Box-Jenkins Method on the S&P 500

Stuart Truax, 2022-06

This notebook demonstrates the use of the Box-Jenkins method for time series model identification and parameter estimation. The time series is the inflation-adjusted monthly S&P 500 index for the years 1900 to 2017.

This application of the Box-Jenkins method suggests that the times series is best modeled by an ARIMA(2,1,0) process. The process is nonstationary, with a autorgressive order of 2 in the differenced series, suggesting that monthly index level changes in the S&P 500 are most strongly dependent on the previous two months of index level changes (i.e. a quarter in the past).

The Box-Jenkins method is composed of two primary tasks:

1. Model identification and determination of the degree of stationarity of the time series.
2. Parameter estimation for the identified model.

A more detailed outline of the method can be given as follows:

1. Model Indentification

   • Determine the degreee of stationarity of the time series by calculating its autocorrelation function (ACF). The behavior of the autocorrelation function indicates the type of time series model to be used.

   ACF Behavior	Model Indication
   Exponential decay	Autoregressive (AR) model
   Multiple impulses, otherwise zero	Moving average (MA) model
   Decay after a few lags	Autoregressive moving average (ARMA) model
   All zero or near zero	Random walk
   Periodic	Posseses seasonal components (e.g. GARCH model)
   No decay or very slow decay	Non-stationary process

   • Many ofthe possible models can be generalized as autoregressive integrated moving average model (ARIMA) models, which are of the form:

$$ \left( 1- \sum^{p}_{i=1} \phi_{i} L^{i} \right) (1-L)^{d} X_t = \left( 1 + \sum^{q}_{i=1} \theta_{i} L^{i} \right)\epsilon_t $$

• Here, $L$ is the lag operator. The shorthand for an ARIMA model is ARIMA(p,d,q), where $p$ is the order of the autorgressive terms, $q$ is the order of the moving average terms, and $d$ is the order of integration (i.e. the number of times the time series has to be differenced to obtain an ARMA model).

• Differencing the time series (i.e. adjusting $d$>0) is a method for removing non-stationary terms in the time series.

• If the ACF indicates an ARIMA(p,d,q) model, determination of the orders of the autoregressive and moving average terms (p and q, respectively) can be determined through the use of the partial autocorrelation function (PACF). The method for this determination follows:

Indicated Model	ACF Behavior on Differenced Data	PACF Behavior on Differenced Data
ARIMA(p,d,0)	Decaying exponential or sinusoidal	PACF shows impulse at lag $p$, but none beyond
ARIMA(0,d,q)	ACF shows impulse at lag $q$, but none beyond	Decaying exponential or sinusoidal

2. Parameter Estimation

   • Determine the model coefficients $\phi_i$ (autoregressive coefficients) and $\theta_i$ (moving average coefficents).

   • For this task, algorithmic techniques are typically employed, including:

     • Maximum likelihood estimation (most common)
     • Nonlinear least squares regression

Visualizations from the Analyses

Index Series Data at Different Levels of Differencing and Corresponding Autocorrelation Functions

Autocorrelation Function of the Differenced Series

Partial Autocorrelation Function of the Differenced Series