In deriving ${Var}(\hat{\beta_1})$ in simple linear regression model, I came across following derivation. Why are the variances of non-random terms zero below?

\begin{align} \text{Var}(\hat{\beta_1}) & = \text{Var} \left(\frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sum_i (x_i - \bar{x})^2} \right) \\ &= \text{Var} \left(\frac{\sum_i (x_i - \bar{x})(\beta_0 + \beta_1x_i + u_i )}{\sum_i (x_i - \bar{x})^2} \right)\;\;\; \\ &= \text{Var} \left(\frac{\sum_i (x_i - \bar{x})u_i}{\sum_i (x_i - \bar{x})^2} \right), \;\;\;\text{noting only $u_i$ is a random variable} \\ \end{align}


closed as unclear what you're asking by Michael Chernick, Tim, kjetil b halvorsen, Nick Cox, Peter Flom Jun 7 '17 at 11:38

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    $\begingroup$ Where did you get the second equation? $\endgroup$ – Michael Chernick Jun 5 '17 at 21:19
  • $\begingroup$ Because they are never different from their mean. $\endgroup$ – AdamO Jun 5 '17 at 21:32
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    $\begingroup$ You don't actually need a definition or concept of variance of "non-random terms". This calculation merely uses the fact that $\operatorname{Var}(X)=\operatorname{Var}(X+c)$ for any random variable $X$ and constant $c$. That fact is readily derived from any of the usual definitions of variance. It is also intuitively obvious: since the variance is supposed to be a measure of the spread of a random variable, and shifting random variables (by adding a constant to them) doesn't change their spread, the variance shouldn't change. $\endgroup$ – whuber Jun 5 '17 at 23:17

In qualitative terms, non-random variables have a single value that does not vary, and thus have zero variance.

If it helps, you can think of scalar variables as having a distribution with a single infinitely sharp peek. This type of distribution can be represented in equations with a delta function. One way to derive the delta function is as the limit of the PDF of a Normal Distribution as its standard deviation goes to zero:

$\delta(t) = \lim_{\sigma\to0} \frac{e^{-t^2/2 \sigma^2}}{\sqrt{2 \pi} \sigma}$.

Since $\text{Var} = \sigma^2$, variance also goes to zero as standard deviation goes to zero. Thus, in this view non-random variables are the same as random variables, but with zero variance.


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