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In my econometrics course, we derived the value of the variance of $\hat \beta$ in an OLS regression. I am slightly confused by the motion, wouldn't $\hat \beta$ be a constant, and therefore $Var(\hat \beta) = 0$?

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If you obtained several samples, $\hat{\beta}$ would be different for each sample.

That is, conditional on the predictors $\hat{\beta}$ is a random variable -- a function of the random variables $Y_1,Y_2,...,Y_n$ of which a particular set of observations ($y_1,y_2,...,y_n$) is a realization.

Since $\hat{\beta}$ is a linear function of the $Y's$ you can infer various of its properties -- for example, under typical assumptions you can determine the expectation and variance of its distribution by basic properties of mean and variance.

In respect of a given sample, you have a realization of $\hat{\beta}$ and yes, that's a fixed number. However, you presumably want to try to understand something about $\beta$ not just your estimate of it, so knowing how $\hat{\beta}$ behaves is important -- e.g. knowing that under certain conditions $E[\hat{\beta}]=\beta$ and knowing how big $\hat{\beta}-\beta$ might tend to be would both be useful.

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