Proxy Exogeneity Test

Proxies

This page illustrates how proxy variables allow to identify the SVAR. Moreover, it shows how endogenous proxy variables affect the estimation and lead to biased estimates.

Proxy SVAR

Let's fix the rotation at $\phi_0 = 0.5$ such that the reduced form shocks are equal to one specific rotation of the structural shocks:

The innovations $e_t(B(\phi))$ are given by rotating the reduced form shocks $u_t$ with an angle of $\phi (180 / \pi)$ degrees.

• Move the $\phi$ slider to adjust the angle of rotation. Observe how different rotations lead to different correlations between the proxy and the innovations $e_{t}$ in the plots below.

Consider a proxy $z_{ t}$ for the target shock $\epsilon_{1t}$ given by $$ z_{ t} = \gamma_1 \epsilon_{1 t} + \gamma_2 \epsilon_{2 t} + \eta_{ t} $$ such that $\gamma_1$ governs the strenght or relevance of the proxy and $\gamma_2$ the exogeneity of the proxy.

• Move the $\gamma_1$ and $\gamma_2$ sliders to adjust the proxy strength and proxy exogeneity. Observe how these changes affect the correlation of the proxy with the structural shocks $\epsilon_t$ and the innovations $e_{t}$ in the plots below.

Proxy relevance and exogeneity

The following two plots show the proxy against the target shock $\epsilon_{1t}$ and against the non-target shock $\epsilon_{2t}$. The left plot illustrates the relevance of the proxy by the correlation with the target shock. The right plot shows the exogeneity of the proxy which should be uncorrelated with the non-target shock.

• Move the $\gamma_1$ and observe how proxy strength affects the correlation of the proxy with the target shock $\epsilon_{1t}$.
• Move the $\gamma_2$ and observe how proxy exogeneity affects the correlation of the proxy with the non-target shock $\epsilon_{2t}$.

In practice, the structural shocks are not observable. However, for a given rotation $\phi$, we can compute the innovations $e_t(B(\phi))$. For the correct rotation $\phi=\phi_0$, the proxy $z_t$ and the non-target innovation $e_{2t}$ should be uncorrelated. Therefore, we can use this correlation to find the correct rotation $\phi_0$. The following two plots illustrate the correlation between the proxy and the innovations for both the target and non-target shocks.

• Move the $\phi$ slider to adjust the rotation. Observe how the correlation between the proxy and the non-target innovation changes. If the proxy is relevant and exogenous, the correlation of the proxy and the non-target innovation $e_2$ should be zero at $\phi=\phi_0=0.5$. For different values of $\phi$ the proxy and the non-target innovation $e_2$ will be correlated.

Interactive Loss Illustration

The following plot shows the loss equal to the covariance of the proxy and non-target innovation $e_{2t}$ for different $\phi$ values.

The loss uses the asymptotically efficient weighting matrix $W=S^{-1}$ where $S$ is the variance covariance matrix of the moment conditions.

• Hit "Estimate" to find the $\phi$ that minimizes the loss.
• Observe how the loss depends on the proxy strength $\gamma_1$. A low relevance leads to a flat loss and a high relevance to a peaked loss.
• Observe how the minima of the loss depends on the proxy exogeneity $\gamma_2$. The orange line indicates the rotation angle of the DGP, $\phi_0=0.5$. For an exogenous proxy, $\gamma_2=0$, the loss is minimized at $\phi=\phi_0$. For an endogenous proxy, $\gamma_2 \neq 0$, the loss is minimized at a value of $\phi$ different from $\phi_0$. This illustrates how proxy endogeneity leads to biased estimates of the SVAR.