What is the difference between E(Y|X) and E(Y|X=x) in linear regression models?

[kingwinner]

1) "In regression models, there are two types of variables:
X = independent variable
Y = dependent variable
Y is modeled as random.
X is sometimes modeled as random and sometimes it has fixed value for each observation."

I don't understand the meaning of the last line. When is X random? When is X fixed? Can anyone illustrate each case with a quick example?


2) "Simple linear regression model: Y = β₀ + β₁X + ε
If X is random, E(Y|X) = β₀ + β₁X
If X is fixed, E(Y|X=x) = β₀ + β₁x"

Now what's the difference between E(Y|X) and E(Y|X=x)? The above is suuposed to be dealing with 2 separate cases (X random and X fixed), but I don't see any difference...
Most of the time, I am seeing E(Y) = β₀ + β₁X instead, how come? This is inconsistent with the above. E(Y) is not the same as E(Y|X=x) and I don't think they can ever be equal.

Thanks for explaining!

 

[Enuma_Elish]

In many cases the question whether X is random is theoretical. A clear-cut case for nonrandom X is the time trend (e.g., seconds into the experiment, or years into the Obama administration, etc.). Two clear cases of random X is (a) when X is co-determined with Y; and (b) when X is measured with random error.

E(Y|X) implies that the random variable X is not assumed to take on a particular value; E(Y|X=x) implies X is assumed to equal the predetermined, nonrandom value x. E[Y] is being used as a shorthand for "E[Y|X] if X is random, E[Y|x] otherwise."

 

[kingwinner]

For example, if we have height v.s. age (Y v.s. X), is X fixed or random?

Also, what does it mean for X to be FIXED? If we have five data points, x1,x2...,x5, and NOT all of them have the same value of X (e.g. x1≠x2), is X fixed in this case?

Thank you!

 

[Enuma_Elish]

"Fixed vs. random" usually depends on your goal. In your example, height vs. age, there may be at least two different contexts:

1. Heights of 10 children are measured at ages 1 through 10. We would like to determine the relationship between height and age for these 10 children.

2. 100 children are selected at random from a population of 10,000; their ages are recorded and their heights are measured. We would like to determine a general relationship between height and age for the entire population, based on this sample.

In case 1, age is fixed. In case 2, it is random.

 

[kingwinner]

"Fixed vs. random" usually depends on your goal. In your example, height vs. age, there may be at least two different contexts:

1. Heights of 10 children are measured at ages 1 through 10. We would like to determine the relationship between height and age for these 10 children.

2. 100 children are selected at random from a population of 10,000; their ages are recorded and their heights are measured. We would like to determine a general relationship between height and age for the entire population, based on this sample.

In case 1, age is fixed. In case 2, it is random.

Thanks for the concrete examples. Things make a lot more sense now!

 

[kingwinner]

2) By definitions,
E(Y)=
∞
∫ y f(y) dy
-∞

E(Y|X)=
∞
∫ y f(y|x) dy
-∞

If X is FIXED, does this ALWAYS imply that X and Y are INDEPENDENT and E(Y)=E(Y|X=x)?? Why or why not?

For simple linear regression model, my textbook typically write
Y= β0 + β1*X + ε as
E(Y) = β0 + β1*X

However, I have seen occasionally that
Y= β0 + β1*X + ε is written as
E(Y|X) = β0 + β1*X which looks a bit inconsistent to the above...how come? The definitions of E(Y) and E(Y|X) are clearly different as I outlined above, but here it seems like they are equal? How come?

Thanks for explaining!

 

[Enuma_Elish]

E(Y) (using shorthand notation) is a function of X: E(Y) = b0 + b1 X. That means E(Y) is never indep. of X; the question is whether it's a dependence on a nonrandom variable ("x"), or a random variable ("X"). As I explained above, E(Y) is a shorthand notation.

 

[Enuma_Elish]

In their Econometric Foundations, Mittelhammer, Judge & Miller hold "E[Y] = E[Y|X] whenever X = x," (i.e. always). [Not an exact quotation.]