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.
Maximizing non-Gaussianity
This page demonstrates that minimizing the innovations' dependencies (as shown on the previous page) is equivalent to maximizing the shocks' non-Gaussianity. Consequently, we can identify $\phi_0$ by finding the most non-Gaussian innovations.
Measuring Non-Gaussianity
The non-Gaussianity of the innovations $ e_t(B(\phi)) $ can be measured using the skewness and excess kurtosis:
Interactive Simulation
Move the $\phi$ slider to see how the rotation angle affects the non-Gaussianity of the innovations displayed in the table below the scatter plot.
Interactive Loss Calculation
The loss value in the table above simply sums up all squared higher-order moments to get an overall measure of the non-Gaussianity.
The plot below shows the loss for all rotation angles $\phi$.
- Push the Maximize non-Gaussianity button to find the rotation angle $\phi$ which leads to the most non-Gaussian innovations.
- Try out different sample sizes $T$ and new data sets using the New Data button to see how the rotation $\phi$ which maximizes the innovations' non-Gaussianity is located around the true rotation $\phi_0 = 0.5$.
Takeaway: Choosing the angle $\phi$ that maximizes the innovations' non-Gaussianity allows us to estimate $\phi_0$.
Next Steps
The next page illustrates how the degree of non-Gaussianity of the structural shocks affects the estimation.