Question

For the following four data sets, your objective is to come up with an appropriate ARIMA model (seasonal or non-seasonal).

Sulfer dioxide series so2

Crude oil prices oil

Global temperature data gtemp

Johnson and Johnson earnings jj

Your answers should include the following components.

Plot of the data.

Box-Cox transformation if necessary, and the plot of the transformed data. Note that if a transformation is necessary, the transformed data must be used throughout.

Use appropriate techniques (if necessary) to remove trend and seasonal variations. Explain clearly what method(s) was used. Also submit the plot.

Plot of ACF and PACF. Explain clearly how you use them to determine a range of ARIMA model. Make sure to use differencing if necessary.

Using certain criterion, determine an optimal ARMA(p, q) model.

Using hypothesis testing methods, check if certain parameters of the ARMA model can be removed.

Performed diagnostic check for the model you obtained. Submit the appropriate plots. Make sure to use Box-Ljung statistics to test for white noise. If diagnostic check failed, adjust your model and start all over. Compare all possible models you considered with AIC values and p-values of the Box-Ljung statistics. Determine the final model.

Write the equation of the final model with clearly indicating the AR and MA coefficients. What is the estimate of the white noise variance? What does it tell you?

Forecast the next 20 values, and submit the plot showing the data with forecast values together with their prediction intervals. State the forecasting values with their standard errors.

Answer 1

The ARIMA models are appropriate for modeling time series with trend characteristics , random walk processes, and seasonal and non seasonal time series.This family includes models that are combinations of autoregressive and moving average processes for stationary and non stationary time series .

More specifically ,the ARIMA (p,d,q) (P,D,Q)model describes the value of a time series as a function of the order of the autoregressive (p.P) ,integrated (d,D) and moving average (p,Q) parts of the model ,where p,q,D,Q are non-negative integers .The number of values (lags and differences)involved inAR,I and MA processes is referred to as the order of the respective effects .The number of parameters, as well as their values ,determines the properties of the given ARIMA model .

The BOX PROCEDURE involves identifying an appropriate ARIMA model ,fitting it to the data and cheaking and diagnostic of the model .THis algorithm is widely used for time series modelling .

The procedure consists of three iterative steps

identification (choice of three model parameters)

Estimation (assessment of the parameters values )

checking and diagnostic of the model going back to the model identification step if the previous assumptions are not satified .on this step the assumption of ARIMA model are checked ,for example that the errors are independently and normally distributed .

The Box procedure can be presented graphically as

The major tools used in the   identification step are autocorrelation function(ACF),and partial autocorrelation function (PACF) .The concept of stationarity and idea of ACF and PACF will be presented further in the course

BOX is subjective in a sense that the result depend to a great degree on the analysts experiemece and background .Because of that it may happen that time series experts will choose different ARIMA models for the same time series .Alternatively it is possiable to apply one of the various automatic modelling procedure .,the automatic procedure implemented in X12 ARIMA which searches through a prederemined list of candidate models to find an satisfactory fit .

Time series simple forecasting

Once an ARIMA model for a time series has been validated ,it may be used to produce forecasts for different periods in the future or in the past .The observation of the past of the series enables to predict ,to some extent ,the future values of the time series .For example crude price strong movement that makes easy to predict the future .