# ADF test suggesting incorrectly that series is stationary

The code below generates series y, which by design is clearly non-stationary. The ADF test below was run with 12 lags to yield (what visually appear to be) uncorrelated residuals and it would have us conclude that y is stationary. What went wrong here?

set.seed(100)
y<-rep(NA,100)
for (i in 1:100) {
y[i]<-rnorm(1,mean=0,sd=i)
}
par(mfrow=c(1,3))
plot(y,type="l",main="y")

u<-urca::ur.df(y=y, type = "none",lags=12)
summary(u)
forecast::Acf(u@res,lag.max=70,type="correlation",main="ACF",xlab="")
forecast::Acf(u@res,lag.max=70,type="partial",main="PACF",xlab="")


The augmented Dickey-Fuller (ADF) test does not have an alternate hypothesis that the data "are stationary." Rather, the ADF tests for evidence that the coefficient $$\beta$$ in the below equation is not equal to 0 (equivalent to testing whether $$\rho=1$$ in the un-augmented Dickey Fuller test):
$$\Delta y_{t} = \alpha + \beta y_{t-1} + \delta t + \zeta_1 \Delta y_{t-1} + \dots + \zeta_{k}\Delta y_{t-k} + \varepsilon_{t}\text{where }\varepsilon \sim \mathcal{N}(0,\sigma^{2})$$
While it is true that a $$\rho = 1$$ (or $$\rho = -1$$) imply nonstationarity of a time series' data, but note that it is possible for $$\beta=0$$, and yet for $$\sigma^{2} = f(t)$$, so the ADF's usefulness in providing evidence that data are stationary is not robust to this issue.