Intercept is the expected mean value of Y when all X=0. If we start with a regression equation with one predictor, X. If X sometimes is equal zero, the intercept is simply the expected mean value of Y at that value. If X never equals zero, then the intercept has no intrinsic meaning. If so, and if X never = 0, there is no interest in the intercept. It doesn’t tell you anything about the relationship between X and Y. You do need it to calculate predicted values, though. When X never =0 is one reason for centering X. If you rescale X so that the mean or some other meaningful value are 0 (just subtract a constant from X), now the intercept has a meaning. It’s the mean value of Y at the chosen value of X.
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