Background
The concept of the order of integration is fundamental in time series analysis, particularly in econometrics. It refers to the minimum number of differencing operations required to transform a non-stationary series into a stationary one.
Historical Context
This concept gained prominence with the development of statistical techniques for time series analysis in econometrics, especially in relation to unit root testing and cointegration in models regularly used in economics and finance.
Definitions and Concepts
Order of Integration (I(n)): The minimum number of times differencing is required to convert a non-stationary time series into a stationary time series. A time series integrated of order \( n \) is denoted as \( I(n) \).
Stationary Series: A time series whose statistical properties such as mean, variance, and autocorrelation are constant over time.
Non-Stationary Series: A time series that has statistical properties that change over time, making it unpredictable and unsuitable for typical time series forecasting models.
Fractionally Integrated Processes: When the order of integration is a fraction, indicating the use of fractional differencing. This process provides a more nuanced characterization of persistence in the series characteristics and can better capture long memory processes.
Major Analytical Frameworks
Classical Economics
Classical economics does not specifically address the order of integration as it primarily deals with long-run economic principles, assuming more stable relationships that do not require differencing for stationarity.
Neoclassical Economics
Neoclassical models assume rational expectations and efficient markets. Time series involved are often assumed to be stationary or made stationary using differencing methods defined by the order of integration.
Keynesian Economics
Keynesian and Post-Keynesian frameworks extensively use time series data to understand macroeconomic phenomena such as inflation and output gaps, often examining the properties of integrated and cointegrated series.
Marxian Economics
Marxian economics typically uses historical and empirical data for its analysis where differencing might be necessary to reveal underlying patterns in long-term series data.
Institutional Economics
Examines the role of institutions over time, often requiring differenced series to study persistence and changes in institutional effects.
Behavioral Economics
Analyzes time series data to understand how psychological, social, and emotional factors impact economic decisions, also requiring differencing to achieve stationarity.
Post-Keynesian Economics
Particularly focuses on the dynamics of economic systems and the long-term relationships represented in time series, requiring an in-depth understanding of orders of integration.
Austrian Economics
Less focused on quantitative analysis but occasionally uses time series to study business cycles requiring stationary data.
Development Economics
Time series analysis in development economics often needs detrending (by differencing) to study growth patterns and causal relationships in developing economies.
Monetarism
Extensively involves time series data, particularly regarding trends in money supply growth, inflation, and output - requiring an understanding of differencing operations and integration.
Comparative Analysis
Time series from different economic schools and frameworks would often need to be compared by transforming non-stationary series into stationary ones through differencing. The frequency and order of such differencing reveal the underlying persistence and trends in the data.
Case Studies
Real-world case studies often involve series such as GDP, inflation, or interest rates, which need to be made stationary through differencing operations where the order of integration is critical. Corns of differencing applied to macroeconomic indicators can reveal deeper insights into long-term economic trends and impacts.
Suggested Books for Further Studies
- “Time Series Analysis” by James D. Hamilton
- “Introduction to Econometrics” by James H. Stock and Mark W. Watson
- “Applied Econometric Time Series” by Walter Enders
- “Econometric Analysis” by William H. Greene
Related Terms with Definitions
Covariance Stationary Process: A time series process where mean, variance, and autocovariance remain constant over time.
Differencing: The process of subtracting the previous observation from the current observation to remove trends and achieve stationarity in a time series.
Unit Root: A characteristic of a non-stationary series that shows a persistent, random walk with no mean reversion.
Cointegration: A statistical relationship between two or more non-stationary series which can be combined in such a way that the resulting series is stationary.
Fractional Differencing: A method for transforming non-stationary data to stationary while retaining long memory properties in a series.
