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Binder

Contents

Multivariate time series

• univariate - temporal dependence between components of a single series

• multivariate - temporal (inter)-dependence between components of different series

Definition: A time series process is a sequence of random vectors indexed by time:

$$\begin{array}{}\text{(1)}& \{{\mathbf{z}}_t:t=...,-2,-1,0,1,2,...\}=\{{\mathbf{z}}_t{\}}_{t=-\mathrm{\infty }}^{\mathrm{\infty }}\end{array}$$

where ${\mathbf{z}}_t$ is $n\ge 1$-dimensional vector

Stationarity

Definition: The process $\{{\mathbf{z}}_t{\}}_{t=-\mathrm{\infty }}^{\mathrm{\infty }}$ is covariance stationary if the first two moments of the joint distribution exist and are time-invariant

• mean

$$\begin{array}{r}\mathrm{E}{\mathbf{z}}_t={\mu }_t=\left[\begin{array}{c}{\mu }_{1t}\\ {\mu }_{2t}\\ ⋮\\ {\mu }_{nt}\end{array}\right]\end{array}$$

• covariance

$$\begin{array}{r}\begin{array}{rl}\mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t-k})& =\mathrm{E}({\mathbf{z}}_t-{\mu }_t)({\mathbf{z}}_{t-k}-{\mu }_{t-k}{)}^{\prime }\\ \\ & =\mathrm{\Gamma }(t,t-k)=\left(\begin{array}{ccc}{\gamma }_{11}(t,t-k)& \cdots & {\gamma }_{1n}(t,t-k)\\ ⋮& \ddots & ⋮\\ {\gamma }_{n1}(t,t-k)& \cdots & {\gamma }_{nn}(t,t-k)\end{array}\right)\end{array}\end{array}$$

• $\{{\mathbf{z}}_t{\}}_{t=-\mathrm{\infty }}^{\mathrm{\infty }}$ is covariance stationary if

$$\begin{array}{r}\begin{array}{rl}{\mu }_t& =\mu \\ \mathrm{\Gamma }(t,t-k)& =\mathrm{\Gamma }(k)\end{array}\end{array}$$

are not functions of $t$

Note:

$$\mathrm{\Gamma }(k)=\mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t-k})=\mathrm{cov}({\mathbf{z}}_{t+k},{\mathbf{z}}_t)\ne \mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t+k})=\mathrm{\Gamma }(-k)$$

• $\mathrm{\Gamma }(k)$ is not symmetric (unless $k=0$)

• but since $\mathrm{cov}({\mathbf{z}}_{t+k},{\mathbf{z}}_t{)}^{\prime }=\mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t+k})$

$$\mathrm{\Gamma }(k)=\mathrm{\Gamma }(-k{)}^{\prime }$$

follows from (set $\mu =0$ w.l.g.)

$$\mathrm{\Gamma }(k)=\mathrm{cov}({\mathbf{z}}_{t+k},{\mathbf{z}}_t)=\mathrm{E}({\mathbf{z}}_{t+k}{\mathbf{z}}_t^{\prime })=\mathrm{E}({\mathbf{z}}_t{\mathbf{z}}_{t+k}^{\prime }{)}^{\prime }={(\mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t+k}))}^{\prime }=\mathrm{\Gamma }(-k{)}^{\prime }$$

If

$$\begin{array}{r}{\mathbf{Z}}_T=\left[\begin{array}{c}{\mathbf{z}}_1\\ {\mathbf{z}}_2\\ ⋮\\ {\mathbf{z}}_T\end{array}\right]\end{array}$$

then

$$\begin{array}{r}\mathrm{E}({\mathbf{Z}}_T)=\left[\begin{array}{c}\mu \\ \mu \\ ⋮\\ \mu \end{array}\right],\mathrm{cov}({\mathbf{Z}}_T)=\left(\begin{array}{cccc}\mathrm{\Gamma }(0)& \mathrm{\Gamma }(1{)}^{\prime }& \cdots & \mathrm{\Gamma }(T-1{)}^{\prime }\\ \mathrm{\Gamma }(1)& \mathrm{\Gamma }(0)& \cdots & \mathrm{\Gamma }(T-2{)}^{\prime }\\ ⋮& ⋮& ⋮& ⋮\\ \mathrm{\Gamma }(T-1)& \mathrm{\Gamma }(T-2)& \cdots & \mathrm{\Gamma }(0)\end{array}\right)(\text{symmetric block Toeplitz matrix})\end{array}$$

