Sample Size: $T$
T represents the sample size or the number of observations in the dataset.
Rotation angle: $\phi_0$
In the bivariate SVAR $u_t = B_0(\phi_0) \epsilon_t$, the reduced-form shocks $u_t$ are a rotation of the structural shocks $\epsilon_t$ by angle $\phi_0$ (in radians).
Rotation angle: $\phi$
In the bivariate SVAR $u_t = B_0(\phi_0) \epsilon_t$, the reduced-form shocks $u_t$ are a rotation of the structural shocks $\epsilon_t$ by an angle of $\phi_0$ (in radians), i.e., $\phi_0 \cdot (180/\pi)$ degrees. On this page, the true rotation is fixed at $\phi_0 = 0.5$. The rotation $\phi$ allows us to compute the innovations as $e_t(B(\phi)) = B(\phi)^{-1} u_t$, which corresponds to rotating the reduced-form shocks by $\phi \cdot (180/\pi)$ degrees.
Non-Gaussianity parameter: $s$
The parameter $\, s$ controls the non-Gaussianity of $\epsilon_{2,t}$. For $s = 0$ the shock is Gaussian; for $s < 0$ the shock is left-skewed; for $s > 0$ it is right-skewed.
Impact of Non-Gaussianity on Estimation
This page illustrates how the degree of non-Gaussianity affects the estimation of structural shocks in our SVAR model.
Shock Distributions
In the simulation below, the first structural shock $\epsilon_{1,t}$ is independently drawn from a standard normal distribution. The second structural shock $\epsilon_{2,t}$ is independently drawn from a mixture of normal distributions
The location parameter $s$ of the second component can be chosen using the slider above:
- For $s = 0$, both shocks are normal.
- For $s < 0$, the second shock is left-skewed.
- For $s > 0$, the second shock is right-skewed.
Interactive Simulation
Move the $\phi$ and $s$ sliders to see how the rotations and non-Gaussianity affect the dependencies of the innovations displayed in the table below the scatter plot.
Interactive Loss Calculation
The loss value in the table again sums up all squared co-moments to get an overall measure of the dependencies.
The plot below shows the loss for all rotation angles $\phi$.
- Push the Minimize Dependencies button to find the rotation angle $\phi$ which leads to the least dependent innovations.
- Vary the degree of non-Gaussianity using $s$ to see how it affects the optimization landscape.
Takeaway: Gaussian shocks lead to a flat optimization landscape without a unique minimum at $\phi_0$. This illustrates why non-Gaussianity is crucial for identifying structural shocks in SVAR models based on independent shocks.
More Resources
- Read Lewis (2024) for a recent literature review on identification approaches based on higher moments.
- Explore SVARpy, a preliminary Python package for estimating non-Gaussian SVAR models.
- Access the code for this dashboard on GitHub.
- I welcome your feedback: sascha.keweloh@tu-dortmund.de