Time Series Analysis

7.

7.1.

Box-Jenkins Methodology

7.1.1.

Model Identification

7.1.2.

Parameter Estimation

7.1.3.

Diagnostic Checking

7.1.4.

7.1.5.

Model Refinement

7.2.

Autoregressive Models

7.2.1.

7.2.2.

Higher-Order AR Models

7.2.3.

Stationarity Conditions

7.2.4.

Characteristic Equation

7.2.5.

Yule-Walker Equations

7.3.

Moving Average Models

7.3.1.

7.3.2.

Higher-Order MA Models

7.3.3.

Invertibility Conditions

7.3.4.

Theoretical Properties

7.4.

7.4.1.

Combined AR and MA Components

7.4.2.

Stationarity and Invertibility

7.4.3.

Theoretical ACF and PACF

7.4.4.

Wold Decomposition

7.5.

Integrated Models

7.5.1.

ARIMA Structure

7.5.2.

Order of Integration

7.5.3.

Differencing Operations

7.5.4.

Unit Root Implications

7.6.

Model Identification

7.6.1.

ACF and PACF Analysis

7.6.2.

Information Criteria

7.6.3.

Parsimony Principle

7.6.4.

Tentative Model Selection

7.7.

Parameter Estimation

7.7.1.

Maximum Likelihood Estimation

7.7.2.

Least Squares Methods

7.7.3.

Method of Moments

7.7.4.

Numerical Optimization

7.8.

Diagnostic Checking

7.8.1.

Residual Analysis

7.8.1.1.

Independence Tests

7.8.1.2.

7.8.1.3.

Normality Tests

7.8.1.4.

Homoskedasticity Tests

7.8.2.

7.8.3.

Overfitting Detection

7.8.4.

Goodness of Fit Measures

7.9.

Forecasting with ARIMA

7.9.1.

Minimum Mean Square Error Forecasts

7.9.2.

Forecast Functions

7.9.3.

Prediction Intervals

7.9.4.

Forecast Updating

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8. Seasonal ARIMA Models