### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

Longitudinal data analysis focused on internal characteristics of a single time series has attracted increasing interest among psychologists. The systemic psychological perspective suggests, however, that many long-term phenomena are mutually interconnected, forming a dynamic system. Hence, only multivariate methods can handle such human dynamics appropriately. Unlike the majority of time series methodologies, the cointegration approach allows interdependencies of integrated (i.e., extremely unstable) processes to be modelled. This advantage results from the fact that cointegrated series are connected by stationary long-run equilibrium relationships. Vector error-correction models are frequently used representations of cointegrated systems. They capture both this equilibrium and compensation mechanisms in the case of short-term deviations due to developmental changes. Thus, the past disequilibrium serves as explanatory variable in the dynamic behaviour of current variables. Employing empirical data from cognitive psychology, psychosomatics, and marital interaction research, this paper describes how to apply cointegration methods to dynamic process systems and how to interpret the parameters under investigation from a psychological perspective.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

Temporal fluctuations of behaviour and performance are one of the main topics in psychological research. Time series analysis has gained increasing attention in psychological research during the last decade. Delignières, Fortes, and Ninot (2004) and Wagenmakers, Farrell, and Ratcliff (2004), for instance, demonstrated that methods investigating internal dependencies of a process offer an innovative perspective on understanding the specifics of psychological functioning. Gilden (2001) revealed that internal process dynamics account for a substantial proportion of experimental variability. According to Molenaar (2007)‘psychological methodology will change profoundly due to the necessity to focus on intra-individual variation’.

Most time series methodologies are restricted to stationary data. Since stationary series have time-invariant first and second moments, implying no trends or changing variability, they often fail to capture some crucial aspects of psychological dynamics. In general, successful interventions are expected to induce trends in the processes under investigation. Considering that a priori stabilizing transformations usually distort interesting characteristics of time series, methods accommodating non-stationary features of the data are required. As many psychological phenomena appear to interact as dynamic systems (Hamaker, Zhang, & van der Maas, 2009), multivariate methods, which model several processes simultaneously in the context of their common relational structure, provide deeper insights into human dynamics and are, therefore, indispensable for psychological research (Vallacher & Nowak, 1994).

The cointegration methodology, introduced by Engle and Granger (1987), accounts for all of these aspects, since it treats non-transformed non-stationary processes as a multivariate system. In cointegrated systems, non-stationary processes are connected by stationary long-run equilibrium relationships. Vector error-correction representations of cointegrated systems offer a convenient way to parameterize and specify cointegrated systems and serve as adequate instruments for the analysis of dynamic interactions of processes participating in a psychological system.

The goal of this paper is to popularize this approach for psychological research. As a first step, we clarify the main conceptual characteristics by pointing out its relevance for psychology. After providing the mathematical models behind integrated and cointegrated processes as well as the vector error-correction representation, we interpret their parameters from a psychological perspective. Applying these methods to psychological data illustrates the steps necessary in a typical research situation. This is done by means of two data sets from cognitive psychology and psychosomatics in a marital interaction framework.

### 2. Conceptual issues and relevance for psychological research

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

A **time series** *Y*_{t} is a sequence of *T* repeated measurements of the same variable in time, *t*= 0, 1, … , *T*. It is called *weakly stationary* if it has constant mean μ and variance σ^{2}. A process with independent and identically distributed measurements is called *white noise*, i.e. *WN*(0, σ^{2}). This process is *strictly stationary* as all of its finite-dimensional distributions (not just mean and variance) are time-invariant. In terms of weakly stationary processes, one important characteristic is stability, implying that these processes have a time-invariant mean and that deviations from this mean are rather small and random. The random deviation ε is due to numerous unknown influences that are levelled out in the long run and are thus regarded as normally distributed with zero mean. Hence, the variable itself is normally distributed with constant mean, i.e. *Y*=μ+ε∼*N*(μ, σ^{2}). Thus, we expect the same value for each measurement with random fluctuations around this value due to measurement error.

A great deal of psychological research is based on the assumption that the phenomena under investigation are **stable over time**. Personality traits, for instance, are defined as stable individual dispositions. Probably every psychologist is familiar with the five-factor model of Costa and McCrae (1992) identifying neuroticism, extraversion, openness, agreeableness, and conscientiousness as the basic structure subsuming all known personality traits. Self-esteem, as a particular example, is regarded as a continuous personality trait not greatly affected by daily events (Mischel, 1969). Concerning Cattell’s concept of fluid and crystallized intelligence (Cattell, 1987), the latter type is assumed to stay relatively stable across most of adulthood. Obviously, some phenomena show stability over a lifetime while others do so over shorter periods.

