Background
The Autoregressive Moving Average (ARMA (p, q)) model is a fundamental tool in statistical analysis and econometrics, widely used for understanding and predicting future points in univariate time series data. Combining the autoregressive (AR) and moving average (MA) components, the ARMA model efficiently captures and encapsulates the linear dependencies in stochastic processes.
Historical Context
The ARMA model’s theoretical roots can be traced back to the work of U. Yule in the 1920s who introduced autoregressive formulations, extended later by G.E.P. Box and G.M. Jenkins in the early 1970s through their pioneering development of Box-Jenkins methodology. This methodology established a systematic way to identify, estimate, and check models for time series data, leading to the widespread adoption of ARMA models in econometric analysis.
Definitions and Concepts
The ARMA model is represented as ARMA(p, q), where:
pdenotes the number of lagged terms (autoregressive order)qrepresents the number of lagged forecast errors or residuals (moving average order)
In the generalized mathematical form, the ARMA model is defined as: \[ X_t = c + \epsilon_t + \sum_{i=1}^{p} \phi_i X_{t-i} + \sum_{j=1}^{q} \theta_j \epsilon_{t-j} \] where:
- \(X_t\) is the value at time \(t\)
- \(c\) is a constant
- \(\phi_i\) are the parameters of the autoregressive terms
- \(\theta_j\) are the parameters of the moving averages
- \(\epsilon_t\) is white noise (a random error term at time \(t\))
Major Analytical Frameworks
Classical Economics
In Classical economics, time series may be examined to identify cyclical behaviors in business cycles or long-term growth trends.
Neoclassical Economics
Neoclassical viewpoints might utilize ARMA models for analyzing efficiency dynamics and rational expectations in market prices.
Keynesian Economics
Keynesians could apply ARMA models to grasp short-term interplay between aggregate demand and business cycles.
Marxian Economics
Marxists might observe time series to understand recurring financial crises and the destabilizing effects of capitalist cycles.
Institutional Economics
The usage in institutional economics often revolves around observing the impact of institutional changes on economic variables over time.
Behavioral Economics
In Behavioral economics, ARMA models help decrypt patterns influenced by psychological factors and investor behavior.
Post-Keynesian Economics
Post-Keynesians focus on the dynamics of financial instability and economic fluctuations using ARMA models.
Austrian Economics
Analyzing purposeful actions and business cycle theories in Austrian economics can also involve ARMA methodology.
Development Economics
Development economists utilize ARMA models to forecast economic growth and identify patterns in underdeveloped regions.
Monetarism
Monetarists employ these models to predict the behavior of monetary aggregates and inflation over time.
Comparative Analysis
Comparing ARMA with other models like ARIMA (which includes differencing for stationarity) and SARIMA (Seasonal ARMA) provides insights into their unique applications and efficiency over different data sets. ARMA specifically excels in capturing underlying linear relationships in time-series data without the complexities of differentiation.
Case Studies
Applications of ARMA models are diverse—from predicting stock prices, analyzing GDP trends to weather forecasting. Case studies illustrate practical application in central banks’ inflation forecasting and production yield predictions based in agricultural studies.
Suggested Books for Further Studies
- Time Series Analysis: Forecasting and Control by George E. P. Box, Gwilym M. Jenkins, and Gregory C. Reinsel
- The Econometric Analysis of Time Series by Andrew C. Harvey
- Introduction to Time Series Analysis and Forecasting by Douglas C. Montgomery, Cheryl L. Jennings, and Murat Kulahci
Related Terms with Definitions
- AR (Autoregressive) Model: A model where current values are based linearly on prior values in time series.
- MA (Moving Average) Model: A model based on linear regressions of current values with past forecast errors.
- ARIMA Model: Differentiation-included ARMA modeling for non-stationary data.
- Box-Jenkins Methodology: Structured methodology for identifying, estimating, and checking time series models.
- Stationarity: A property where the statistical properties of a time series are constant over time.
