Consider sheep , the sheep population of England and Wales...

Consider sheep , the sheep population of England and Wales from 18671939. a. Produce a time plot of the time series. b. Assume you decide to fit the following model: $$ y_t=y_{t-1}+\phi_1\left(y_{t-1}-y_{t-2}\right)+\phi_2\left(y_{t-2}-y_{t-3}\right)+\phi_3\left(y_{t-3}-y_{t-4}\right)+\varepsilon_t, $$ where $\varepsilon_t$ is a white noise series. What sort of ARIMA model is this (i.e., what are $p, d$, and $q$ )? c. By examining the ACF and PACF of the differenced data, explain why this model is appropriate. d. The last five values of the series are given below: $\begin{array}{llllll}\text { Year } & \mathbf{1 9 3 5} & \mathbf{1 9 3 6} & \mathbf{1 9 3 7} & \mathbf{1 9 3 8} & \mathbf{1 9 3 9} \\ \text { Millions of sheep } & 1648 & 1665 & 1627 & 1791 & 1797\end{array}$ The estimated parameters are $\phi_1=0.42, \phi_2=-0.20$, and $\phi_3=-0.30$. Without using the forecast function, calculate forecasts for the next three years (1940-1942). e. Now fit the model in R and obtain the forecasts using forecast . How are they different from yours? Why?

Consider sheep , the sheep population of England and Wales from 18671939.
a. Produce a time plot of the time series.
b. Assume you decide to fit the following model:
$$
y_t=y_{t-1}+\phi_1\left(y_{t-1}-y_{t-2}\right)+\phi_2\left(y_{t-2}-y_{t-3}\right)+\phi_3\left(y_{t-3}-y_{t-4}\right)+\varepsilon_t,
$$
where $\varepsilon_t$ is a white noise series. What sort of ARIMA model is this (i.e., what are $p, d$, and $q$ )?
c. By examining the ACF and PACF of the differenced data, explain why this model is appropriate.
d. The last five values of the series are given below:
$\begin{array}{llllll}\text { Year } & \mathbf{1 9 3 5} & \mathbf{1 9 3 6} & \mathbf{1 9 3 7} & \mathbf{1 9 3 8} & \mathbf{1 9 3 9} \\ \text { Millions of sheep } & 1648 & 1665 & 1627 & 1791 & 1797\end{array}$
The estimated parameters are $\phi_1=0.42, \phi_2=-0.20$, and $\phi_3=-0.30$. Without using the forecast function, calculate forecasts for the next three years (1940-1942).
e. Now fit the model in R and obtain the forecasts using forecast . How are they different from yours? Why?

Key Concepts

Time Series Visualization

This concept involves plotting data points in time order, typically on a line graph, to visually inspect underlying patterns, trends, seasonal variations, and potential outliers. Such a plot is one of the first steps in time series analysis as it provides intuitive insight into the data's temporal structure and can help guide subsequent modeling decisions.

ARIMA Model Identification

ARIMA models combine autoregressive (AR) components, differencing (I for integration), and moving average (MA) components, denoted by the parameters (p, d, q). Identifying the appropriate model involves determining how many past values influence the current value (p), the order of differencing needed to achieve stationarity (d), and the number of lagged forecast errors to include (q).

Differencing and Stationarity

Stationarity is a key assumption in many time series models, where statistical properties such as mean and variance remain constant over time. Differencing transforms a non-stationary time series into a stationary one by subtracting previous observations from the current observation, effectively removing trends and stabilizing the mean.

Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF)

ACF and PACF are essential diagnostic tools used to examine correlations in time series data. The ACF measures the correlation of the series with its own lagged values, while the PACF controls for the correlations attributable to the intervening lags. These tools help in identifying the order of AR and MA terms in an ARIMA model by revealing the lag structures that significantly contribute to the model.

Time Series Forecasting

Time series forecasting uses established models, such as ARIMA, to predict future values based on historical data. Accurate forecasting involves applying the fitted model’s parameters to project future behavior, and comparing manual calculations with automated forecasts can help validate the model’s adequacy or highlight differences due to methodological or computational nuances.

Model Evaluation and Forecast Comparison

Evaluating a model requires comparing manual calculations with those produced by computational tools, such as the forecast function. Differences may arise from factors like computational approximations, the treatment of initial conditions, or the inclusion of additional adjustments in the automated methods. This concept emphasizes the importance of understanding both theoretical and practical implementation aspects in forecasting.

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