Mathematically, **trends** are either deterministic or stochastic. For a *deterministic trend*, the development of the process follows a straight course. Deterministic trends are usually linear, implying that the process moves along a straight line. Due to measurement errors or random deviations from this path, the process fluctuates around this line. It has a stable variance and a changing (e.g., linear) mean. If we subtract this mean from the measured value at each point in time and this difference is stationary, the resulting series is called *trend stationary*. Reaction tasks with permanently increasing complexity of the stimuli over time (i.e., mental rotation) might serve as an example of a linear trend as we expect the reaction time to increase linearly with a constant slope during the course of the experiment. Obviously, this is easy to predict and to interpret. In contrast to this, variables following a *stochastic trend* do not display such a straight development. Here, both mean and variance change over time. The process is called *difference stationary* as it is stationary after transformation into a series of period-to-period differences. This is possible because the change (i.e., difference) between periods is stationary. Such difference stationary processes are also called *integrated*. As will be described later, the aim of time series analysis in psychology is usually to identify and interpret trends. It is especially challenging if the trends are stochastic as they are not predictable at all. A great number of psychological processes exhibit stochastic trends. Glass, Willson, and Gottman (1975) found, for instance, that 44 out of 95 psychological time series (i.e., approx. 46%) were integrated.

Undoubtedly, the volume of **non-stationary psychological processes** cannot be neglected. To give some examples, children’s vocabulary permanently increases. In general, the development of children’s cognitive skills in their first years displays permanent progress. Apart from their psychic development, this is also true for their physiological functioning. With increasing age, trends seem to be the opposite for many cognitive functions. The ongoing Seattle Longitudinal Study – which has been in progress since 1956 (for an overview, see Schaie, 1996) – reports that personality remains relatively stationary over the adult lifespan, while cognitive abilities change if they are not trained. The identification of such trends in neuropsychological data may be relevant to the detection and treatment of dementia. Concerning the development of coping (i.e., the ability to deal with stressors), Skinner and Zimmer-Gembeck (2007) investigated age differences or changes in coping from infancy to adolescence. Apart from lifespan psychology, symptoms of disease may change in short order. Concerning the stationarity of self-esteem mentioned above, it has been widely recognized, however, that some specific life events such as professional success or failure can cause meaningful short-term instabilities in self-esteem (Rosenberg, 1995). Note that time series analysis was introduced to social and behavioural sciences to evaluate psychotherapy (Glass *et al.*, 1975). Obviously, the purpose of therapeutic interventions would need to be questioned if the client’s symptoms constantly fluctuated around the same value. Time series analysis in psychotherapy research is applied with the goal of studying mechanisms of change in psychotherapy process (e.g., Tschacher & Ramseyer, 2009). In general, research in social and behavioural sciences is interested in treatment effects. Self-evidently, the development of these psychological variables is not stationary (i.e., fluctuating around a constant mean) but trends can be clearly identified. Such trends are indicators of development and change – in short: human dynamics.

For social scientists, it is even more interesting to compare the temporal course of several variables. Obviously, the insight into the functioning of psychological variables can be improved if several phenomena are modelled in the context of their common relational structure. Therefore, the **dynamic systems perspective** is increasingly popular in psychology and has led to many studies – for example, in research on self-regulation (Carver & Scheier, 1998), psychotherapy (Schiepek, 2003; Tschacher, Baur, & Grawe, 2000), personality (Shoda, Tiernan, & Mischel, 2002), addiction (Witkiewitz, Van der Maas, Hufford, & Marlatt, 2007) and development (Van der Maas & Molenaar, 1992). The notion can be sustained by *systems* or even *chaos theory*, thus having implications for many domains of psychological research such as neuropsychology, psychopathology, psychotherapeutic processes as well as group dynamics in social and organizational psychology (Guastello, Koopmans, & Pincus, 2009). In fact, this approach has led to a new movement of *nonlinear systems science* in psychology (e.g., the *Journal of Nonlinear Dynamics, Psychology and Life Sciences*).^{1} Additionally, systemic approaches have a long applied tradition in psychology: systemic therapy addresses the individual as a member of a system identifying interactional patterns and dynamics.

It is possible, for instance, that two non-stationary series share the same stochastic trend, that is, they move synchronously, each of them on its particular scale. Processes with a stochastic trend are **integrated**. If several processes share a common stochastic trend, they are called **cointegrated**. It is even possible that a system consisting of more than two processes contains more than one common stochastic trend. In this case, not all series need to contain the same combination of trends. In a three-dimensional process system, for instance, one process may display two trends characterizing the remaining series separately, which means that two of the series do not move synchronously. Still, the three-dimensional process system is driven by two shared trends, indicating that its dynamic is not purely random. Thus, the systemic character is only obvious if the variables are appropriately combined. The aim of psychological time series research is to identify cointegrated systems and interpret these complex dynamics due to stochastic trends. The relationship between cointegrated variables can be represented as **vector error-correction models** (VECM). These are the focus of this paper as they allow for insights into the short- and long-term dynamics of complex process systems.

