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\begin{enumerate} | ||
\item There are \(K\) variables and \(K\) structural shocks; hence, there are \(K^2\) IRFs each of length \(H+1\). | ||
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\item Let's derive an expression for \(Y_{t+h}\): | ||
\begin{align*} | ||
Y_t &= A Y_{t-1} + U_t | ||
\\ | ||
Y_{t+1} &= A Y_{t} + U_{t+1} = A^2 Y_{t-1} + A U_t + U_{t+1} | ||
\\ | ||
Y_{t+2} &= A Y_{t+1} + U_{t+2} = A^3 Y_{t-1} + A^2 U_t + A U_{t+1} + U_{t+2} | ||
\\ | ||
Y_{t+h} &= A^{h+1} Y_{t-1} + \sum_{j=0}^h A^j U_{t+h-j} | ||
\end{align*} | ||
Left-multiply by \(J\) (also note that \(J'J=I\)): | ||
\begin{align*} | ||
J Y_{t+h} = y_{t+h} = J A^{h+1} Y_{t-1} + \sum_{j=0}^h J A^j \textcolor{red}{J'J} U_{t+h-j} = J A^{h+1} Y_{t-1} + \sum_{j=0}^h J A^j J' u_{t+h-j} | ||
\end{align*} | ||
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\item From the previous exercise: | ||
\begin{align*} | ||
y_{t+h} = J A^{h+1} Y_{t-1} + \sum_{j=0}^h J A^j J' \underbrace{u_{t+h-j}}_{B_0^{-1} \varepsilon_{t+h-j}} | ||
\end{align*} | ||
Taking the derivative: | ||
\begin{align*} | ||
\frac{\partial y_{t+h}}{\partial \varepsilon_{t}'} = \Theta_h = J A^h J' B_0^{-1} | ||
\end{align*} | ||
\item Note the following identity: \(p_{t+h}=p_{t-1}+\Delta p_t +\Delta p_{t+1} + \cdots +\Delta p_{t+h}\). | ||
That is we simply need to cumulate (\texttt{cumsum}) the impulse responses of the inflation rate to get the impulse response for the price level. | ||
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\item A possible implementation: | ||
\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/irfPlots.m} | ||
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\end{enumerate} |
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\begin{enumerate} | ||
\item[1.] The model is identified recursively with the real price of oil ordered first: | ||
\begin{align*} | ||
y_t = \begin{pmatrix}\Delta rpoil_t \\ \Delta p_t \\ \Delta gdp_t \end{pmatrix} | ||
\end{align*} | ||
such that the real price of oil is \textbf{predetermined} with respect to the U.S. economy. | ||
In other words, only the structural oil price shock (ordered first) has an immediate effect on the real price of oil, | ||
the other two structural shocks affect the real price of oil with a delay and not on impact. | ||
The ordering is thus very important here, | ||
as our focus is on the effect of an \textbf{unanticipated (exogenous)} increase in the real price of oil. | ||
Moreover, as we are only interested in one shock, | ||
the model is only partially identified in that only the oil price shock can be given an economic interpretation. | ||
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\item[2./3./4./5.] \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/USOil.m} | ||
Here is the helper function for the more general approach: | ||
\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/USOil_fSR.m} | ||
\item[6.] In the period under consideration, crude oil is a commodity that is considered vulnerable to negative supply shocks | ||
due to the political and social volatility of its (Middle East) source. | ||
In macroeconomics a negative supply shock is an unexpected event that changes the supply of a product, | ||
resulting in an increase in prices and a decrease in output. | ||
From the IRFs we see exactly this pattern: | ||
an unexpected oil price shock creates inflationary pressure on the GDP deflator and a reduction in real GDP in the US\@. | ||
In this sense, a positive oil price shock (originating e.g.\ from the oil producing countries) indeed acts like a negative domestic supply shock for the U.S. economy. | ||
\end{enumerate} |
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\begin{enumerate} | ||
\item The OLS estimate of the reduced-form covariance matrix \(\Sigma_u\) is | ||
\begin{align*} | ||
\widehat{\Sigma}_u = \begin{pmatrix} | ||
0.061054 & -0.015282 & 0.042398 & 0.0037836 \\ | ||
-0.015282 & 0.52305 & 0.07967 & 0.030602 \\ | ||
0.042398 & 0.07967 & 0.71689 & -0.2451 \\ | ||
0.0037836 & 0.030602 & -0.2451 & 1.1093 \\ | ||
\end{pmatrix} | ||
\end{align*} | ||
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\item In order to identify the structural shocks from \(\Sigma_u = B_0^{-1} \Sigma_\varepsilon B_0^{-1'}\), | ||
we require at least \(4(4-1)/2=6\) additional restrictions on \(B_0\) or \(B_0^{-1}\). | ||
In this exercise, we'll do this on \(B_0\) and not its inverse; that is, we can rewrite the equations in matrix notation: | ||
