12-09-2026
Effect of monetary policy. Consider these events:
identification is the interpretation of historically observed variation in data in a way that allows the variation to be used to predict the consequences of an action not yet undertaken.
Suppose we have the following **Structural VAR(1): \(AX_t=\beta_0+\beta_1 X_{t-1}+u_t\) Structural shocks u are independent
If \(X\) variables GDP gap (\(y\)) and the interest rate (\(r\)): \[X_t=\begin{bmatrix} y \\ r \\ \end{bmatrix}\] the system will be \(y_t+a_{12}r_t=\beta_{10}+\beta_{12}y_{t-1}+\beta_{12}r_{t-1}+u_{yt}\)
\(a_{21}y_t+r_t=\beta_{20}+\beta_{21}y_{t-1}+\beta_{22}r_{t-1}+u_{rt}\)
In martrix form : \[\begin{bmatrix} 1 & a12\\ a21 & 1 \\ \end{bmatrix}\begin{bmatrix} y_t\\ r_t\\ \end{bmatrix}=\begin{bmatrix} \beta_{10}\\ \beta_{20}\\ \end{bmatrix}+\begin{bmatrix} \beta_{11} & \beta_{12}\\ \beta_{21} & \beta_{22} \\ \end{bmatrix}\begin{bmatrix} y_{t-1}\\ r_{t-1}\\ \end{bmatrix}+\begin{bmatrix} u_{yt}\\ u_{rt}\\ \end{bmatrix}\]
If we pre multiply this VAR by \(A^{-1}\) we get the reduced-form VAR
\(A^{-1}A X_t=A^{-1}\beta_0+A^{-1}\beta_1 X_{t-1}+A^{-1}u_t\) \(X_t=G_0+G_1 X_{t-1}+e_t\) \(A^{-1}A=I\)
\(e_t=A^{-1}u_t\) \(e_{yt}=(u_{yt}-a_{12}u_{rt})/(1-a{12}a_{21})\)
\(e_{rt}=(u_{-a_{12}u_{yt}+rt})/(1-a{12}a_{21})\)
The structural VAR cannot be estimated directly.
Start from the reduced-form VAR: \(X_t=G_0+G_1 X_{t-1}+u_{t}\) and get to the structurala one: \(AX_t=\beta_0+\beta_1 X_{t-1}+u_t\)
To do that, we need to restrict matrix A: IDENTIFICATION
\(AX_t=AG_0+AG_1 X_{t-1}+A e_t\)
\(AX_t=\beta_0+\beta_1 X_{t-1}+y_t\)
\(AA^{-1}\beta_0=\beta_0\)
Usual approach: impose restrictions on matrix A which is equivalent to:
\[\begin{bmatrix} 1 & 0\\ a21 & 1 \\ \end{bmatrix}\begin{bmatrix} y_t\\ r_t\\ \end{bmatrix}=\begin{bmatrix} \beta_{10}\\ \beta_{20}\\ \end{bmatrix}+\begin{bmatrix} \beta_{11} & \beta_{12}\\ \beta_{21} & \beta_{22} \\ \end{bmatrix}\begin{bmatrix} y_{t-1}\\ r_{t-1}\\ \end{bmatrix}+\begin{bmatrix} u_{yt}\\ u_{rt}\\ \end{bmatrix}\]
This is what identification is about: 0 in top left corner of A.
