I am reading a textbook on econometrics. in this textbook the distribution of beta-hat and alpha-hat are computed. I have a question. if the beta-hat and alpha-hat are computed by OLS, shouldn't they be just two numbers? what is the distribution for? is it because of other methods or other data sets? I am confused. ps: I am completely new to statistics and econometrics...:)
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2$\begingroup$ They are parameters that are estimated. Once estimated they indeed are read as plain numbers. Since they are estimated, there is a variance to the estimate that is also estimated based on the noise in the data. $\endgroup$– TonyCommented Jan 21, 2017 at 13:06
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1$\begingroup$ @Tony You could add a sentence or two and make that an answer. $\endgroup$– Peter FlomCommented Jan 21, 2017 at 13:16
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The $\hat \alpha $ and $\hat \beta$ commonly seen in regression equations are parameters that are estimated. They indeed represent numbers, but we don't know their value. Once they are estimated the estimated numbers can be substituted in their place.
The distribution of the parameters comes from the fact that there is uncertainty in the estimation process (due to possible noise and random sampling). If a different set of samples was drawn from the population, the estimates would vary. The distribution of the parameter estimates is generally assumed to be normal, with the estimate as the mean, and variance that is generally rather easy to estimate as well, based on the residuals.