Apart from their econometric origin, cointegration methods have been increasingly applied to empirical data in the domain of sociology as well as political science. They have been used to clarify the relation between divorce and female labour force participation (Bremmer & Kesselring, 2004), between population and economic growth (Darrat & Al-Yousif, 1999), between age-specific fertility and female labour supply (McNown, 2003), between crime and immigration (Lin & Brannigan, 2003), between crime and its economic determinants (Luiz, 2001), between crime arrest rates for males and females (O’Brien, 1999) as well as between crime, prison and female labour supply (Witt & Witte, 2000).

Until now, the cointegration methodology has for the most part been unknown in psychological research. An adaptation from econometrics to psychology has been undertaken by Stroe-Kunold and Werner (2008, 2009). In this paper, we aim to provide methodological details not considered in the publications mentioned. We employ two psychological data sets for empirical demonstration. They are described first in order to stimulate the generation of hypotheses in the course of reading.

### 4. Defining integration and cointegration

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

Before defining *cointegration*, it is necessary to explain what *integration* means. A time series *Y*_{t} is called *integrated* of order *d*, *Y*_{t}∼*I*(*d*), if it can be transformed into a stationary process by differencing *d* times. For first differencing (*d*= 1), we write Δ*Y*_{t}=*Y*_{t}−*Y*_{t− 1}. Generally, a process is called integrated if *d*≠ 0. Note, however, that *Y*_{t}∼*I*(0) implies a stationary process as it is stationary without differencing (i.e., zero times).

The simplest integrated process is a *random walk*

- (1)

where ε_{t}∼*WN*(0, σ^{2}) denotes *white noise*. The first difference of this process Δ*RW*_{t}=ε_{t} is obviously stationary. The unpredictable walk of a drunk is a popular metaphor for a random walk. For understanding cointegration, the mathematical distinction between the properties of stationary and integrated processes is useful (see also Engle & Granger, 1987). For *Y*_{t}∼*I*(0), the variance is finite, shocks have only a temporary effect on *Y*_{t} and the expected length of intervals between mean-crossings is finite, that is, the process is mean-reverting. For *Y*_{t}∼*I*(1) processes, however, the variance is growing to infinity with increasing *t*, and shocks have a permanent effect on the development of *Y*_{t}. *Y*_{t}∼*I*(1) is the most frequent case of integration in applied research.

Including a constant term *c* in equation (1), *RW*_{t}=*c*+*RW*_{t− 1}+ε_{t}, leads to a model called *random walk with drift* exhibiting a linear (deterministic) trend in the mean. Figure 1 displays a white noise process as well as a random walk with and without drift.

As linear combinations play a crucial role in cointegration methodology, it is important to see how *I*(0) and *I*(1) processes can be combined.

Generally, the following rules hold:

- 1
A linear transformation does not change the degree of integration, i.e.

*Y*_{t}∼

*I*(

*d*)

*aY*_{t}+

*b*∼

*I*(

*d*).

- 2
The linear combination of two stationary processes is stationary.

- 3
The linear combination of a stationary and an integrated process is integrated of the same order.

- 4
The linear combination of two integrated processes is (mostly) integrated of the same order.

As a rule, two *I*(1) processes are controlled by stochastic mechanisms totally independent of each other. Nevertheless, different processes may share a *common stochastic trend*. In these cases, there exists a stationary linear combination in which the common trend is eliminated. Such processes are called cointegrated. Their common trend is the motor of the system’s dynamic.

As defined by Engle and Granger (1987; see also Lütkepohl, 2005), *K* univariate processes *Y*_{1t}, … , *Y*_{Kt} are *cointegrated* of order (*d*, *b*) if (i) all of them are integrated of the same order *d*, and (ii) there exists a linear combination β_{1}*Y*_{1t}+…+β_{K}*Y*_{Kt} with lower degree of integration, ∼*I*(*d*−*b*), *b* > 0. A multivariate process *Y*_{t} consisting of cointegrated variables is called a *cointegrated process*, denoted *Y*_{t}∼*CI*(*d*, *b*). In general, the *equilibrium process Z*_{t}=β′*Y*_{t} is a multivariate *r*-dimensional process with *r*×*K* matrix β called the *cointegration matrix*. In practice, the case *d*=*b*= 1 is especially relevant, where all the component series are integrated (*I*(1)) and the equilibrium process is stationary, that is, *Z*_{t}∼*I*(0).

