Assumptions for multiple regression

Question

I have a question about some assumptions in multiple regression. Based on the theory in my university we have been taught that before running regression we have to analyze the following assumptions:

1. Simple random sampling ( Pretty clear)
2. Independency ( pretty clear)
3. Trustworthiness of data (pretty clear)
4. Errors are normally distributed for every condition of X
5. Error means are equal to 0 for every condition of x
6. error std.dev is constant for every condition of X
7. Linearity
8. Probably something i forgot
9. Variation in x

There's nothing written in any of the materials/books provided by university about assumption of variation in x. I have not found anything by googling it. Our T.A. would always mention it, but he would go through it so fast that noone even understood what it is.

Anyone could explain what assumption of ''variation in x'' could be relevant for multiple regression ?

Answer 1

I am not exactly sure about what "variation of $x$" means, but if it is that different observations should have different values for $x$, it is pretty obvious: you can't predict how $y$ changes as a function of $x$ if you only have data on a single $x$

This assumption also appears on simple regression and pretty much any model you can possibly conceive. From a purely mathematical perspective, if you take a look at the formulas, you will see that the variance of the regressors appears a few times in the denominator. What would happen if it were 0?

Answer 2

A condition you haven't mentioned is that there is no exact multicollinearity. (I.e. the design matrix is of full column rank with probability 1. In simpler terms: no explanatory variable is a linear combination of the other explanatory variables or the constant 1, if you have an intercept.) Sometimes it is said that ''there is no explanatory variable with zero variance'', which is of course true, but is only a special case: if an explanatory variable's variance is zero, then it is constant (say $c$), and then it can be created as a linear combination - simply $c$ times the constant 1 column. I suspect this might be what you're talking about: ''variation in x'' is pretty close to ''there is no x with zero variance''.

A few sidenotes: simple random sampling and independency is not strictly speaking needed (although makes life easier, and some notations are less complicated). ''Trustworthiness of data'' is not a statistical assumption I think. And one last thing: normality of error is not needed for the OLS estimator to be BLUE (unbiased, consistent and efficient), but only for the inferential statistical questions, namely that $t$ and $F$-distributed things indeed have $t$ and $F$ distribution (thus $p$-values and confidence intervals are valid), and even those are only relevant for small samples where you can't rely on central limit theorem.