RDP 2022-04: The Unit-effect Normalisation in Set-identified Structural Vector Autoregressions Appendix A: Derivations for Bivariate SVAR

A.1 Sign restrictions on impulse responses

This appendix derives the identified sets for the impulse responses to a unit shock under the sign restrictions on impulse responses presented in Section 3.

In the absence of any identifying restrictions, the identified set for $A_0^{-1}$ (the matrix of impact impulse responses) is

and the identified set for A₀ is

The impact response of the second variable to a shock that raises the first variable by one unit is

Consider the sign restrictions that the impulse response of the first variable to the first shock is non-negative $\left({\eta }_{1,1,0}\equiv {e^{\prime }}_{1,2}{A}_0^{-1}{e}_{1,2}\ge 0\right)$ and the impact response of the second variable to the first shock is non-positive $\left({\eta }_{2,1,0}\equiv {e^{\prime }}_{2,2}{A}_0^{-1}{e}_{1,2}\le 0\right)$, plus the sign normalisation $\text{diag}\left(A_0\right)\ge 0_{2\times 1}.$ Under this set of restrictions, $\theta$ is restricted to lie within the following set:

There are two cases to consider depending on the sign of ${\sigma }_{21}$. If ${\sigma }_{21}<0,$ the second set is empty. The first set is equivalent to

This set of inequalities implies that the identified set for $\theta$ is

which lies within the interval $\left(-\pi /2,\pi /2\right).$ The impact response of the first variable to the first shock is ${\eta }_{1,1,0}={\sigma }_{11}\cos \theta .$ The lower bound of the identified set for $\theta$ is negative and the upper bound is positive, so zero lies within this identified set and $\cos \theta$ attains its maximum of one. The upper bound of the identified set for ${\eta }_{1,1,0}$ is therefore ${\sigma }_{11}$. The lower bound is attained at one of the end points of the identified set for $\theta$ and therefore satisfies

where: the second line uses the fact that arctan is an odd function and cos is an even function; and the third line uses the fact that cos is increasing over $\left(-\pi /2,0\right).$ Since arctan is an increasing function, it follows that:

This identified set excludes zero. Since ${\tilde{\eta }}_{2,1,0}$ is strictly increasing in $\theta$ over the interval $\left(-\pi /2,\pi /2\right),$ its lower and upper bounds are attained at the end points of the identified set for $\theta$. Plugging the end points of the identified set for $\theta$ into the expression for ${\tilde{\eta }}_{2,1,0}$ yields the identified set for ${\tilde{\eta }}_{2,1,0}$:

which is bounded.

Similarly, if ${\sigma }_{21}>0,\theta$ is restricted to lie in the set

The second inequality implies that $\tan \theta \le 0,$ so the last inequality never binds. The identified set for $\theta$ is therefore

The upper bound of the identified set for $\theta$ is negative. ${\eta }_{1,1,0}$ is therefore strictly increasing over the identified set for $\theta$, and the bounds of the identified set for ${\eta }_{1,1,0}$ are attained at the end points of the identified set for $\theta$:

If ${\sigma }_{21}=0,\theta$ is restricted to the set

The first set implies $\theta \in \left[-\pi /2,0\right]$ and the second implies $\theta =-\pi /2,$ so ${\eta }_{1,1,0}\in \left[0,{\sigma }_{11}\right].$ The expression for the identified set for ${\eta }_{1,1,0}$ when ${\sigma }_{21}\ge 0$ therefore also applies when ${\sigma }_{21}=0$. $\tan \theta \to -\infty$ as $\theta$ approaches$-\pi /2$ from above. $\tan \theta$ is strictly increasing over the identified set for $\theta$, so the upper bound for the identified set for ${\tilde{\eta }}_{2,1,0}$ is obtained by evaluating ${\tilde{\eta }}_{2,1,0}$ at the upper bound of the identified set for $\theta$. Consequently, ${\tilde{\eta }}_{2,1,0}\in \left(-\infty ,0\right].$

A.2 Sign restrictions on impulse responses to multiple shocks

If we additionally impose the sign restrictions that ${\eta }_{1,2,0}\equiv {e^{\prime }}_{1,2}{A}_0^{-1}{e}_{2,2}\ge 0$ and ${\eta }_{2,2,0}\equiv {e^{\prime }}_{2,2}{A}_0^{-1}{e}_{2,2}\ge 0,$ the parameter $\theta$ is restricted to lie within the following set:

Using working similar to that in Appendix A.1, the identified sets for $\theta ,{\eta }_{1,1,0}$ and ${\tilde{\eta }}_{2,1,0}$ are given by:

As in the case where there are sign restrictions on the impulse responses to the first shock only, the identified set for ${\eta }_{1,1,0}$ includes zero when ${\sigma }_{21}\ge 0$ and the identified set for ${\tilde{\eta }}_{2,1,0}$ is unbounded.^[24] In the case where ${\sigma }_{21}<0$, the additional sign restrictions tighten the identified set. In particular, the upper bound is now strictly less than zero (and is a differentiable function of $\varphi$, as discussed in Section 3.2.1).

