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Suppose an estimated simple linear regression equation is given as $$E[y| \textbf{x}] = \hat{\beta}_{0}+ \hat{\beta}_{1}\textbf{x}$$

Then the interpretation of the slope is as follows: For a unit increase in $\textbf{x}$, $E[y| \textbf{x}]$ increases by $\hat{\beta}_{1}$. Can this same equation be used for predicting $y$ from some given $\textbf{x}$?

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Yes.

The difference between regression to find an association, and regression to provide prediction (for the scenario you've given), comes largely from how variables are selected and the like. The regression equation you just gave is usable for both purposes.

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  • $\begingroup$ So I don't have to change it to $\hat{y}_{i} = \hat{\beta}_{0}+\hat{\beta}_{1} \textbf{x}_i$? $\endgroup$
    – Laurent
    Jan 22, 2012 at 22:14
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    $\begingroup$ It is true that you can use the least squares estimates of $\beta$ to predict, I think it is worthwhile to mention that, for example, a ridge regression estimator (or another regularized regression estimate) of $\beta$ may provide better prediction than the least squares estimator and would also not satisfy $E(y|x) = \beta_{0} + \beta_{1}x$. $\endgroup$
    – Macro
    Jan 22, 2012 at 22:17
  • $\begingroup$ You'd just be calculating the expectation of y given a number of constants. So no, though your suggested rewrite might make it more clear. $\endgroup$
    – Fomite
    Jan 22, 2012 at 22:18
  • $\begingroup$ @Macro Fair - in an extended sense, there are other ways to provide predictions beyond what was provided. My answer is restricted to what happens with this particular scenario. And has been edited to reflect that. $\endgroup$
    – Fomite
    Jan 22, 2012 at 22:21

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