If we impose \(a_{12}=0\), we impose that \(y\) is only affected with a lag by a shock to \(r\) but that a shock to \(y\) affects \(r\) contemporaneously
\(y_t=\beta_{10}+\beta_{12}y_{t-1}+\beta_{12}r_{t-1}+u_{yt}\)
\(a_{21}y_t+r_t=\beta_{20}+\beta_{21}y_{t-1}+\beta_{22}r_{t-1}+u_{rt}\)
\(A^{-1}A X_t=A^{-1}\beta_0+A^{-1}\beta_1 X_{t-1}+A^{-1}u_t\)
\[\begin{bmatrix} y_t\\ r_t\\ \end{bmatrix}=\begin{bmatrix} 1 & 0\\ a21 & 1 \\ \end{bmatrix}^{-1}\begin{bmatrix} \beta_{10}\\ \beta_{20}\\ \end{bmatrix}+\begin{bmatrix} 1 & 0\\ a21 & 1 \\ \end{bmatrix}^{-1}\begin{bmatrix} \beta_{11} & \beta_{12}\\ \beta_{21} & \beta_{22} \\ \end{bmatrix}\begin{bmatrix} y_{t-1}\\ r_{t-1}\\ \end{bmatrix}+\\ \begin{bmatrix} 1 & 0\\ a21 & 1 \\ \end{bmatrix}^{-1}\begin{bmatrix} u_{yt}\\ u_{rt}\\ \end{bmatrix}\]
The matrices product makes the reduced-form VAR:
\[\begin{bmatrix} y_t\\ r_t\\ \end{bmatrix}=\begin{bmatrix} \beta_0\\ -a_{21}\beta_{10}+\beta_{20} \\ \end{bmatrix}+\begin{bmatrix} \beta_{11} & \beta_{12}\\ -a_{21}\beta_{11}+\beta_{12} & -a_{21}\beta_{12}+\beta_{22} \\ \end{bmatrix}\begin{bmatrix} y_{t-1}\\ r_{t-1}\\ \end{bmatrix}+\\ \begin{bmatrix} u_{yt}\\ -a_{21}u_{yt}+u_{rt}\\ \end{bmatrix}\]
\[\begin{bmatrix} y_t\\ r_t\\ \end{bmatrix}=\begin{bmatrix} g_{10}\\ g_{20} \\ \end{bmatrix}+\begin{bmatrix} g_{11} & g_{12}\\ g_{21} & g{22} \\ \end{bmatrix}\begin{bmatrix} y_{t-1}\\ r_{t-1}\\ \end{bmatrix}+ \begin{bmatrix} e_{yt}\\ e_{rt}\\ \end{bmatrix}\]
\(a_{21}, \beta_{10},\beta_{20},\beta_{11},\beta_{21},\beta_{22}, \sigma_{vy}^2 and \sigma_{vr}^2\) - The structural shocks may also be recovered.
Summary of SVAR Identification up to now: - Get the structural model, - to do that, we need to get A, so - impose restrictions on A, based on economics, to recover the structural shocks and the structural parameters, using the reduced-form estimation.
The minimum number of restrictions required: difference between the number of unknown and known elements. Let n be the number of variable in the VAR.
Unknown: as the elements of the diagonal of A are equal 1’s, there are \(n(n-1)\) unknown elements in A. There are also n unknown variances of \(u\). The total is: \(n(n-1)+n=n^2\).
Known elements: the estimation allows us to get \(n(n-1)/2+n=n(n+1)/2\) distinct elements contained in the symmetric var-cov matrix of the errors: \(Ee_t e_t'=\Sigma_e\).
Why the known elements are \(n(n+1)/2\)? The distinct elements contained in the symmetric var-cov matrix of the forecast errors: \(Ee_t e_t'=\Sigma_e\)
Because there are n distinct elements in the diagonal of \(\Sigma_e\) and \(n(n-1)/2\) distinct elements in the upper triangular part of \(\Sigma_e\). The total number of known elements is: \(n+n(n-1)/2=n(n+1)/2\).
Thus, impose \(n^2-n(n+1)/2=n(n-1)/2\) restrictions on A.A VAR with 3 variables requires 3(3-1)/2=3 restrictions on A; A VAR with 4 variables requires 4(4-1)/2=6 restrictions on A.
Impulse responses trace the effects of structural shocks on the endogenous variables over time.
Each response includes the effect of a specific shock on one of the endogenous variables of the system at impact t, then on t+1, and so on.
Transform our structural autoregressive vector \(AX_t=\beta_0+\beta_1 X_{t-1}+u_t\) into a moving average representation: \(X_t=\mu+\sum_{i=0}^{\infty}C_i u_t\)
Forecast errors and the relationships among the variables.
Propotion of the movements of a variable due to shocks to itself and shocks to other variables in the system.
This forecast error variance decomposition and the foreces behind business cycles.
Are all business cycles alike? Are GDP surprises mostly due to technology shocks or to monetary policy shocks?
Are demand shocks the main source of GDP forecast errors at long hirzons?
In our recursive ordering for \(y\) and \(r\), all of the one period forecast-error variance of \(y\) is due to shocks to \(u_{yt}\). At longer horizons the explanatory share of shocks to \(u_{yt}\) will diminish.