Engle and Granger (1987) prove that *r* cointegrating relations indicate *K*−*r* common trends. Therefore, between two series *Y*_{1t} and *Y*_{2t} (i.e., *K*= 2), only one cointegrating relation is possible (*r*= 1), meaning that they share one common stochastic trend. Let us assume that each of the series consists of a non-stationary, integrated part and an error term which is white noise, ε_{1t}, ε_{2t}∼*I*(0). If they share the same integrated part, it is assumed to be their common trend *CT*_{t}∼*I*(1). *CT*_{t} does not necessarily have the same influence on both series and is thus weighted by λ_{1} and λ_{2}. According to rule 1 given above, both *Y*_{1t} and *Y*_{2t} are *I*(1) as *CT*_{t}∼*I*(1):

Obviously, there exists a linear combination where the common trend *CT*_{t} is eliminated and which is thus stationary:

The common trend is the reason why these series move synchronously as a system in the long run, that is, why they share a long-run equilibrium relation (which is described by the linear combination). This is surprising because this trend is stochastic, implying that the series do not move on a predictable course. Murray (1994) illustrates cointegration with the metaphor of a drunk and her dog, each individually representing a random walk. In spite of the non-stationarity of each individual’s path one would say: ‘if you find the drunk, the dog is unlikely to be very far away’. In his banquet speech, Nobel Prize laureate Clive Granger describes cointegration as a way ‘to discover that two large boats are drifting with the same current’.^{3} For psychological intervention, it is interesting to identify this common trend and to interpret its function for the dynamics of the system. Obviously, this is not trivial. The drunk and the dog move randomly as each of them has different interests. Perhaps the drunk is looking for places to rest while the dog is following impressions stimulating his nose. Perhaps something like their feeling of mutual everyday commitment is what makes them move synchronously in the long run. Concerning marital therapy, for example, Willi (1984) defines the unconscious aspects of the complementary defensive patterns in couples as *couple collusion*. They are supposed to stabilize neurotic relationships. If cointegration and thus common trend(s) are identified in couple settings, this psychodynamic concept might serve as an explanation. Probably every student of psychology is amazed when first hearing about the results of twin research, especially about the fact that individuals who do not share any aspects of their life display similarities in the development of personal preferences, traits, etc. The only explanation is their identical genetic material. The common trend determines a part of their lifespan dynamics. The fact that brothers and sisters develop similarly in some respects after moving away from home might be explained by their common socialization.

Cointegrating relations (i.e., stationary linear combinations) define the stationary long-run equilibrium inherent in the system. Equilibrium relationships are typical for many psychological variables and are an interesting subject for researchers studying conditions maintaining the system. Their identification often means a first step of (e.g., therapeutic) intervention or even prevention. Still, the participating series in such a cointegrated system are non-stationary as they follow a stochastic trend. It is interesting to find out how the system minimizes these instabilities, that is, how the drunk and the dog maintain their synchronicity in spite of diverging interests. In addition to the identification of cointegrating relations, vector error-correction models give insights into these short-term dynamics.

### 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

In general, systems consisting of several process variables can be modelled by means of vector autoregression (VAR), where *K* processes are analysed as a system. In other words, they are modelled as a multivariate or vector process of dimension *K*. Autoregression means that current behaviour is modelled on the basis of its previous characteristics, that is, that the processes consist of stochastic component series that can be described by the weighted sum of their previous values and a white noise error. In general, a *K*-dimensional vector autoregressive process of order *p* (VAR(*p*) process) is written as

- (2)

Therefore, each component series *Y*_{it} depends not only on its own but also on past values of other components of the system. By means of VAR models, Tschacher, Haemmig, and Jacobshagen (2003) investigated how differences in the perceived effects of morphine and heroin influence the acceptance of the drugs by intravenous drug users in opiate maintenance programmes. For this purpose, the authors measured the desired and adverse effects of high doses of these drugs in order to determine the causal interactions between both types of effects.

