Factor-augmented VARs
Session 09 · Worked solutions
A two-variable example makes the factor normalization and reconstruction explicit. The remaining solutions derive an observable recursive rotation, recover level responses from differenced variables, and show exactly how a metadata row can contaminate the estimation sample.
1 Normalization and reconstruction
The sample covariance is
\frac{Z'Z}{3-1}=\begin{pmatrix}1&1\\1&1\end{pmatrix}.
Its eigenvalues are two and zero. Choose v_1=(1,1)'/\sqrt2, whose Euclidean norm is one. The first factor is F=Zv_1=(-\sqrt2,0,\sqrt2)'. Its mean is zero and its sample variance is (2+0+2)/2=2, not one.
The common component Fv_1' equals Z exactly, so the idiosyncratic component is the zero matrix. If instead the factor were divided by \sqrt2 to have unit sample variance, its loading vector would need to be multiplied by \sqrt2. Normalization changes coordinates, not the fitted common component, provided both sides of the decomposition change together.
2 An observable rotation
First,
B_0B_0'=\begin{pmatrix}5&5\\5&10\end{pmatrix},\qquad L=\begin{pmatrix}\sqrt5&0\\\sqrt5&\sqrt5\end{pmatrix}.
Solving the two linear systems B_0h_j=L_{\cdot j} gives
H=\frac1{\sqrt5}\begin{pmatrix}1&2\\2&-1\end{pmatrix}.
Indeed H'H=\frac15\begin{pmatrix}5&0\\0&5\end{pmatrix}=I and B_0H=L. The first observable has zero impact response to the second shock, because L_{12}=0. The second observable may respond immediately to the first shock. Reversing the observable order would impose a different economic restriction, even though each ordering reproduces the same reduced-form covariance.
3 Cumulating responses
The first cumulative sum of (1,0,0) is (1,1,1), the response of the first difference. The second cumulative sum is (1,2,3), the level response. A one-time acceleration therefore produces a persistent change in growth and a growing level effect under this simple response path.
For the first difference of a log series, one cumulative sum gives (0.01,0.008,0.008). Multiplying by 100 yields approximate percentage responses (1,0.8,0.8). The exact proportional level change is 100[\exp(b_h^{log})-1], approximately (1.0050,0.8032,0.8032) percent. The figure uses the explicitly labelled log approximation, not the exact nonlinear transformation.
4 A metadata row
The preferred procedure first removes the metadata row, leaving 756 observations. Dropping two unavailable initial differences then gives 756-2=754 usable months. Counting metadata as an observation creates 757 rows, after which dropping two leaves 755.
For real values x_1,x_2 and metadata value m, the erroneous second difference attached to x_2 is x_2-2x_1+m. It survives the deletion of the first two rows because it is in the third row of the augmented array. The valid second difference first exists at x_3, where it equals x_3-2x_2+x_1. Thus the wrong rule introduces both an extra observation and an artificial initial transformed value for second-differenced series. The preferred and compatibility samples must be stored separately.