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+

+
+

+
    +
  • monitorowanie postÄ™pu
  • +
+

+
+ +
+

strukturalne modele VAR

+
+
+

strukturalne modele VAR

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    +
  • podstawowe dla modelowania efektów polityki ekonomicznej
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  • analiza dynamicznych efektów przyczynowych dobrze izolowanej przyczyny
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  • stosunkowo proste w pracy z danymi i dostarczajÄ… empirycznych dowodów na propagacjÄ™ szoków przez gospodarki i rynki
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  • dostarczajÄ… empirycznych faktów do uwzglÄ™dnienia w modelach teoretyczne
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  • szeroko stosowane w: polityce pieniężnej i fiskalnej, rynku finansowym, …
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  • rozszerzalne: wiele wariantów specyfikacji +
      +
    • nieliniowość
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    • heteroskedastyczność
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    • zmienne parametry w czasie
    • +
    • modelowanie hierarchiczne bayesowskie
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  • +
  • zaproponowane przez Sims (1980)
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-
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Structural Vector Autoregressions

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+

strukturalne modele VAR

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The model.

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model.

\[\begin{align} -\text{VAR equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \epsilon_t\\[1ex] -\text{structural equation: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex] -\text{structural shocks: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\mathbf{I}_N\right) +\text{równanie VAR: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \mathbf{A}_d x_{t} + \epsilon_t\\[1ex] +\text{równanie structuralne: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex] +\text{structuralne szoki: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\text{diag}\left(\boldsymbol\sigma_t^2\right)\right) \end{align}\]

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Notation.

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notacja.

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    +
  • \(y_t\) - wektor \(N\) zmiennych na okres \(t\)
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  • \(\mathbf{A}_i\) i \(\mathbf{B}\) - \(N\times N\) macierze parametrów autoregresyjnych i strukturalnych
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  • \(\epsilon_t\) i \(u_t\) - wektory \(N\) bÅ‚Ä™dów statystycznych i szoków strukturalnych
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  • \(\boldsymbol\sigma_t^2\) - wektor \(N\) wariancji szoków strukturalnych
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SVAR: hierarchiczne rozkłady a priori

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+
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    +
  • normalny-uogólniony normalny rozkÅ‚ad a priori dla \(\mathbf{A}\) i \(\mathbf{B}\)
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  • wielopoziomowa estymacja wariancji a priori
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  • rozkÅ‚ad a priori z Minnesoty dla niestacjonarnych szeregów czasowych
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  • bardziej precyzyjne estymacja i prognozowanie
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+
+ +
+

+
    +
  • rozkÅ‚ad a priori normalny i odwrócony Wisharta dla \(\mathbf{A}\) i \(\mathbf{\Sigma} = (\mathbf{B}'\mathbf{B})^{-1}\)
  • +
  • estymacja wariancji a priori
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  • rozkÅ‚ad a priori z Minnesoty dla niestacjonarnych szeregów czasowych
  • +
  • bardziej precyzyjne estymacja i prognozowanie
  • +
+
+
+
+
+

SVAR: modelowanie zmienności

+ +
+
+

+
    +
  • homoskedastyczność \(\boldsymbol\sigma_{n.t}^2 = 1\)

  • +
  • zmienność stochastyczna

  • +
  • stacjonarny proces Markova dla zmiennoÅ›ci

  • +
  • nieparametryczny proces Markova dla zmiennoÅ›ci

  • +
  • rozkÅ‚ady szoków

    +
      +
    • normalny
    • +
    • skoÅ„czona mieszanka rozkÅ‚adów normalnych
    • +
    • nieparametryczna mieszanka rozkÅ‚adów normalnych
    • +
    • rozkÅ‚ad t-Studenta
    • +
  • +
+
+ +
+

+
    +
  • homoskedastyczność
  • +
  • normalny rozkÅ‚ad szoków
  • +
+
+
+
+
+

SVAR: identyfikacja

+ +
+
+

+
    +
  • restrykcje zerowe
  • +
  • heteroskedastyczność
  • +
  • nienormalne rozkÅ‚ady szoków
  • +
+
+ +
+

+
    +
  • restrykcje znaków
  • +
  • restrykcje zerowe
  • +
  • restrykcje narracyjne
  • +
+
+
+
+
+

strukturalne modele VAR

+ +

błędy statystyczne.

+

\[\begin{align} +&&&\\ +\text{równanie strukturalne: }&& \epsilon_t &= \mathbf{B}^{-1}u_t = \mathbf{\Theta}_0 u_t\\[1ex] +\text{błędy statystyczne: }&& \epsilon_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\Sigma\right)\\[1ex] +\text{kowariancja: }&& \mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime} = \Theta_0\Theta_0' +\end{align}\]

+

Notacja.

