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I am trying to understand the correlations among three explanatory variables of commercials and response variable sales through the scatter plot matrix. It seems like there are non linear relationships.
Can anyone please show me some insights more specifically which possible regression models should be applied?
Are your variables 'stationary'? Otherwise, what you see may be just spurious correlation. For example, if you simulate two totally independent random walks, but which happen to be both increasing on the sample horizon, then you will measure a spurious correlation of something like 0.5 instead of 0 correlation between the underlying 'stationary' (in fact, i.i.d.) independent variables which are the increments. Cf. this recent post Why are random walks intercorrelated?
If you want to look at non-linear correlation using scatterplot, I would suggest you to use the scatterplot of the ranked variables (empirical copulas). That is to say, for each variable, you sort its values from the smallest (rank 1) to the biggest (which has rank nb_obversations), and you plot the scatterplot of these.
If it is diagonal, then there is a perfect comonotonic relation between the variables, cf. for example this paper (no need for all the geometrical sophistication, just plot the scatterplots of the ranks :)
x is ecdf(x)(x). Plot that against ecdf(y)(y). This has the nice property of removing the effects of the marginal variables, which can appear to be part of the dependence structure.
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