Box–Jenkins method

Box–Jenkins Method

The Box–Jenkins method refers to a systematic approach to identifying, fitting, and checking time series models. It was developed by George Box and Gwilyn Jenkins in the early 1970s and is primarily used for forecasting. The method focuses on the use of autoregressive integrated moving average (ARIMA) models.

Overview

The Box–Jenkins method involves three main stages:

1. Identification: Determining the appropriate form of the model by analyzing the time series data.
2. Estimation: Estimating the parameters of the model using statistical techniques.
3. Diagnostic Checking: Evaluating the model's adequacy by checking the residuals and making necessary adjustments.

Identification

The identification stage involves:

• Stationarity: Checking if the time series is stationary. If not, differencing the series may be necessary.
• Autocorrelation Function (ACF): Analyzing the ACF to identify the order of the moving average (MA) part.
• Partial Autocorrelation Function (PACF): Analyzing the PACF to identify the order of the autoregressive (AR) part.

Estimation

Once the model form is identified, the next step is to estimate the parameters. This is typically done using the method of maximum likelihood estimation or least squares.

Diagnostic Checking

After estimating the parameters, the model is checked for adequacy by:

• Residual Analysis: Checking the residuals for randomness. The residuals should resemble white noise.
• Ljung-Box Test: A statistical test to check for lack of fit in the residuals.

ARIMA Models

The ARIMA model is a combination of:

• Autoregressive (AR) part: A model that uses the dependency between an observation and a number of lagged observations.
• Integrated (I) part: Differencing of raw observations to make the time series stationary.
• Moving Average (MA) part: A model that uses dependency between an observation and a residual error from a moving average model applied to lagged observations.

The general form of an ARIMA model is denoted as ARIMA(p, d, q), where:

• p is the order of the AR part.
• d is the degree of differencing.
• q is the order of the MA part.

Applications

The Box–Jenkins method is widely used in various fields such as:

• Economics: For forecasting economic indicators like GDP, inflation, etc.
• Finance: For predicting stock prices and market trends.
• Meteorology: For weather forecasting.

Limitations

While powerful, the Box–Jenkins method has limitations:

• It assumes linearity in the time series data.
• It may not perform well with non-stationary data that cannot be made stationary through differencing.
• It requires a large amount of historical data for accurate forecasting.

See Also

• Time Series Analysis
• Forecasting
• Statistical Model

References

• Box, G. E. P., & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.
• Brockwell, P. J., & Davis, R. A. (2002). Introduction to Time Series and Forecasting. Springer.

External Links

• Box–Jenkins Method on Wikipedia