The *Granger representation theorem* says that for a system of cointegrated *I*(1) variables (written as a VAR process) a vector error-correction model (VECM) exists and vice versa (Engle & Granger, 1987)^{4}. In mathematical terms, the VAR process is the function and the VECM its derivation. VECMs simultaneously allow insights into the short- as well as long-term dynamics of cointegrated systems. This commonly used representation of cointegrated systems was originally stated by Johansen (1995). To explain the role of all parameters, we first define the general VECM. Let *Y*_{t} be a *K*-dimensional process integrated of order *d* (i.e., *Y*_{t}∼*I*(*d*)). For its first difference, the VECM is written as

- (3)

where

- •
*c* _{0} and

*c*_{1} are

*K*-dimensional constant vectors;

^{5} - •
α is the *K* × *r* loading matrix describing error-correction mechanisms in the cointegrated system;

- •
β, a matrix of cointegration weights, characterizes the long-run equilibrium relation between the cointegrated processes and is the

*K*×

*r* cointegration matrix normed to

with

denoting the identity matrix;

^{6} - •
Γ_{1}, … , Γ_{p− 1} are *K*×*K* matrices attaching weight to the lags, thus representing the autocorrelated structure of the cointegrated system;

- •
*p* is the number of lags in the corresponding VAR model;

- •
*r* is the rank of the *K* × *r* matrix Π=α · β′ and indicates the number of linearly independent cointegration relations and thus the number of stationary linear combinations characterizing the cointegrated system;

Recall that error-correction and thus vector error-correction modelling are only possible if the processes are cointegrated. How the particular function of the modelling parameters α, β and Γ can be interpreted will be described in detail in the following section.

For *r*= 0 and *r*=*K*, cointegration is impossible. Note that *r*= 0 indicates that there are no cointegration relations as the series are integrated but not sharing a common stochastic trend. On the other hand, *r*=*K* indicates *K* independent linear combinations, implying that all component processes are stationary themselves. The latter can be excluded by means of stationarity tests at the beginning of the analysis (described later in the framework of the empirical examples). For 0 < *r* < *K*, there may be cointegrating relations between the participating processes in a multivariate system. In the VECM, they are summarized by the *K*-dimensional vector β′*Y*_{t} which is a stationary process called ‘equilibrium error’ (Engle & Granger, 1987):

- (4)

### 6. Interpreting VECMs for bivariate cointegrated systems (*K*= 2)

- Top of page
- Abstract
- 1. Introduction
- 2. Conceptual issues and relevance for psychological research
- 3. Psychological data for empirical demonstration
- 4. Defining integration and cointegration
- 5. Short- and long-term system dynamics at a glance: Vector error-correction modelling
- 6. Interpreting VECMs for bivariate cointegrated systems (
*K*= 2) - 7. A drunk, her dog and a boyfriend: Interpreting VECMs for
*K* > 2 - 8. Summary and concluding remarks
- Acknowledgements
- References
- Appendices

In the following, we describe how VECMs estimated for bivariate cointegrated process systems can be interpreted. It is the nature of the beast that this part is rather technical and mathematical. Still, it can be a helpful guide to interpretation for psychological researchers. The section is concluded by vector error-correction (VEC) modelling for both data sets described at the beginning of this paper, including a detailed psychological interpretation.

First, we consider the impact of α_{1}, α_{2} and β on cointegrated systems. For this purpose, we assume a VECM of order *p*= 1 without constant vectors. Note that in this case β is a matrix of rank 1 and can therefore be reduced to a vector normed to (1, β*). Thus, we get the simplified model

- (5)

- (6)

Changes in both variables Δ*Y*_{it}, *i*= 1, 2, result from deviations from the equilibrium relation (Lütkepohl, 2005). The process *Y*_{t} is in equilibrium if β′*Y*_{t}= 0. Most of the time, however, the participating series deviate from equilibrium, with *Z*_{t}=β′*Y*_{t}=*Y*_{1t}+β**Y*_{2t} as equilibrium error.

#### *6.1* . *Characterizing the long-run equilibrium relation by means of β*

Let us assume that *Y*_{1t} and *Y*_{2t} are both *I*(1). In this case, only one stationary cointegration relation is possible (i.e., *r*= 1): [*Y*_{1t}+β**Y*_{2t}] ∼*I*(0). For the population, *Y*_{1t}+β**Y*_{2t} is a stationary process. In the case of an equilibrium relation, there is no systematic change as *Y*_{1t}+β**Y*_{2t}= 0. In the bivariate case, the equilibrium relation between the two processes depends on β*:

- •
if β*= 0, then *Y*_{1t}∼*I*(0) – no cointegration relation;

- •
if β* < 0, then *Y*_{1t} and *Y*_{2t} have the same direction;

- •
if β* > 0, then *Y*_{1t} and *Y*_{2t} have opposite directions, that is, are mirrored at the *x*-axis.

