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I have a question that emerged from my previous post. If you look at my previous post, the dependent variable there is a ratio variable and thus is bounded, i.e. it can only take the values between 0 and 1 (and there is a great bunch of 1s in the data set).

As I understand, one of the issues is that such dependent variables do not meet the normality assumption of OLS, which states that the error term would be distributed normally.

Given that the normality of errors is the weakest assumption of the OLS, I wonder if there are any other reasons for why it is problematic to use an OLS regression for a bounded dependent variable?

Could you share any useful sources?

Appreciate it.

Ken Lee
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    The biggest problem is the spurious correlation induced by the ratio. If values of the dependent variables are not too near 0 or 1, then it can be argued a normal approximation is possible in that range – Firebug May 30 '21 at 19:03
  • @Firebug Could you expand on what you mean by "correlation induced by the ratio"? – Ken Lee May 30 '21 at 21:49

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Another common problem with OLS for a bounded variable is heteroscedasticity; as the expected value approaches the boundary, the variance generally shrinks (this is true of error distributions such as beta and binomial; it need not be true, e.g. if the error distribution simultaneously becomes much more skewed). Heteroscedasticity generally has more serious consequences (in terms of inefficiency, undercoverage, etc.) than violations of normality.

(An answer to the previous post comments "this isn't heteroscedasticity, it's truncation". They're right that the ultimate problem is truncation, but it does cause heteroscedasticity in your example (see the decreasing trend line in your scale-location plot).)

This may be too obvious to state, but naive model predictions for extreme values of the predictor variables would generally be biased (because the model would predict values outside the allowable range). You can see some evidence of such bias in the fitted vs residual plot in your original post, specifically the non-constant trend line.

More generally, if you do web searches for "linear probability model vs. logistic regression" you'll find lots of discussion, e.g. here. Econometricians generally prefer LPM, statisticians prefer logistic regression. If you search for "linear probability model" on CrossValidated, you'll mostly find statisticians telling you that LR is better (e.g. here). (This is a big rabbit hole which I have chosen not to go down, so I can't lay out the arguments in favour of the LPM for you.)

Ben Bolker
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  • Thanks, Ben, that's very helpful. Is there a term for the phenomenon that you describe in your third paragraph, i.e. that the independent variables would be predicting values outside of the range of the dependent variable? I would like to read more about this. Any sources, of course, would be greatly appreciated. Could you also tell me how the red line in the residuals v fitted plot suggests that this is the case in my example? – Ken Lee May 30 '21 at 22:15
  • I don't know a name for it: people [joke about it a lot on twitter](https://twitter.com/search?q=linear%20probability%20model%20110%25). The red line suggests that there is some form of bias (error is not constant with respect to predicted value), and this truncation or clipping or censoring or whatever is the most obvious culprit. – Ben Bolker May 30 '21 at 22:28