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The question is to fill out the missing numbers (A-L) of a simple linear regression model. I am having problems with converting and interpreting the given table in terms of variables. Would it be possible for someone to confirm and clarify things for me.

The first table represents regression statistics

True model $$ Y_t = \beta_o + \beta_1X_t + \mu_t $$

Estimated model $$ \hat Y_t = \hat\beta_0 + \hat\beta_1x_t $$

This is what I am confused about

  • Does the first standard error ($12.8478$) mean $\sum\hat\mu_t^2$ ?
  • Does the standard error for the intercept in last table ($14.6208$) mean $\sum\mu_t^2$ ?
  • Does $3.8508$ equal $\hat\beta_1$ ?
  • In order to calculate RSS (for J) I need $\sum \hat\mu_t^2$ does this confirm that my first two points are incorrect
  • I know $G=\hat\beta_1^2\sum x_t^2$, how do I find $\sum x_t^2$

If I am wrong, would it be possible to know what those numbers mean in terms of variables

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Let us first calculate all the unknowns:

D = 1 (as it has only one variable)

E = n-1-1 = 13

F = 15-1 = 14

MSE = (Standard Error of Estimate)^2 = 12.8478^2 = 165.06

So J = 165.06

I/J = F-statistic = 24.15

I = 24.15*165.06 = 3986.34

G = I*k = 3986.34

H = J*(n-k-1) = 165.06*13 = 2145.78

K/14.6208 = 1.3081

K = 19.125

3.8508/L = 4.9146

L = 3.8508/4.9146 = .7835

R-Squared = B = 3986.34/6132.85 = .649

Adj R-Squared =

1-[(n-1)/(n-k-1)*(1-R^2)] = .622

To give explanation

Sum of (Mu-Hat)^2 = Sum of Squared Errors

Standard Esimate of Error = SQRT(MSE)

Mean Squared Error = SSE/(n-k-1)

14.4208 means the Estimated Standard Error of Beta1-hat

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  • $\begingroup$ I can't follow the calculation for J, H & I. Could you point me to the formulas and definition that deals with calculating J. From my understanding, doesn't 12.8478 equal $\sum\hat\mu^2$? $\endgroup$
    – Kartik
    Oct 29, 2013 at 8:20
  • $\begingroup$ 12.8478 is the Standard Estimate of Error and it is not sigma(mu-ht)^2. Sigma (Mu-hat)^2 is the Standard Sum of Errors(Unexplained Variance of the Regression). Mean Sum of Errors (MSE) = SSE/(n-k-1) $\endgroup$ Oct 29, 2013 at 9:19
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    $\begingroup$ There are two ways to find the MSE, 1) Since SEE is given, you can square it and find it. Other way around is SST is given, you are given the F-Statistic = [(SSR/k)/(SSE/(n-k-1))] = 13*(SSR/SSE) = 13*[SST-SSE]/SSE = 13*[SST/SSE - 1] => 24.15 = 13*[(6132.84/SSE)-1]. From this we can find SSE = 2145. and SSR = SST-SSE = 3986.34. Now you have found H and I, to calculate J MSE = 2145/13 = 165 MSR = SSR = 3986.34. SEE, on the other hand is the square root of MSE. Since SSE is given in the table you can find MSE and subsequently find SSE and SSR and MSR. Either way you will get same answer. $\endgroup$ Oct 29, 2013 at 9:26
  • $\begingroup$ cs.gmu.edu/cne/modules/dau/stat/regression/multregsn/see.gif $\endgroup$ Oct 29, 2013 at 9:49
  • $\begingroup$ awesome! that explains it. Thank you $\endgroup$
    – Kartik
    Oct 29, 2013 at 19:13

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