Introduction Forecasting Basics: The basic idea behind self-projecting time series forecasting models is to find a mathematical formula that will approximately generate the historical patterns in a time series. Time Series: A time series is a set of numbers that measures the status of some activity over time. It is the historical record of some activity, with measurements taken at equally spaced intervals (exception: monthly) with a consistency in the activity and the method of measurement. Approaches to time Series Forecasting: There are two basic approaches to forecasting time series: the self-projecting time series and the cause-and-effect approach. Cause and effect methods attempt to forecast based on underlying series that are believed to cause the behavior of the original series. The self-projecting time series uses only the time series data of the activity to be forecast to generate forecasts. This latter approach is typically less expensive to apply and requires far less data and is useful for short to medium-term forecasting. Box-Jenkins Forecasting Method: The univariate version of this methodology is a self- projecting time series forecasting method. The underlying goal is to find an appropriate formula so that the residuals are as small as possible and exhibit no pattern. The model- building process involves four steps. Repeated as necessary, to end up with a specific formula that replicates the patterns in the series as closely as possible and also produces accurate forecasts. Box-Jenkins Methodology Box-Jenkins forecasting models are based on statistical concepts and principles and are able to model a wide spectrum of time series behavior. It has a large class of models to choose from and a systematic approach for identifying the correct model form. There are both statistical tests for verifying model validity and statistical measures of forecast uncertainty. In contrast, traditional forecasting models offer a limited number of models relative to the complex behavior of many time series with little in the way of guidelines and statistical tests for verifying the validity of the selected model. Data: The misuse, misunderstanding, and inaccuracy of forecasts is often the result of not appreciating the nature of the data in hand. The consistency of the data must be insured and it must be clear what the data represents and how it was gathered or calculated. As a rule of thumb, Box-Jenkins requires at least 40 or 50 equally-spaced periods of data. The data must also be edited to deal with extreme or missing values or other distortions through the sue of functions as log or inverse to achieve stabilization. Preliminary Model Identification Procedure: A preliminary Box-Jenkins analysis with a plot of the initial data should be run as the starting point in determining an appropriate model. The input data must be adjusted to form a stationary series, one whose values vary more or less uniformly about a fixed level over time. Apparent trends can be adjusted by having the model apply a technique of "regular differencing," a process of computing the difference between every two successive values, computing a differenced series which has overall trend behavior removed. If a single differencing does not achieve stationarity, it may be repeated, although rarely if ever, are more than two regular differencings required. Where irregularities in the differenced series continue to be displayed, log or inverse functions can be specified to stabilize the series such that the remaining residual plot displays values approaching zero and without any pattern. This is the error term, equivalent to pure, white noise. Pure Random Series: On the other hand, if the initial data series displays neither trend nor seasonality and the residual plot shows essentially zero values within a 95% confidence level and these residual values display no pattern, then there is no real-world statistical problem to solve and we go on to other things. Model Identification Background Basic Model: With a stationary series in place, a basic model can now be identified. Three basic models exist, AR (autoregressive), MA (moving average) and a combined ARMA in addition to the previously specified RD (regular differencing) combine to provide the available tools. When regular differencing is applied together with AR and MA, they are referred to as ARIMA, with the I indicating "integrated" and referencing the differencing procedure. Seasonality: In addition to trend, which has now been provided for, stationary series quite commonly display seasonal behavior where a certain basic pattern tends to be repeated at regular seasonal intervals. The seasonal pattern may additionally frequently display constant change over time as well. Just as regular differencing was applied to the overall trending series, seasonal differencing (SD) is applied to seasonal nonstationarity as well. And as autoregressive and moving average tools are available with the overall series, so too, are they available for seasonal phenomena using seasonal autoregressive parameters (SAR) and seasonal moving average parameters (SMA). Establishing Seasonality: The need for seasonal autoregression (SAR) and seasonal moving average (SMA) parameters is established by examining the autocorrelation and partial autocorrelation patterns of a stationary series at lags that are multiples of the number of periods per season. These parameters are required if the values at lags s, 2s, etc. are nonzero and display patterns associated with the theoretical patterns for such models. Seasonal differencing is indicated if the autocorrelations at the seasonal lags do not decrease rapidly.
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Referring to the above chart, know that, the variance of the errors of the underlying model must be invariant (i.e. constant). This means that the variance for each subgroup of data is the same and does not depend on the level or the point in time. If this is violated then one can remedy this by stabilizing the variance. Make sure that, that there are no deterministic patterns in the data. Also one must not have any pulses or one-time unusual values. Additionally there should be no level or step shifts. Also no seasonal pulses should be present.
The reason for all of this is that if they do exist then the sample autocorrelation and partial autocorrelation will seem to imply ARIMA structure. Also the presence of these kind of model components can obfuscate or hide structure. For example a single outlier or pulse can create an effect where the structure is masked by the outlier.
Improved Quantitative Identification Method
Relieved Analysis Requirements: A substantially improved procedure is now available for conducting Box-Jenkins ARIMA analysis which relieves the requirement for a seasoned perspective in evaluating the sometimes ambiguous autocorrelation and partial autocorrelation residual patterns to determine an appropriate Box-Jenkins model for use in developing a forecast model.
ARMA (1, 0): The first model to be tested on the stationary series consists solely of an autoregressive term with lag 1. The autocorrelation and partial autocorrelation patterns are examined for significant autocorrelation often early terms and to see whether the residual coefficients are uncorrelated, that is the coefficient values are zero within 95% confidence limits and without apparent pattern. When fitted values as close as possible to the original series values are obtained, the sum of the squared residuals will be minimized, a technique called least squares estimation. The residual mean and the mean percent error should not be significantly nonzero. Alternative models are examined comparing the progress of these factors, favoring models which use as few parameters as possible. Correlation between parameters should not be significantly large and confidence limits should not bracket zero. When a satisfactory model has been established a forecast procedure is applied.
ARMA (2, 1): Absent a satisfactory ARMA (1, 0) condition with residual coefficients approximating zero, the improved model identification procedure now proceeds to examine the residual pattern when autoregressive terms with order 1 and 2 are applied together with a moving average term with an order of 1.
Subsequent Procedure: To the extent that the residual conditions described above remain unsatisfied, the Box-Jenkins analysis is continued with ARMA (n, n-1) until a satisfactory model is arrived at. In the course of this iteration, when an autoregressive coefficient (phi) approaches zero, the model is reexamined with parameters ARMA (n-1, n-1). In like manner whenever a moving average coefficient (theta) approaches zero, the model is similarly reduced to ARMA (n, n-2). At some point, either the autoregressive term or moving average term may fall away completely and the examination of the stationary series is continued with only the remaining term until the residual coefficients approach zero within the specified confidence levels.
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