0013704   .0131245    -0.10   0.917    -.0274123    .0246715
       _cons |   3.260085   1.574913     2.07   0.041     .1351165    6.385054
------------------------------------------------------------------------------

Here FD is the first difference of subsequent value (so $FD.p_t = (p_{t+1}-p_t)$).

The $R^2$ (aka R-squared) of both models is around zero, so very little of the variation in price changes tomorrow can be explained by the price today. This illustrates the intuition that knowing what you know today, you cannot act on this correlation to make money tomorrow.

You can play around with variations on this approach (using the lagged price change as a predictor, non-linear models, adding more data, more noise, or adding trends), with identical results.

You might object that my toy example is flawed because the high correlation is contemporaneous, so if you knew p1 today, you could predict p2 today. I think that is wrong for the following reason. Suppose the DGP is as above, but unknown to you. You are an executive at company 1, and you learn that your CEO had been falsifying earnings and pinching bottoms. The news will become public shortly and lower p1. You can’t short your own stock without a vacation at Club Fed. Should you short the stock of company 2 if you know the correlation between p1 and p2 is ~1? I think that would be a terrible idea. This is what makes the correlation spurious and why that matters.

You could also have a causal relationship, but no correlation. When a house has air-conditioning with a preset desired temperature, there will be a strong positive non-spurious correlation between the amount of electricity used by the AC and the temperature outside. But there will be no correlation between the amount of electricity consumed and the inside temperature. The outside temperature and the inside temperature will also be uncorrelated. The last two are spurious non-correlations in my mind. But all three correlation are valid (though that has no formal definition in statistics) since a correlation is just a transformation of the data.

This is all to say that a strong correlation is not necessary for a causal dependence to exist. And it is certainly not sufficient. Even the sign on the causal relationship could be different from the sign of the correlation. This matters for using correlations to do things out in the real world (i.e., interventions). This is not just an issue with time series data, but can happen with observational data.

Answer 2

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 measures the tendency for things to vary together (usually measured linearly), irrespective of the cause of this tendency. The only thing that can be spurious is if we go further than this and interpret the correlation in a way that is not justified. This can occur if a person uses correlation to infer a causal relationship between variables, or it can occur if a person uses marginal correlation to infer conditional correlation. In either case the larger inference can be "spurious" insofar as it does not follow from the correlation. As I've noted in another answer, I've always hated the term "spurious correlation" because it it is not the correlation that is spurious, but the inference to some stronger result. If it were up to me we would never use this term, and would instead just state what we actually mean --- e.g., "spurious inference of cause", "spurious inference to conditional correlation", etc.

Now, with that little rant out of the way, let me address your specific concern. Since you are only interested in describing the past statistical relationships between the stock prices (as you say in your comments), you can report the correlation, but it should come with a number of important caveats on interpretation. Firstly, you should note that strong correlation between time-series can occur even for purely deterministic series with no statistical variation, so it often does not reflect any stochastic dependency between the series. This is something that has been recognised in the statistical community for over a century (see e.g., Yule 1926 and see this related answer). Secondly, even if changes in the stock prices are correlated, the ability to predict one stock from the other will depend on the cross-correlation in changes in the stock prices at sufficient lag values to allow use of one series to predict changes in the other. In large part, analysis of stock prices is best done by looking at lagged cross-correlation of changes in price, rather than correlation of the price series themselves.

Answer 3

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?

Answer 4

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 relationship between the two. In such case, if a causal relationship between two variables is inferred based on a correlation between two variables then people use the term 'spurious relationship' (the inference is not correct).

• Correlation in a sample when statistical relationship for the population is absent. Another concern is that the correlation might be likely found in data, even in the absence of an underlying statistical relationship. Time series with autocorrelation have a tendency to go up/down for short periods of time and so they tend to correlate with each other within short windows of time. But, this correlation is not significant. Yes, if you would compute the significance assuming that the datapoints are independently distributed according to a bivariate normal distribution (for which you can compute the exact sample distribution for the correlation coefficient), then it will turn out to be significant, but that assumption of independence is not correct when the time series follow trends or are autocorrelated.