• Multivariate Gaussian tme series

$${\mathbf{Z}}_T\sim \mathcal{N}(\mathit{\mu },\mathbf{\Sigma })$$

$$$$

$$\begin{array}{r}\left[\begin{array}{c}{\mathbf{z}}_1\\ {\mathbf{z}}_2\\ ⋮\\ {\mathbf{z}}_T\end{array}\right]\sim \mathcal{N}(\left[\begin{array}{c}\mu \\ \mu \\ ⋮\\ \mu \end{array}\right],\left(\begin{array}{cccc}\mathrm{\Gamma }(0)& \mathrm{\Gamma }(1{)}^{\prime }& \cdots & \mathrm{\Gamma }(T-1{)}^{\prime }\\ \mathrm{\Gamma }(1)& \mathrm{\Gamma }(0)& \cdots & \mathrm{\Gamma }(T-2{)}^{\prime }\\ ⋮& ⋮& ⋮& ⋮\\ \mathrm{\Gamma }(T-1)& \mathrm{\Gamma }(T-2)& \cdots & \mathrm{\Gamma }(0)\end{array}\right))\end{array}$$

Number of unique parameters

$$\begin{array}{r}\begin{array}{rl}\mathrm{\Gamma }(0)& :n(n+1)/2\\ & +\\ \mathrm{\Gamma }(1)& :n^2\\ & +\\ & ⋮\\ & +\\ \mathrm{\Gamma }(T-1)& :n^2\end{array}\end{array}$$

with $n=5,T=200=>4990$

• time series models allow to represent temporal (inter)dependence parsimoniously

• by imposing restrictions - reducing the number of unique parameters

• making estimation feasible

VARMA(p, q) model

$$\begin{array}{r}\begin{array}{rl}A(L){\mathbf{z}}_t& =B(L){\mathit{\epsilon }}_t,{\mathit{\epsilon }}_t\sim \mathrm{WN}(0,\mathbf{\Sigma })(\text{vector white noise, i.e. }\mathrm{E}({\mathit{\epsilon }}_t,{\mathit{\epsilon }}_{t-k}^{\prime })=0)\\ \\ A(L)& =I-A_{1}L-\cdots -A_p{L}^p\\ B(L)& =I+B_{1}L+\cdots +B_q{L}^q\end{array}\end{array}$$

zero auto and cross-auto correlations of the innovations

$$\begin{array}{r}{A}_i=\left(\begin{array}{ccc}{a}_{11,i}& \cdots & a_{1n,i}\\ ⋮& \ddots & ⋮\\ a_{n1,i}& \cdots & a_{nn,i}\end{array}\right),B_j=\left(\begin{array}{ccc}{b}_{11,j}& \cdots & b_{1n,j}\\ ⋮& \ddots & ⋮\\ b_{n1,j}& \cdots & b_{nn,j}\end{array}\right)\end{array}$$

• $n^2(p+q)+n(n+1)/2$ unknown parameters

• in general, difficult to estimate

  • not all parameters are identified

  • have to use numerical optimization

VAR§ model

$$\begin{array}{r}\begin{array}{c}{\mathbf{z}}_t=A_1{\mathbf{z}}_{t-1}+\cdots +A_p{\mathbf{z}}_{t-p}+{\mathit{\epsilon }}_t,{\mathit{\epsilon }}_t\sim \mathrm{WN}(0,\mathbf{\Sigma })\\ \end{array}\end{array}$$

$$\begin{array}{r}\begin{array}{rl}A(L){\mathbf{z}}_t& ={\mathit{\epsilon }}_t,{\mathit{\epsilon }}_t\sim \mathrm{WN}(0,\mathbf{\Sigma })\\ \\ A(L)=I& -A_{1}L-\cdots -A_p{L}^p\end{array}\end{array}$$

• ${\mathbf{\Sigma }}_{ij}$ captures all contemporaneous (time $t$) relationships between $z_i$ and $z_j$

• $\mathbf{\{}{A}_k{\}}_{ij}$ captures all dynamic interactions between $z_{it}$ and $z_{j,t-k}$

Stationarity

VAR§ is stationary if all roots of the equation

$$|A(x)|=|I-A_{1}x-\cdots -A_p{x}^p|=0$$

are outside the unit circle ($|x|>1$)

