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I have the following Y outcomes distribution with the normal density function represented by the superimposed red line:

enter image description here

I need to develop a regression methodology to predict $Y$ given a number of predictors $X_n$. OLS doesn't seem to fit the bill because of the non-normal distribution of $Y$.

Given that the outcomes are non-continuous increments of 0.5, would an ordered probit regression be the right approach? As I understand it, a probit regression allows to specify the distribution of the outcomes. What other alternatives do I have?

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Which distribution of the dependent variable is relevant?

First, there is no reason to think that OLS will do badly on this data. At least none from looking at your dependent variable. However...

Second, Looking at the distribution of your dependent variable is not so helpful. Even regression models that assume Normally distributed errors only expect them to be Normal after conditioning on X. This does not imply that the marginal distribution is Normal, which is what you plot here, I think. So seeing fat tails here is not necessarily a problem.

Models and distributional assumptions

OLS does not describe a model but a procedure (minimise the sum of squared errors). OLS generated coefficient estimates have various properties depending on what you are happy to assume by way of an actual statistical model of the the data generating process. The more you assume, the more you get to say. Any good econometrics text will go into the gory details of this, since econometricians always want to know what they can infer from as few assumptions as possible. But in short, you can go a long way without explicitly assuming they are even conditionally Normal.

Probit regression and the alternatives

Probit does allow you to specify a distribution. Indeed, all generalised linear model, a class which includes homoskedastic linear models for which OLS is the maximum likelihood estimation procedure and probit models allow you to specify the distribution of the dependent variable. You can always do that. And you can often only partially specify it too and get some more general results. The class of generalised linear models provides a good starting point when looking for 'alternatives'.

The histogram

I don't know if I understand the graph properly because I can't see anything that varies at 0.5 intervals (except the histogram bins, but they are designed to do that sort of thing). So thus far, I see no reason to consider probit models.

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  • $\begingroup$ The outcomes are discrete in 0.5 as I stated in the questions. $\endgroup$ Mar 26 '12 at 19:08
  • $\begingroup$ so how did you make the graph in the question? $\endgroup$ Mar 26 '12 at 19:13
  • $\begingroup$ The problem I have with a linear solution is relative to the kurtosis of the Y distribution, not the skewness. Isn't the normality of outcomes a requirement of OLS regression? $\endgroup$ Mar 26 '12 at 19:22
  • $\begingroup$ The graph is an histogram of the results I need to predict. I did use a continuous variable to produce the graph but ultimately I am only interested in the discretized version of the continuous variable. $\endgroup$ Mar 26 '12 at 19:25
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    $\begingroup$ Not quite ok for 2. The 'non-linearities' in the predictive probabilities are due to having unevenly spaced cut points slicing the probit's (conditionally Normal) latent variable. Some slices are just bigger than others. The question for you is whether your data are genuinely ordinal (cut points could be very unevenly spread) or just coarsely measured (effective cut points are 0.5 intervals and do not need to be estimated). In the latter case, treating the dv as continuous will be ok. Re 3. the nonlinearity can be dealt with via probit if you've got it, but the graph leaves that open. $\endgroup$ Mar 26 '12 at 20:47
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This is an old thread but I just came across it. Just one quick comment - when your range is so large, 0.5 increments are fairly minute. That means it approaches being a continuous variable, and linear regression would be far easier to interpret than an ordered logit. Ordered logit coefficients are easy to look at, but not so a) simple to explain to a non-technical audience and b) not very detailed. They tell you the proportional odds above and below a line (odds ratio of moving up the chain, but don't know how far up you go).

As conjugateprior implied, I would use this histogram to say "Yeah, this looks close enough to normal to me let's model with OLS or GLS or something linear", and then look at the residuals you get. In my experience normality tests are very sensitive, so if you have high kurtosis like this in the residuals, it may not be the end of the world for you. Social scientists eyeball these things all the time, it's kind of common procedure (for better or worse).

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