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suppose that I want to use linear regression on a data where independent variables x1,x2,...xn are all more or less normally distributed, while the dependent variable y is almost exponentially distributed, what should I do to y (or the whole dataset) so that it is okay to use linear regression model? I applied log transformation to y and then used ols, which seems fine to me, until a friend argued strongly against using log transform under such condition, but he did not provide any solution. So I am here to ask for help.

if not limit to the ordinary linear regression model, what model can I use to have a better fit?

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I want to use linear regression on [...] independent variables x1,x2,...xn [...]
while the dependent variable y is almost exponentially distributed

If you expect the relationship between y and the x's to be linear, then a nonlinear transformation of y will make the relationship between it and the x's nonlinear. It will also alter the spread about the model (if the data had constant variance before transformation, it won't have it afterward).

Note further that in regression, there's no assumption about the distribution of the dependent variable itself (unconditionally). That is, there's little value in looking at say a histogram of the $y$ values -- it doesn't directly relate to any regression assumption. The assumption of normality applies when you're using normal based tests or intervals, and applies to the conditional distribution, which you can't usually assess until you look at residuals.

If you're not interested in hypothesis tests or confidence intervals, an ordinary regression with non-normal conditional distribution may in some situations be reasonable (non-constant variance may be more of an issue than distribution-shape anyway). If you do want to perform inference as well, there are several ways of going about it (some approximate) that may be suitable.

If you thought that the conditional distribution $Y|x1,x2,...$ was distributed as exponential, and that the relationship between $Y$ and the $x$'s was linear, you could use a GLM with identity link. There's advice relating to fitting exponential models in this way on site.

independent variables [...] are all more or less normally distributed, while

The distribution of the independent variables doesn't matter, since you condition on them in regression. No assumption about their distribution is made. The only way it's relevant is that sometimes the joint distribution can help inform us how to interpret the marginal distribution of the dependent variable, y (e.g. jointly normal x's would not produce an exponential y from conditionally normal y, so it would lead us to doubt the y's were conditionally normal).

I applied log transformation to y and then used ols, which seems fine to me, until a friend argued strongly against using log transform under such condition, but he did not provide any solution.

If it makes sense to model $E(\log(y))$ as a linear function of the predictors, that may be fine, but note that if you exponentiate such a fit, you don't get a suitable estimate of $E(Y|X=x)$ out (unless there's almost no variation about the model, in which case the bias may sometimes be small enough to ignore). An alternative to that would be to use a GLM with log link (in which case you'd be modelling $\log(E(y))$ as a linear function of parameters -- and expected values do come straight out of that model.

You should consider the spread about the relationship; if you know something about that it already it may help inform your choice of model (but beware your inferences if you're using the same data to identify the model as to make inferences about it)

There are many alternative ways than least squares to fit linear relationships, and some might be more suitable in the case of some non-normal conditional distributions.

You should clarify your expectations about what it is that will be linearly related to the x's and how you understand the variability about the line would behave (say as a function of the mean for example -- would it tend to spread more as the mean increased, or not?) on whatever that scale is.

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  • $\begingroup$ Thank you for the great answer. I have following questions: 1. I understand that whether linear regression is valid has no direct link to the distribution of y and x, but if both x and y are normal, the residual should be normal right? this is why I transform all x_s and y so that the histogram of them looks more or less normal. 2. If linear regression (ols or glm) is chosen for the sake of being interpretable, and I only assume there is a linear relation between f(y) and [g1(x1),g2(x2),gn(xn)], are there any guidelines on how to treat x_s and y (again, embarrassingly general )? $\endgroup$ – user6396 May 12 '17 at 3:03
  • $\begingroup$ 1. Not necessarily. If they were jointly normal, yes, but that's not a good reason to transform since you may already satisfy the requirements of regression and transformation may screw up more important things. 2. I'm not quite sure I follow. Perhaps you could write a longer explanation in a new question. $\endgroup$ – Glen_b May 12 '17 at 3:09
  • $\begingroup$ question2: suppose I have a dataset with dependent variable y, and 1000 features (or independent variables) x_s, after removing some feature which are highly correlated with other features, I have, saying 300 x_s left, for whatever reasons, I decide to use linear regression (ols or glm, with or without regularization), I don't assume y is linearly dependent on x_s, but I assume the linear relation holds between f(y) and [g1(x1),g2(x2),...], where f and g1,g2... are just some transformation function, such as log. how should I proceed? $\endgroup$ – user6396 May 12 '17 at 3:36
  • $\begingroup$ Are the f's and g's known? (Note that when I said "in a new question" I meant click the "ASK QUESTION" button and post a new question) $\endgroup$ – Glen_b May 12 '17 at 3:42
  • $\begingroup$ right, I am posing another question. $\endgroup$ – user6396 May 12 '17 at 3:45

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