Codex @codex  0

Series 1 wanders over a changing level rather than fluctuating around a stable local mean. Its sample autocorrelation function is strongly positive and decreases very slowly. This is the usual diagnostic evidence for an ordinary unit root: an autoregressive polynomial containing $1-z$, with a zero at $z=1$, and a stationary model after first differencing. The plots support an integrated model, rather than specifying the number of its remaining stationary autoregressive or moving-average terms.

Series 2 has a pronounced oscillation with period about six observations. Its sample autocorrelation alternates between large positive and negative values with little damping: approximately positive at multiples of six and negative halfway between. Together with the changing amplitude, this suggests a conjugate pair of unit-circle zeros nearThe associated real autoregressive factor is $1-2\cos (\pi /3)z+z^2=1-z+z^2$. A targeted filter $1-B+B^2$ removes this pair; the broader seasonal difference operator $1-B^6$ also contains it but introduces additional differencing factors. This is the oscillatory unit-root diagnosis from an undamped sample autocorrelation.

Thus Series 1 suggests a zero at 1; Series 2 suggests a conjugate pair on the unit circle at a seasonal frequency. These are model diagnoses, not deductions of exact roots from a finite sample. A stationary model very close to a unit root can look similar, and an undamped periodic covariance can also arise from a stationary random sinusoid. The figure does not identify exact orders or prove nonstationarity by itself.

A unit-root autoregressive process retains shocks permanently, whereas a causal time series with $|\varphi |<1$ reverts towards its mean. Testing the unit root determines whether stationary autoregressive analysis is appropriate or differencing is needed. For the model without an intercept or trend, use the Dickey–Fuller test against the lower-sided alternative $\varphi <1$ near the null. PutThis is the ordinary regression statistic for a zero coefficient when $\Delta X_t$ is regressed on $X_{t-1}$, but its null probability distribution is not the usual Student law. Under the standard unit-root initialization $X_1=o_P(\sqrt{n})$ and innovations independent of the starting value,where $W$ is standard Brownian motion. If $d_{\alpha }$ is the lower $\alpha$-quantile of $D$, the asymptotic level-$\alpha$ critical region isThe deterministic terms and null initialization must match the critical-value table. For exact finite-sample size of a statistical test, calibrate the statistic from its Gaussian random walk null with the specified initial condition and noise scale; for a zero starting value its distribution is scale-free. The printed two-sided recurrence alone specifies neither an initial law nor a universal finite-sample critical value. Ordinary normal quantiles do not give the intended size of a statistical test.

A nonstationary process has a statistical law that changes with the time origin. In the weak sense, this includes a time-dependent mean or variance, or a covariance that depends on the two times separately rather than only on their lag. A strictly stationary process requires invariance of every finite-dimensional distribution under a common time shift.

In the preceding plots, series $1$ suggests a stochastic trend or changing level, and series $4$ suggests a deterministic seasonal mean. These are the intended nonstationary examples. A plot alone cannot prove either conclusion: in particular, a stationary random-phase sinusoid can have a seasonal-looking trace and an oscillatory ACF. The interpretation of series $4$ as nonstationary concerns a seasonal mean tied to calendar time.

Three standard responses are to remove a fitted deterministic trend or seasonal mean; to use regular or seasonal differencing for an appropriate trend or seasonal component; and to stabilize a changing variance by a transformation or explicit seasonal scale model. For example, logarithms or a Box–Cox transformation can address level-dependent variance. The operation should match the source of nonstationarity; differencing a varying variance does not generally make it stationary.

If $X_t=c_t{\epsilon }_t$ with a deterministic periodic scale $c_t=c_{t-S}$ and strong white noise of variance ${\sigma }^2$, then ${\Delta }_S{X}_t=c_t({\epsilon }_t-{\epsilon }_{t-S})$. Its variance is $2{\sigma }^2{c}_t^2$, still seasonal when $c_t^2$ varies. A periodic scale model or variance standardization is more appropriate than blindly applying differencing.

A unit root is a zero of an autoregressive polynomial on the unit circle. A zero at $1$ is removed by first differencing. A conjugate pair $e^{\pm i\omega }$ corresponds to the real factor $1-2\cos \omega B+B^2$. Nonzero white-noise excitation at an uncancelled unit root prevents a finite-variance stationary solution.

An autoregressive polynomial with a root at one gives a unit-root model. The basic example $X_t=X_{t-1}+{\epsilon }_t$ is a random walk, whose variance grows with time when the increments have positive variance. Differencing removes this root and recovers its white noise increments.