Proxy Exogeneity Test

Joint proxy exogeneity test for multiple proxies

This page shows how the proxy exogeneity assumption can be tested if two proxies are available. The exogeneity test using two proxies is well known in the literature. The next page introduces the exogeneity test in Bruns and Keweloh (2024) using only one proxy.

Proxy SVAR

Consider a proxy SVAR with two proxies for the target shock $\epsilon_{1t}$.

The first proxy $z_{ t}$ is given by $ z_{ 1t} = \gamma_1 \epsilon_{1 t} + \gamma_2 \epsilon_{2 t} + \eta_{ 1t} $, such that $\gamma_1$ governs the strength or relevance of the proxy and $\gamma_2$ the exogeneity of the proxy.

• Move the $\gamma_1$ and $\gamma_2$ slider to adjust the strength and exogeneity of the first proxy $z_{1t}$.

The second proxy $z_{2t}$ is given by $ z_{2t} = \rho_1 \epsilon_{1 t} + \rho_2 \epsilon_{2 t} + \eta_{ 2t} $, such that $\rho_1$ governs the strength or relevance of the proxy and $\rho_2$ the exogeneity of the proxy.

• Move the $\rho_1$ and $\rho_2$ slider to adjust the strength and exogeneity of the second proxy $z_{2t}$.

Proxy relevance and exogeneity

The following two plots show the covariance of the proxies with the non-target shock $\epsilon_{2t}$.

• Move the $\gamma_2$ and $\rho_2$ sliders to adjust the exogeneity of $z_1$ and $z_2$. Observe how the covariance of the proxies with the non-target shock $\epsilon_{2}$ changes.

The following two plots show the covariance of the proxies with the non-target innovation $e_{2t}$.

If both proxies are exogenous, the correlation of both proxies with the non-target innovation $e_2$ should be zero at the same rotation angle $\phi$. If at least one proxy is endogenous, the correlation of the proxy with the non-target shock will be different from zero at the same rotation angle $\phi$. We can use this intuition to construct a joint proxy exogeneity test based on a J-test, see Hansen (1982).

Define the loss measuring the correlation of the proxies with the non-target shock at a given rotation angle $\phi$:

Under the null hypothesis of two exogenous proxies

the J-test statistic $ J = T \cdot loss(\hat{\phi})$ where $\hat{\phi}$ is the value minimizing the loss function follows a Chi-Squared distribution with $J \sim \chi^2(r-q)$ where $r=2$ is equal to the number of moment conditions and $q=1$ is equal to the number of free parameters in $\phi$.

Consequently, the null hypothesis of two exogenous proxies can be rejected at a significance level of $\alpha$ if $J$ exceeds the critical value of the Chi-Squared distribution with $r-q$ degrees of freedom at the $\alpha$ quantile.

Interactive Loss Calculation

The following plot shows three different loss functions based on the proxy variables $z_1$ and $z_2$.

• The first loss corresponds to the correlation of the first proxy $z_1$ with the non-target innovation $e_{2t}$:
  $ loss_{z_1}(\phi) = \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{1t} e_{2t}(\phi) \end{bmatrix} W \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{1t} e_{2t}(\phi) \end{bmatrix}' $
• The second loss corresponds to the correlation of the second proxy $z_2$ with the non-target innovation $e_{2t}$:
  $ loss_{z_2}(\phi) = \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{2t} e_{2t}(\phi) \end{bmatrix} W \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{2t} e_{2t}(\phi) \end{bmatrix}' $
• The third loss is the joint loss function and corresponds to the weighteds sum of the correlations of both proxies $z_1$ and $z_2$ with the non-target innovation $e_{2t}$:
  $ loss(\phi) = \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{1t} e_{2t}(\phi) \\ \frac{1}{T} \sum_{t=1}^{T} z_{2t} e_{2t}(\phi) \end{bmatrix} W \begin{bmatrix} \frac{1}{T} \sum_{t=1}^{T} z_{1t} e_{2t}(\phi) \\\frac{1}{T} \sum_{t=1}^{T} z_{2t} e_{2t}(\phi) \end{bmatrix}' $

If both proxies are exogenous, all loss functions should (asymptotically) exibit a minimum at the same rotation angle $\phi$. Additionally, the loss of the third loss function measuring the weighted sum of the correlations of both proxies with the non-target innovation $e_{2t}$ should be smaller than the critical value of the Chi-Squared distribution.

• Hit "Estimate" to find the $\phi$ that minimizes the joint loss function.
• Move the $\gamma_2$ and $\rho_2$ sliders to adjust exogeneity of the first and second proxy. Observe how the minima of the loss functions change. For example, an endogenous proxy $z_1$ leads to a minima of the loss function $loss_{z_1}$ at a value of $\phi$ different from the DGP, $\phi_0$. In this case, the minimum of the loss joint loss function $loss(\phi)$ is moves above the critical value of the Chi-Squared distribution and indicates that the null hypothesis of both proxies being exogenous can be rejected.
• Move the $\rho_1$ and $\rho_2$ slider to zero to generate an exogenous but irrelevant second proxy. Observe how the exogeneity test looses its ability to detect endogeneity if one of the proxies is exogenous and irrelevant.