\begin{align*} | ||
\underbrace{\begin{pmatrix} 1 & 0 & 0 & 0 \\ b_{21,0} & 1 & b_{23,0} & b_{24,0} \\ 0 & 0 & 1 & b_{34,0} \\ b_{41,0} & b_{41,0} & b_{43,0} & 1\end{pmatrix}}_{B_0} | ||
\underbrace{\begin{pmatrix}u_t^p\\u_t^{gnp}\\u_t^{i}\\u_t^{m} \end{pmatrix}}_{u_t} | ||
= | ||
\underbrace{\begin{pmatrix}\varepsilon_t^{AS}\\\varepsilon_t^{IS}\\\varepsilon_t^{MS}\\\varepsilon_t^{MD} \end{pmatrix}}_{\varepsilon_t} | ||
\end{align*} | ||
Note that as the diagonal elements of \(B_0\) equal unity, we don't assume that \(E(\varepsilon_t \varepsilon_t')=\Sigma_\varepsilon \) is the identity matrix. | ||
So this is a different normalization rule as before. | ||
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Economically, the above restrictions embody to some extent a baseline IS-LM model: | ||
\begin{enumerate} | ||
\item The first equation provides three restrictions by assuming that the price level is predetermined | ||
except that producers can respond immediately to aggregate supply shocks (e.g.\ an unexpected increase in oil or gas prices). | ||
This is basically a horizontal aggregate supply (AS) curve; that is why we label the shock with AS\@. | ||
\item The second equation provides no restrictions as it is assumed that real output responds to all other model variables contemporaneously. | ||
This equation can be interpreted as an aggregate demand curve, or an IS curve, that is why we label the shock IS\@. | ||
\item The third equation provides two restrictions by assuming that the interest rate does not react contemporaneously to aggregate measures of output and prices. | ||
This represents a simple money supply function, according to which the central bank adjusts the rate of interest in relation to the money stock | ||
and does not immediately observe aggregate output and aggregate prices. | ||
Of course, this is in contrast to modern monetary policy theory (and practice). | ||
\item The fourth equation provides one additional restriction by assuming that the first two entries in the last row are identical. | ||
The underlying idea is that this equation represents a money demand function in which short-run money holdings rise in proportion to NOMINAL GNP\@. | ||
Moreover, money holdings are allowed to be dependent on the interest rate. | ||
\end{enumerate} | ||
In sum, we have \(3+0+2+1=6\) restrictions, that is we have an exactly identified SVAR model, | ||
identified by short-run exclusion restrictions on \(B_0\). | ||
As the structure is not recursive, a Cholesky decomposition is not valid and hence we need to rely on a numerical optimizer (or an algorithm for short-run exclusion restrictions). | ||
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\item The code might look like this: | ||
\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/keatingSR.m} | ||
Here is the helper function to impose the restrictions: | ||
\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/keatingSR_f.m} | ||
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\item An unexpected upward shift of the aggregate supply curve | ||
\begin{itemize} | ||
\item raises the price deflator | ||
\item lowers real GNP | ||
\item raises the federal funds rate | ||
\item lowers the money supply at first, but ultimately raises M1 | ||
\end{itemize} | ||
This response is not really appropriate for the U.S. economy during the considered sample period, | ||
making the identifying restrictions rather questionable. | ||
\end{enumerate} |
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% ------------------------------------------------------------------------- | ||
% Estimates a SVAR(4) model to identify an oil price shock for the US economy | ||
% ------------------------------------------------------------------------- | ||
% Willi Mutschler, December 8, 2022 | ||
% [email protected] | ||
% ------------------------------------------------------------------------- | ||
clearvars; clc;close all; | ||
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%% data handling | ||
USOil_data = importdata('../../data/USOil.csv'); | ||
ENDO = USOil_data.data; % note that it already has correct order dlog(poil), dlog(p), dlog(gdp) for recursive identification | ||
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%% estimate reduced-form | ||
nlag = 4; | ||
opt.const = 1; | ||
VAR = VARReducedForm(ENDO,nlag,opt); | ||
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%% structural identification with Cholesky decomposition | ||
ENDO = ENDO(:,[1 2 3]); % order dlog(oilprice) first, then dlog(price deflator) and dlog(GDP) | ||
