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Introduction To Stationary And Non-Stationary Processes

Introduction To Stationary And Non-Stationary Processes

Financial institutions and corporations as well as individual investors and researchers often use financial time series data (such as asset prices, exchange rates, GDP, inflation and other macroeconomic indicators) in economic forecasts, stock market analysis or studies of the data itself.

But refining data is key to being able to apply it to your stock analysis. In this article, we'll show you how to isolate the data points that are relevant to your stock reports.

Cooking Raw Data
Data points are often non-stationary or have means, variances and covariances that change over time. Non-stationary behaviors can be trends, cycles, random walks or combinations of the three.

Non-stationary data, as a rule, are unpredictable and cannot be modeled or forecasted. The results obtained by using non-stationary time series may be spurious in that they may indicate a relationship between two variables where one does not exist. In order to receive consistent, reliable results, the non-stationary data needs to be transformed into stationary data. In contrast to the non-stationary process that has a variable variance and a mean that does not remain near, or returns to a long-run mean over time, the stationary process reverts around a constant long-term mean and has a constant variance independent of time.

Figure 1

Types of Non-Stationary ProcessesBefore we get to the point of transformation for the non-stationary financial time series data, we should distinguish between the different types of the non-stationary processes. This will provide us with a better understanding of the processes and allow us to apply the correct transformation. Examples of non-stationary processes are random walk with or without a drift (a slow steady change) and deterministic trends (trends that are constant, positive or negative, independent of time for the whole life of the series).

Figure 2

• Pure Random Walk (Y_t = Y_t-1 + ε_t )Random walk predicts that the value at time "t" will be equal to the last period value plus a stochastic (non-systematic) component that is a white noise, which means ε_t is independent and identically distributed with mean "0" and variance "σ²". Random walk can also be named a process integrated of some order, a process with a unit root or a process with a stochastic trend. It is a non mean reverting process that can move away from the mean either in a positive or negative direction. Another characteristic of a random walk is that the variance evolves over time and goes to infinity as time goes to infinity; therefore, a random walk cannot be predicted.
• Random Walk with Drift (Y_t = α + Y_t-1 + ε_t )If the random walk model predicts that the value at time "t" will equal the last period's value plus a constant, or drift (α), and a white noise term (ε_t), then the process is random walk with a drift. It also does not revert to a long-run mean and has variance dependent on time.
• Deterministic Trend (Y_t = α + βt + ε_t )Often a random walk with a drift is confused for a deterministic trend. Both include a drift and a white noise component, but the value at time "t" in the case of a random walk is regressed on the last period's value (Y_t-1), while in the case of a deterministic trend it is regressed on a time trend (βt). A non-stationary process with a deterministic trend has a mean that grows around a fixed trend, which is constant and independent of time.
• Random Walk with Drift and Deterministic Trend (Y_t = α + Y_t-1 + βt + ε_t ) 
  Another example is a non-stationary process that combines a random walk with a drift component (α) and a deterministic trend (βt).It specifies the value at time "t" by the last period's value, a drift, a trend and a stochastic component. (To learn more about random walks and trends, see our Financial Concepts tutorial.)

Trend and Difference StationaryA random walk with or without a drift can be transformed to a stationary process by differencing (subtracting Y_t-1 from Y_t, taking the difference Y_t - Y_t-1) correspondingly to Y_t - Y_t-1 = ε_t or Y_t - Y_t-1 = α + ε_t and then the process becomes difference-stationary. The disadvantage of differencing is that the process loses one observation each time the difference is taken.

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Figure 3

A non-stationary process with a deterministic trend becomes stationary after removing the trend, or detrending. For example, Yt = α + βt + εt is transformed into a stationary process by subtracting the trend βt: Yt - βt = α + εt, as shown in Figure 4 below. No observation is lost when detrending is used to transform a non-stationary process to a stationary one.

Figure 4

In the case of a random walk with a drift and deterministic trend, detrending can remove the deterministic trend and the drift, but the variance will continue to go to infinity. As a result, differencing must also be applied to remove the stochastic trend.

ConclusionUsing non-stationary time series data in financial models produces unreliable and spurious results and leads to poor understanding and forecasting. The solution to the problem is to transform the time series data so that it becomes stationary. If the non-stationary process is a random walk with or without a drift, it is transformed to stationary process by differencing. On the other hand, if the time series data analyzed exhibits a deterministic trend, the spurious results can be avoided by detrending. Sometimes the non-stationary series may combine a stochastic and deterministic trend at the same time and to avoid obtaining misleading results both differencing and detrending should be applied, as differencing will remove the trend in the variance and detrending will remove the deterministic trend.

