Quantile SVARs and growth risk
Session 08 · Worked solutions
The exercises derive the quantile check loss and its dual bound, distinguish a width innovation from its impact vector, impose contemporaneous zeros, and check the dates of the growth forecast. Each solution shows the intermediate calculation and its interpretation.
1 The forecast origin
There are 4(2019-1960)+4=240 predictor quarters. To use both t and t-1, the first origin is 1960 Q2. The last remains 2019 Q4, giving 239 observations. Adding one quarter to each origin produces targets from 1960 Q3 to 2020 Q1. Growth at the last target is 100[\log GDP_{2020Q1}-\log GDP_{2019Q4}], so the first-quarter 2020 GDP level is needed even though it is never a current predictor.
The count can be checked directly from the target calendar: 4(2020-1960)+(1-3)+1=239. Plotting the origins as if they were target dates would shift the fitted growth series back by one quarter.
2 A quantile certificate
The sample median is b=3, with residuals (-2,-1,0,1,6)'. At \tau=1/2, the objective is
P=\tfrac12(2+1+0+1+6)=5.
Take a=(-1/2,-1/2,0,1/2,1/2)'. Every weight is in the allowed box and \mathbf1'a=0, which is the dual equality constraint for an intercept. The dual objective is
D=y'a=-\tfrac12-1+0+2+\tfrac92=5.
Weak duality gives D\leq P^*\leq P for the unknown optimal primal value P^*. Since both bounds equal five, P^*=5. No differentiability or local curvature argument is needed. The automated test repeats this calculation and checks the returned dual weights and objective gap.
3 A normalized distribution innovation
First, \gamma\Sigma\gamma'=4, so the unrestricted impact vector is \Sigma\gamma'/2=(2,1/2)'. The scalar shock u_1/2 has variance one. Its second-variable impact is not zero because the reduced-form innovations have covariance one.
For D=(0,1), D\Sigma D'=1 and \gamma\Sigma D'=1. Hence
\alpha=(1,0)-1\cdot1^{-1}(0,1)=(1,-1).
Its variance is \alpha\Sigma\alpha'=4-1-1+1=3, while \Sigma\alpha'=(3,0)'. The normalized restricted impact is therefore (\sqrt3,0)'. The zero is exact under the restriction, not a small estimated coefficient. Renormalization matters: using the old standard deviation two would no longer produce a unit-variance shock.
4 The covariance denominator
Three lags leave 239-3=236 residual observations. There are 1+7\times3=22 regressors in each equation. The unbiased divisor is 236-22=214. The classroom expression instead gives 240-3-21-1=215. For the same residual cross product, \widehat\Sigma_{preferred}=(215/214)\widehat\Sigma_{source}.
For any fixed \gamma, replacing \Sigma by c\Sigma yields
\frac{c\Sigma\gamma'}{\sqrt{\gamma c\Sigma\gamma'}} =\sqrt c\frac{\Sigma\gamma'}{\sqrt{\gamma\Sigma\gamma'}}.
The preferred response is larger by \sqrt{215/214}, approximately 1.002334. The projected direction is unchanged because the scalar covariance factor cancels from its regression projection. The benchmark removes this known scale change before comparing the two implementations.