VAR(1) representation of VAR§ process

$${\mathbf{Z}}_t=\mathbf{\Phi }{\mathbf{Z}}_{t-1}+{\mathit{E}}_t$$

$$\begin{array}{r}\underset{{\mathbf{Z}}_t}{\underbrace{\left[\begin{array}{c}{\mathbf{z}}_t\\ {\mathbf{z}}_{t-1}\\ ⋮\\ {\mathbf{z}}_{t-p+1}\end{array}\right]}}=\underset{\mathbf{\Phi }}{\underbrace{\left[\begin{array}{ccccc}{A}_1& A_2& \cdots & A_{p-1}& A_p\\ I& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& \cdots & I& 0\end{array}\right]}}\underset{{\mathbf{Z}}_{t-1}}{\underbrace{\left[\begin{array}{c}{\mathbf{z}}_{t-1}\\ {\mathbf{z}}_{t-2}\\ ⋮\\ {\mathbf{z}}_{t-p}\end{array}\right]}}+\underset{{\mathit{E}}_t}{\underbrace{\left[\begin{array}{c}{\epsilon }_t\\ 0\\ ⋮\\ 0\end{array}\right]}}\end{array}$$

• $\mathbf{\Phi }$ - companion matrix

${\mathbf{Z}}_t$ is stationary if all roots $x$ of

$$|\mathit{I}-\mathbf{\Phi }x|=0$$

are outside the unit circle ($|x|>1$), which is equivalent to all solutions of

$$|\mathit{I}\lambda -\mathbf{\Phi }|=0$$

being $|\lambda |<1$, i.e. all eigenvalues of $\mathbf{\Phi }$ being less than 1 in absolute value.

Eigenvalue decomposition

• $\mathrm{\Lambda }$ - diagonal matrix of the eigenvalues of $A$

• $V$ - matrix of the eigenvectors of $A$

from

$$AV=V\mathrm{\Lambda }$$

we have

$$\begin{array}{r}\begin{array}{rl}A& =V\mathrm{\Lambda }{V}^{-1}\\ A^2& =AA=V\mathrm{\Lambda }{V}^{-1}V\mathrm{\Lambda }{V}^{-1}=V{\mathrm{\Lambda }}^2{V}^{-1}\\ A^h& =V{\mathrm{\Lambda }}^h{V}^{-1}\end{array}\end{array}$$

if for all eigenvalues $|{\lambda }_i|<1$,

$${\mathrm{\Lambda }}^h⟶0\text{ and }{A}^h⟶0\text{as }h⟶\mathrm{\infty }$$

• similar to $|\alpha |<1$ in $z_t=\alpha z_{t-1}+{\epsilon }_t$

VAR(1) process

$${\mathbf{z}}_t=A{\mathbf{z}}_{t-1}+{\mathit{\epsilon }}_t$$

From VAR parameters to moments of ${\mathbf{z}}_t$

• what are the VAR parameters?

• mean

$$\begin{array}{r}\begin{array}{rl}\mathrm{E}{\mathbf{z}}_t& =A\mathrm{E}{\mathbf{z}}_{t-1}+\mathrm{E}{\mathit{\epsilon }}_t\\ & =0\end{array}\end{array}$$

• covariance

$${\mathbf{z}}_t=A{\mathbf{z}}_{t-1}+{\mathit{\epsilon }}_t\Rightarrow {\mathbf{z}}_t{\mathbf{z}}_t^{\prime }=(A{\mathbf{z}}_{t-1}+{\mathit{\epsilon }}_t){(A{\mathbf{z}}_{t-1}+{\mathit{\epsilon }}_t)}^{\prime }$$

Since $\mathrm{E}({\mathbf{z}}_{t-1}{\epsilon }_t^{\prime })=0$

$$\mathrm{E}({\mathbf{z}}_t{\mathbf{z}}_t^{\prime })=A\mathrm{E}({\mathbf{z}}_{t-1}{\mathbf{z}}_{t-1}^{\prime })A^{\prime }+\mathrm{E}({\epsilon }_t{\epsilon }_t^{\prime })$$

or

$$\begin{array}{r}\begin{array}{rl}\mathrm{\Gamma }(0)=& A\mathrm{\Gamma }(0)A^{\prime }+\mathrm{\Sigma }\\ \\ & \text{and}\\ \\ \mathrm{vec}(\mathrm{\Gamma }(0))=& {(I-A\otimes A)}^{-1}\mathrm{vec}(\mathrm{\Sigma })\end{array}\end{array}$$

