6.2 Moving average (MA) models

Moving Average (MA) models are a powerful tool in time series analysis. They capture relationships between observations and past forecast errors, providing insights into short-term dependencies in data. MA models are stationary, meaning their statistical properties remain constant over time.

Understanding MA models is crucial for forecasting and interpreting time series data. From model order to parameter estimation and invertibility conditions, these concepts help analysts make accurate predictions and draw meaningful conclusions from temporal data patterns.

Moving Average (MA) Models

Moving average models in time series

• MA models capture the relationship between an observation and past forecast errors (residuals)
  • Forecast errors represent the difference between observed and predicted values
• MA models are stationary, meaning the mean, variance, and autocovariance remain constant over time
• MA models account for short-term dependencies between an observation and a finite number of lagged forecast errors
• The impact of a shock (unexpected event) on the time series diminishes over time due to the model's finite memory

Order of MA models

• The order of an MA model, denoted as $MA(q)$, is determined by the number of lagged error terms ($q$) in the model equation
• The general form of an $MA(q)$ model is:
  • $y_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + ... + \theta_q \varepsilon_{t-q}$
    • $y_t$: Observed value at time $t$
    • $\mu$: Mean of the time series
    • $\varepsilon_t$: Error term at time $t$
    • $\theta_1, \theta_2, ..., \theta_q$: Moving average coefficients

Parameter estimation for MA models

• Least squares estimation minimizes the sum of squared residuals between observed and predicted values
  • Provides estimates for moving average coefficients and error term variance
• Maximum likelihood estimation finds parameter values that maximize the likelihood function
  • Measures the probability of observing the given data under the assumed model
  • Requires an iterative optimization process to estimate parameters
• The choice of estimation method depends on time series properties and error term assumptions

Invertible vs non-invertible MA models

• Invertibility ensures a unique model representation and allows expression as an equivalent autoregressive (AR) model
• Invertibility conditions:
  • Absolute values of the roots of the characteristic equation $1 + \theta_1 z + \theta_2 z^2 + ... + \theta_q z^q = 0$ must be greater than 1
  • Absolute values of moving average coefficients must sum to less than 1
• Non-invertible MA models occur when invertibility conditions are not satisfied
  • May lead to multiple model representations, making interpretation and forecasting challenging
  • Can be transformed into invertible models by adding or removing common factors from the characteristic equation

Application of MA models to data

• Steps to apply MA models:
  1. Identify the order of the MA model based on the autocorrelation function (ACF) and partial autocorrelation function (PACF)
  2. Estimate MA model parameters using an appropriate estimation method
  3. Assess model fit using diagnostic tools (residual analysis, AIC, BIC)
  4. Use the fitted MA model for forecasting and interpret results in the problem domain context
• Interpretation of MA model results:
  • Moving average coefficients represent the impact of past forecast errors on the current observation
  • Positive coefficients indicate positive errors in the past likely followed by positive errors in the present, and vice versa for negative coefficients
  • Coefficient magnitudes determine the strength and duration of past error impact on the current observation