Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 40 / 2

Difference the local-level state-space model to eliminate its latent level:Independence and Gaussianity give a stationary Gaussian process withFor $V,W>0$, letThese satisfy $q(1+{\vartheta }^2)=W+2V$ and $q\vartheta =-V$. Therefore $D$ has the covariance of the moving-average model ${\epsilon }_t+\vartheta {\epsilon }_{t-1}$, where $\epsilon$ is Gaussian white noise of variance $q$. This is an actual representation, not just covariance matching: define ${\epsilon }_t={\sum }_{j\ge 0}(-\vartheta {)}^j{D}_{t-j}$. The convergent inverse filter has constant spectral density $q/(2\pi )$, so its Gaussian coordinates are independent, and multiplication by $1+\vartheta B$ recovers $D$.

ThusThis is the formal ARMA equation with autoregressive coefficient one. Its unit root means that the levels are generally not stationary: if the initial state has finite variance and is independent of future noises, $\mathrm{Var}(S_t)=\mathrm{Var}(S_0)+tW$. In stationary-model terminology the correct classification is ARIMA(0,1,1). The requested ARMA description must therefore allow a unit-root equation.

When $V=0$, the differences are just $w_t$; when $W=0$ and $V>0$, they are $v_t-v_{t-1}$, a noninvertible MA(1) with coefficient $-1$. If both variances vanish, the model is deterministic apart from a possible initial random level. These limits explain the qualifications on the invertible representation above.

Conditionally on past observations, adding the independent state increment gives the predictionThe new observation then has conditional distribution $N({\hat{S}}_{t-1},R_t+V)$, and its conditional covariance with $S_t$ is $R_t$. Equivalently, multiplying the state prediction probability density function by the observation likelihood and completing the square givesFor positive variances, rearrangement yields the scalar Gaussian Kalman recursionHere the innovation is the observation minus its predicted mean. The displayed covariance formula also handles zero variances whenever its denominator is nonzero; a completely deterministic prediction and observation require no probabilistic update. Independence of the new noises from the past and the initial state is the usual state-space assumption needed for this conditional calculation.

A constant posterior variance must be a fixed point of the recursion in part (b). Multiplying $P=V(P+W)/(P+W+V)$ by its positive denominator givesConversely, if a nonnegative $P$ solves this equation and the filter is initialized with $P_0=P$, the recursion preserves $P_t=P$ at every step. The fixed-point equation alone does not force an arbitrary initial covariance to be constant.

For $W,V>0$, the admissible root and its steady-state local-level Kalman gain areThe state estimate then obeys exponential smoothing:Iteration makes the weights explicit:Thus older observations receive geometrically decreasing weights. Consistently with part (a), the innovations variance is $q=P+W+V$ and the differenced MA coefficient is $\vartheta =-(1-K)$. The degeneracies $W=0,P=0$ and $V=0,P=0$ give gain zero and gain one respectively when the denominator is positive.