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Linear Regression Models (1)

  1. May 14, 2009 #1
    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!
     
    Last edited: May 14, 2009
  2. jcsd
  3. May 18, 2009 #2
    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."
     
  4. May 20, 2009 #3
    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!
     
  5. May 20, 2009 #4
    "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.
     
  6. May 20, 2009 #5
    Thanks for the concrete examples. Things make a lot more sense now!
     
  7. May 21, 2009 #6
    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!
     
    Last edited: May 22, 2009
  8. May 22, 2009 #7
    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.
     
  9. May 29, 2009 #8
    In their Econometric Foundations, Mittelhammer, Judge & Miller hold "E[Y] = E[Y|X] whenever X = x," (i.e. always). [Not an exact quotation.]
     
    Last edited: May 29, 2009
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