Proxy Exogeneity Test

Strong Proxy Exogeneity Test

This page illustrates the proxy exogeneity test in Bruns and Keweloh (2024). While the proxy exogeneity test from the previous always requires two proxies, the test on this page allows to test the proxy exogeneity of a single proxy.

Note that for illustration purposes, this page slightly deviates from the setup in Bruns and Keweloh (2024). This page uses a test is build on moment conditions around the correlation of the proxy and the innovations $e_t(\phi) = B(\phi)^{-1} u_t$. In Bruns and Keweloh (2024), the test does not require to compute innovations and it is build on moment conditions closely related to the the proxy SVAR estimator, $\hat{\beta} = \frac{1/T \sum_{t=1}^{T} u_t z_t }{1/T \sum_{t=1}^{T} u_{1t} z_t} $.

Strong proxy exogeneity

If only a single proxy variable is avaiable, the identifying assumption of an exogenous proxy, meaning a proxy which is uncorrelated with the non-target shock such that $E[\epsilon_{2t} z_t]=0$, cannot be tested without further assumptions. Instead, most applications using proxy variables rely on economic arguments to justify the exogeneity condition. Bruns and Keweloh (2024), argue that these economic arguments typically imply a stronger form of exogeneity.

Definition: Strong Exogeneity

A proxy variable $z_t$ for a target shock $\epsilon_{1t}$ is strongly exogenous if it contains no information on the expected value of the non-target shock, i.e., $E[\epsilon_{2t} | z_t]=0$.

Without further assumptions, the exogeneity assumption $E[\epsilon_{2t} z_t]=0$ cannot be tested. However, the strong exogeneity assumption $E[\epsilon_{2t} | z_t]=0$ is a stronger assumption and can be tested. Before introducing the strong exogeneity test, it is important to discuss whether the strong exogeneity assumptions is a reasonable economic assumption and what are the economic consequences of rejecting the strong exogeneity?

Intuivily, the econmic reasoning used to justify the exogeneity of a proxy typically relies on the argument, that the proxy is a function of the target shock and not affected by the non-target shock. For example, consider the tax proxy in Mertens and Ravn (2014), constructed as a series of tax shocks based on narrative documents. Intuitvely, if the proxy is equal to a series of tax shocks not containing information on the expected value of other non-target shocks, then the tax proxy contains no information on the expected value of other non-target shocks. Consequently, the economic reasoning used to justify the exogeneity of the tax proxy infact justifies the strong exogeneity assumption and rejecting the strong exogeneity assumption idicates evidence against the economic reasoning used to justify the exogeneity of the proxy.

The strong proxy exogeneity test in Bruns and Keweloh (2024) is based on a simple idea. If a proxy $z_t$ is strongly exogenous, i.e. $E[\epsilon_{2t} | z_t]=0$, we can generate additional synthetic proxies $z_{2t} = h(z_t)$ as functions of the original proxy. Strong exogeneity immediatly implies that the synthetic proxy $z_{2t}$ is exogenous, i.e. $E[\epsilon_{2t} z_{2t}]=0$. Consequently, the original proxy and the synthetic proxy should provide a system of two exogenous proxies and we can apply the proxy exogeneity test described in the previous page. The only difference is that the second proxy is now a synthetic proxy and generated based on the first proxy.

Proxy SVAR

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 strength or relevance of the proxy and $\gamma_2$ the exogeneity of the proxy.

Consider a simple synthetic proxy $z_{2t} = z_{1t}^2 $ equal to the square of the original proxy.

$ z_{2t} = z_{1t}^2 $

It is easy to see that the synthetic proxy $z_{2t}$ is exogenous, i.e. $E[\epsilon_{2t} z_{2t}]=0$, if the original proxy $z_{1t}$ is exogenous, i.e. if $\gamma_2=0$. However, as seen on the previous page, the exogeneity test only has power if both proxies are relevant. Consequently, the strong exogeneity test only has power if the synthetic proxy $z_{2t}$ is relevant. For a linear proxy $z_{1t}$, relevancy of the synthetic proxy $z_{2t}$ depends on the non-Gaussianity of the shocks. Specifically, a skewed target shock leads to a relevant synthetic proxy $z_{2t}$.

$E[\epsilon_{1t} z_{2t}] = E[\epsilon_{1t} z_{1t}^2] = E[\epsilon_{1t} ( \gamma_1 \epsilon_{1 t} + \gamma_2 \epsilon_{2 t} + \eta_{ t})^2 ] = \gamma_1 E[\epsilon_{1t}^3] $

Proxy relevance and exogeneity

The following two plots show the covariance of the original proxy $z_{1t}$ and the synthetic proxy $z_{2t}$ with the target shock $\epsilon_{2t}$.

The following two plots show the covariance of the original proxy $z_{1t}$ and the synthetic proxy $z_{2t}$ with the non-target shock $\epsilon_{2t}$.

The following two plots show the covariance of the original proxy $z_{1t}$ and the synthetic proxy $z_{2t}$ with the non-target innovation $e_{2t}$.

Strong Proxy Exogeneity Test

If the original proxy $z_1$ is strongly exogenous, the correlation of the original proxy with the non-target innovation $e_2$ as well as the correlation of the synthetic proxy $z_2$ with the non-target innovation $e_2$ should be zero at the same rotation angle $\phi$. If one of both correlations differs from zero, the proxy is not strongly exogenous. We can use this intuition to construct a strong proxy exogeneity test based on a J-test, see Hansen (1982).

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

$ 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}' $
where $W$ is the asymptotic efficient weighting matrix.

Under the null hypothesis of strong proxy exogeneity

$ H_0: E[z_{1t} e_{2t}(\phi_0)] = E[z_{2t} e_{2t}(\phi_0)] = 0 $

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 strong proxy exogeneity 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 original proxy $z_1$ and the syntehtic proxy $z_2$.

If the original proxy $z_1$ is strongly 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 the original and synthetic proxy with the non-target innovation $e_{2t}$ should be smaller than the critical value of the Chi-Squared distribution.

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