# In linear regression, do the errors overall have a normal distribution, or do the errors at each value of x have a normal distribution?

In linear regression with fixed effects (i.e. with constant $$x_i$$, not random $$X_i$$), the model states that

$$\epsilon_i \sim N(0, \sigma^2), \ \ i = 1, 2, ..., n$$

Does this say that

1. the overall set of errors have a normal distribution?

or

1. the errors at each value of $$x_i$$ have a normal distribution?

I think that the answer is #2. I know that the responses ($$Y_i$$) are normal at each value of $$x_i$$, not overall. Therefore, it only seems right that the errors are normal at each value of $$x_i$$, too.

• It is important to understand mathematically what #1 even means.
– usul
Commented Mar 23 at 20:06
• A pedantic note, but "fixed effects" does not mean the same as "fixed regressors". You seem to refer to the latter. Commented Mar 24 at 16:06
• Fitting a model using least squares involves no distributional assumptions, it is a pure exercise in calculus. The resulting residuals can be tested for normality, and if that's plausible you can go on to use statistical inference to assess significance, but while assuming i.i.d. normality is great for writing statistics texts it's not a given for real-world data and it's not a necessity for doing least squares regression.
– pjs
Commented Mar 24 at 17:16

You are correct that this notation addresses each individual error term. Given an $$i$$, $$\varepsilon_i$$ is normal.

However, with an added assumption of independence of the error terms, the errors are jointly multivariate normal.

• "with an added assumption of independence of the error terms, the errors are jointly multivariate normal." Could you please elaborate on what you mean by that? Commented Apr 28 at 18:44

The second answer is one of the assumptions made in linear regression. This assumption is typically written as

$$\epsilon|X \sim~ N(0,\sigma^2I)$$

where $$\epsilon|X$$ denotes the error given the values of the X variables. See e.g. William Green, Econometric Analysis, Fourth Edition page 222. It is not necessarily true that a normal distribution of all error terms considered together would guarantee that normality also holds for each separate X value! To give an example, suppose there is only one X variable with two values, 0 and 1.

For $$X=0$$ the error terms come from a truncated standard normal with lower bound -1 and upper bound +1, which could be written as "truncnorm(-1,+1)". So the mean of these error terms is zero.

For $$X=1$$, half of the error terms come from truncnorm(-Infinity, -1) and the other half comes from truncnorm(+1, +Infinity). The mean of these error terms is also zero.

So the error terms of both X values do not have the same distribution and variance for both X values, but that is not the issue of this question.

The point is that the entire set of errors does have a normal distribution, whereas for each X value this is NOT true.

Here is an R script which generates data as in this example.

library(truncnorm)
library(ggplot2)

set.seed(12345)

n <- 10000

e0 <- rtruncnorm(0.68*n, a=-1, b=1)
e1 <- c(rtruncnorm(0.16*n, a=-Inf, b=-1), rtruncnorm(0.16*n, a=1, b=Inf))
e <- c(e0, e1)

x <- c(rep(0,0.68*n), rep(1,0.32*n))

y <- 1 + x + e

model <- lm(y ~ x)
resid <- residuals(model)
hist(resid, breaks=50)


The histogram of the estimated errors (residuals) is:

For $$X=0$$ we get:

hist(resid[x==0], breaks=20)


For $$X=1$$ we get:

hist(resid[x==1], breaks=20)


The idea that the errors for each separate X value should be normally distributed is sometimes graphically shown as follows:

For each separate (combination of) X value(s) there is one and the same normal distribution from which the data are randomly and independently drawn. That is the idea behind "ordinary" regression. So, it is not enough to formulate the condition as: "across all cases the errors should be normally distributed".

I would say that both answers are correct, but I would interpret it as statement #1 and #2 is perhaps not fully "appropriate".

Note that $$y_i \sim N(x_i'\beta, \sigma^2)$$ for some fixed $$x_i$$ and this indeed holds at each value $$x_i$$, as you correctly point out. On the other side, $$\epsilon_i \sim N(0, \sigma^2)$$ which does not depend on $$x_i$$. Here, $$x_i$$ is fixed so it is independent of $$\epsilon_i$$. Therefore, you are right the errors at each value of $$x_i$$ have a normal distribution, but this is because this holds in general and therefore also for any value of $$x_i$$. In that sense, it is "superfluous" to say that the normality holds for each value of $$x_i$$. For $$y_i$$, this is not the case, as this actually depends on $$x_i$$.

• Sorry, I don't follow. How do you know that "the errors at each value of xi have a normal distribution"? Commented Mar 23 at 2:54
• The errors are normally distributed. This is what the equation above says. Moreover, they are independent of $x_i$. Thus, this holds for every value of $x_i$. Think about the equation 1+1=2, this holds in general. This is independent of the value of $x_i$. As 1+1=2 holds in general, it also holds for every value of $x_i$.
– Stan
Commented Mar 23 at 9:09