# Understanding Fixed Regressors and Conditional Expectation on Fixed Regressors $E(Y|X_i)$

I'm having trouble with the statistical idea of a fixed regressor, it seems that our $$X_i's$$ are not treated as random variables, but we are still able to meaningfully condition $$Y$$ on them in a way that we couldn't if $$X_i$$ was just the constant $$c$$. The fixed regressor seems to be occupying a middle ground between a random variable and a constant. I have scoured the forums for all the questions related to this but have been unable to come to a conclusion.

• I think part of my confusion might stem from different econometrics books loose definition of the simple regression e.g. some use $$Y_i = \beta_0 + \beta_1 X_i + u_i$$, and make no mention of $$\beta_0 + \beta_1 X_i = E(Y|X_i)$$.
• I think the mixed use between capital letters $$Y, X$$ and lowercase $$y, x$$ across texts is also not helping. (Given the very specific random variable vs observed value use in statistics generally).
• I think i'm also getting slightly confused as to when subscript i refers to a fixed value e.g. $$X_i$$ vs what i assume is a particular but still random value e.g. $$Y_i$$

Let's take our hypothetical population regression as $$Y_i = \beta_0 + \beta_1 X_i + u_i$$, where $$\beta_0 + \beta_1 X_i = E(Y|X_i)$$ and $$X_i$$ is fixed, $$E(u_i) = E(u_i|X_i) = 0$$

1) What does it mean to take conditional expectation on a fixed $$X_i$$?

• Are we saying $$E(Y|X = X_i)$$? i.e. $$E(Y|X = x)$$ where $$X_i = x$$ an observed value
• i.e. $$\int_{-\infty}^{\infty} y f_{Y|X}(y|X_i) dy$$ where $$f_{Y|X}(y|X_i) = \frac{f_{Y, X}(y, X_i)}{f_{X}(X_i)}$$
• But then if we are told our regressor $$X$$ is non-random how can we give it a distribution?

2) How do we take the expectation of $$Y_i$$ in this form?

$$E(Y_i) = E[E(Y|X_i)+u_i]$$

• If $$X_i$$ were stochastic we would get $$E(Y_i) = E(E(Y|X_i)) + E(u_i) = E(Y)$$

• If $$X_i$$ is non-stochastic then $$E(Y|X_i)$$ is just a constant? $$E(Y_i) = E(E(Y|X_i)) = E(Y|X_i)$$

• However, Gujarati & Porter's Basic Econometrics 5th edition Page 34-35 goes to great lengths to highlight why this isn't the case.

• But then in what feels like a contradiction on page 40 footnote 8 (Screenshot below) they mention "Note that $$E(Y|X_i)$$, once the value of $$X_i$$ is fixed is treated as a constant."

• They seem to avoid this issue by taking the conditional expectation rather than non-conditional expectation.

$$Y_i = E(Y|X_i) + u_i$$

$$E(Y_i | X_i) = E(E(Y|X_i)|X_i) + E(u_i | X_i) = E(Y|X_i) + 0$$

3) Understanding $$Y$$ vs $$Y_i$$

• Is the random variable $$Y$$ fully defined by the fandom process $$Y = \beta_0 + \beta_1X + u$$ of which particular values are written as $$Y_i = \beta_0 + \beta_1X_i + u_i$$?

• Is this analogous to how we might say a sample of $$Z_1...Z_n$$ from a random variable $$Z$$ but where $$Z_1...Z_n$$ are still random until they become observed values $$z_1...z_n$$.

• If so given that $$X$$ and hence $$\beta_0 + \beta_1X_i$$ is completely non stochastic how would we define this probability distribution $$Y = f(u; X, \beta_0, \beta_1)$$ I don't think this is correct because $$X$$ isn't really a parameter, but it's also not a random variable? So again i don't know how to deal with the fixed regressor.

• When authors write $$y_i = \beta_0 + \beta_1x_i + u_i$$ using lower case $$y_i, x_i$$ generally are they referring to particular observed values or not e.g. Chapter 2 p.g. 27 Woolridge

The $$X$$s are not treated as random variables.

Take the formulation $$Y_i=\beta_0+\beta_1 x_i+U_i$$ (I am changing $$x$$ to lowercase to emphasise it is not a random variable, and $$U$$ to uppercase to emphasise it is). There are only two random variables - $$U_i$$ (usually assumed to be a Gaussian$$(0,\sigma^2)$$, and $$Y_i$$ - a Gaussian$$(\beta_0+\beta_1 x_i, \sigma^2)$$ by definition.

But then if we are told our regressor $$x$$ is non-random how can we give it a distribution?

Example: if you have a random variable $$A$$ defined as $$A=B+2$$, where $$B\sim \text{Gaussian}(0, 1)$$, what is the distribution of $$A$$?

Now, apply the same logic to $$Y_i=\beta_0+\beta_1 x_i+U_i$$, keeping in mind what is random, and what is not.

Regarding the subscript $$i$$, you can just think of it as "some value". E.g. if you have $$n$$ data points total, $$i$$ can refer to any one of those $$n$$ points. So $$Y_1$$ is the first, $$Y_2$$ the second, ..., $$Y_i$$ the $$i$$-th. There's not really such thing as $$Y$$ in this case, it's more just notational laziness.

As a more concrete example, let's say we're regressing weight $$Y$$ against height $$x$$. It is true that $$X$$ has some distribution. But it doesn't matter what that distribution is, because a linear regression model specifies the distribution of $$Y$$ given the predictors. So the model says "given a person is 170cm tall, what is their weight? Given a person is 153cm tall, what is their weight?" etc.

• Thank you for the answer i will digest and get back if i have any comments! Commented Mar 5 at 9:44
• The thing that is confusing me about your answer, is when you say A = B + 2, isn't that analogy just proving that Y has a distribution? Which I understand. What's confusing me is that we condition on X, i.e. E(Y|X), but if x were a constant this would make no sense, but X is also not a random variable so what is going on? i.e to take the conditional expectation (conditional on X) X must have a distribution for this to make sense... Thanks in advance if you're able to provide any more insight Commented Mar 7 at 19:20
• Linear models are always modelling $E[Y|X]$, or more explicitly, $E[Y|X=x]$. Since we are conditioning on $X$, this means $X$ is known and not a random variable. It doesn't matter how $X$ is distributed, because in the context of linear modelling $X$ is always known to be a value, and thus can be treated as a constant. Commented Mar 7 at 22:25
• If we were modelling say $Y_i=\beta_0 + \beta_1 X_i + U_i$ where $X$ was not conditioned on, that would be another case. But $X$ is always known in linear modelling. Commented Mar 7 at 22:26