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I am trying to run a Multivariate Regression Model where I am trying to explain House Prices by some given data such as the number of baths, location of the house, square feet and so on.

While playing with the variables I decided to model the price per square feet variable to explain the House Price (obviously just a linear modification of the variable I am trying to explain). Running the Regression on this, all my previous variables that were significant all became insignificant (all p-values above 0,10). Nonetheless my adjusted R-Squared skyrocketed up to 0,98. I obviously have a gut-feeling that this result is wrong and may be prone to multicollinearity or some other statistical sin I am not aware of. I would like to understand if this result is viable or not and why?

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    $\begingroup$ I think it would help us give an answer if you could tell us exactly what are the independent variables $\endgroup$
    – famargar
    Commented Jan 2, 2018 at 11:13

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With a single response (outcome, dependent variable) your procedure is best called just regression. The term multivariate regression is better used if and only if you have several response variables. It's not a synonym for multiple regression, a term itself slowly fading away, as having several predictor variables is long since utterly routine.

That said, if all your houses had the same area, then price and price per unit area would be just be the same variable, up to a constant factor, and regression on price and price per unit area would give you $R^2$ of 1.

You don't quite get that, which is not surprising if only because house area is genuinely variable, but the way to understand exactly why not is to plot price versus price per unit area.

This may well show you other features of your data, such as nonlinearity, skewness, and/or possible outliers. For those and other reasons it is often advisable to work with house prices on a logarithmic scale. (One of the other reasons is that if you don't do that, it is all too easy to end with a regression model which predicts negative prices for the cheapest houses.)

It's hard to give precise advice without more details of your data, but the significance levels for the other predictors are indeed just side effects of the basic fact that they depend on what else is in the model. If one predictor dominates, then the others will necessarily look puny by comparison.

But I agree with your instinct. I don't think price is "explained" well, or at all, by price per unit area. It's fine to use area as another predictor. As with price itself, area may be easier to work with on logarithmic scale.

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