Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 1

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.

Among the supplied candidates, choose ARMA(2,1). It has the smallest Akaike information criterion. Relative to ARMA(2,2), its AIC improvement is $1.71$; the additional second moving-average estimate is only half a standard error from zero and increases the log likelihood function by only about $0.14$. Dropping it is a reasonable parsimony choice, although that AIC gap is small. ARMA(1,1) has AIC larger by $17.55$, a much clearer loss of fit. In the chosen model the second autoregressive term is about seven standard errors from zero, so it should not be dropped merely to obtain order one. The first autoregressive term is less precisely estimated; this does not justify automatically deleting it without fitting and comparing the reduced candidate.

Using the usual positive-sign moving-average convention, the fitted autoregressive moving-average model isThese are plug-in values rounded as given, not exact population parameters. The model has zero mean. Its polynomials are $\varphi (z)=1+0.30z-0.28z^2=(1-0.4z)(1+0.7z)$ and $\theta (z)=1+0.5z$.

With angular frequency $\omega \in [-\pi ,\pi ]$, so that the autocovariance is $\gamma (h)={\int }_{-\pi }^{\pi }{e}^{ih\omega }{f}_X(\omega )d\omega$, the spectral density of a stationary process isThe frequency convention makes the normalization unambiguous.

The autoregressive polynomial factors as $(1-0.4z)(1+0.7z)$, with zerosBoth have modulus greater than one. The causality root criterion for an autoregressive model therefore gives a causal stationary solution. The moving-average polynomial has its only zero at $-2$, also outside the unit circle, so the invertibility of a moving-average model holds. There is no common root to cancel.

To use unit-variance white noise, put $W_t=Z_t/2$. One suitable pair isThen $\tilde{\varphi }(B)X_t=\tilde{\theta }(B)W_t$ with $W\sim \mathrm{WN}(0,1)$. The factor two changes the innovation scale, not the zero of the moving-average polynomial.

Write $X_t={\sum }_{j\ge 0}{\psi }_j{Z}_{t-j}$, with innovation variance four. The generating function of this linear process isEquating coefficients gives ${\psi }_0=1$, ${\psi }_1=0.2$ and ${\psi }_j=-0.3{\psi }_{j-1}+0.28{\psi }_{j-2}$ for $j\ge 2$. Thus the first five coefficients, including lag zero, areFor all remaining lags, partial fractions giveThe coefficients are absolutely summable, so the series converges in mean square and defines the causal moving-average expansion. In the unit-variance convention of part (c), $X_t={\sum }_{j\ge 0}{c}_j{W}_{t-j}$ with $c_j=2{\psi }_j$; its first five coefficients are $(2,0.4,0.44,-0.02,0.1292)$. This is the two-geometric-coefficient expansion of a causal ARMA(2,1) process.

White noise orthogonality gives, for any integer $h$,To see this directly, expand the covariance of the two convergent series. Only matching noise indices contribute. Cauchy-Schwarz inequality makes the coefficient-product sum finite, justifying the covariance limit.

There is also a closed expression. Set $a=9/11$, $d=2/11$, $r=0.4$, $s=-0.7$. For $h\ge 0$,and negative lags follow by symmetry. Dividing by the expression at zero gives the same autocorrelation function.