Module 10: Advanced multiple regression

(The attached PDF file has better formatting.)

Homework assignment: Two correlated independent variables

Do this homework assignment after module 12, which gives the equations for the standard errors of the least squares estimators.

We regress the Y values on the X₁ and X₂ values in the table below.

X1	X2	Y	X1	X2	Y
1	1	1.016	6	8	-1.076
2	6	-3.429	7	4	3.461
3	2	0.049	8	9	-2.525
4	7	-3.099	9	5	4.195
5	3	0.359	10	10	-0.746

What is the correlation of X₁ and X₂?

What is the least squares estimator of

á?

What is the least squares estimator of

â₁, the coefficient of X₁?

What is the least squares estimator of

â₂, the coefficient of X₂?

What is the standard error of the least squares estimator of

â₁, the coefficient of X₁?

What is the standard error of the least squares estimator of

â₂, the coefficient of X₂?

Show the formulas and the computations. You can check your work with Excel or other statistical software.

I need some help with letters E and F. For E, it asks for the standard error of the least squares estimator of B1, the coefficient of X1. From the readings, I only saw a formula for the standard error of the regression. Should I use this formula but use the X1 values instead of the Y values for the residuals?

[NEAS: The t-vaue is the estimator divided by its standard error. Look at the formulas for the t-values.]

Parts B,C,D: Why are we being asked for alpha first, when we need beta1 and beta2 to find alpha? Am I misunderstanding the question?

Parts E,F: Are you refering to the formula in the gray box at the top of page 105?
SE(B) = S-sub-E/sqrt(sum(x-sub-i - x-bar)^2))


EDIT: Nope. It's the forumla in the gray box on page 107. Wish someone had told me a lot sooner.

Did anyone else get SE(beta1) = SE(beta2) = 0.108?

[NEAS: See next post]

Okay, I finally found out how to do the regression in excel, so I checked my answers and I definitely did E & F wrong.

1)For multiple regression, degrees of residual freedom should be n-k-1. Here that's 7.
2)For multiple regression, we want the std error formula on pg. 107, not from pg. 105.

Now I have SE(B1) = SE(B2) = 0.1498.

[NEAS: Yes, this is correct.]

In the standard error of slope coefficient formula, how do you calculate Rj^2? I thought it was just R^2=RegSS/TSS which in this case would be 52.16/59.87=.871

[NEAS: Yes, this is correct.]

The equation for r is actually at the bottom of page 88:

r=∑(x₁-x₁bar)(x₂-x₂bar)/sqrt(∑(x₁-x₁bar)²∑( x₂-x₂bar)²)

Then you just want to square this result to get R².

Hope this helps!

If I understand correctly, when we have only one explanatory variable, r is equal to the correlation between the explanatory variable and the response variable.  When we have two explanatory variables, r is equal to the correlation between the explanatory variables.

Could someone explain to me why there is no consideration of the response variable in the case where we have two explanatory variables?

[NEAS: To determine the standard error of the betas, we consider the correlation of the two explanatory variables. For the R-squared of the regression equation, we use the correlation of the response variable to the explanatory variables. These are two separate items.]

Also, what the formula for r be when we have more than 2 explanatory variables?

[NEAS: Same as the r for a multiple regression equation. For the standard error of a beta with N explanatory variables, we ask: what is the r value if this explanatory variable were the response variable and the other explanatory variables were the explanatory variables.]

Thanks in advance.

Did you use the same value of Rj in calculating the values of SE(B1) and SE(B2)?
I am getting SE(B1) = SE(B2) = 0.19196 and r=0.636363 which is what I used as Rj

Some one please elaborate on the calculation of Rj part? Will it be the same for calculating for both SE(B1) and SE(B2)?

[NEAS: X₁ and X₂ have the same ten values, though arranged differently, so they have the same standard errors, which are 0.14978. The correlation of the two independent variables is 0.636364, as noted above.]

Please help... I am getting different numbers between manual calc and Excel regression add-in for questions E and F.

Manual Calc SE = .116, Excel SE = .1497.

I am using formula to do my manual calc.

[NEAS: The excel regression is correct. Use the formula on page 107 for the computation.]