Proxy Exogeneity Test

Introduction

Proxy variables are among the most popular tools to identify Structural Vector Autoregression (SVAR) models. This short interactive guide illustrates how endogenous proxy variables lead to biased SVAR estimates and it illustrates the test for proxy exogeneity in Bruns and Keweloh (2024).

The structure of the guide

The guide is structured as follows:

The SVAR model

In a SVAR a vector of time series $$ y_t = [y_{1t},...,y_{nt}]'$$ is explained by its own past and a vector of structural shocks $$\epsilon_t =[\epsilon_{1t},...,\epsilon_{nt}]'$$ using

$$ y_t = A_1 y_{t-1} + ... + A_p y_{t-p} + u_t$$ $$ \quad u_t =B_0 \epsilon_t. $$

The following assumptions allow an easy visualization of the identification problem.

Simplifying assumptions

The VAR(p)

$ y_t = A_1 y_{t-1} + ... + A_p y_{t-p} + u_t $
can be estimated by OLS. To simplify, consider an SVAR(0) such that the reduced form shocks $u_t$ can be thought of as observable random variables equal to an unknown linear combination $B_0$ of unknown structural form shocks $\epsilon_t$.

Assumption 1: No Lags
The SVAR has zero lags:

$ y_t = u_t \quad \text{and} \quad u_t =B_0 \epsilon_t $$

To further simplify, let's consider an SVAR with only two variables and shocks such that each of the two reduced form shocks is equal to an unknown linear combination of the two structural shocks.

Assumption 2: Bivariate SVAR

The SVAR contains only two variables and shocks:

$ \begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \begin{bmatrix} \epsilon_{1,t} \\ \epsilon_{2,t} \end{bmatrix} $

Finally, assume that $B_0$ is orthogonal.i Without the orthogonality assumption the matrix $B_0$ contains four unknown parameters. However, an orthogonal matrix $B_0$ essentially only depends on a single unknown parameter, which simplifies the visualizations.

Assumption 3: Orthogonal $B_0$

The matrix $B_0$ is orthogonal and can be written as:

$B_0 = B(\phi_0) = \begin{bmatrix} \cos(\phi_0) & -\sin(\phi_0) \\ \sin(\phi_0) & \cos(\phi_0) \end{bmatrix} $
where $\phi_0$ is the unknown rotation angle.

The relationship between reduced and structural form shocks

In the simplified SVAR model, the reduced form shocks $u_t$ can be thought of a rotation with an angle of $\phi_0 (180 / π)$ degree of the structural form shocks $\epsilon_t$:

$ \begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix} = \begin{bmatrix} \cos(\phi_0) & -\sin(\phi_0) \\ \sin(\phi_0) & \cos(\phi_0) \end{bmatrix} \begin{bmatrix} \epsilon_{1,t} \\ \epsilon_{2,t} \end{bmatrix} $

Structural shocks and innovations

In a given application, we neither know the rotation matrix $B(\phi_0)$ nor do we observe the structural shocks $\epsilon_t$. Instead, we only observe the reduced form shocks $u_t$ (in practice we only observe the macroeconomic variables $y_t$ and need to estimate the VAR to get the reduced form shocks $u_t$. However, with the simplifying no-lags assumption it holds that $u_t=y_t$). Consequently, the question is how can we identify $B(\phi_0)$ and recover the structural shocks $\epsilon_t$ from the reduced form shocks $u_t$?

We can always choose an angle $\phi$, calculate $B(\phi)$:

$ B(\phi) = \begin{bmatrix} \cos(\phi) & -\sin(\phi) \\ \sin(\phi) & \cos(\phi) \end{bmatrix} $

and get the corresponding innovations $ e_t(B(\phi)) = B(\phi)^{-1} u_t $ such that

$ \begin{bmatrix} e_{1,t} \\ e_{2,t} \end{bmatrix} = \begin{bmatrix} \cos(\phi) & \sin(\phi) \\ -\sin(\phi) & \cos(\phi) \end{bmatrix} \begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix} $

These innovations represent the shocks that would yield $u_t$ if the true angle of the data-generating process $\phi_0$ were equal to the chosen angle $\phi$. Specifically, for $\phi=\phi_0$ we get $e_t(B(\phi_0)) = \epsilon_t$.

Interactive simulation

Note that all rotations $\phi$ lead to uncorrelated innovations. Therefore, the assumption of unccorelated shocks can not be used to identify the SVAR. Meaning the assumption of uncorrelated shocks is not sufficient to use the reduced form shocksto recover the rotation $\phi_0$ and the structural shocks $\epsilon_t$.

Next steps

The next page illustrates how a proxy allows to identify the SVAR.