ARIMA Time Series Analysis

Models in Time Series Analysis enable the user to generate forecasts of a (dependent) time series that is based upon the information of its own past, explain events that occurred in the past, and provide insight into the dynamical interrelationships between variables. For obvious reasons these methodologies can only apply to time series. Above that, the steps or intervals of the time series under investigation are always supposed to be equally spaced in time (which is an important restriction). Furthermore we assume that each observation of the time series has the same expectation function, standard deviation, and probability distribution function.

Before discussing ARIMA Time Series Analysis, let’s see what Time Series Analysis is: The analysis of time series provides valuable information for current economic research and forecasts. It presents the economic data observed over time without those fluctuations that make it difficult to recognize the medium to long-term trends in a series. Disturbing factors may appear in the form of seasonal effects, calendar effects (e.g. differing composition of weekdays in a month or quarter, and public holidays), outliers (e.g. caused by strikes) or other irregularities.

For this reason, the analytical procedures are aimed at decomposing time series into individual components, namely a trend cycle component, a seasonal component, a calendar component and a residual component. The trend cycle component and the seasonally and calendar adjusted time series are particularly important for monitoring the economic development

ARIMA Time Series Analysis modeling and forecasting procedures discussed in the Identifying Patterns in Time Series Data, involved knowledge about the mathematical model of the process. However, in real-life research and practice, patterns of the data are unclear, individual observations involve considerable error, and we still need not only to uncover the hidden patterns in the data but also generate forecasts.

The ARIMA methodology developed by Box and Jenkins (1976) allows us to do just that; it has gained enormous popularity in many areas and research practice confirms its power and flexibility (Hoff, 1983; Pankratz, 1983; Vandaele, 1983). However, because of its power and flexibility, ARIMA is a complex technique; it is not easy to use, it requires a great deal of experience, and although it often produces satisfactory results, those results depend on the researcher's level of expertise (Bails & Peppers, 1982).

There are two main goals of ARIMA Time Series Analysis one is identifying the nature of the phenomenon represented by the sequence of observations, and second is  forecasting (predicting future values of the time series variable). Both of these goals require that the pattern of observed time series data is identified and more or less formally described. Once the pattern is established, they can interpret and integrate it with other data.

The Multiplicative seasonal ARIMA Time Series Analysis is a generalization and extension of the method introduced in the previous paragraphs to series in which a pattern repeats seasonally over time. In addition to the non-seasonal parameters, seasonal parameters for a specified lag (established in the identification phase) need to be estimated. Analogous to the simple ARIMA parameters, these are: seasonal autoregressive (ps), seasonal differencing (ds), and seasonal moving average parameters (qs).

The ARIMA Time Series Analysis method is appropriate only for a time series that is stationary (i.e., its mean, variance, and autocorrelation should be approximately constant through time) and it is recommended that there are at least 50 observations in the input data. It is also assumed that the values of the estimated parameters are constant throughout the series.