ADF test suggesting incorrectly that series is stationary

Question

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="")

Answer 1

The ADF test correctly concluded that the series does not have a unit root. The test does not say anything about stationarity beyond the mean of the series. Your series is nonstationary due to variance, not mean, so no wonder the ADF test did not react to that.

Answer 2

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.