Mergers and Acquisition Research Material

Journal Of Management (JOM)

Association of Corporate Growth

e-journal on Mergers & Acquisitions

Journal of Business Strategy

Oxford Journal

Houston Business Journal

Wall Street Journal

Books and Journals-Michigan University

Elsevier

Oil & Gas Financial Journal


Marketing Research


American Marketing Association

International Journal of Marketing Research


Papers


Merger Movements in American Industry, 1895-1956

Markup pricing in mergers and acquisitions

Thirty Years of Mergers and Acquisitions Research: Recent Advances and Future Opportunities

Acculturation in Mergers and Acquisitions

The Post-merger Performance Puzzle

The effect of mergers and acquisitions on the efficiency and profitability of EC credit institutions

Integrating Strategic, Organizational, and Human Resource Perspectives on Mergers and Acquisitions: A Case Survey of Synergy Realization

Mergers and Acquisitions of Financial Institutions: A Review of the Post-2000 Literature

Returns to bidding firms in mergers and acquisitions: Reconsidering the relatedness hypothesis

Making mergers and acquisitions work: What we know and don't know – Part I

List of Nobel Memorial Prize laureates in Economics

Year	Laureate		Country	Schools of economic thought	Rationale
1969		Ragnar Frisch	Norway		"for having developed and applied dynamic models for the analysis of economic processes"[2]
		Jan Tinbergen	Netherlands
1970		Paul Samuelson	United States	Keynesian	"for the scientific work through which he has developed static and dynamic economic theory and actively contributed to raising the level of analysis in economic science"[8]
1971		Simon Kuznets	United States	Institutional	"for his empirically founded interpretation of economic growth which has led to new and deepened insight into the economic and social structure and process of development"[9]
1972		John Hicks	United Kingdom	Keynesian	"for their pioneering contributions to general economic equilibrium theory and welfare theory."[10]
		Kenneth Arrow	United States	Neoclassical
1973		Wassily Leontief	Soviet Union  United States		"for the development of the input-output method and for its application to important economic problems"[11]
1974		Gunnar Myrdal	Sweden	Monetarism Keynesian Stockholm	"for their pioneering work in the theory of money and economic fluctuations and for their penetrating analysis of the interdependence of economic, social and institutional phenomena."[12]
		Friedrich Hayek	Austria  United Kingdom	Austrian Chicago
1975		Leonid Kantorovich	Soviet Union		"for their contributions to the theory of optimum allocation of resources"[13]
		Tjalling Koopmans	Netherlands  United States
1976		Milton Friedman	United States	Chicago Monetarism	"for his achievements in the fields of consumption analysis, monetary history and theory and for his demonstration of the complexity of stabilisation policy"[14]
1977		Bertil Ohlin	Sweden	Stockholm	"for their pathbreaking contribution to the theory of international trade and internationalcapital movements"[15]
		James Meade	United Kingdom	Keynesian
1978		Herbert A. Simon	United States	Carnegie	"for his pioneering research into the decision-making process within economic organizations"[16]
1979		Theodore Schultz	United States	Chicago	"for their pioneering research into economic development research with particular consideration of the problems of developing countries."[17]
		Arthur Lewis	Saint Lucia  United Kingdom
1980		Lawrence Klein	United States	Keynesian	"for the creation of econometric models and the application to the analysis of economic fluctuations and economic policies"[18]
1981		James Tobin	United States	Keynesian	"for his analysis of financial markets and their relations to expenditure decisions, employment, production and prices"[19]
1982		George Stigler	United States	Chicago	"for his seminal studies of industrial structures, functioning of markets and causes and effects of public regulation"[20]
1983		Gérard Debreu	France	Neoclassical	"for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium"[21]
1984		Richard Stone	United Kingdom		"for having made fundamental contributions to the development of systems of national accounts and hence greatly improved the basis for empirical economic analysis"[22]
1985		Franco Modigliani	Italy	Keynesian	"for his pioneering analyses of saving and of financial markets"[23]
1986		James M. Buchanan	United States	Chicago Constitutional	"for his development of the contractual and constitutional bases for the theory of economic and political decision-making"[24]
1987		Robert Solow	United States	Keynesian	"for his contributions to the theory of economic growth"[25]