\(e_{yt}=u_{yt}\) and \(e_{rt}=u_{rt}-a_{21}u_{yt}\)
\[\begin{bmatrix} e_{yt}\\ e_{rt}\\ \end{bmatrix}=\begin{bmatrix} 1 & 0\\ -a_{21} & 1 \\ \end{bmatrix}\begin{bmatrix} u_{yt}\\ u_{rt}\\ \end{bmatrix}\]
Shocks to \(r\) only affects output gap \(y\) with a lag: \(y_t=\beta_0+\beta_1 y_{t-1}+\beta_2 r_{t-1}+u_{yt}\)
Both shocks have contemporaneous effects on the interest rate\(r\): \(r_t+a_{21}y_t=\beta_0+\beta_1 y_{t-1}+\beta_2 r_{t-1}+u_{rt}\)
If \(r\) is the policy rate, then the recursive ordering is consistent with the view that monetary policy responds to output gap \(y\) with a lag, but responds to the interest rate \(r\) contemporaneously.
\(e_{yt}=u_{yt}\) \(e_{\pi t}=-a_{21}u_{yt}+u_{\pi t}\) \(e_{rt}=(-a_{31}+a_{21}a_{32})u_{yt}-a_{32}u_{\pi t}+u_{rt}\)
\[\begin{bmatrix} e_{yt}\\ e_{\pi t}\\ e_{rt}\\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0\\ -a_{21} & 1 & 0\\ -a_{31}+a_{21}a_{32} & -a_{32} & 1\\ \end{bmatrix}\begin{bmatrix} u_{yt}\\ u_{\pi t}\\ u_{rt}\\ \end{bmatrix}\]
If movements of \(r\) are monetary policy(MP) decisions, policy is reacting to output \(y\) and inflation \(\pi\) within the period. All shocks have contemporaneous effects on \(r\).
The policy rate \(r\) affects output \(y\) with a lag, and inflation \(\pi\) with a lag.
Indeed, when one has a recursive odering.
\(A^{-1}\) is the Choleski factor of the variance-covariance matrix.
Why? substitute \(e_t=A^{-1}u_t\) into \(Ee_t e_t'=\Sigma_e\) and get \(A^{-1}\Sigma_u A^{-1'}=\Sigma_e\).
Assuming that \(u_tu_t'\) is diagonal, then \(A^{-1}I (A^{-1})'=\Sigma_e\) same as \(\Sigma_e=A^{-1}A^{-1'}\). The inverse of \(A\) is lower triangular consistent with a recursive ordering.
Not that assuming \(\Sigma_u=1\) implies that \(A\) will not have \(1s\) on the diagonal.
Interepreting the macroeconomic time series facts European Economic Review
Sims estimated a VAR to trace the effect of a shock to monetary policy (Central Bank Interest Rate) using specific ordering.
To assess the empirical support for alternative views (models) about monetary policy: VAR agenda.
He included:
Shocks to the nominal interest rate ff, represent monetary policy shocks.
A shcok to the policy variable affects all other variables contemporaneously.
The variable that is affected by all the others, within the period, is ordered last.
Authorities only observe non-policy variables with a lag.
Empirical results:
“An empirical characterization of the dynamic effects of changes in government spending and taxes on output.” Quarterly Journal of Economics, 117(4):1329–1368, 2002.
Use a semi-recursive VAR to econometrically identify fiscal policy shocks; i.e. exogenous movements in public expenditure/taxes. - They estimate a VAR: - Using contemporaneous identification restrictions on variables. - Imposing, on matrix A, coefficients estimated outside the VAR. - They pay careful attention to institutional procedures and periods of tax collection and public spending.
The VAR is \(X_t=G(L)X_{t-1}+e_t\) where \(X_t=[S_t,T_t,Y_t]\) includes the logs of quarterly real per capita government spending, taxes, and GDP, respectively.
And \(e_t=[e_t^S,e_t^T,e_t^Y\) is the vector of reduced form residuals or forecast errors of spending, taxes, and GDP, respectively.
Spending: government consumption and investment.
Revenue: total taxes minus transfers and minus interest payments.
An empirical characterization of the dynamic effects …
\[\begin{bmatrix} 1 & 0 & -b_1\\ 0 & 1 & -a_1\\ -c_1 & -c_2 & 1\\ \end{bmatrix}\begin{bmatrix} e_t^{S}\\ e_t^{T}\\ e_t^{y}\\ \end{bmatrix}=\begin{bmatrix} 1 & b_2 & 0\\ a_2 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}\begin{bmatrix} u_t^{S}\\ u_t^{T}\\ u_t^{y}\\ \end{bmatrix}\] These are matrices \(A\) and \(B\) to be estimated.
An empirical characterization of the dynamic effects …
An empirical characterization of the dynamic effects …
SPENDING SHOCKS VAR 4 lags 1960:1 1997:4 (\(a_2neq0\),\(b_2=0\))