AR-, MA-, and ARMA-models

Suppose that an AR(1)-process, $\left\{ X_{t}, t = 0, \pm 1, \ldots \right\}$, has the covariance function $r(k) = 10 \cdot 0.5^{\left|k\right|}$. Which AR(1)-process is it? What is the variance of the innovations?

Find the spectral density, $R(f)$, and the covariance function, $r(k)$, at time lag zero, one and two, of the AR(2)-process

where $\left\{ e_{t}\right\}$ is a white innovation process with variance $\operatorname {Var}\left[e_{t}\right] = 4$.

The covariance function of a signal, $\left\{ Y_{t}\right\}$, in discrete time, has been well estimated at a few time lags:

We decide to use the following AR(2)-process to model the signal,

Find the innovation variance, the second AR-parameter, and the covariance function at time lag two.

The MA(q)-process

where $\left\{ e_{t}\right\}$ is a white innovation process with unit variance, has covariance function $r(0) = \frac{5}{4}$, $r(1) = 0$, $r(2) = \frac{1}{2}$, and $r(k) = 0$, $\left|k\right| \geq 3$. Determine the order of the process, the MA-coefficients, and the spectral density.

Let $e_{t}, t \in \mathbb {Z}$, be independent random variables with expectation $m$ and variance $\sigma^{2}$. Define, for a constant $M$,

Let $\left\{ X_{t}, t = 0, \pm 1, \ldots \right\}$ be a stationary AR(1)-process,

and let $\left\{ Y_{t}, t = 0, \pm 1, \ldots \right\}$ be an MA(2)-process,

where $\left\{ e_{t}\right\}$ and $\left\{ u_{t}\right\}$ are independent white Gaussian processes with unit variance. Realizations and estimated covariance functions of $\left\{ X_{t}\right\}$ and $\left\{ Y_{t}\right\}$ are shown in Figure 7.13, but which subplot shows what?

Figure 7.13 Realizations and estimated covariance functions of one AR(1)- and one MA(2)-process in Exercise 7:6.

Figure 7.16 shows pole-zero plots, spectral densities, and covariance functions for three AR(2)-processes. Determine which figures that describe the same process.

Figure 7.16 Pole-zero plots, covariance functions, and spectral densities of two processes in Exercise 7:8.

(This is a classical example of time series analysis in hydrology.) The Wabash river water level, $W_{t}$ (meter), is measured once every month. The following AR-type model has been suggested:

where $\left\{ e_{t}\right\}$ is a white noise process with variance 4. What is the expected water level according to this model? What is the standard deviation?

Let $\left\{ X_{k}, k \in \mathbb {Z}\right\}$ be an AR(1)-process, $X_{k} + a_{1} X_{k-1} = e_{k}$, with $\operatorname {Var}\left[e_{k}\right] = \sigma^{2}$. Construct a new process, $\left\{ Y_{m}, m \in \mathbb {Z}\right\}$, by using every second element of $\left\{ X_{k}\right\}$, that is: $Y_{m} = X_{2m}$. Prove that $\left\{ Y_{m}\right\}$ is an AR(1)-process and that it thus can be written in the following form:

Also, compute $a$ and $\operatorname {Var}\left[u_{m}\right]$.

Compute the cross-covariance, $\operatorname {Cov}\left[ X_{t+k}, e_{t} \right]$, $k = \ldots , -1, 0, 1, \ldots$, where $\left\{ X_{t}\right\}$ is an AR(1)-process: $X_{t} = 0.4 X_{t-1} + e_{t}$, and $\operatorname {Var}\left[e_{t}\right] = 2$.

A random process, $\left\{ X_{t}\right\}$, is generated by filtering white noise:

where $a_{1}$ and $\sigma^{2}$ are unknown. A simulation gives the following values:

Let $\left\{ X_{t}\right\}$ be stationary with mean value $m$ and assume

where $\left\{ e_{t}\right\}$ is Gaussian white noise with mean value zero and unit variance. The following values of $X_{t}$ are observed:

Estimate $m$ and compute a 95 confidence interval for your estimate.

Find the best one-step ahead predictor of the ARMA(1,1)-process defined by $X_{t} - 0.5 X_{t-1} = e_{t} + 0.5 e_{t+1}$.

Figure 7.17 shows the mean wave height, $X_{t}$, outside the Pakistan coast, as measured by a satellite passing the Indian Ocean on May 21, 1993. The process is well described by an AR(3)-model:

where $e_{t}$ is white noise with unit variance. Compute $r_{X}(4)$ and $r_{X}(5)$ from the first four covariances:

Figure 7.17 Wave height, spectrum, and covariance function estimated using data, and spectrum and covariance function of the model in Exercise 7:15.