Non-Gaussian SVAR: Minimizing Dependencies

Minimizing Dependencies

This page explores how we can verify whether we've correctly identified the structural shocks by minimizing dependencies between innovations.

Independent Shocks

While we cannot observe the structural shocks $\epsilon_t$, we do know something about their stochastic properties: The structural shocks $\epsilon_t$ are independent, meaning a given shock $\epsilon_{1,t}$ contains no information about the shock $\epsilon_{2,t}$ and vice versa. The same property should apply to the innovations $e_t(B(\phi))$, meaning a given innovation $e_{1,t}$ should contain no information on the innovation $e_{2,t}$ and vice versa.

We can use the stochastic property of independent shocks to verify whether we have chosen $\phi = \phi_0$ and whether our innovations equal the structural shocks. We can do this in two steps:

  1. Propose a rotation angle $\phi$ and calculate the corresponding innovations $e_t(B(\phi))$.
  2. Measure the dependency of the innovations and rule out rotation angles $\phi$ that lead to dependent innovations.

Moment Conditions for Independent Shocks

The dependency of the innovations $ e_t(B(\phi)) $ can be measured using moment conditions. Specifically, independent shocks with mean zero and unit variance will satisfy the following covariance-, coskewness-, and cokurtosis- conditions.

Covariance condition: $$E[\varepsilon_{1,t}\varepsilon_{2,t}] = 0 \implies \ E[e_{1,t}(\phi)e_{2,t}(\phi)] \overset{!}{=} 0$$ Coskewness condition: $$E[\varepsilon_{1,t}^2\varepsilon_{2,t}] = 0 \implies \ E[e_{1,t}(\phi)^2e_{2,t}(\phi)] \overset{!}{=} 0$$ $$E[\varepsilon_{1,t}\varepsilon_{2,t}^2]=0 \implies \ E[e_{1,t}(\phi)e_{2,t}(\phi)^2] \overset{!}{=} 0$$ Cokurtosis condition: $$E[\varepsilon_{1,t}^3\varepsilon_{2,t}] = 0 \implies \ E[e_{1,t}(\phi)^3 e_{2,t}(\phi)] \overset{!}{=} 0$$ $$E[\varepsilon_{1,t}\varepsilon_{2,t}^3] = 0 \implies \ E[e_{1,t}(\phi) e_{2,t}(\phi)^3] \overset{!}{=} 0$$ $$E[\varepsilon_{1,t}^2\varepsilon_{2,t}^2] = 1 \implies \ E[e_{1,t}(\phi)^2 e_{2,t}(\phi)^2] \overset{!}{=} 1$$

Interactive Simulation

Explore the reduced-form shocks $u_t$ and the innovations $e_t(B(\phi))$ for the current rotation. The tables below each plot show the corresponding co-moments used in the dependency loss.

Observations

Note that any rotation $\phi$ yields uncorrelated shocks. However, even though all innovations are uncorrelated, some are clearly dependent as measured by the higher-order moment conditions.

The loss value simply sums up all squared co-moments to get an overall measure of the dependencies:

$$loss(\phi) = mean( e_{1,t}(\phi) e_{2,t}(\phi) )^2 + \cdots + mean( e_{1,t}(\phi)^2 e_{2,t}(\phi)^2 - 1)^2$$

Interactive Loss Calculation

Use the button to numerically minimize dependencies between innovations by searching over rotation angles $\phi$.

Next Steps

Proceed to Maximizing non-Gaussianity to see how maximizing the non-Gaussianity of the innovations can be used to identify the SVAR.