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I'm trying to do an exercise but I can't figure out how to proceed. OLS is run for this model, with 100 observations. $$ y_i=b_0+b_1x_i+\epsilon_i $$ The results are $$ \hat\beta =\begin{pmatrix} 9 \\1 \end{pmatrix} $$ $$ V(\hat\beta)=\hat\sigma^2 (X'X)^{-1}=\begin{pmatrix} 3 &0.12 \\ 0.12 &0.064 \end{pmatrix} $$ It asks me to calculate $$ e'e$$ aka the sum of squared residuals. I don't know what to do. I know in simple regressione the coefficients and variances have formulas that have to do with averages and variances of y and x. But I don't have any info on the individual observations. Could anyone tell me what I'm missing or how to solve this?

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2 Answers 2

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Notice that

$X'X = \begin{pmatrix} n & \sum x_i\\ \sum x_i & \sum x_i^2 \end{pmatrix}$

since you know $n$, there are three unknowns that determine the elements of the matrix $\text{Var}(\hat \beta)$ - $\sum x_i, \sum x_i^2$ and $\hat\sigma^2$. Use the three indpendent elements of the matrix to find those quantities, and then assuming

$$\hat\sigma^2 = \frac{e'e}{n-2}$$

you can find $e'e.$

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  • $\begingroup$ Thanks, this definitely makes sense. However I don't really know the formulas for the single entries of the variance covariance matrix in function of the three elements you listed. How can I derive them? $\endgroup$
    – Francesco
    May 28, 2022 at 20:34
  • $\begingroup$ Simply invert the 2-by-2 matrix $\endgroup$
    – J. Delaney
    May 28, 2022 at 20:54
  • $\begingroup$ Thanks, I managed to solve it now! $\endgroup$
    – Francesco
    May 29, 2022 at 15:06
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This is how you would calculate the sum of squared residuals -

  1. Use the value of your coefficients to come up with a prediction for all 100 values, $\hat{y}$

  2. Next compute the difference, for all observations, between the real $y$ and predicted value $\hat{y}$, $y - \hat{y}$

  3. Sum the square of those differences $\sum_{i=1}^{i=100}(y_i - \hat{y_i})^2$

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  • $\begingroup$ I know the formula, but I don't have the values for x to calculate the predictions $\endgroup$
    – Francesco
    May 28, 2022 at 20:02

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