Solved – Can we calculate the standard error of prediction just based on simple linear regression output

predictionregressionstandard errorstata

The standard error of prediction in simple linear regression is $\hat\sigma\sqrt{1/n+(x_j-\bar{x})^2/\Sigma{(x_i-\bar{x})^2}}$.

My question is to calculate the standard error of prediction for $pop=1029$ just based on the following regression output. I can get all except $\bar{x}$. And I also know how to calculate the approximate standard error of prediction based on the standard errors of intercept and coefficient of $pop$, ignoring their correlation.

Stata regression output

Best Answer

The question is to calculate the following statistic from the above regression output:

$$s.e.(\hat\mu|x_j)=\hat\sigma\sqrt{1/n+(x_j-\bar{x})^2/\Sigma{(x_i-\bar{x})^2}}.$$

The answer is inspired by @whuber:

  • get $\hat\sigma$ from $\hat\sigma^2=SS_{Residual}/(n-p-1)$, where $p=1$;
  • $n$ and $x_j$ are known;
  • obtain $\bar{x}$ from $\hat{Var}(β_{cons})=\hatσ^2(1/n+\bar{x}^2/\Sigma{(x_i-\bar{x})^2})$;
  • $\Sigma{(x_i-\bar{x})^2}=SS_{Model}/\hat{\beta}_{pop}^2$.
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