A.3 Alternative parameterisation

Consider an alternative parameterisation of the bivariate model that directly imposes the unit-effect normalisation: $y_t=H{\epsilon }_t,$ where ${\epsilon }_t\sim N\left(0_{2\times 1},\Omega \right)$,

Let $E\left(y_t{y^{\prime }}_t\right)=\Sigma$ and $\text{vech}\left(\Sigma \right)={\left({\text{Σ}}_{11},{\text{Σ}}_{21}{\text{,Σ}}_{22}\right)}^{\prime }.$ The structural parameters ${\left(H_{12},H_{21},{\omega }_{11},{\omega }_{22}\right)}^{\prime }$ and reduced-form parameters $\left({\text{Σ}}_{11},{\text{Σ}}_{21}{\text{,Σ}}_{22}\right)$ are related via $\Sigma =H\Omega H^{\prime }$. Eliminating ${\omega }_{11}$ and ${\omega }_{22}$ from the system of equations yields a single equation in (H₁₂, H₂₁):

Solving for H₂₁ as a function of H₁₂ (using the quadratic formula) yields:

Figure A1 plots the two solutions. Under the sign restrictions $H_{21}\le 0$ and $H_{12}\ge 0,$ the identified set for (H₁₂, H₂₁) lies in the lower-right quadrant. When ${\text{Σ}}_{21}<0$ (Panel A), the identified set for H₂₁ is bounded. When ${\text{Σ}}_{21}\ge 0$ (Panel B), the identified set for H₂₁ is unbounded, so unbounded identified sets may also arise when the unit-effect normalisation is directly imposed.

A.4 Magnitude restrictions

In addition to the sign restrictions considered in Section 3 and Appendix A.1, consider the restriction that ${\eta }_{1,1,0}\ge \lambda$ for some $\lambda >0.$ Under this set of restrictions, $\theta$ is restricted to lie within the set:

The second set is always empty, since ${\sigma }_{11}\cos \theta \ge \lambda$ and $-{\sigma }_{11}\cos \theta \ge 0$ cannot hold simultaneously when $\lambda >0.$ The identified set for $\theta$ is empty if $\lambda >{\sigma }_{11},$ since $\cos \theta \le 1$ for all $\theta$.

If ${\sigma }_{21}\ge 0$, the first set is equivalent to

The last inequality never binds and the identified set for $\theta$ is

which is contained within the interval $\left(-\pi /2,0\right].{\tilde{\eta }}_{2,1,0}$ is strictly increasing over this interval, so the bounds of the identified set for ${\tilde{\eta }}_{2,1,0}$ are attained at the end points of the identified set for $\theta$. The identified set for ${\tilde{\eta }}_{2,1,0}$ is therefore

The lower bound of this identified set, $\ell \left(\varphi ,\lambda \right),$ can be expressed as^[25]

which converges to $-\infty$ as $\lambda$ approaches zero from above. The derivative of $\ell \left(\varphi ,\lambda \right)$ with respect to $\lambda$ is

In the limit as $\lambda$ approaches zero from above, this derivative approaches $\infty$, which implies that the lower bound is extremely sensitive to small changes in $\lambda$ when $\lambda$ is close to zero.

A.5 Bounds on the FEVD

The FEV of y_1t is ${\sigma }_{11}^2$ and the contribution of ${\epsilon }_{1t}$ to the FEV of y_1t is ${\sigma }_{11}^2{\cos }^2\theta .$ The FEVD of y_1t with respect to ${\epsilon }_{1t},FEV{D}_{{\epsilon }_{1t}}^{y_{1t}},$ is therefore ${\cos }^2\theta$. Consider imposing the restriction that $FEV{D}_{{\epsilon }_{1t}}^{y_{1t}}\ge \kappa$ for some $0<\kappa \le 1$ in addition to the sign restrictions considered in Section 3 and Appendix A.1. Under this set of restrictions, $\theta$ is restricted to lie within the following set:

When ${\sigma }_{21}\ge 0,$ the first set is equivalent to

The inequalities $\tan \theta \le {\sigma }_{22}/{\sigma }_{21}$ and $\theta \le \arccos \sqrt{\kappa }$ never bind and the identified set for $\theta$ is

which is contained within the interval $\left(-\pi /2,0\right].{\tilde{\eta }}_{2,1,0}$ is strictly increasing over this interval, so the bounds of the identified set for ${\tilde{\eta }}_{2,1,0}$ are attained at the end points of the identified set for $\theta$. The identified set for ${\tilde{\eta }}_{2,1,0}$ is therefore

The lower bound of this identified set, $\ell \left(\varphi ,\kappa \right)$, can be expressed as

The lower bound converges to $-\infty$ as $\kappa$ approaches zero from above. The derivative of $\ell \left(\varphi ,\kappa \right)$ with respect to $\kappa$ is

In the limit as $\kappa$ approaches zero from above, this derivative approaches $\infty$, which implies that the lower bound is extremely sensitive to small changes in $\kappa$ when $\kappa$ is close to zero.

To summarise, under the additional restriction on the FEVD, the identified set is bounded; in the absence of this restriction (or as $\kappa$ converges to zero from above), the identified set is $\left(-\infty ,0\right].$

However, as in the case where the normalising impulse response is directly bounded away from zero, the lower bound of the identified set is sensitive to the choice of $\kappa$, particularly for small values of $\kappa$; the derivative of the lower bound tends to $\infty$ as $\kappa$ approaches zero from above. Setting $\kappa$ to some small positive number to rule out an unbounded identified set for ${\tilde{\eta }}_{2,1,0}$ will therefore yield an identified set that is highly sensitive to the choice of $\kappa$

Footnotes

The identified set for ${\tilde{\eta }}_{1,2,0}$ is unbounded when ${\sigma }_{21}<0$ and bounded when ${\sigma }_2\ge 0.$ [24]

This follows from the fact that $\tan \left(\arccos x\right)=x^{-1}\sqrt{1-x^2.}$ [25]