• follows from $\mathrm{vec}(ABC)=(C^{\prime }\otimes A)\mathrm{vec}(B)$

• autocovariances

$$\mathrm{E}({\mathbf{z}}_t{\mathbf{z}}_{t-1}^{\prime })=A\mathrm{E}({\mathbf{z}}_{t-1}{\mathbf{z}}_{t-1}^{\prime })+\mathrm{E}({\epsilon }_t{\mathbf{z}}_{t-1}^{\prime })$$

• lag 1

$$\mathrm{\Gamma }(1)=A\mathrm{\Gamma }(0)$$

• lag 2

$$\mathrm{\Gamma }(2)=A\mathrm{\Gamma }(1)=A^2\mathrm{\Gamma }(0)$$

• lag h

$$\mathrm{\Gamma }(h)=A^h\mathrm{\Gamma }(0)$$

From moments of ${\mathbf{z}}_t$ to VAR parameters

$$\begin{array}{r}\begin{array}{rl}\mathrm{\Gamma }(1)=A\mathrm{\Gamma }(0)& \Rightarrow A=\mathrm{\Gamma }(1)\mathrm{\Gamma }(0{)}^{-1}\\ \\ \mathrm{\Gamma }(0)=A\mathrm{\Gamma }(0)A^{\prime }+\mathrm{\Sigma }& \Rightarrow \mathrm{\Sigma }=\mathrm{\Gamma }(0)-\mathrm{\Gamma }(1)\mathrm{\Gamma }(0{)}^{-1}\mathrm{\Gamma }(1{)}^{\prime }\end{array}\end{array}$$

Non-zero mean $\mathrm{E}{\mathbf{z}}_t=\mu \ne 0$

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_t& ={\mathit{a}}_0+A{\mathbf{z}}_{t-1}+{\epsilon }_t\\ \\ \mu & ={\mathit{a}}_0+A\mu \\ \\ \mu & ={(I-A)}^{-1}{\mathit{a}}_0\end{array}\end{array}$$

and

$$\mathrm{E}{\overline{\mathbf{z}}}_t=\mathrm{E}({\mathbf{z}}_t-\mu )=0$$

Note For the moments of a VAR§ model, use the VAR(1) representation, and apply the selection matrix

$$\mathit{s}=[I,0,\cdots ,0]$$

to obtain the autocovariances of ${\mathbf{z}}_t$ from the autocovariances of ${\mathbf{Z}}_t$

since

$${\mathbf{z}}_t=\mathit{s}{\mathbf{Z}}_t$$

• $\mathrm{E}{\mathbf{z}}_t=\mathit{s}\mathrm{E}{\mathbf{Z}}_t$

• $\mathrm{var}({\mathbf{z}}_t)=\mathit{s}\mathrm{var}({\mathbf{Z}}_t){\mathit{s}}^{\prime }$

• $\mathrm{cov}({\mathbf{z}}_t,{\mathbf{z}}_{t-k})=\mathit{s}\mathrm{cov}({\mathbf{Z}}_t,{\mathbf{Z}}_{t-k}){\mathit{s}}^{\prime }$

Estimation

We can write VAR§

$$\begin{array}{r}{\mathbf{z}}_t={\mathit{a}}_0+A_1{\mathbf{z}}_{t-1}+\cdots +A_p{\mathbf{z}}_{t-p}+{\mathit{\epsilon }}_t,{\mathit{\epsilon }}_t\sim \mathrm{WN}(0,\mathbf{\Sigma })\end{array}$$

as

$${\mathbf{z}}_t=\mathit{A}{\mathit{x}}_{t-1}+{\mathit{\epsilon }}_t$$

where $\mathit{A}=[{\mathit{a}}_0,A_1,\cdots ,A_p]$, and ${\mathit{x}}_{t-1}=[1,{\mathbf{z}}_{t-1}^{\prime },\cdots ,{\mathbf{z}}_{t-p}^{\prime }{]}^{\prime }$

Assume a pre-sample ${\mathbf{z}}_0,{\mathbf{z}}_1,\cdots ,{\mathbf{z}}_{-p+1}$ is given. (alternatively, re-define $T$)

Then, we have

$$\mathit{Z}=\mathit{A}\mathit{X}+\mathit{U}$$

where

• $\mathit{Z}=[{\mathbf{z}}_1,\cdots ,{\mathbf{z}}_T]$ is $n\times T$

• $\mathit{X}=[{\mathbf{x}}_0,\cdots ,{\mathbf{x}}_{T-1}]$ is $n(p+1)\times T$

• $\mathit{U}=[{\mathit{\epsilon }}_1,\cdots ,{\mathit{\epsilon }}_T]$ is $n\times T$

