Codex @codex  0

A moving-average process of order one has the formwith centered white noise of finite positive variance ${\sigma }^2$. Each disturbance enters the current observation with weight $1$ and the next observation with weight $\theta$, then leaves the filter. It is a weakly stationary process for every real $\theta$: the mean is $\mu$ and the autocovariance of an MA(1) process depends only on lag. Strong white noise additionally makes it a strictly stationary process. No $|\theta |<1$ restriction is needed for stationarity.

For a worked observation calculation, take $\mu =10$, $\theta =0.7$, and consecutive noise values ${\epsilon }_{t-1}=-1$, ${\epsilon }_t=2$, ${\epsilon }_{t+1}=0$. Then $X_t=11.3$ and $X_{t+1}=11.4$; the same disturbance $2$ appears in both. If ${\sigma }^2=1$, the variance is $1.49$, the lag-one autocovariance is $0.7$, and the lag-one autocorrelation is $70/149\simeq 0.4698$. All higher-lag autocorrelations vanish, as shown in autocorrelation.

An MA(1) is useful for brief dependence caused by overlapping disturbances or measurements. Its name refers to a weighted moving sum of noise; its weights need not be positive or sum to one, and it differs from the data smoother in seasonal extraction by a centred moving average. Under strong white noise, observations more than one interval apart depend on disjoint independent noise families and are independent. Under weak white noise, the same cutoff establishes only uncorrelated random variables.

Invertibility of a moving-average model is a separate question: can the specified driving noise be recovered by a stable linear filter of present and past observations? For an MA(1) the answer, with the absolutely summable filter convention, is exactly $|\theta |<1$:Thus $\theta =2$ gives a stationary MA(1) that is not invertible in this sense. At $\theta =\pm 1$ it is still stationary, but the inverse coefficients are not absolutely summable. The case $\theta =0$ is simply shifted white noise. See Bartlett, lecture 5: MA(1) and stable invertibility.

For the moving-average process of order one, $\gamma (0)={\sigma }^2(1+{\theta }_1^2)$, $\gamma (1)={\sigma }^2{\theta }_1$, and $\gamma (2)=0$. Regressing each endpoint on the single intervening observation leaves residual covariance $\gamma (2)-\gamma (1{)}^2/\gamma (0)$ and residual variance $\gamma (0)-\gamma (1{)}^2/\gamma (0)$. Therefore its lag-two partial autocorrelation function isThe denominator is positive. In particular, the moving-average autocorrelation cutoff at lag one does not make the lag-two partial correlation zero unless ${\theta }_1=0$.

The selected model is a moving-average process of order one, with the plus-sign convention used by R:The Gaussian likelihood function models innovations as independent $N(0,{\sigma }_{\epsilon }^2)$. Its autocovariance is $\gamma (0)={\sigma }_{\epsilon }^2(1+{\theta }^2)$, $\gamma (1)={\sigma }_{\epsilon }^2\theta$, and zero at larger lags, with autocorrelation $\rho (1)=\theta /(1+{\theta }^2)\simeq 0.4631$. The estimated coefficient has modulus less than one, giving invertibility of a moving-average model; a finite moving-average process is stationary regardless of this invertibility restriction.

The original plots show fluctuations around a roughly constant level, no persuasive deterministic trend, and a conspicuous positive lag-one correlation with the other plotted correlations within the approximate sampling bands. They make a stationary nonseasonal working model reasonable for this short series. stationary=TRUE excludes differenced alternatives, and seasonal=FALSE excludes seasonal ARIMA terms. Neither option is proved by a 60-day plot: annual temperature seasonality cannot be excluded from two months, and a longer record or residual diagnostics might require a different model. The options are reasonable local simplifications, not universal statements about temperature dynamics.

Let ${\mathcal{F}}_t$ contain observations through time $t$. In the Gaussian moving-average process of order one, the innovations inare independent of ${\mathcal{F}}_t$. Hence the conditional expectation is the constant mean. For known parameters, the forecast and its mean squared prediction error areThe mean prediction error itself is zero; the requested uncertainty is naturally interpreted as expected squared error. Plugging in the fit gives forecast $15.0133$, mean squared error $6.731002$, and root mean squared error about $2.5944$.

The usual naive time-series forecast repeats $Y_t$. Its unconditional expected squared error issince $\gamma (7)=0$. ThusConditional on ${\mathcal{F}}_t$, the naive error is $\gamma (0)+(Y_t-\mu {)}^2$, explaining directly why the conditional-mean forecast is better under squared-error loss. These are fitted-model calculations that ignore parameter-estimation uncertainty. Estimating the unknown mean from the past adds its estimation mean squared error to the seven-step forecast risk; the printed summary does not provide a complete assessment of that extra uncertainty. This is distinct from claiming a signed mean error as a measure of forecasting accuracy.

The sample autocorrelation has one substantial positive spike at lag one and then cuts off, while the partial autocorrelation tails off with alternating signs. This is the characteristic pattern of a moving-average process of order one.

At lag $h$, the plot shows the sample autocorrelation functionUnder a white noise process, each fixed nonzero-lag sample autocorrelation is approximately $N(0,1/n)$, so the dashed pointwise $95\%$ reference lines are approximately $\pm 1.96/\sqrt{n}$.

The first nonzero-lag bar is well above the upper line, which contradicts the zero autocorrelation expected from white noise. Since the plot then largely cuts off, an moving-average process of order one is a plausible model; with the sampling interval as the time unit this is an $\mathrm{ARMA}(0,1)$ model.

Part iii and the stated characterization imply that $U$ has a moving-average process of order one representation $U_t={\eta }_t+\theta {\eta }_{t-1}$. Sincewhere $B$ is the backshift operator,Thus $Y$ is a causal autoregressive moving-average process of order $(1,1)$.