## What Is the Box-Jenkins Model?

The Box-Jenkins Model is a mathematical model designed to forecast data ranges based on inputs from a specified time series. The Box-Jenkins Model can analyze several different types of time series data for forecasting purposes.

Its methodology uses differences between data points to determine outcomes. The methodology allows the model to identify trends using autoregresssion, moving averages, and seasonal differencing to generate forecasts.

Autoregressive integrated moving average (ARIMA) models are a form of Box-Jenkins model. The terms ARIMA and Box-Jenkins are sometimes used interchangeably.

### Key Takeaways

- The Box-Jenkins Model is a forecasting methodology using regression studies on time series data.
- The methodology is predicated on the assumption that past occurrences influence future ones.
- The Box-Jenkins Model is best suited for forecasting within time frames of 18 months or less.
- Autoregressive integrated moving average (ARIMA) models are a form of Box-Jenkins model.

## Understanding the Box-Jenkins Model

Box-Jenkins Models are used for forecasting a variety of anticipated data points or data ranges, including business data and future security prices.

The Box-Jenkins Model was created by two mathematicians: George Box and Gwilym Jenkins. The two mathematicians discussed the concepts that comprise this model in a 1970 publication called "Time Series Analysis: Forecasting and Control."

Estimations of the parameters of the Box-Jenkins Model can be very complicated. Therefore, similar to other time-series regression models, the best results will typically be achieved through the use of programmable software. The Box-Jenkins Model is also generally best suited for short-term forecasting of 18 months or less.

## Box-Jenkins Methodology

The Box-Jenkins Model may be one of several, time series analysis models a forecaster will encounter when using programmed forecasting software. In many cases, the software will be programmed to automatically use the best fitting forecasting methodology based on the time series data to be forecasted. Box-Jenkins is reported to be a top choice for data sets that are mostly stable and have low volatility.

The Box-Jenkins Model forecasts data using three principles: autoregression, differencing, and moving average. These three principles are known as p, d, and q, respectively. Each principle is used in the Box-Jenkins analysis; together, they are collectively shown as ARIMA (p, d, q).

The autoregression (p) process tests the data for its level of stationarity. If the data being used is stationary, it can simplify the forecasting process. If the data being used is non-stationary it will need to be differenced (d). The data is also tested for its moving average fit (which is done in part q of the analysis process). Overall, initial analysis of the data prepares it for forecasting by determining the parameters (p, d, and q), which are then applied to develop a forecast.

A one-time shock will affect subsequent values of a Box-Jenkins model infinitely into the future. Therefore, the legacy of the financial crisis lives on in today’s autoregressive models.

## Autoregressive Integrated Moving Average (ARIMA)

Box-Jenkins is a type of autoregressive integrated moving average (ARIMA) model that gauges the strength of one dependent variable relative to other changing variables. The model's goal is to predict future securities or financial market moves by examining the differences between values in the series instead of through actual values.

An ARIMA model can be understood by outlining each of its components as follows:

- Autoregression (AR): refers to a model that shows a changing variable that regresses on its own lagged, or prior, values.
- Integrated (I): represents the differencing of raw observations to allow for the time series to become stationary, i.e., data values are replaced by the difference between the data values and the previous values.
- Moving average (MA): incorporates the dependency between an observation and a residual error from a moving average model applied to lagged observations.

## Forecasting Stock Prices

One use for Box-Jenkins Model analysis is to forecast stock prices. This analysis is typically built out and coded through R software. The analysis results in a logarithmic outcome, which can be applied to the data set to generate the forecasted prices for a specified period of time in the future.

ARIMA models are based on the assumption that past values have some residual effect on current or future values. For example, an investor using an ARIMA model to forecast stock prices would assume that new buyers and sellers of that stock are influenced by recent market transactions when deciding how much to offer or accept for the security.

Although this assumption will hold under many different circumstances, it is not always true. For example, in the years prior to the 2008 Financial Crisis, most investors were not aware of the risks posed by the large portfolios of mortgage-backed securities (MBS) held by many financial firms.

During those times, an investor using an autoregressive model to predict the performance of U.S. financial stocks would have had good reason to predict an ongoing trend of stable or rising stock prices in that sector. However, once it became public knowledge that many financial institutions were at risk of imminent collapse, investors suddenly became less concerned with these stocks' recent prices and far more concerned with their underlying risk exposure.

Therefore, the market rapidly revalued financial stocks to a much lower level, a move that would have utterly confounded an autoregressive model.