OLS estimation

$$\begin{array}{r}\begin{array}{rl}\hat{\mathit{A}}& =\mathit{Z}{\mathit{X}}^{\prime }(\mathit{X}{\mathit{X}}^{\prime }{)}^{-1}\\ \\ \hat{\mathbf{\Sigma }}& =\frac{1}{T-np-1}\hat{\mathit{U}}{\hat{\mathit{U}}}^{\prime }\\ \\ \hat{\mathit{U}}& =\mathit{Z}-\hat{\mathit{A}}\mathit{X}\end{array}\end{array}$$

Asymptotic distribution

$$\mathrm{vec}\left(\hat{\mathit{A}}\right)\stackrel{a}{\sim }\mathcal{N}(\mathrm{vec}\left(\mathit{A}\right),(\mathit{X}{\mathit{X}}^{\prime }{)}^{-1}\otimes \hat{\mathbf{\Sigma }})$$

Notes:

• the rows of $\mathit{A}$ can be estimated with OLS equation by equation

• also equivalent to conditional MLE, assuming that ${\mathit{\epsilon }}_t\sim \mathcal{N}(0,\mathbf{\Sigma })$

• $\hat{\mathit{A}}$ has a small-sample bias, which can be corrected for analytically (when the VAR has only intercept) or using bootstrap (when a deterministic trand is included)

Choice of $p$

• define a set of models - select $p_{min}$ and $p_{max}$

• estimate each one and compute IC§

• pick the one with lowest IC§ value

Most commonly used ICs:

$$\begin{array}{r}\begin{array}{rl}AIC& =\ln |{\hat{\mathbf{\Sigma }}}^{ml}(p)|+\frac{2}{T}(p{n}^2+n)\text{Akaike’s Information Criterion}\\ BIC& =\ln |{\hat{\mathbf{\Sigma }}}^{ml}(p)|++\frac{\ln (T)}{T}(p{n}^2+n)\text{Bayesian Information Criterion}\end{array}\end{array}$$

Notes

• For ICs to be comparable for different $p$, the sample has to be the same (set $t=p_{max}+1,\cdots ,T$)

• ${\hat{\mathbf{\Sigma }}}^{ml}(p)=\frac{1}{T}\hat{\mathit{U}}{\hat{\mathit{U}}}^{\prime }=\frac{T-np-1}{T}{\hat{\mathbf{\Sigma }}}^{ols}(p)$

• typically, $p_{min}=12$ for monthly and $p_{min}=4$ for quarterly data

Forecasting

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_t& =A{\mathbf{z}}_{t-1}+{\mathit{\epsilon }}_t,\\ \\ {\mathbf{z}}_{t+1}& =A{\mathbf{z}}_t+{\mathit{\epsilon }}_{t+1}\\ \\ {\mathbf{z}}_{t+2}& =A^2{\mathbf{z}}_t+A{\mathit{\epsilon }}_{t+1}+{\mathit{\epsilon }}_{t+2}\\ & ⋮\\ {\mathbf{z}}_{t+h}& =A^h{\mathbf{z}}_t+A^{h-1}{\mathit{\epsilon }}_{t+1}+\cdots +{\mathit{\epsilon }}_{t+h}\end{array}\end{array}$$

Optimal forecast given information at $T$:

$$\begin{array}{r}\begin{array}{rl}\mathrm{E}({\mathbf{z}}_{T+1}|{\mathbf{z}}_T)& =A{\mathbf{z}}_T\\ \mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)& ={\mathit{A}}^h{\mathbf{z}}_T\end{array}\end{array}$$

Optimal forecast given information at $T+1$:

$$\begin{array}{r}\begin{array}{rl}\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_{T+1})& ={\mathit{A}}^{h-1}{\mathbf{z}}_{T+1}\\ & ={\mathit{A}}^{h-1}(A{\mathbf{z}}_T+{\mathit{\epsilon }}_{T+1})\\ & ={\mathit{A}}^h{\mathbf{z}}_T+A^{h-1}{\mathit{\epsilon }}_{T+1}\\ & =\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)+A^{h-1}({\mathbf{z}}_{T+1}-A{\mathbf{z}}_T)\\ & =\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)+A^{h-1}({\mathbf{z}}_{T+1}-\mathrm{E}({\mathbf{z}}_{T+1}|{\mathbf{z}}_T))\end{array}\end{array}$$

