We have estimated the model $wage=\beta _{0}+\beta _{1}$ experience $+u$ using OLS based on a sample of 4 observations. We know that the matrix of explanatory variables is $$ \mathbf {X=}\begin {pmatrix} 1 & 1 \ 1 & 2 \ 1 & 3 \ 1 & 4\end {pmatrix}$$and we are told that the OLS residuals are: $$ \mathbf {\hat {u}=}\begin {pmatrix} 1 \ 1 \ -1 \ -1\end {pmatrix}$$We can immediately say that the residual vector:

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BlackTom AI · Accepted Answer

The question concerns the OLS normal equations, which require that the residual vector û be orthogonal to every column of X, i.e., X'û = 0.
First, examine the residual vector given: û = [1, 1, -1, -1]'.
Next, compute the orthogonality with the first column of X, which is a vector of ones: X1 = [1, 1, 1, 1]'. The inner product X1'û = 1*1 + 1*1 + 1*(-1) + 1*(-1) = 1 + 1 - 1 - 1 = 0. This part is satisfied, as the residuals sum to zero, which is consistent with OLS.
Then check orthogonality with the second column of X, X2 = [1, 2, 3, 4]'. The inner product X2'û = 1*1 + 2*1 + 3*(-1) + 4*(-1) = 1 + 2 - 3 - 4 = -4. This is not zero, so û is not orthogonal to the second regressor.
Since the OLS residuals must be orthogonal to all columns of X, failing orthogonality with X2 means the residual vector provided cannot come from the OLS solution for this X. Put differently, the reported residuals violate the normal equations, and the statement that the residual vector is reported incorrectly because it is not orthogonal to the second column of X is correct.
If there were other answer choices, we would contrast them similarly: any claim asserting orthogonality to X2 or claiming a valid OLS solution despite this nonzero inner product would be incorrect, because the math of OLS requires X'û = 0 for all columns of X.

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