I have the following question and I couldn't figure out how to solve it.

The given is the following:

  • The model is: Yi = B0 + B1 Xi + Ei
  • I have the following dependent variables 2, 4 and 8 (3 observations)

After estimating the model by OLS, we know that the summation of the estimated (y hat) squared is = 80.

how can I calculate R square with the above given

the answer is supposed to be .7857

Any help?

I added the exact question:

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  • $\begingroup$ You need to apply several formulas for variances and $R^2$ as well as know how the mean of $\hat Y_i$ is related to the mean of $Y_i.$ At what step do you need help? $\endgroup$
    – whuber
    Commented Jan 27, 2020 at 20:55
  • $\begingroup$ @whuber can you list the steps please, I'm stuck on how to calculate R-square without having the independent variables listed. $\endgroup$
    – Ali Kleit
    Commented Jan 27, 2020 at 21:09
  • $\begingroup$ If this is homework, you need the self-study tag along with the others: regression etc. $\endgroup$
    – Carl
    Commented Jan 28, 2020 at 0:31

1 Answer 1


Because this problem is most instructive when you work out the answer yourself, I will suggest a logical sequence of attack but not provide a complete answer.

Begin at the end: you want to find $R^2,$ which is defined as a ratio of variances. The numerator is the variance of the predicted values $\hat y_i$ while the denominator is the variance of the response values $y_i.$

Although the problem does not specify any variances, it does gives us the sum of squares of the $\hat y_i$ and the individual $y_i.$ Obviously you can compute any property you need from the $y_i,$ which are $(2,4,8).$ You need the variance and most computations of the variance first obtain the mean, so you might as well compute these two statistics right away.

That leaves you with a conundrum: how to find the variance of the $\hat y_i$ when all you have is their sum of squares? There must be some connection between properties of the $\hat y_i$ and properties of the $y_i.$ That's the key: apart from exercising your skills in algebra, this problem invites you to think about any necessary relationships between the predicted values and the original responses in an ordinary least squares regression (which includes a constant term: this is crucial).

If you cannot think of any relationship, take this opportunity to review your textbook and notes for relationships. Focus on the one missing piece: since you have the sum of squares of the $\hat y_i,$ all you need in order to find their variance is their mean (and, of course, their count): how is the mean of the predicted values related to the original responses?

The rest is arithmetic.


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