# 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

(A1) $A 0 −1 ∈{ [ σ 11 cosθ − σ 11 sinθ σ 21 cosθ+ σ 22 sinθ σ 22 cosθ− σ 21 sinθ ] } ∪{ [ σ 11 cosθ σ 11 sinθ σ 21 cosθ+ σ 22 sinθ σ 21 sinθ− σ 22 cosθ ] }$

and the identified set for A0 is

(A2) $A 0 ∈{ 1 σ 11 σ 22 [ σ 22 cosθ− σ 21 sinθ σ 11 sinθ − σ 21 cosθ− σ 22 sinθ σ 11 cosθ ] } ∪{ 1 σ 11 σ 22 [ σ 22 cosθ− σ 21 sinθ σ 11 sinθ σ 21 cosθ+ σ 22 sinθ − σ 11 cosθ ] }$

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

(A3) $η ˜ 2,1,0 = η 2,1,0 η 1,1,0 = σ 21 cosθ+ σ 22 sinθ σ 11 cosθ = σ 21 σ 11 + σ 22 σ 11 tanθ$

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 $\text{diag}\left({A}_{0}\right)\ge {0}_{2×1}.$ Under this set of restrictions, $\theta$ is restricted to lie within the following set:

(A4) $θ∈{θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ} ∪{θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ,− σ 11 cosθ≥0}$

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

(A5) ${ θ:cosθ>0,tanθ≤− σ 21 σ 22 ,tanθ≥ σ 22 σ 21 }$

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

(A6) $θ∈[ arctan( σ 22 σ 21 ),arctan( − σ 21 σ 22 ) ]$

which lies within the interval $\left(-\pi /2,\pi /2\right).$ The impact response of the first variable to the first shock is ${\eta }_{1,1,0}={\sigma }_{11}\mathrm{cos}\theta .$ 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 $\mathrm{cos}\theta$ 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

(A7) $ℓ( ϕ )=min{ σ 11 cos( arctan( σ 22 σ 21 ) ), σ 11 cos( arctan( − σ 21 σ 22 ) ) } = σ 11 min{ cos( arctan( σ 22 σ 21 ) ),cos( arctan( σ 21 σ 22 ) ) } = σ 11 cos( min{ arctan( σ 22 σ 21 ),arctan( σ 21 σ 22 ) } )$

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:

(A8) $η 1,1,.0 ∈[ σ 11 cos( arctan( min{ σ 22 σ 21 , σ 21 σ 22 } ) ), σ 11 ]$

This identified set excludes zero. Since ${\stackrel{˜}{\eta }}_{2,1,0}$ 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 ${\stackrel{˜}{\eta }}_{2,1,0}$ yields the identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$:

(A9) $η ˜ 2,1,0 ∈[ σ 21 σ 11 + σ 22 2 σ 11 σ 21 ,0 ]$

which is bounded.

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

(A10) $θ∈{ θ:cosθ>0,tanθ≤− σ 21 σ 22 ,tanθ≤ σ 22 σ 21 }∪{ − π 2 }$

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

(A11) $θ∈[ − π 2 ,arctan( − σ 21 σ 22 ) ]$

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 $\theta$:

(A12) $η 1,1,0 ∈[ 0, σ 11 cos( arctan( − σ 21 σ 22 ) ) ]$

If ${\sigma }_{21}=0,\text{\hspace{0.17em}}\theta$ is restricted to the set

(A13) $θ∈{ θ:cosθ≥0,0≤− σ 22 sinθ }∪{ θ:0≤− σ 22 sinθ,cosθ≥0,− σ 11 cosθ≥0 }$

The first set implies $\theta \in \left[-\pi /2,0\right]$ and the second implies $\theta =-\pi /2,\text{\hspace{0.17em}}$ 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 ${\sigma }_{21}=0$. $\mathrm{tan}\theta \to -\infty$ as $\theta$ approaches$-\pi /2$ from above. $\mathrm{tan}\theta$ is strictly increasing over the identified set for $\theta$, so the upper bound for the identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$ is obtained by evaluating ${\stackrel{˜}{\eta }}_{2,1,0}$ at the upper bound of the identified set for $\theta$. Consequently, ${\stackrel{˜}{\eta }}_{2,1,0}\in \left(-\infty ,0\right].$

## 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:

(A14) $θ∈{ θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ,−σ 11 sinθ≥0 } ∪{ θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ,− σ 11 cosθ≥0, σ 21 sinθ≥ σ 22 cosθ, σ 11 sinθ≥0 }$

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

(A15) $θ∈{ [ arctan( σ 22 σ 21 ),0 ] if σ 21 <0 [ − π 2 ,arctan( − σ 21 σ 22 ) ] if σ 21 ≥0$
(A16) $η 1,1,0 ∈{ [ σ 11 cos( arctan( σ 22 σ 21 ) ), σ 11 ] if σ 21 <0 [ 0, σ 11 cos( arctan( − σ 21 σ 22 ) ) ] if σ 21 ≥0$
(A17) $η ˜ 1,1,0 ∈{ [ σ 21 σ 11 + σ 22 2 σ 11 σ 21 , σ 21 σ 11 ] if σ 21 <0 ( −∞,0 ] if σ 21 ≥0$

