Box–Jenkins method

Builds on Autoregressive integrated moving average Autoregressive–moving-average model Autocorrelation function Partial autocorrelation function Maximum likelihood estimation Least squares

Definition

Let $x_1, \dots, x_n$ be an observed time series. The Box–Jenkins method fits an ARIMA( $p$ , $d$ , $q$ ) model to the series and then uses it to forecast. It repeats the following three stages until the checks of stage 3 are passed:

Identification: choose $d$ so that the differenced series $\nabla^d x_t$ looks stationary. Then choose tentative orders $p$ and $q$ from the sample autocorrelation function and partial autocorrelation function of $\nabla^d x_t$ .
Estimation: estimate the parameters of the ARMA( $p$ , $q$ ) model for the differenced series by maximum likelihood or by nonlinear least squares.
Diagnostic checking: test whether the residuals of the fitted model behave like white noise, for instance with a portmanteau test. If they do not, return to stage 1 with a modified model.

Remarks

The identification stage uses typical patterns. For an AR( $p$ ) model the partial autocorrelation vanishes beyond lag $p$ while the autocorrelation decays gradually. For an MA( $q$ ) model the autocorrelation vanishes beyond lag $q$ while the partial autocorrelation decays gradually.

Example: suppose the sample autocorrelation of a series decays very slowly. Suppose that of its first differences has a single significant value, at lag $1$ . Then stage 1 suggests $d = 1$ , $p = 0$ and $q = 1$ , that is ARIMA( $0$ , $1$ , $1$ ).

Non-example: fitting one fixed AR( $1$ ) model by least squares without examining the residuals is not the Box–Jenkins method, because stage 3 and the loop back to stage 1 are missing.

The method prefers parsimonious models, with as few parameters as give an adequate fit. Seasonal ARIMA models are handled in the same way, with seasonal differences and seasonal orders.

Reading path

To understand this page from the ground up, read these 97 pages in order. The foundations come first.

Autocorrelation function concept
Autoregressive integrated moving average concept
Autoregressive–moving-average model concept
Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Matrix definition
Ordered field definition
Partial autocorrelation function concept
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Vector space definition
Set definition
Ring of integers statement
Transpose definition
Disjunction definition
Implication definition
Predicate definition
Inner product definition
Linear map definition
Linear subspace definition
Norm definition
Ordered pair definition
Union definition
Existential quantifier definition
Sample space definition
Universal quantifier definition
Inner product space definition
Orthogonal vectors definition
The rationals form a field statement
Endomorphism definition
Sum of subspaces definition
Cartesian product definition
Subset definition
Uniqueness quantifier definition
Norm induced by an inner product definition
Orthogonal complement definition
The rationals form an ordered field statement
Direct sum of subspaces definition
Binary relation definition
Universal set definition
Power set definition
Order on the reals definition
A space is the direct sum of a subspace and its orthogonal complement statement
Projection definition
Function definition
Complement definition
Interval definition
Orthogonal projection definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Orthogonal projection is the closest point statement
Countable set definition
Event definition
Least squares concept
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Convergence in probability definition
Random variable with a density definition
Joint probability distribution definition
Probability mass function definition
Mutual independence of events definition
Expected value definition
Identically distributed random variables definition
Independence of two random variables definition
Probability density function definition
Marginal distribution definition
Independent sigma-algebras definition
Variance definition
Linearity of expectation statement
Independence of random variables definition
Independence means the joint law is the product of the marginals statement
Covariance definition
Independent and identically distributed random variables definition
Variance of a sum statement
Expectation of a product of independent random variables statement
Statistic definition
Likelihood function definition
Independent random variables are uncorrelated statement
Estimator definition
Variance of a sum of independent random variables statement
Sample mean definition
Consistent estimator definition
Maximum likelihood estimation concept
Box–Jenkins method definition

Tag	`V192HFRX5A`
URI	`m2://V192HFRX5A`
Kind	definition
Writing	W0 stub
MSC	62M10, 62M20
Defines	Box–Jenkins method
Uses	Autoregressive integrated moving average, Autoregressive–moving-average model, Autocorrelation function, Partial autocorrelation function, Maximum likelihood estimation, Least squares

Box–Jenkins method

Definition

Remarks

Reading path

Further reading

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