Orthogonalized shocks fundamentals

• We mentioned that a one time shock on a given variable may have inmediate consequences on others.

• Applying a shock on a past \(\varepsilon_{jt}\) may correlate with \(\varepsilon_{kt}\)

• The errors’ variance covariance matrix (or correlation matrix) may be used to shock the whole residual vector: \(\bf \varepsilon_t = \Sigma\) for all \(t\).

• Let’s recall the infinite MA representation of a VARX, and bring \(\bf Y_t = a + \sum_{j=1}^s \tilde B_j X_{t-j} + \sum_{j=0}^{\infty} \mathbf \phi_j \varepsilon_{t-j}\)

• Now we need a underlying residual (behind \(\varepsilon_t\)) scaled to have variance 1, so that every shock is standardized and diffused accross equations. We define a unit variance new residual vector \(V u_t = I\)

• Then we apply a cholesky decomposition of the variance matrix means \(\Sigma = PP'\), so that:
• \(I = P^{-1} \Sigma P'^{-1}\), where P is a lower triangular matrix.

\[\dots+ \sum_{j=0}^{\infty} \mathbf \phi_j \varepsilon_{t-j}\] * One can rewrite \(V(\bf \varepsilon_{t}) = \Sigma\), as \(V(\bf I \varepsilon_{t}) = V(\underbrace{PP^{-1}}_{I}\varepsilon_t)\).

• then \(P^{-1}\varepsilon_t\) can be shown to have unit variance : \(V (P^{-1}\varepsilon_t)= I\),
• Thus we can rewrite the innovations (residuals) as:
• \(\dots +\sum_{j=0}^{\infty} \mathbf \phi_j PP^{-1}\varepsilon_{t-j}\)

• \(\dots + \sum_{j=0}^{\infty} \underbrace{\mathbf \phi_j P}_{\omega_j} \underbrace{P^{-1}\varepsilon_{t-j}}_{u_{t-j}}\)

• \(\omega_j\) is a new vector of (IRF) orthogonalized coefficients.

• Now you can apply a shock to \(\Delta u_t\) that is known to have 0 mean and variance 1, while accounting for the contemporaneous effects at the time of the impulse.

• This specification (Sims, 1980) led to wide economic applications and to a Nobel prize.

• Today’s, orthogonal effect of a shock on \(t-h\) is

  • \(\Delta Y_t=\omega_h\Delta u_{t-h}\). The OIRF matrix is given by \(\omega\)

Structural VAR

• Now think in terms of a linear structural equations model which may include contemporaneous relationships:

• So that the Structural model (with lags) - Structural VAR - comes from :
• \(\bf B Y_t = \sum_{j=1}^p A_j Y_{t-j} + \bf \varepsilon_t \quad (1)\)
  
• Thus,
• \(\bf B Y_t = \sum_{j=1}^p A_j Y_{t-j} + \bf P^{-1} u_t\)
  
• the \(B\) matrix contains the structural relationships between endogenous variables, while \(P\) provides the recursive structure between shocks \(u_t\).

• Then the reduced form VAR can be reinterpreted vis-a-vis an structural VAR model:

• \(\bf Y_t = \sum_{j=1}^p \underbrace{B^{-1}A_j}_{\Pi_j} Y_{t-j} + B^{-1} \bf P u_t \quad (2)\)

• Its reduced form counterpart (easy to estimate from a VAR):

• \(\bf Y_t = \sum_{j=1}^p {\Pi_j} Y_{t-j} + \varepsilon_t \quad (3)\)

• The error variance matrix \(\Sigma\) is:

• \(V(B^{-1} P u_t) = B^{-1} P P' {B^{-1}}'\,\)

Exercise 1

• Show that when there are no contemporaneous relationships between endogenous variables, \(\Sigma = P P'\)

Exercise 2

• Consider the elements of \(\Sigma\) are known or estimated from a (reduced-form) VAR. Show that if B and \(P\) are full matrices of a system with n-endogeneous variables, identification of the elements of B and P requires imposing \(n(3n+1)/2\) constraints across B and \(P\) .

• There might be many ways to specify constraints over both matrices.
• The simplest is to let \(P\) to be a triangular one (SIMS, 1980) and to let B to be the identity matrix. Note that this was already done by the OIRFs

Stata application

Let’s assume that investment shocks affects GDP growth only. Consumption may increase given investment and gdp growth. This is a recursive structure that must be reflected in the B matrix (next class…).