Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 208 / 3

The labels in PDF Figure 2 match as follows:Series $1$ has a wandering level and an overall upward drift. Its empirical autocorrelation function stays close to $1$ and declines very slowly, matching panel $c$; this is characteristic of a trend or a record from a unit-root autoregressive process. Such an empirical ACF is not a population stationary ACF if the underlying model is a random walk.

Series $2$ shows relatively smooth runs of adjacent observations on the same side of its level. Positive short-lag dependence emphasizes low frequencies, matching the decreasing spectrum $d$.

Series $3$ is rougher, with much more rapid fluctuation. Negative short-lag dependence emphasizes high frequencies, matching the increasing spectrum $a$. A negative-coefficient AR(1), for example, has this type of spectrum; the figure does not uniquely identify its exact order or parameters.

Series $4$ has conspicuous seasonality. Its autocorrelation function should alternate between positive peaks at full seasonal periods and negative troughs between them, matching $b$. Here “characteristic function” is used descriptively for an ACF or spectrum, rather than for the probability-theoretic characteristic function of a random variable.

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.

Put $c_t=\cos (2\pi t/S)$ and assume ${\sigma }^2>0$. ThenThus $Y$ generally has a periodic mean, and $Z$ generally has a periodic variance. The edge cases allowed by $S\in {\mathbb{N}}^{\ast }$ matter: $Y$ is stationary when $S=1$, because the mean is then constant, and $Z$ is stationary for $S=1$ or $S=2$. In the latter case multiplying iid centered Gaussian white noise by $(-1{)}^t$ leaves its iid distribution unchanged. For every $S\ge 3$, $c_0^2=1$ but $c_1^2<1$, so $Z$ is nonstationary. For $S>1$, $Y$ is nonstationary because $c_t$ is not constant.

Use the seasonal difference operator ${\Delta }_S=1-B^S$. Since $c_t=c_{t-S}$,This is a stationary moving-average model of iid Gaussian white noise. Its covariance is $2{\sigma }^2$ at lag $0$, $-{\sigma }^2$ at lags $\pm S$, and zero otherwise.

For the variance-modulated process the same operation givesIts variance is $2{\sigma }^2{c}_t^2$ and its covariance at lag $S$ is $-{\sigma }^2{c}_t^2$. Therefore it remains nonstationary for $S\ge 3$, and the operation doubles the marginal variance at nonzero seasons. Seasonal differencing does not remove periodic variance: it removes a periodic mean, but the periodic variance generally remains. Practical alternatives are a periodic model or seasonal variance standardization; at seasons with $c_t=0$ the observations are deterministically zero, so division by $c_t$ is not possible there. For the already stationary special cases $S=1,2$, filtering preserves stationarity.

A periodically correlated process with period $S$ has its mean and two-time covariance unchanged when both times are shifted by $S$. For $Z$, the mean is zero andHence $Z$ is a periodically correlated process. Its covariance period can be smaller than $S$: when $S$ is even, $c_{t+S/2}^2=c_t^2$, so $S/2$ is already a covariance period.

Use the usual normalization that ${\epsilon }_t$ is mean-zero unit-variance white noise orthogonal to the past, and extend $\varphi (\nu ),\sigma (\nu )$ periodically in $\nu$. A nondegenerate causal periodic autoregressive model of order one requires $|{\prod }_{\nu =1}^T\varphi (\nu )|<1$. Write $V_{\nu }={\gamma }_{\nu }(0)$, with season indices understood modulo $T$.

For $h\ge 1$, multiply the recursion by $X_{nT+\nu -h}$ and use orthogonality of the current innovation to the past. The periodic Yule-Walker equations areAt lag zero, expansion of the squared recursion givesIn particular, ${\gamma }_{\nu }(1)=\varphi (\nu )V_{\nu -1}$, so an equivalent variance equation is $V_{\nu }=\varphi (\nu ){\gamma }_{\nu }(1)+\sigma (\nu {)}^2$. Negative lags are obtained from the given symmetry ${\gamma }_{\nu }(-h)={\gamma }_{\nu +h}(h)$.

Replacing covariances by their sample versions givesTake the nonnegative scale and require a positive estimated predecessor variance. Using matched seasonal sample covariance matrices makes the quantity under the square root nonnegative by Cauchy-Schwarz inequality; with unmatched covariance estimates one can instead fit seasonal ordinary least squares and use the nonnegative residual variance. Statistical consistency of the covariance estimates gives statistical consistency of these plug-in estimators. If the driving noise variance were not normalized to $1$, the equations would identify $\sigma (\nu {)}^2\mathrm{Var}({\epsilon }_t)$, not the two scales separately.