Portfolio Decisions and Brain Reactions via the CEAD method

Abstract

Decision making can be a complex process requiring the integration of several attributes of choice options. Understanding the neural processes underlying (uncertain) investment decisions is an important topic in neuroeconomics. We analyzed functional magnetic resonance imaging (fMRI) data from an investment decision study for stimulus-related effects. We propose a new technique for identifying activated brain regions: cluster, estimation, activation, and decision method. Our analysis is focused on clusters of voxels rather than voxel units. Thus, we achieve a higher signal-to-noise ratio within the unit tested and a smaller number of hypothesis tests compared with the often used General Linear Model (GLM). We propose to first conduct the brain parcellation by applying spatially constrained spectral clustering. The information within each cluster can then be extracted by the flexible dynamic semiparametric factor model (DSFM) dimension reduction technique and finally be tested for differences in activation between conditions. This sequence of Cluster, Estimation, Activation, and Decision admits a model-free analysis of the local fMRI signal. Applying a GLM on the DSFM-based time series resulted in a significant correlation between the risk of choice options and changes in fMRI signal in the anterior insula and dorsomedial prefrontal cortex. Additionally, individual differences in decision-related reactions within the DSFM time series predicted individual differences in risk attitudes as modeled with the framework of the mean-variance model.

Appendix

See (Figures 10–17 and Tables 4, 5).

1.1 Simulation Study

Fig. 10

Stimulus time series derived as a convolution of double Gamma hemodynamic response function and uncorrelated portfolio stimulus $\times 64$ plotted against time (each $2$ s).

Fig. 11

Simulated spatially correlated Gaussian noise for $2$ vertical neighbor voxels (red and blue) plotted against time (each $2$ s); $\mathop {\hbox {Corr}}_t(\varepsilon _{t,1},\varepsilon _{t,2})=0.97$.

Fig. 12

Simulated stimulus time series as the AR($2$) process: $\widetilde{Z}_{t}=0.5\widetilde{Z}_{t-1}+0.2\widetilde{Z}_{t-2}+\varepsilon _{AR,t}$, plotted against time (each $2$ s).

1.2 Clustering and Sensitivity Analysis

Fig. 13

Sensitivity analysis of the risk attitude $\phi$: estimates $\widehat{\phi }_i, i=1,\ldots ,19$ with $95\,\%$ confidence intervals.

Fig. 14

The derived risk attitude of subject $1$ in a rolling window exercise ($\widehat{\phi }_i$ estimated from past $100$ ID answers).

Fig. 15

Time series of the correlation coefficient derived by the rolling window (250 top, 500 bottom) for the center voxel and: horizontal, vertical diagonal neighboring voxel for aINS(right) of subject 1.

Fig. 16

Contour plots of derived aINS(left), aINS(right) and DMPFC (upper, middle lower panel) clusters for subjects $1$ (left) and $19$ (right), respectively; derived by the NCUT algorithm with $C=1,000$. $x$, $y$ $z$ axis denote the $3D$ space given in millimeters.

1.3 Factor Loadings

Fig. 17

Sample autocorrelation function of aINS(left), aINS(right), and DMPFC $\widehat{Z}_t$ (top left, top right, bottom panel, respectively) for subjects $1$ (top) and $19$ (bottom), respectively.

Table 4 KPSS, ADF test statistics for estimated factor loadings aINS(left), aINS(right), and DMPFC $\widehat{Z}_t$; subject $1$ (left panel), subject $19$ (right panel) (KPSS: $H_0$: weak stationarity, critical values at $0.10$, $0.05$, $0.01$ are $0.119$, $0.146$, and $0.216$; ADF: $H_0$: unit root, critical values at $0.01$, $0.05$, $0.10$ are $-1.61$, $-1.94$, and $-2.58$).

Table 5 The position of the cluster local maximum, denoted in the Montreal Neurological Institute (MNI) standard at $2$mm resolution, corresponding $Z$-score (middle) and $p$ value (bottom) of activated “risk” clusters during the ID stimuli.