Optimal forecast update:

$$\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_{T+1})-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)=A^{h-1}\underset{\text{1-step ahead forecast error}}{\underbrace{({\mathbf{z}}_{T+1}-\mathrm{E}({\mathbf{z}}_{T+1}|{\mathbf{z}}_T))}}$$

$h$-step-ahead forecast error:

$${\mathbf{z}}_{T+h}-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)=A^{h-1}{\mathit{\epsilon }}_{t+1}+\cdots +A{\mathit{\epsilon }}_{t+h-1}+{\mathit{\epsilon }}_{t+h}$$

with

$$\begin{array}{r}\begin{array}{rl}\mathrm{E}({\mathbf{z}}_{T+h}-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T))& =0\\ \mathrm{cov}({\mathbf{z}}_{T+h}-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T))& =A^{h-1}\mathrm{\Sigma }(A^{h-1}{)}^{\prime }+\cdots +A\mathrm{\Sigma }{A}^{\prime }+\mathrm{\Sigma }\end{array}\end{array}$$

Note: as in the univariate case, we can write VAR$(p)$ as VMA$(\mathrm{\infty })$. For VAR(1)

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_t& =A(L{)}^{-1}{\mathit{\epsilon }}_t\\ & ={\epsilon }_t+A{\epsilon }_{t-1}+A^2{\epsilon }_{t-2}+\cdots \end{array}\end{array}$$

and

$$\mathrm{cov}({\mathbf{z}}_t)=\mathrm{\Gamma }(0)=\mathrm{\Sigma }+A\mathrm{\Sigma }{A}^{\prime }+A^2\mathrm{\Sigma }(A^2{)}^{\prime }+\cdots$$

As $h⟶\mathrm{\infty }$

$$\begin{array}{r}\begin{array}{rl}\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T)& ⟶0(\text{unconditional mean of }{\mathbf{z}}_t)\\ \mathrm{cov}({\mathbf{z}}_{T+h}-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{z}}_T))& ⟶\mathrm{\Gamma }(0)(\text{unconditional covariance of }{\mathbf{z}}_t)\end{array}\end{array}$$

For VAR§ - use the VAR(1) representation

$${\mathbf{Z}}_t=\mathbf{\Phi }{\mathbf{Z}}_{t-1}+{\mathit{E}}_t$$

• forecast of ${\mathbf{z}}_{T+h}$ (using the selection matrix $\mathit{s}$)

$$\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{Z}}_T)=\mathit{s}\mathrm{E}({\mathbf{Z}}_{T+h}|{\mathbf{Z}}_T)$$

• variance of forecast errors

$$\mathrm{cov}({\mathbf{z}}_{T+h}-\mathrm{E}({\mathbf{z}}_{T+h}|{\mathbf{Z}}_T))=\mathit{s}\mathrm{cov}({\mathbf{Z}}_{T+h}-\mathrm{E}({\mathbf{Z}}_{T+h}|{\mathbf{Z}}_T)){\mathit{s}}^{\prime }$$

Impulse response functions

From the VMA representation of of VAR(1) model

$$\begin{array}{r}{\mathbf{z}}_{t+h}={\epsilon }_{t+h}+A{\epsilon }_{t+h-1}+A^2{\epsilon }_{t+h-2}+\cdots +A^h{\epsilon }_t+\cdots \end{array}$$

we have

$$\frac{\partial {\mathbf{z}}_{t+h}}{\partial {\epsilon }_t}=A^h$$

• Note that ${\epsilon }_t$ is a vector

• typically, we want to know the effect of a shock on a variable (for example monetary policy on inflation)

• here $\mathrm{cov}({\epsilon }_t)=\mathrm{\Sigma }$, i.e. ${\epsilon }_{it}$ and ${\epsilon }_{jt}$ are correlated

• ${\epsilon }_t$ are not shocks (statistical innovations, residuals, forecast errors)

(orthogonalized ) impulse response functions

Since $\mathrm{\Sigma }$ is positive definite matrix, there exists a matrix $B_0$ such that

$$B_0^{-1}(B_0^{-1}{)}^{\prime }=\mathrm{\Sigma }\Rightarrow B_0\mathrm{\Sigma }{B}_0^{\prime }=I$$

