# Does regression coefficient variance reduce with increased amount of data points?

For a model, $$y = \beta_0 + \beta_1 x + \epsilon$$, does the variance for $$\hat{\beta_0}$$ and $$\hat{\beta_1}$$ always decrease when more datapoints are added?

I've decuded the variance expressions as: $$\begin{equation} Var[\hat{\beta_0}] = \sigma^2 (\frac{1}{n} + \frac{\bar{x}^2}{S_{xx}}), \end{equation}$$ and $$\begin{equation} Var[\hat{\beta_1}] = \frac{\sigma^2}{S_{xx}}. \end{equation}$$

But I'm still yet to draw a conclusion, I can think of examples where $$S_{xx}$$ would remain constant if adding $$x_i = 0$$ to a set that already has $$\bar{x} = 0$$, so I don't think $$Var[\hat{\beta_1}]$$ has to decrease, however I'm not entirely sure. What can be said about $$Var[\hat{\beta_0}]$$? Obviously, it would decrease considering the same example as for $$\hat{\beta_1}$$ since $$1/n$$ naturally does, but can something be said in general?

Thank you.

Here is an example for when the estimated variance of the two coefficients is larger for a larger sample. (Your notation $$\sigma^2$$ might suggest that you are asking the question for the true error variance - here, of course, the additional data point also affects the error variance estimate.)

set.seed(9)
n <- 5
y <- rnorm(n)
y <- 3
x <- rnorm(n)

plot(x[-n], y[-n], ylim=c(-1,4), pch=19, col="red")
points(x[n], y[n], col="blue", pch=19)
abline(lm(y~x), lwd=2, col="blue")
abline(lm(y[-n]~x[-n]), lwd=2, col="red")
summary(lm(y[-n]~x[-n]))
summary(lm(y~x))


Essentially, the blue additional point is an outlier in the y direction that will not affect the estimate for $$\bar x$$, but will have a large residual and hence affect the error variance estimate. 