B0inv_chol = chol(VAR.SigmaOLS([1 2 3],[1 2 3]),'lower'); | ||
% note that the Cholesky decomposition always yields positive diagonal elements, so no normalization needed | ||
table(B0inv_chol) | ||
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%% structural identification with numerical optimization, identification restrictions are put into auxiliary function USOil_fSR | ||
f = str2func('USOil_fSR'); | ||
StartValueMethod = 1; %0: Use identity matrix, 1: use square root, 2: use cholesky as starting value | ||
% options for fsolve | ||
TolX = 1e-4; % termination tolerance on the current point | ||
TolFun = 1e-9; % termination tolerance on the function value | ||
MaxFunEvals = 50000; % maximum number of function evaluations allowed | ||
MaxIter = 1000; % maximum numberof iterations allowed | ||
OptimAlgorithm = 'trust-region-dogleg'; % algorithm used in fsolve | ||
options = optimset('TolX',TolX,'TolFun',TolFun,'MaxFunEvals',MaxFunEvals,'MaxIter',MaxIter,'Algorithm',OptimAlgorithm); | ||
if StartValueMethod == 0 | ||
B0inv = eye(size(ENDO,2)); % use identity matrix as starting value | ||
elseif StartValueMethod == 1 | ||
B0inv = VAR.SigmaOLS^.5; % use square root of vcov of reduced form as starting value | ||
elseif StartValueMethod == 2 | ||
B0inv = chol(VAR.SigmaOLS,'lower'); % use Cholesky decomposition of vcov of reduced form | ||
end | ||
f(B0inv,VAR.SigmaOLS)' % test whether function works at initial value (should give you no error) | ||
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% call optimization routine fsolve to minimize f | ||
[B0inv_opt,fval,exitflag,output] = fsolve(f,B0inv,options,VAR.SigmaOLS); | ||
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% normalize sign of B0inv such that diagonal elements are positive | ||
if any(diag(B0inv_opt)<0) | ||
idx_negative = diag(B0inv_opt)<0; | ||
B0inv_opt(:,(idx_negative==1)) = -1*B0inv_opt(:,(idx_negative==1)); % flip sign | ||
end | ||
table(B0inv_opt) | ||
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%% compute and plot structural impulse response function | ||
nSteps = 30; | ||
cumsumIndicator = [1 0 1]; % the variables in the SVAR are in differences, we want to plot IRFs of the first and last variable by using cumsum | ||
varNames = ["Real Price of Oil","GDP Deflator Inflation","Real GDP"]; | ||
epsNames = ["Oil Price Shock", "eps2 Shock", "eps3 Shock"]; | ||
IRFpoint_chol = irfPlots(VAR.Acomp,B0inv_chol,nSteps,cumsumIndicator,varNames,epsNames); | ||
IRFpoint_opt = irfPlots(VAR.Acomp,B0inv_opt,nSteps,cumsumIndicator,varNames,epsNames); | ||
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%% both approaches are (numerically) equivalent | ||
norm(abs(B0inv_chol-B0inv_opt),'Inf') % max abs error | ||
norm(abs(IRFpoint_chol(:) - IRFpoint_opt(:)),'Inf') % max abs error |
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function f = USOil_fSR(B0inv,hatSigmaU) | ||
% f = USOil_fSR(B0inv,hatSigmaU) | ||
% ------------------------------------------------------------------------- | ||
% Evaluates the system of nonlinear equations | ||
% vech(hatSigmaU) = vech(B0inv*B0inv') | ||
% subject to specified short-run restrictions | ||
% ------------------------------------------------------------------------- | ||
% INPUTS | ||
% - B0inv : candidate for short-run impact matrix. [nvars x nvars] | ||
% - hatSigmaU : covariance matrix of reduced-form residuals. [nvars x nvars] | ||
% ------------------------------------------------------------------------- | ||
% OUTPUTS | ||
% - f : function value, see below | ||
% ------------------------------------------------------------------------- | ||
% Willi Mutschler, December 8, 2022 | ||
% [email protected] | ||
% ------------------------------------------------------------------------- | ||
f = [vech(B0inv*B0inv' - hatSigmaU); | ||
B0inv(1,2) - 0; | ||
B0inv(1,3) - 0; | ||
B0inv(2,3) - 0; | ||
]; | ||
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% % more general way to implement the restrictions | ||
% % nan means unconstrained, 0 (or any other number) restricts to this number | ||
% Rshort = [nan 0 0; | ||
% nan nan 0; | ||
% nan nan nan; | ||
% ]; | ||
% selSR = ~isnan(Rshort); % index for short-run restrictions on impact | ||
% f = [vech(B0inv*B0inv'-hatSigmaU); | ||
% B0inv(selSR) - Rshort(selSR); | ||
% ]; | ||
end |
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% ------------------------------------------------------------------------- | ||
% Short-run restrictions in the Keating (1992) model. | ||
