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I understand, in a correlation, r just signifies the relationship between two variables, and we CANNOT deduce that there is a causal relationship in the case of a correlation.
By contrast, we use a linear Regression analysis to predict y from x, using the equation y= mx+c. Can we discuss that there is a causal effect in this case?
You are correct by saying you cannot deduce causal relationships when there is a statistically significant test of r=0, r being the Pearson correlation coefficient.
Statistical testing of the least squares regression slope (what you call m) is equivalent to tests of Pearson correlation coefficient: if one is non-zero, the other is non-zero. Tests of these hypotheses are asymptotically equivalent.
A necessary but insufficient condition to infer causality: you must either 1) randomize a cohort of participants to receive an experimental treatment or 2) control for confounding in a pseudoexperimental design using a multivariate model (estimating partial correlation or adjusted regression coefficients).
Suggested reading: Causality by Pearl 2nd ed, Causal Inference by Hernan, Robbins, Causal Diagrams for Epidemiologic Research by Greenland, Robins, Pearl.
By contrast, we use a linear Regression analysis to predict y from x, using the equation y= mx+c. Can we discuss that there is a causal effect as well?
No. Consider the univariate regression where your coefficient m is intimately related to the correlation coefficient r.
You can study causality using linear regression, but in and of itself it doesn't say anything about causality. You can run and use regression when there's no causal relationship.
First of all, as Aksakal pointed out as well, regression coefficients and (partial) correlation coefficients are linked, with the latter basically being a normalized version of the former.
Having noted that, you reason (I think) that because we can use a regression to "predict" y from x, this implies a kind of causality flowing from x to y. The problem is that prediction says nothing about causality. For example, the height of the sun in the sky is causally related to how light it is outside. Clearly, the sun being higher in the sky causes more illumination, and not the other way around. Yet, we can use our observation of how light it is outside to predict the height of the sun in the sky. I guess this can seem a bit misleading since the word "prediction" has a connotation of one event (causally) preceding another, but in this context the term is used more broadly to mean something similar to "inference".
Say your x variable is the dollar value of goods imported from year 2007 - 2017, and your y variable is private health spending for a person in year 2007 - 2017 as well. You plot x, y, and see a clear increasing and linear trend between x and y, and you can definitely fit a regression model to get the m. But there is no causal relationship in this case, rather a lurking variable (the common year). The association between x and y are just due to the fact that both imports and health spending grew rapidly in those years.
Reference: Introduction to the practice of statistics, chapter 2.4.