Thus, β characterizes the long-run equilibrium between the cointegrated component series, illustrating the relation of the variables necessary to maintain it. For a psychologist, this implies hints as to where to intervene. If the equilibrium (i.e., the status quo) is disturbed, the system has chances to change. Figure 2 illustrates the influence of β on the equilibrium relation of two cointegrated series by means of simulated processes.^{7}

Obviously, β levels out differences in the scales of the participating variables. Additionally, the inclusion of an additive constant (*K*× 1) vector *v* in the [β′*Y*_{t− 1}] term compensates differences in the mean of the component series: Δ*Y*_{t}=α[β′*Y*_{t− 1}+*v*′] +ε_{t}. Thus, the processes can be compared regardless of differences in scale and mean.

#### *6.2* . *Interpreting error-correction mechanisms by means of α*

This part of the paper analyses the consequences of previous deviations from the equilibrium relation. In the terminology of VEC modelling, α indicates how system errors (i.e., deviations) are corrected or minimized. In the simplest case, these deviations result from the fact that only one of the component processes has changed. α_{i} weights the influence of the equilibrium error on changes Δ*Y*_{it} in the process system. If α_{i}= 0, changes are exclusively random: Δ*Y*_{it}=ε_{it}.

First, |α_{i}| describes the intensity of the influence. For |α_{i}| < 1 the impact of deviations from the equilibrium in the previous period is decreased, while for |α_{i}| > 1 it is enhanced. Second, whether α_{i} is negative or positive has different implications for the system of cointegrated variables. It is helpful to look again at equation (5). Obviously, there is no systematic change in the case of an equilibrium (i.e., *Z*_{t− 1}= 0). For α_{i} > 0, the change in the process *Y*_{it} behaves like the deviation from the equilibrium. For α_{i} < 0, the opposite behaviour occurs. The question is: which values of α_{1} and α_{2} indicate cointegration between *Y*_{1t} and *Y*_{2t} with the equilibrium error (as a stationary process) permanently being corrected in the next period?^{8} All possible variations of α_{1} and α_{2} and their implications for the system dynamics are summarized in Table 2.

Table 2. Error-correction mechanisms in bivariate cointegrated systems for variations of α_{1} and α_{2} α_{1}≠ 0, α_{2}= 0 | α_{1} < 0 | cointegration |

| | error correction possible |

| α_{1} > 0 | no cointegration |

| | no error correction |

α_{1}= 0, α_{2}≠ 0 | α_{2}β* < 0 | cointegration |

| | error correction possible |

| α_{2}β* > 0 | no cointegration |

| | no error correction |

α_{1}≠ 0, α_{2}≠ 0 | α_{1} < 0, α_{2}β* < 0 | cointegration |

| | error correction possible |

| α_{1} < 0, α_{2}β* > 0 | cointegration possible |

| α_{1} > 0, α_{2}β* < 0 | cointegration possible |

| α_{1} > 0, α_{2}β* > 0 | no cointegration |

| | no error correction |

First, we consider the case where α_{1}≠ 0 and α_{2}= 0, implying Δ*Y*_{2t}=ε_{2t}. For α_{1} < 0, deviations from the equilibrium can be minimized. The system’s dynamic strives to restore the equilibrium. The linear combination of the participating processes is stationary and the processes are cointegrated. Note that for α_{2}= 0 only *Y*_{1t} can minimize short-term deviations. For α_{1} > 0, the disequilibrium is enhanced. The distance between *Y*_{1t} and *Y*_{2t} permanently increases to infinity, and the system explodes. A return to equilibrium is impossible. The error is non-stationary. Thus, no error correction takes place and the processes are not cointegrated.

Second, we are interested in the consequences of deviations from equilibrium for α_{1}= 0 and α_{2}≠ 0, meaning

where ε_{2t} is set to zero for the purpose of demonstration. As the cointegration vector is normed to (1, β*), it is relevant for the interpretation of α_{2} whether β* is positive or negative. For α_{2}β* < 0, error correction is possible and the processes are cointegrated. Note that here (for α_{1}= 0) only dynamics in *Y*_{2t} can restore the equilibrium. On the contrary, the disequilibrium is enhanced for α_{2}β* > 0 and the processes are not cointegrated.

Finally, we examine the deviations for α_{1}≠ 0 and α_{2}≠ 0. Here, the full model has to be considered:

Again ε_{1t}=ε_{2t}= 0 for demonstration. For α_{1} < 0 and α_{2}β* < 0 error correction is possible and the series are cointegrated. If both α_{1} and α_{2}β* are greater than zero, the processes cannot be cointegrated as the distance between *Y*_{1t} and *Y*_{2t} increases to infinity. Thus, a stationary equilibrium between the series is excluded. For α_{1} < 0 and α_{2}β* > 0 as well as for α_{1} > 0 and α_{2}β* < 0 the picture is rather unclear. Here, both processes have opposite tendencies: while one series drifts apart, the other series tries to minimize this deviation.^{9} Readers interested in how the consequences of the deviations can be concluded mathematically may find these explanations in Appendix C. Figure 3 illustrates the function of α in bivariate cointegrated systems.