1988		Maurice Allais	France	Neoclassical	"for his pioneering contributions to the theory of markets and efficient utilization of resources"[26]
1989		Trygve Haavelmo	Norway	Keynesian	"for his clarification of the probability theory foundations of econometrics and his analyses of simultaneous economic structures"[27]
1990		Harry Markowitz	United States	Chicago	"for their pioneering work in the theory of financial economics"[28]
		Merton Miller		Chicago
		William Forsyth Sharpe
1991		Ronald Coase	United Kingdom	Chicago New institutional	"for his discovery and clarification of the significance of transaction costs and property rights for the institutional structure and functioning of the economy"[29]
1992		Gary Becker	United States	Chicago	"for having extended the domain of microeconomic analysis to a wide range of human behaviour and interaction, including non-market behaviour"[30]
1993		Robert Fogel	United States	Chicago	"for having renewed research in economic history by applying economic theory and quantitative methods in order to explain economic and institutional change"[31]
		Douglass North		New institutional
1994		John Harsanyi	United States		"for their pioneering analysis of equilibria in the theory of non-cooperative games."[32]
		John Forbes Nash
		Reinhard Selten	Germany
1995		Robert Lucas, Jr.	United States	New classical Chicago	"for having developed and applied the hypothesis of rational expectations, and thereby having transformed macroeconomic analysis and deepened our understanding of economic policy"[33]
1996		James Mirrlees	United Kingdom		"for their fundamental contributions to the economic theory of incentives under asymmetric information"[34]
		William Vickrey	Canada  United States	Keynesian
1997		Robert C. Merton	United States		"for a new method to determine the value of derivatives."[35]
		Myron Scholes	Canada  United States	Chicago
1998		Amartya Sen	India	Capability approach	"for his contributions to welfare economics"[36]
1999		Robert Mundell	Canada	Keynesian Supply-side	"for his analysis of monetary and fiscal policy under different exchange rate regimes and his analysis of optimum currency areas"[37]
2000		James Heckman	United States	Chicago	"for his development of theory and methods for analyzing selective samples"[38]
		Daniel McFadden	United States		"for his development of theory and methods for analyzing discrete choice"[38]
2001		George Akerlof	United States	Keynesian	"for their analyses of markets with asymmetric information"[39]
		Michael Spence
		Joseph E. Stiglitz		Keynesian
2002		Daniel Kahneman	Israel  United States	Behavioral	"for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty"[40]
		Vernon L. Smith	United States	New classical	"for having established laboratory experiments as a tool in empirical economic analysis, especially in the study of alternative market mechanisms"[40]
2003		Robert F. Engle	United States		"for methods of analyzing economic time series with time-varying volatility (ARCH)"[41]
		Clive Granger	United Kingdom		"for methods of analyzing economic time series with common trends (cointegration)"[41]
2004		Finn E. Kydland	Norway	New classical	"for their contributions to dynamic macroeconomics: the time consistency of economic policy and the driving forces behind business cycles."[42]
		Edward C. Prescott	United States	New classical
2005		Robert J. Aumann	United States  Israel		"for having enhanced our understanding of conflict and cooperation through game-theoryanalysis."[43]
		Thomas C. Schelling	United States
2006		Edmund S. Phelps	United States		"for his analysis of intertemporal tradeoffs in macroeconomic policy"[44]
2007		Leonid Hurwicz	Poland  United States		"for having laid the foundations of mechanism design theory"[45]
		Eric S. Maskin	United States
		Roger B. Myerson
2008		Paul Krugman	United States	Keynesian	"for his analysis of trade patterns and location of economic activity"[46]
2009		Elinor Ostrom	United States	New institutional	"for her analysis of economic governance, especially the commons"[47]
		Oliver E. Williamson		New institutional	"for his analysis of economic governance, especially the boundaries of the firm"[47]
2010		Peter A. Diamond	United States		"for their analysis of markets with search frictions"[48]
		Dale T. Mortensen
		Christopher A. Pissarides	Cyprus
2011		Thomas J. Sargent	United States	Neoclassical	"for their empirical research on cause and effect in the macroeconomy"
		Christopher A. Sims		Neoclassical
2012		Alvin E. Roth	United States		"for the theory of stable allocations and the practice of market design."
		Lloyd S. Shapley
2013		Eugene F. Fama	United States	Chicago	"for their empirical analysis of asset prices."
		Lars Peter Hansen		Chicago
		Robert J. Shiller		Keynesian
2014		Jean Tirole	France		"for his analysis of market power and regulation".