# In a linear regression, should I include independent variables that is already known to be predictive of the dependent variable?

I own an online shop and I'm trying to find the factors that would predict the profitability of a merchandise. To do this, I ran a linear regression with profit as the dependent variable. For the independent variables, I used all the data that I have recorded about each merchandise, which included things like the number of competitors, competitors' prices, year of production, etc. The idea is to create a list of criteria to determine if I should or should not sell a certain merchandise.

However, the result is always obvious: the selling price and buying cost are the only significant predictors (there are other statistically significant variables, but their beta is very low). Since they would obviously affect the profit, should I exclude them?

I did try to exclude them and the result is a lot more interesting, but I don't know if they are valid. I've also tried swapping the dependent variable from profit to the selling price and their results are identical.

• Trial and error is essential. Try splitting up your original data into 'training' and 'testing' sets, so you can use the $\beta$ from your model to predict future out-of-set outcomes. Then you can use both models to see which one is better at prediction. As an extension, look into Partial Least Squares regression (PLS), Elastic Net regression, and Principal Components regression (PCR) as ways of dealing with grouping, and LASSO regression (a corner-case of EN) for covariate selection.
– ERT
Jul 10, 2018 at 21:51
• How do you calculate profit? Is it simply profit = selling - buying? Jul 11, 2018 at 2:04
• Without reference to the specifics of your question, in general it's a bad idea to omit known predictors because of the effect of omitted variable bias also see Simpson's paradox (though properly a different name is used for the continuous y-continuous x case). However if you have an obvious and direct functional relationship with some predictors, it may be that you want to to consider adjusting your response (looking at what's left after you remove the known effect of those variables) Jul 11, 2018 at 2:55
• @MarkWhite Yes, but the buying cost has also included any other cost associated with each respective merchandise (shipping, processing, tax, etc). These costs have been optimized as much as I could. Jul 11, 2018 at 18:15

$$\text{profit} = (\text{price}-\text{cost})\times\text{sales}.$$

I don't think it makes sense to regress profit on price or cost. Or sales, for that matter. We know the relationship above.

Instead, work on understanding the three drivers above. For instance, as a first approximation, you can treat cost as fixed, since you may only get to negotiate it in the medium term. That leaves us with your selling price and sales.

And of course price will have an impact on sales.

So I would recommend forecasting future sales, using price as a driver, e.g., using regression. Play around with different possible price points and find the one that maximized profit per the formula above. Feel free to include other drivers like your competitors' prices in the sales forecast (but note that you will of course need to forecast future competitors' prices).

The problem: if you are running this analysis to determine if you should or should not sell a certain merchandise, then you will most likely not yet have any data on the particular product you are looking at. So a straightforward regression approach won't help. Instead, look for "similar" products with known historical sales to train your models.

• I already have historical sales for similar products. In fact, the majority of my merchandise are books and I could check that specific book on Amazon for historical sales. I have also included them wherever possible in the regression. Does that mean a straightforward regression would be valid in this case? Jul 11, 2018 at 19:16
• Well, you would still need to carry over the effects from an "established" book to a new one. There might also be lifecycle patterns. Jul 11, 2018 at 19:27

I think something that might help is slightly rephrase the question. I agree for profitability, the largest difference in buying vs selling price would be a good indicator (or perhaps the only indicator you might need), but that's hardly insightful. What I would do is to normalize the profit by the buying or selling prices. For example for each product, compute something like

And regress on the other facts. So a large coefficient would now tell you what is important for getting more profit for each dollar in the buying/selling price.

• I believe Profit/(Buying Price) equals Return on Investment while Profit/(selling price) equals Profit Margin. I've tried both in my original calculation and the effect of buying price and selling price became even stronger. I'll still use them in my next attempts though, in case something interesting pops up. Jul 11, 2018 at 18:47
• Sorry, I might not have been clear. Once you compute Return on Investment or Profit Margin, I would remove both Buying and Selling Price from your data and redo the regression. Since buying and selling price and highly correlated, I'm not surprised at all, that you had an even stronger relation. The original goal of computing these values is to normalize out the buying or selling price and studying the relative effect of the other variables. Jul 12, 2018 at 1:04