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 ${\stackrel{˜}{\eta }}_{2,1,0}$ is unbounded. 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 ${\epsilon }_{t}\sim N\left({0}_{2×1},\Omega \right)$,

(A18)

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):

(A19) $( Σ 21 H 12 2 − Σ 11 H 12 ) H 12 2 +( Σ 11 − Σ 22 H 12 2 ) H 21 + Σ 22 H 12 − Σ 21 =0$

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

(A20) $H 21 = −( Σ 11 − Σ 22 H 12 2 )± ( Σ 11 − Σ 22 H 12 2 ) 2 −4( Σ 21 H 12 2 − Σ 11 H 12 )( Σ 22 H 12 − Σ 21 ) 2( Σ 21 H 12 2 − Σ 11 H 12 )$

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:

(A21) $θ∈{ θ: σ 11 cosθ≥λ, σ 21 cosθ≤− σ 22 sinθ, σ 22 θ≥ σ 21 sinθ } ∪{ θ: σ 11 cos≥λ, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ,− σ 11 cosθ≥0 }$

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

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

(A22) $θ∈{ θ:cosθ≥ λ σ 11 ,tanθ≤− σ 21 σ 22 ,tanθ≤ σ 22 σ 22 }$

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

(A23) $θ∈[ −arccos( λ σ 11 ),arctan( − σ 21 σ 22 ) ]$

which is contained within the interval $\left(-\pi /2,0\right].\text{\hspace{0.17em}}{\stackrel{˜}{\eta }}_{2,1,0}$ is strictly increasing over this interval, so the bounds of the identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$ are attained at the end points of the identified set for $\theta$. The identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$ is therefore

(A24) $η ˜ 2,1,0 ∈[ σ 21 σ 11 + σ 22 σ 11 tan( −arccos( λ σ 11 ) ),0 ]$

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

(A25) $ℓ( ϕ,λ )= σ 21 σ 11 − σ 22 λ ( 1− ( λ σ 11 ) 2 )$

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

(A26) $∂ℓ( ϕ,λ ) ∂λ = λ −2 ( 1− ( λ σ 11 ) 2 ) − 1 2$

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 y1t is ${\sigma }_{11}^{2}$ and the contribution of ${\epsilon }_{1t}$ to the FEV of y1t is ${\sigma }_{11}^{2}{\mathrm{cos}}^{2}\theta .$ The FEVD of y1t with respect to ${\epsilon }_{1t},FEV{D}_{{\epsilon }_{1t}}^{{y}_{1t}},$ is therefore ${\mathrm{cos}}^{2}\theta$. 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:

(A27) $θ∈{ θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ, cos 2 θ≥κ } ∪{ θ: σ 11 cosθ≥0, σ 21 cosθ≤− σ 22 sinθ, σ 22 cosθ≥ σ 21 sinθ,− σ 11 cosθ≥0, cos 2 θ≥κ }$

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

(A28) $θ∈{ θ:cosθ>0,tanθ≤− σ 21 σ 22 ,tanθ≤ σ 22 σ 21 ,−arccos κ ≤θ≤arccos κ }$

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

(A29) $θ∈[ −arccos κ ,arctan( − σ 21 σ 22 ) ]$

which is contained within the interval $\left(-\pi /2,0\right].\text{\hspace{0.17em}}{\stackrel{˜}{\eta }}_{2,1,0}$ is strictly increasing over this interval, so the bounds of the identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$ are attained at the end points of the identified set for $\theta$. The identified set for ${\stackrel{˜}{\eta }}_{2,1,0}$ is therefore

(A30) $η ˜ 2,1,0 ∈[ σ 21 σ 11 + σ 22 σ 11 tan( −arccos( κ ) ),0 ]$

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

(A31) $ℓ( ϕ,κ )= σ 21 σ 11 − σ 22 σ 11 1−κ κ$

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

(A32) $∂ℓ( ϕ,κ ) ∂κ = 1 2 σ 22 σ 11 κ − 3 2 ( 1−κ ) − 1 2$

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 ${\stackrel{˜}{\eta }}_{2,1,0}$ will therefore yield an identified set that is highly sensitive to the choice of $\kappa$

## Footnotes

The identified set for ${\stackrel{˜}{\eta }}_{1,2,0}$ is unbounded when ${\sigma }_{21}<0$ and bounded when ${\sigma }_{2}\ge 0.$ 

This follows from the fact that $\mathrm{tan}\left(\mathrm{arccos}\text{\hspace{0.17em}}x\right)={x}^{-1}\sqrt{1-{x}^{2}.}$