Time Series Analysis

For the AR(1) model of the form (17.17), obtain representation (17.22) and find the covariance function $R_{\varepsilon }(k)$ of the form (17.23).

Table 19.7 presents statistical data on lynx trapping in Canada for the period 1821-1922.

Table 19.7

Show that the corresponding time series is described by the following model:

where $s_{t}$ is the seasonal component and $\varepsilon_{t}$ is the random component described by an AR(2) model.

Under the conditions of Example 19.3, show that the consumption volume $C_{t}$ is described quite accurately by an additive linear model with seasonal and random components:

where $s_{t}$ is the seasonal component and $\varepsilon_{t}$ is an AR(1) model. Construct estimates of the parameters and variables in the model for $C_{t}$ from the observations $t = 1, \ldots , 40$ and construct a forecast of $C_{t}$ for $t = 41, \ldots , 44$. Compare the constructed forecast with the actual values of $C_{t}$ and with the forecast $\widetilde{C}_{t}$ constructed in Example 19.3.

Table 19.8 presents data on the annual number of strikes in the USA for the period 1951-1980.

Table 19.8

It is required to construct a mathematical model of the corresponding time series and to construct a forecast of the number of strikes for the three years 1981— 1983.

Table 19.9 presents statistical data on the number of accidents per month in the USA from 1973 to 1978.

Table 19.9

Construct a mathematical model of the time series of the form

where $\varphi_{t} = \theta_{1}+\theta_{2} t+\theta_{3} t^{2}$, $s_{t}$ is the seasonal component, and $\varepsilon_{t}$ is the random component described by an AR(1) model.