Question

When asked to state the simple linear regression model, a student wrote it as follows: E(Yi) = βo + β1Xi + ei . Do you agree? Explain.

Answer 1

Answer:

No. See the explanation below.

Step-by-step explanation:

No. When we have the lineal model given by:

$Y_i = \beta_0 +\beta_1 X_i +\epsilon_i , i = 1,....,n$

For n observations, where y represent the dependent variable, X represent the independent variable and $\beta_0, \beta_1$ are the parameters of the model, we are assuming that $\epsilon$ is and independent and identically distrubuted variable that follows a normal distribution with the following parameters $e\sim N(\mu=0,\sigma^2)$.

So then the expected value for any error term is $E(\epsilon_i) =0, i =1,...,n$

So then if we find the expected value for any observation we have this:

$E(Y_i) = E(\beta_0 +\beta_1 X_i +\epsilon_i) , i = 1,....,n$

Now we can distribute the expected value on the right by properties of the expected value like this:

$E(Y_i) = E(\beta_0) +E(\beta_1 X_i) +E(\epsilon_i), i =1,...,n$

By properties of the expected value $E(aX) =aE(X)$ if a is a constant and X a random variable, so then if we apply this property we got:

$E(Y_i) = \beta_0 +\beta_1 E(X_i) +0 ,i=1,...,n$

$E(Y_i) =\beta_0 +\beta_1 X_i, i=1,.....,n$

And if we see that'ts not the result supported by the claim for this reason is FALSE the statement.