Structural identification
Session 02 · Worked solutions
Derivations of recursive and long-run impact matrices, an incompatible mixed restriction, and the conversion from annualized growth to levels.
1 A recursive shock
Write B=\begin{pmatrix}a&0\\b&c\end{pmatrix}. Multiplication gives
BB'=\begin{pmatrix}a^2&ab\\ab&b^2+c^2\end{pmatrix}.
The positive-diagonal convention yields a=2, then b=2/a=1, and finally c=\sqrt{5-1}=2. Hence B=\begin{pmatrix}2&0\\1&2\end{pmatrix}. A unit second shock affects variable two by two units on impact. Multiplying its column by .5/2=.25 gives (0,.5)'.
If that column replaces the original one without changing its variance, the implied covariance becomes \begin{pmatrix}4&2\\2&1.25\end{pmatrix}, not the observed \Sigma. The normalized response is a different-sized experiment, not a new unit-variance decomposition of the same residual covariance. Keep the original factor for variance accounting.
2 The infinite-horizon zero
The exact multiplier is
C(1)=(I-A)^{-1}=\operatorname{diag}(2,4/3).
Consequently,
C(1)\Sigma C(1)'= \begin{pmatrix}16&16/3\\16/3&80/9\end{pmatrix}.
Its lower-triangular factor is D=\begin{pmatrix}4&0\\4/3&8/3\end{pmatrix}: the bottom-right covariance is (4/3)^2+(8/3)^2=80/9. Premultiply by I-A:
B=\begin{pmatrix}1/2&0\\0&3/4\end{pmatrix} \begin{pmatrix}4&0\\4/3&8/3\end{pmatrix} =\begin{pmatrix}2&0\\1&2\end{pmatrix}.
The direct check gives C(1)B=\begin{pmatrix}4&0\\4/3&8/3\end{pmatrix}, whose upper-right element is zero. In this diagonal-dynamics example the short-run and long-run restrictions select the same factor. That is a feature of this fixture, not a general equivalence: cross-equation dynamics usually make the long-run impact matrix nontriangular.
3 Conflicting restrictions
Write q_1=(a,b)'. The impact restriction gives a=0. The second row of C(1) gives the long-run restriction a+2b=0. Substituting a=0 yields b=0. The only solution is therefore the zero vector, which violates q_1'q_1=1.
The stacked restriction matrix \begin{pmatrix}1&0\\1&2\end{pmatrix} has rank two and no nontrivial null space. There is no admissible shock direction. More restrictions do not necessarily produce a better-identified model; they may produce an inconsistent model. A single impact restriction here would leave direction (0,1)' up to sign. The additional long-run restriction rules it out.
4 Annualized growth
Annualized quarterly log growth is four times quarterly log growth. The underlying quarterly responses are therefore 1, .5, and 0. Cumulating yields
L_0=1,\qquad L_1=1+.5=1.5,\qquad L_{10}=1.5.
The return of growth to zero means that the log-level difference stops increasing; it does not imply that the accumulated level difference vanishes. An offsetting negative growth response would be needed for that.
If the supplied 4 and 2 were already nonannualized quarterly growth, no division by four would be appropriate. The level responses would be 4, 6, and 6. The discrepancy is exactly a units error, even though both calculations use the same cumulative-sum command.