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During my research for a class, I came across a paper that said they estimated an equation using OLS. But the parameter they were estimating appeared to be an expression that looked like this (not the actual equation just something I made up to illustrate):

$$ y = \frac{\beta-1}{2}\chi $$

Rather than what I am used to seeing for OLS:

$$ y = \beta_{0}X_{0} + \beta_{1}X_{1} $$

How does one estimated the equation that way? Or am I missing something?

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  • $\begingroup$ Could it be that they estimated the expression in front of $X$ as a whole by OLS and then obtained $\beta$ by multiplying the obtained value by two and adding one? $\endgroup$ – Richard Hardy Feb 25 '15 at 6:58
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    $\begingroup$ It would be impossible to tell without a reference or a description of what $\chi$ here is. $\endgroup$ – StatsStudent Feb 25 '15 at 7:09
  • $\begingroup$ What was the paper? $\endgroup$ – Glen_b Feb 25 '15 at 10:25
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As Richard Hardy said in his comment, in this case you can simply estimate $y=\delta X + \epsilon$. Since $\delta = \frac{\beta - 1}{2}$, you can calculate $\beta = 2\delta + 1$. You do this kind of thing when your regression equation comes from a theoretical model.

Standard errors are simple to calculate in a similar way, using the rules for linear transformations of variances.

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