Mathematical Intuition of VAR Equations
VAR models are mathematically represented as a system of simultaneous equations, where each equation describes the behavior of one variable as a function of its own lagged values and the lagged values of all other variables in the system.
Mathematically, a VAR(p) model with ‘p’ lags can be represented as:
[Tex]Y_t = c + \Phi_1 Y_{t-1} + \Phi_2 Y_{t-2} + \dots + \Phi_p Y_{t-p} + \varepsilon_t [/Tex]
Here,
- [Tex]Y_t[/Tex]: This represents the value of the time series at time t.
- c: This represents the constant intercept term in the model.
- [Tex]\Phi_1, \Phi_2, …, \Phi_p[/Tex]: These represent the autoregressive coefficients for lags 1, 2, …, p, respectively.
- [Tex]Y_{t-1}, Y_{t-2}, …, Y_{t-p}[/Tex]: These represent the values of the time series at lags 1, 2, …, p before time t.
- [Tex]\varepsilon_t[/Tex]: This represents the error term at time t.
To ensure the validity and trustworthiness of the results from VAR analysis, various assumptions and requirements must be met.
Vector Autoregression (VAR) for Multivariate Time Series
Vector Autoregression (VAR) is a statistical tool used to investigate the dynamic relationships between multiple time series variables. Unlike univariate autoregressive models, which only forecast a single variable based on its previous values, VAR models investigate the interconnectivity of many variables. They accomplish this by modeling each variable as a function of not only its previous values but also of the past values of other variables in the system. In this article, we are going to explore the fundamentals of Vector Autoregression.
Table of Content
- What is Vector Autoregression?
- Mathematical Intuition of VAR Equations
- Assumptions underlying the VAR model
- Steps to Implement VAR on Time Series Model
- Step 1: Importing necessary libraries
- Step 2: Generate Sample Data
- Step 3: Function to plot time series
- Step 4: Function to check stationarity
- Step 5: VAR analysis
- Output Explanation
- Applications of VAR Models