Will the coefficients of an error correction or lasso-penalized model usually reveal spurious correlation? As time goes by I have learned of more and more ways that correlations can be spurious and more and more tests and correction procedures intended to avoid taking such correlations as meaningful. My question concerns whether either of two common correction procedures are sufficient as applied to economic time series with the usual characteristics of such series.
Suppose I have two highly correlated economic time series each of which is approximately stationary (after differencing if need be) but with some internal time structure such as auto correlation. Suppose, moreover, that there is a plausible story suggesting a casual relationship between the two, but,  unknown to me, there is no real causal relationship between these series, direct or indirect. Will either of the following procedures, without more, generally reveal the spurious nature of the relationship?


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*If I estimate a simple linear model of one on the other, but subject it to a lasso penalty with a cross-validated shrinkage coefficient, will the penalty usually  shrink the coefficient to roughly zero?

*If I run a standard error-correction model of one variable 
on the other, can I assume that the coefficients on the level and change of the dependent variable will show up as insignificant?
I am not asking about pathological cases. Obviously any test can be defeated by a sufficient coincidence in the random components of the variables. My question is, can I trust such results to the (admittedly limited) extent that I should generally accept significance levels as evidence of a true relationship? Or are there additional tests beyond these that are required before I should take an apparent relationship between two time series seriously?
 A: The qualifier spurious in spurious correlation comes from the subject-matter interpretation of the observed relationship, not the probabilistic one. Probabilistically, spurious correlation is as good as nonspurious correlation. As Ben writes in this thread,

it is not the correlation that is spurious, but the inference of an underlying (false) causal relationship. So-called "spurious correlation" arises when there is evidence of correlation between variables, but the correlation does not reflect a causal effect from one variable to the other. If it were up to me, this would be called "spurious inference of cause", which is how I think of it. 

Within a given non-causal model, you will not be able to distinguish one type of correlation from another. Thus attempting to get rid of it via penalized estimation or a non-causal modification of the model does not make sense. What could make sense is building a causal model and making inference on causal relationships rather than probabilistic ones.
See also "Spurious relationships: flavours, terminology" for a brief overview of types of spurious regressions.
