22th February, 2019

Context

  • Most psychological theories are within-subject, but measurements are between-subject (Luce, 1997)

  • There is no simple mapping between the two (Molenaar, 2004)

  • Ubiquity of smartphones enables collecting intensive longitudinal data (Miller, 2012)
    • Understanding humans as dynamical systems
    • Goes in hand in hand with the network paradigm (Borsboom, 2017)

Vector Autoregression

VAR Model

  • We observe some realization of the (discrete-time) \(p\)-dimensional stochastic process

\[ \{Y(t): t \in T\} \]

  • with state space \(Y(t) \in \mathbb{R}^p\)
  • Vector autoregressive (VAR) model assumes

\[ \mathbf{y}_t = \mathbf{\nu} + \mathbf{A}_1 \mathbf{y}_{t-1} + \ldots + \mathbf{A}_l \mathbf{y}_{t-1} + \mathbf{\epsilon}_t \]

  • \(\nu \in \mathbb{R}^p\) is the intercept
  • \(A_l \in \mathbb{R}^{p \times p}\) describes the coefficients at lag \(l\)
  • \(\mathbf{\epsilon}_t\) are the stochastic innovations at time \(t\), for which

\[ \begin{align} \mathbf{\epsilon}_t \sim \mathcal{N}(0, \Sigma) \\[0.5em] \mathbb{E}[\mathbf{\epsilon}_t\mathbf{\epsilon}_{t-1}] = 0 \end{align} \]

  • VAR process is covariance-stationary, i.e. its first and second moment are time-invariant

AR Model

  • Lag \(l = 1\) VAR model requires estimation of \(A^{p \times p}\)
  • This may lead to high variance of the estimates when data is poor
  • Reduce variance by introducing bias: set all off-diagonal elements to zero
  • This saves estimating \(p \times (p - 1)\) parameters

\[ A_{\text{VAR}} = \begin{pmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} & \alpha_{14} & \alpha_{15} & \alpha_{16} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \alpha_{24} & \alpha_{25} & \alpha_{26} \\ \alpha_{31} & \alpha_{32} & \alpha_{33} & \alpha_{34} & \alpha_{35} & \alpha_{36} \\ \alpha_{41} & \alpha_{42} & \alpha_{43} & \alpha_{44} & \alpha_{45} & \alpha_{46} \\ \alpha_{51} & \alpha_{52} & \alpha_{53} & \alpha_{54} & \alpha_{55} & \alpha_{56} \\ \alpha_{61} & \alpha_{62} & \alpha_{63} & \alpha_{64} & \alpha_{65} & \alpha_{66} \end{pmatrix} \]

\[ A_{\text{AR}} = \begin{pmatrix} \alpha_{11} & 0 & 0 & 0 & 0 & 0 \\ 0 & \alpha_{22} & 0 & 0 & 0 & 0 \\ 0 & 0 & \alpha_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \alpha_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \alpha_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & \alpha_{66} \\ \end{pmatrix} \]

Bias-Variance Trade-off

  • Bulteel et al. (2018) compared performance of (mixed) AR and (mixed) VAR models
    • Find equal predictive performance on 3 data sets
    • "[…] it is not meaningful to analyze the presented typical applications with a VAR model" (p. 14)
  • We re-evaluate this claim (Dablander⋆, Ryan⋆, & Haslbeck⋆, under review)