Is it possible for a statistical model to have explanatory power but no predictive power?
I'm going to offer the dissenting opinion on @Sympa's answer, and go with, No.
It's not really well defined what you mean by explanatory power, so I'll offer a definition. A model has explanatory power if it can be used to draw qualitative inferences about the underlying statistical process. A qualitative inference could be, for example, the sign or direction of an association between two variables of interest (a sign or direction of casuation is a separate issue which I am setting aside for the moment).
A statistical model is an approximation to an unseen process of interest to the researcher. If a model has no predictive power, i.e. no ability to correctly anticipate unseen data, then fundamentally the model is not representing the underlying statistical process being studied. If this is so, then any qualitative inferences drawn from said model can not be confidently generalized to inferences about the underlying process.
So, I would say, some level of predictive power should be a necessary condition for statistical explanation.
Yes, I think it happens quite often. Let's say you could have a model estimating GDP growth (as the dependent variable) using two independent variables: change in national home prices and change in the S&P 500. That model may have good explanatory power. It may have a high R Square. The two independent variables may have the correct positive sign and be very statistically significant. And, most important the model's explanatory power could be very well supported by economic theory (Hyman Minsky's credit cycles associated with home prices, etc.). But, when you test this model using Hold Out sample it may not perform that well. It may not be predictive. In other words, change in home prices and the stock market can explain a significant percentage of economic growth (after all that is one of the meanings of R Square). But, such a model may be mediocre at actually predicting the future path of the economy. That's almost a summary of the quasi-tragedy of a large body of econometrics models. Many are or appear to be very good at "explaining", so far I am not sure we have found many or any that are good at "predicting."
The correct answer is yes. A statistical model can have explanatory value without being a good predictor. But it depends upon what you are trying to predict.
The classic example? Sex differences in body weights.
Studies show that on average males are heavier than females. I can model the effect, estimate the difference and attach estimates of uncertainty to that estimate.
But can I use it for prediction?
Well it depends upon what I'm trying to predict.
If I'm trying to predict the sex of an individual based on their body weight then my statistical model is unlikely to perform much better than chance.
If I'm trying to separate two groups of same-sex individuals into the male and female groups then I'm in good shape. Providing we have an unbiased sample of a reasonable size then I should be able to do this with some confidence.