#### *6.3* . *Disentangling the function of the participating processes by means of Γ*

In some cases, additional lags have to be included in order to have white noise errors. Here Γ has to be considered, too (see equation (3)). The general bivariate VECM is written

Significant elements in the main diagonal of Γ indicate self-regulation mechanisms within a certain series itself, meaning that changes in the series depend on previous changes in the series themselves. Additional significant elements in the counterdiagonal imply inter-regulation, that is, that previous changes in the series have an influence on current changes in other participating processes. If |γ_{it}| > 1 these changes are enhanced, while they decrease for |γ_{it}| < 1. Thus, short-term adjustment dynamics can be identified by means of Γ.

Note, however, that whether a multivariate system is explosive or dissipative does not only depend on the absolute value of every single element of the weight matrices. As in the standard interpretation of VAR processes, researchers need to consider, for instance, whether the roots of the associated characteristic equation lie outside of the unit circle. For details, consult Stadnytska (2011) and Stadnytska and Gruber (2011).

#### *6.4* . *Empirical demonstration: Psychological examples for K= 2*

The methods described are applied to bivariate process systems from both the temporal estimation and the marital interaction data. In general, cointegration analysis consists of three steps (see also Beck, 1993; Stroe-Kunold & Werner, 2008): (1) separate analysis of the participating processes by means of stationarity tests; (2) cointegration tests; and (3) VEC modelling.^{10} Recall that the data analysed in the following are described at the beginning of the paper. Interested readers may find additional practical information on how to work with multivariate time series data in Stadnytska (2011) and Stadnytska and Gruber (2011).

##### 6.4.1. Temporal estimation task

As expected, the temporal estimation series are stationary for the baseline condition with correct feedback. Concerning the treatment condition, we confine our presentation to a randomly chosen case since the results were similar for all subjects. Note that the length of the series allows for fractional (i.e., long memory) analysis. [*SIM*]_{t} was simulated with the memory parameter *d*= 0.75, while estimation procedures indicate (fdSperio *CI*_{0.95} [0.67, 0.89] and fdGPH *CI*_{0.95} [0.59, 0.97]) for [*RT*]_{t}. As *d* is close to 1, full cointegration analysis (i.e., for *I*(1) processes) is appropriate for the purpose of demonstration.

Stationarity of a series can be tested by means of so-called unit root procedures checking the null hypothesis that ‘there is a unit root’ in the VAR representation against the alternative of stationarity. The most popular unit root test is the *augmented Dickey–Fuller* or *ADF test* (Dickey & Fuller, 1979), described in detail in Stadnytska (2010). It can be conducted with three types of specifications: zero mean, non-zero mean and trend (in the first difference of the process). If the null hypothesis is rejected, it means in the first case that the process is a stationary series with zero mean; in the second case, it is stationary with a non-zero mean; and in the third case, the series is stationary around a deterministic trend. The test statistic is not normally distributed. Thus, the critical values implemented in the software were computed by means of Monte Carlo simulations and differ for the three cases. The number of lags is determined empirically using the Akaike and Bayesian information criteria (AIC and BIC), the idea being to include enough lagged terms so that the error part is serially uncorrelated. According to the test, [*RT*]_{t} is non-stationary. As expected, the same is indicated for [*SIM*]_{t}.

For testing the number of cointegrating relations *r*, the *Johansen trace test* (Johansen, 1995) is commonly applied (for details see, for instance, Hamilton, 1994). Stadnytska and Gruber (2011) outline how to work with this test on psychological data. This *likelihood ratio* (LR) cointegration test subsequently compares different null hypotheses. For example, if *H*_{0}[*r*≤ 0] is rejected and *H*_{0}[*r*≤ 1] is not, there is exactly one cointegrating relation. The distribution of the test statistic under the respective null hypothesis depends on the deterministic terms. Again, the number of lags is determined employing the AIC and BIC. Notice that the number of lags included may have a strong impact on the test decision. Here, for instance, AIC recommends three lags while BIC proposes two lags. For the former case, the *LR* statistic for *H*_{0}[*r*≤ 0] is 112.72 (*p* < 0.001) and for *H*_{0}[*r*≤ 1] *LR*= 5.61 (*p*= 0.2314). For the latter case, *H*_{0}[*r*≤ 0] is rejected (*LR*= 166.49, *p* < 0.001) and *H*_{0}[*r*≤ 1] is not (*LR*= 7.29, *p*= 0.1147). Clearly, one cointegrating relation is indicated. This means that – as expected – the estimated time and the simulated series share a common stochastic trend and are in long-run equilibrium. Thus, the manipulated feedback (which includes the simulated series) has a clear impact on the subject’s performance in temporal estimation, contradicting the hypothesis of an inner clock but underlining that feedback has a great impact on temporal estimation, no matter whether it is correct or false. For this example, the simulated series can be easily identified as the common trend or the motor of the systems’ dynamic. Thus, we find no interaction but a causal relation.

