In many real-world applications, e.g., brain imaging and or weather patterns, data are captured over particular periods or intervals, which we call time series. Time series analysis comprises statistical techniques to explore and analyze time-series data to extract meaningful characteristics from data. In this work, we are interested in determining brain connectivity from time-series data. Our emphasis of this work is on functional magnetic resonance imaging (fMRI) data. fMRI is one of the most widely used techniques used to detect brain activity associated with blood flow and study human cognition. It can measure blood-oxygen-level-dependent (BOLD) responses from activity due to tasks/stimuli (task-related fMRI) or resting condition (resting-state fMRI) which shows the subjects' baseline BOLD variance. The goal of this work is to study brain connectivity and its relation associated with external variables, e.g. external tasks or demographic information. Motivated by the fact that fMRI measures the indirect and convoluted signals of brain neuronal activities and neuronal activities can be modeled by ordinary differential equations (ODE), for task-related fMRI data, we propose a causal dynamic network model and algorithm to estimate causal connections of convoluted signals (effective brain connectivity). To study dynamic functional brain connectivity of coupled brain regions, we propose time-dependent canonical correlation analysis (TDCCA) method, which extends canonical correlation analysis (CCA) to general time-series data. Lastly, we explore a projection regression model for connecting covariance matrices (static functional brain connectivity) with auxiliary demographic information of each subject. Our results support the hypothesis that brain connectivity is associated with each subject's demographic information. The methods proposed in this work can also be applied to datasets from other topics such as finance and economics.