% ------------------------------------------------------------------------- | ||
% Willi Mutschler, December 27, 2022 | ||
% [email protected] | ||
% ------------------------------------------------------------------------- | ||
clearvars; clc; close all; | ||
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% data handling | ||
Keating1992 = importdata('../../data/Keating1992.csv'); | ||
ENDO = Keating1992.data; | ||
[obs_nbr,var_nbr] = size(ENDO); | ||
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% estimate reduced-form | ||
nlag = 4; | ||
opt.const = 1; | ||
VAR = VARReducedForm(ENDO,nlag,opt); | ||
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% identification restrictions are set up in keatingSR_f_SR | ||
f = str2func('keatingSR_f'); | ||
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% options for fsolve | ||
TolX = 1e-4; % termination tolerance on the current point | ||
TolFun = 1e-9; % termination tolerance on the function value | ||
MaxFunEvals = 50000; % maximum number of function evaluations allowed | ||
MaxIter = 1000; % maximum numberof iterations allowed | ||
OptimAlgorithm = 'trust-region-dogleg'; % algorithm used in fsolve | ||
options = optimset('TolX',TolX,'TolFun',TolFun,'MaxFunEvals',MaxFunEvals,'MaxIter',MaxIter,'Algorithm',OptimAlgorithm); | ||
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% initital guess, note that we need candidates for both B_0 and diag(SIG_eps) stored into a single candidate matrix | ||
B0_diageps= [eye(var_nbr) ones(var_nbr,1)]; % the first 4 columns belong to B_0, the last column are the diagonal elements of SIG_eps | ||
f(B0_diageps,VAR.SigmaOLS)' % test whether function works at initial value (should give you no error) | ||
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% call optimization routine fsolve to minimize f | ||
[B0_diageps,fval,exitflag,output] = fsolve(f,B0_diageps,options, VAR.SigmaOLS); | ||
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% display results | ||
B0 = B0_diageps(:,1:var_nbr); % get B0 | ||
SIGeps = B0_diageps(:,var_nbr+1); % these are just the variances | ||
impact = inv(B0)*diag(sqrt(SIGeps)); % compute impact matrix used to plot IRFs, i.e. inv(B0)*sqrt(SigEps) | ||
table(impact) | ||
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% compute and plot structural impulse response function | ||
nSteps = 12; | ||
cumsumIndicator = [1 1 0 1]; | ||
varNames = ["Deflator", "Real GNP", "Federal Funds Rate", "M1"]; | ||
epsNames = ["AS", "IS", "MS", "MD"] + " shock"; | ||
irfPoint = irfPlots(VAR.Acomp,impact,nSteps,cumsumIndicator,varNames,epsNames); |
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function f = keatingSR_f(B0_diageps,hatSigmaU) | ||
% f = keatingSR_f(B0_diageps,hatSigmaU) | ||
% ------------------------------------------------------------------------- | ||
% Evaluates the system of nonlinear equations vech(hatSigmaU) = | ||
% vech(B0inv*SIGeps*B0inv') where SIGeps has only values on the diagonal | ||
% given a candidate for B0 and the diagonal elements of SIGeps | ||
% subject to the short-run restrictions in the Keating (1992) model. | ||
% ----------------------------------------------------------------------- | ||
% INPUTS | ||
% - B0_diageps [nvars x (nvars+1)] candidate matrix for both short-run impact matrix [nvars x nvars] and diagonal elements of SIGeps (nvar x 1) | ||
% - hatSigmaU [nvars x nvars] covariance matrix of reduced-form residuals | ||
% ----------------------------------------------------------------------- | ||
% OUTPUTS | ||
% - f : function value, see below | ||
% ------------------------------------------------------------------------- | ||
% Willi Mutschler, December 14, 2022 | ||
% [email protected] | ||
% ------------------------------------------------------------------------- | ||
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nvars = size(hatSigmaU,1); % number of variables | ||
B0 = B0_diageps(:,1:nvars); % get B0 matrix from candidate matrix | ||
diageps = diag(B0_diageps(1:nvars,nvars+1)); % get diagonal elements of SIG_eps from candidate matrix and make it a full diagonal matrix | ||
B0inv = inv(B0); % compute short-run impact matrix | ||
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f = [vech(B0inv*diageps*B0inv'-hatSigmaU); | ||
B0(1,1) - 1; | ||
B0(3,1) - 0; | ||
B0(4,1) - B0(4,2); | ||
B0(1,2) - 0; | ||
B0(2,2) - 1; | ||
B0(3,2) - 0; | ||
B0(1,3) - 0; | ||
B0(3,3) - 1; | ||
B0(1,4) - 0; | ||
B0(4,4) - 1; | ||
]; |
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