Cointegration implies that error-correction mechanisms may exist due to short-term deviations of the participating processes from the equilibrium relation. Again, the number of lags in the VECM is determined by means of AIC and BIC. The estimated VECM for *p*= 1 and the corresponding equilibrium relation are summarized in Table 3.

Table 3. Experiment: estimated VECM (*K*= 2) As expected, the linear combination indicates that [*RT*] and [*SIM*] move in the same direction as [*RT*]_{t}= 1.003[*SIM*]_{t}+ 958.861. Differences in the mean of both processes are levelled out by the constant ≈1000 ms. Obviously, the equilibrium between both series is very strong as the series move extremely synchronously. The equilibrium ratio of the linear combination characterizes the system of both variables. If at one point in time, however, this ratio is not true, deviations from the equilibrium take place. As |α_{1}| < 1, the influence of previous deviations from the equilibrium on changes in [*RT*] is decreased in the next period. Note that the estimated α_{2} is not significantly different from zero. According to Table 2, this is consistent with the above finding of cointegration as α_{1} < 0 and α_{2}= 0, meaning that only [*RT*] can minimize deviations from the equilibrium and confirming the experimental set-up, which implies that [*SIM*] is an artificial simulated series unable to adapt to external changes. This is additionally underlined by the fact that only elements in the main diagonal of Γ are significantly different from zero. Hence, changes in the series depend on previous changes in the series themselves but not on previous changes in other processes. As the absolute values of these Γ coefficients are <1, previous changes in the series decrease in the next period. Figure 4, showing both series and their linear combination, illustrates these findings.

##### 6.4.2. Marital interaction

Stroe-Kunold and Werner (2008) already analysed these data using the cointegration methodology. Their study was, however, restricted to the separate analysis of the bivariate relations of [*BUL*] and [*AH*] as well as [*DERM*] and [*AW*]. In this paper, we are interested in understanding the dynamics of different constellations as well as examining more complex systems of higher order.

The estimated VECMs and the equilibrium relations are listed in Table 4. Only for the equilibrium relation between [*AH*] and [*DH*] the inclusion of one lag is it necessary to get white noise residuals. This decision is based on the indications of AIC and BIC in combination with residual analyses (e.g., the Portmanteau test). For [*DH*] and [*DW*] as well as [*BUL*] and [*DH*], AIC and BIC as well as the residual analyses indicate that *p*= 0 is sufficient. Therefore, all terms with lagged differences weighted by Γ can be excluded.

Table 4. Clinical study: estimated VECMs for *K*= 2 For [*DH*] and [*DW*], β* < 0 indicates that both series have the same direction. This implies, for instance, that the husband’s depressiveness increases if the wife gets more depressive. Additionally, |β*| is close to 1, suggesting the intensity of their depressive moods is approximately equal. If at a certain moment she is much more depressive than he is, the system is temporarily not in equilibrium. For [*AH*] and [*DH*] as well as [*BUL*] and [*DH*], β* > 0 implies that the processes have opposite directions. This discrepancy is larger for [*AH*] and [*DH*] (β*= 1.772) than for [*BUL*] and [*DH*] (β*= 0.519). It seems logical that he is not aggressive and depressive at the same time, but that his aggressiveness is low if his depressive mood is intense and vice versa. In the latter case, her bulimic symptoms are not triggered but rather decrease when his depressiveness is intense. From the systemic perspective, this is functional as he probably needs her attention when his depression is acute while she needs his care when her bulimic symptoms are intense. Otherwise, their psychosomatic couple system is not in equilibrium.

We find α_{1} < 0 and α_{2}β* < 0 for all of the three bivariate process systems. This means that changes in none of the variables result only from randomness but that error correction takes place. Thus, deviations are corrected by the dynamics of both series. As |α_{1}| < 1 and |α_{2}| < 1 for all constellations, the influence of previous deviations from the equilibrium on current changes in the variables decreases.

For [*AH*] and [*DH*], only elements in the main diagonal of Γ are significantly different from zero, implying self-regulation mechanisms within the series. As these coefficients are negative, a negative change yesterday is followed by a positive change today, etc. Figure 5 plots the original series as well as their linear combination.