Then

$${\mathbf{u}}_t=B_0{\epsilon }_t\Rightarrow {\mathbf{u}}_t\sim \mathrm{WN}(0,I)$$

• $u_{it}$ and $u_{jt}$ are uncorrelated (orthogonal) for all $i\ne j$

Using ${\epsilon }_t=B_0^{-1}{\mathbf{u}}_t$ in the MA representation

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_{t+h}& ={\epsilon }_{t+h}+A{\epsilon }_{t+h-1}+A^2{\epsilon }_{t+h-2}+\cdots +A^h{\epsilon }_t+\cdots \\ \\ & =B_0^{-1}{\mathbf{u}}_{t+h}+A{B}_0^{-1}{\mathbf{u}}_{t+h-1}+A^2{B}_0^{-1}{\mathbf{u}}_{t+h-2}+\cdots +A^h{B}_0^{-1}{\mathbf{u}}_t+\cdots \end{array}\end{array}$$

and therefore

$$\frac{\partial {\mathbf{z}}_{t+h}}{\partial {\mathbf{u}}_t}=A^h{B}_0^{-1}\equiv {\mathrm{\Psi }}_h$$

Since $u_{it}$ and $u_{jt}$ are uncorrelated, the $k,l$ element of ${\mathrm{\Psi }}_h$ gives the response of $z_{k,t+h}$ to a (one standard deviation shock to $u_{l,t})$

$$\frac{\partial z_{k,t+h}}{\partial u_{lt}}={\psi }_{kl,h}$$

and the $l$ column of ${\mathrm{\Psi }}_h$ gives the response of ${\mathbf{z}}_{t+h}$ to a one standard deviation shock to $u_{l,t}$

$$\frac{\partial {\mathbf{z}}_{t+h}}{\partial u_{lt}}={\mathit{\psi }}_{l,h}$$

${\mathit{\psi }}_{l,h}$ is the $l$-th column of $A^h{B}_0^{-1}$

From VAR§ to Structural VAR§ (and vice versa)

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_t& ={\mathit{a}}_0+A_1{\mathbf{z}}_{t-1}+\cdots +A_p{\mathbf{z}}_{t-p}+{\mathit{\epsilon }}_t,{\mathit{\epsilon }}_t\sim \mathrm{WN}(0,\mathbf{\Sigma })\\ \\ {\mathbf{z}}_t& ={\mathit{a}}_0+A_1{\mathbf{z}}_{t-1}+\cdots +A_p{\mathbf{z}}_{t-p}+B_0^{-1}{\mathit{u}}_t(\text{using }{\mathbf{u}}_t=B_0{\epsilon }_t)\\ & \downarrow \\ B_0{\mathbf{z}}_t& =B_0{\mathit{a}}_0+B_0{A}_1{\mathbf{z}}_{t-1}+\cdots +B_0{A}_p{\mathbf{z}}_{t-p}+{\mathbf{u}}_t,(\text{pre-multiply by }{B}_0)\\ \\ B_0{\mathbf{z}}_t& ={\mathit{b}}_0+B_1{\mathbf{z}}_{t-1}+\cdots +B_p{\mathbf{z}}_{t-p}+{\mathbf{u}}_t,{\mathit{u}}_t\sim \mathrm{WN}(0,I)\end{array}\end{array}$$

• $B_0$ captures contemporaneous (time $t$) interactions among variables

• ${\mathbf{u}}_t$ are orthogonal shocks: $u_{i,t}$ only affects $z_{i,t}$ contemporaneously

• impulse responses

$$\frac{\partial {\mathbf{z}}_{t+h}}{\partial u_{it}}$$

Identification

• We can estimate the reduced-form coefficients $a_0$, $A_1$, … $A_p$ and $\mathrm{\Sigma }$

• and compute reduced-form MA representation of $\mathbf{z}$

• to compute impulse responses to structura shocks we need $B_0^{-1}$

• $B_0^{-1}$ is not identified from

$$B_0^{-1}(B_0^{-1}{)}^{\prime }=\mathrm{\Sigma }$$

by symmetry $\mathrm{\Sigma }$ has $n(n+1)/2$ unique elements $\Rightarrow n(n+1)/2$ equations, but $B_0$ has $n^2$ unknown elements

• need to impose restrictions on either $B_0$ or $B_0^{-1}$ in order to identify it

• the restrictions must be implied by economic theory.

