Purpose
If two non stationary series are regressed, the t stat increases

> set.seed(1977)
> N <- seq(1000, 1e+05, 5000)
> results <- numeric(0)
> for (i in seq_along(N)) {
+     y <- cumsum(rnorm(N[i]))
+     x <- cumsum(rnorm(N[i]))
+     fit.sum <- summary(lm(y ~ x))
+     results <- c(results, coef(fit.sum)[2, 3])
+ }
> plot(N, results, pch = 19, col = "blue", ylab = "tstat", xlab = "N")

C3_1-001.jpg

clearly t stat diverges

If two stationary series are regressed, the t stat increases

> set.seed(1977)
> N <- seq(1000, 2 * 1e+05, 5000)
> results <- numeric(0)
> for (i in seq_along(N)) {
+     y <- arima.sim(n = N[i], model = list(ar = 0.9))
+     x <- arima.sim(n = N[i], model = list(ar = 0.95))
+     fit.sum <- summary(lm(y ~ x))
+     results <- c(results, coef(fit.sum)[2, 3])
+ }
> plot(N, results, pch = 19, col = "blue", ylab = "tstat", xlab = "N")

C3_1-002.jpg

clearly t stat converges when you regress 2 stationary processes.