How are $n$ and $Var(\varepsilon)$ affecting to Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression

Once I have derived the variance of $$\hat{\beta_1}$$ as:

$$\text{Var}(\hat{\beta_1})= \frac{\sigma^2}{\sum(x_i-\overline{x})^2}$$

I would like to know how are affecting to this formula:

• the size of the dataset $$n$$ and
• the variance of residuals $$\text{Var}(\varepsilon)$$.

Any idea about how to check it? I mean i.e.: A higher $$n$$ will make $$\text{Var}(\hat{\beta_1})$$ increase/decrease, so on. I am really stuck on finding any kind of relationship between them.

• If $\sigma^2$ isn't defined as $\operatorname{Var}(\varepsilon),$ then what is it?
– whuber
Jan 2 '20 at 23:42

Nick is technically right. But note that if you assume that you have a series of random samples from the underlying population, larger sample sizes (larger $$n$$) cause $$Var(\hat\beta)$$ to decrease asymptotically.
To see this, note that $$\frac{1}{n} \sum (x_i - \bar{x})^2 \overset{p}{\to} Var(x)$$ so $$\sum (x_i - \bar{x})^2 \overset{p}{\to} n \,Var(x) = \infty$$ (where $$\overset{p}{\to}$$ denotes convergence in probability). So the denominator goes to infinity as the sample size increases. You can similarly show that the numerator converges to a constant (namely the variance of the errors). So the variance of $$\hat\beta$$ goes to $$0$$ as the sample size gets large.
Just look at the formula. A larger n won't definitely make $$Var(\hat{\beta}_1)$$ increase or decrease. The variance only depends how spread out the data is from the mean. You could have a very large n, with data points that are close to $$\bar{x}$$ (and therefore, a very large variance) or you could have a very large n with data points that are spread out, and have a small variance. It's not directly influenced by the number of data points.
Then as whuber mentioned, $$\sigma^2$$ is the variance of the errors (which we usually estimate with the variance of the residuals). According to the formula, a larger error variance will result in a larger variance in $$Var(\hat{\beta}_1)$$.