Types of identifying restrictions

• short-run restrictions

• long-run restrictions

• sign restrictions

• combinations of short/long-run restrictions, sign restrictions

• …

Short-run restrictions:

• time $t$ impact of structural shocks ${\epsilon }_t=B_0^{-1}{\mathbf{u}}_t$

$$\begin{array}{r}\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\\ ⋮\\ {\epsilon }_{n,t}\end{array}\right]=\underset{B_0^{-1}}{\underbrace{\left[\begin{array}{cccc}{b}_0^{1,1}& b_0^{1,2}& \cdots & b_0^{1,n}\\ b_0^{2,1}& b_0^{2,2}& \cdots & b_0^{2,n}\\ ⋮& ⋮& ⋮\\ b_0^{n,1}& b_0^{n,2}& \cdots & b_0^{n,n}\end{array}\right]}}\left[\begin{array}{c}{u}_{1,t}\\ u_{2,t}\\ ⋮\\ u_{n,t}\end{array}\right]\end{array}$$

• time $t$ interactions among variables $B_0{\mathbf{z}}_t=\cdots +{\mathbf{u}}_t,$

$$\begin{array}{r}\underset{B_0}{\underbrace{\left[\begin{array}{cccc}{b}_{0,11}& b_{0,12}& \cdots & b_{0,1n}\\ b_{0,21}& b_{0,22}& \cdots & b_{0,2n}\\ ⋮& ⋮& ⋮\\ b_{0,n1}& b_{0,n2}& \cdots & b_{0,nn}\end{array}\right]}}\left[\begin{array}{c}{z}_{1,t}\\ z_{2,t}\\ ⋮\\ z_{n,t}\end{array}\right]=\cdots +\left[\begin{array}{c}{u}_{1,t}\\ u_{2,t}\\ ⋮\\ u_{n,t}\end{array}\right]\end{array}$$

Long-run restrictions:

MA representation of ${\mathbf{z}}_t$

$$\begin{array}{r}\begin{array}{rl}{\mathbf{z}}_{t+h}& ={\epsilon }_{t+h}+A{\epsilon }_{t+h-1}+A^2{\epsilon }_{t+h-2}+\cdots +A^h{\epsilon }_t+\cdots \\ \\ & =B_0^{-1}{\mathbf{u}}_{t+h}+A{B}_0^{-1}{\mathbf{u}}_{t+h-1}+A^2{B}_0^{-1}{\mathbf{u}}_{t+h-2}+\cdots +A^h{B}_0^{-1}{\mathbf{u}}_t+\cdots \end{array}\end{array}$$

The cumulative impuse responses of shocks in $t$ on ${\mathbf{z}}_t$, ${\mathbf{z}}_{t+1}$, … are given by

$$(I+A+A^2+A^3+\cdots )B^{-1}=A(1{)}^{-1}{B}_0^{-1}$$

• if ${\mathbf{z}}_t$ contains growth rates (e.g. of GDP), the cumulative response givs the permanent effect on the level

• common long-run restrictions: some shocks don’t have permanent effect on some variables (nominal shocks on real variabls)

  • some elements of $A(1{)}^{-1}{B}_0^{-1}$ are 0

Sign restrictions:

If $B_0$ is such that

$$B_0^{-1}(B_0^{-1}{)}^{\prime }=\mathrm{\Sigma }$$

then for any orthogonal matrix $Q$ ($Q{Q}^{\prime }=Q^{\prime }Q=I$)

we have

$$(B_0^{-1}Q)(B_0^{-1}Q{)}^{\prime }=\mathrm{\Sigma }$$

There are inifitely many such matrices.

• find the set of solutions that satisfy sign restrictions implied by theory (monetary policy shock raises $i_t$ and lowers ${\pi }_t$ and $y_t$)

• find all matrices $Q$ such that $B_0^{-1}=PQ$ meets those restrictions, where $P$ is the Cholesky factor of $\mathrm{\Sigma }$, and $Q$ is orthogonal matrix.

  • every real-valued symmetric positive-definite matrix has a unique Cholesky decomposition:

$$\mathrm{\Sigma }=P{P}^{\prime }$$

There are different ways to generate $Q$

For example, for $n=2$ candidates $Q$ can be generated using

$$\begin{array}{r}Q=\left[\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right]\end{array}$$

and $\theta \in (0,2\pi )$

Can also use the QR decomposition of a random matrix $H$ such that $H_{ij}\sim N(0,1)$

• with sign restrictions we get a set of impulse responses for each shock and variable (set vs point identfication)

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