From these equations we can see that the mean, variance, and covariance structures all depend on time, making the process nonstationary. Note that nonstationary pro- cesses are not necessarily more complicated than stationary processes. However, many time series modeling tools are designed to specifically handle stationary pro- cesses.
1.2 Initial Condition and Stationarity.
Thus far I have discussed the substantive importance of initial condition in psy- chological theories and defined stationarity with respect to time series processes. I will now give a more detailed explanation of an initial condition distribution and de-
may be formally defined as
xi0 ∼ N(µ0,P0) (1.22)
whereµ0 is the mean vector andP0 the covariance matrix of the initial condition. To
better understand the role of initial condition, consider a multiple-subject univariate AR(1) model:
xit =α1xi,t−1+vit vit ∼ N(0,σv2). (1.23)
Since the current process variables are affected by previous process variables, the model can be iterated backwards:
xit =α1xi,t−1+vit (1.24)
xi,t−1 =α1xi,t−2+vi,t−1
xit =α1(α1xi,t−2+vi,t−1) +vit
xit =α1(α1(α1xi,t−3+vi,t−2) +vi,t−1) +vit.
This process can recur indefinitely if one continues to go back in time. Going back in
time b times leads to the following,
xit =αb1xi,t−b+ b−1
∑
j=0α1jvi,t−j. (1.25)
Now we can clearly seen that if |α| > 1, the process will explode, i.e., the effect of
the process variable will be magnified with time, and thus will not be stationary. Also, the model cannot go back in time indefinitely, thus there needs to be an initial starting point. In order to initiate this otherwise infinite recursion, a distribution for
xi0 needs to be specified.
may be used as the initial condition distribution (Harvey, 1991). This is because a stationary process states that the mean and covariance structures do not change with time. For example, a stationary AR(1) process may have an initial condition distribution of xi0 ∼ N 0, σv2 1−α21 ! . (1.26)
This holds because for the stationary AR(1) process the mean is equal to zero and the
variance is equal to σv2
1−α21. If a process is nonstationary, or even if it is hypothesized
that the process prior to the first observation has a different structure, the uncon- ditional mean and covariance matrix may not be the best choice for specifying the initial condition distribution.
There are a number of methods for specifying initial condition (the unconditional mean vector and covariance matrix specification is just one), some of which apply to stationary models and some that apply to nonstationary models. As psychologists are currently implementing models incorporating time series processes (Hamaker & Dolan, 2009), further research regarding the specification of initial condition becomes important. In time series analysis, nonstationary processes may be made stationary by either detrending or differencing (Shumway & Stoffer, 2006). Detrending removes a trend by subtracting an estimated trend component from the original process and working with the subsequent residuals. Differencing involves subtracting a previous observation from the current observation a specified number of times. If the trend is linear, then performing this calculation once, also called first differencing, will remove the trend. A second difference removes a quadratic trend, a third difference removes a cubic trend, and so forth. While these approaches work well with respect to making a nonstationary process stationary, in many psychological theories it is the trend that is of actual interest. For example, a trend in psychological data may
Since the parameters that describe the development over time may be integral in understanding the psychological process, it is not always helpful to remove the trend. Therefore, when psychologists implement time series models it may be important to retain the nonstationarity of a process, rendering the proper specification of initial condition also important.
Browne and Nesselroade (2005) and Du Toit and Browne (2001, 2007) have dis- cussed how multivariate time series models may be estimated as a longitudinal model in the SEM framework. This modeling framework is often used by psy- chologists as it can readily and flexibly model relations among both manifest and latent variables. Furthermore, latent variable measurement models may be easily es- timated. However, when estimating any times series model in the SEM framework, several methodological issues must be addressed, including the proper specification of an initial condition distribution. While some methodologists have addressed this issue (Browne & Nesselroade, 2005; Du Toit & Browne, 2001, 2007), the focus has been on stationary processes.
A particularly flexible way to represent a time series model is to formulate them within the state-space modeling framework (Harvey, 1991). State-space models were originally derived for single-subject time series analysis but may also be used to estimate intensive repeated measures or panel data within the structural equation modeling framework. In this thesis I will formulate structural equation models con- taining time series processes within the state-space modeling framework. Given this set-up, I will describe how to incorporate different types of initial condition distribu- tions. To facilitate a more technical discussion of initial condition specification given this set-up, I will first describe the state-space modeling framework in more detail followed by a description of the SEM framework.