+
    +
  • \(\mathbf\Sigma\) - \(N\times N\) kowariancja bÅ‚Ä™dów statystycznych
  • +
  • \(\Theta_0 = \mathbf{B}^{-1}\) - \(N\times N\) macierz efektów strukturalnych
  • +
+
+
+

strukturalne modele VAR

+ +

Wstaw równanie VAR w równanie strukturalne:

+

\[\begin{align} +\mathbf{B}y_t &= \mathbf{B}\mathbf{A}_1 y_{t-1} + \dots + \mathbf{B}\mathbf{A}_p y_{t-p} + \mathbf{B}\boldsymbol\mu_0 + u_t\\[1ex] +&\\ +\end{align}\]

+

relacje strukturanlne.

+

Niech \(N=2\)

+

\[\begin{align} +\mathbf{B}y_t &= \begin{bmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\end{bmatrix}\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix} +\end{align}\]

+
+
+

strukturalne modele VAR

+ +

Wstaw równanie strukturalne dla \(\epsilon_t\) w równanie VAR:

+

\[\begin{align} +y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{B}^{-1}u_t\\[1ex] +y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{\Theta}_0 u_t +\end{align}\]

+

efekty strukturane.

+

Niech \(N=2\)

+

\[\begin{align} +\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix} &= \dots + +\begin{bmatrix}\Theta_{11}&\Theta_{12}\\\Theta_{21}&\Theta_{22}\end{bmatrix}\begin{bmatrix}u_{1t}\\ u_{2t}\end{bmatrix} +\end{align}\]

+
+
+

identyfikacja modeli strukturalnych

+
+
+

identyfikacja modeli strukturalnych

+ +

kowariancja i relacje strukturalne.

+

\[\begin{align} +&\\ +\mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime}\\[1ex] +\end{align}\]

+
    +
  • \(\mathbf\Sigma\) może być estymowana z danych
  • +
  • ukÅ‚ad równaÅ„ strukturalnych do rozwiÄ…zanie dla \(\mathbf{B}\)
  • +
  • \(\mathbf\Sigma\) jest macierzÄ… symetrycznÄ… \(N\times N\)
  • +
  • \(\mathbf\Sigma\) ma \(N(N+1)/2\) unikalnych elementów, tj. równaÅ„
  • +
  • \(\mathbf{B}\) jest \(N\times N\) macierzÄ… z \(N^2\) elementami do estymacji
  • +
  • niewystarczajÄ…ca liczba równaÅ„ do estymacji \(\mathbf{B}\)
  • +
  • \(\mathbf{B}\) nie jest indentyfikowalna
  • +
+
+
+

identyfikacja modeli strukturalnych

+ +

restrykcje zerowe.

+

\[\begin{align} +&\\ +\mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime}\\[1ex] +\end{align}\]

+

identyfikacja.

+
    +
  • jedynie \(N(N+1)/2\) elementów w \(\mathbf{B}\) może być wyestymowanych
  • +
  • naÅ‚ożenie \(N(N-1)/2\) restrykcji na \(\mathbf{B}\) uÅ‚atwia rozwiÄ…zanie
  • +
  • wiersze w \(\mathbf{B}\) (i kolumny w \(\mathbf\Theta_0\)) identyfikowane co do znaku
  • +
  • zmiana znaków wierszy w \(\mathbf{B}\) nie zmienia wartoÅ›ci \(\mathbf\Sigma\)
  • +
  • czÄ™sto zakÅ‚adamy trójkÄ…tnÄ… macierz \(\mathbf{B}\)
  • +
+
+
+

identyfikacja modeli strukturalnych

+ +

restrykcje zerowe.

+

Niech \(N=2\)

+

\[\begin{align} +\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad +\begin{bmatrix}B_{11}&B_{12}\\ B_{21}&B_{22}\end{bmatrix}\\[1ex] +\end{align}\]

+
    +
  • 3 unikalne elementy w \(\mathbf\Sigma\) - 3 równania
  • +
  • 4 elemnty w \(\mathbf{B}\) do rozwiÄ…zania
  • +
+

identyfikacja.

+

\[\begin{align} +\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad +\begin{bmatrix}B_{11}& 0\\ B_{21}&B_{22}\end{bmatrix}\\[1ex] +\end{align}\]

+
    +
  • 3 równania pozwalaja rozwiÄ…zać 3 niewiadome w \(\mathbf{B}\)
  • +
+
+
+

identyfikacja modeli strukturalnych

+ +

identyfikacja przez heteroskedastyczność.

+

Rozważ:

+
    +
  • dwie kowariancje, \(\mathbf\Sigma_1\) and \(\mathbf\Sigma_2\),
  • +
  • macież \(\mathbf{B}_0\) niezmienna w czasie
  • +
  • kowariancje heteroskedastycznych szoków strukturalnych \(\text{diag}\left(\boldsymbol\sigma_1^2\right)\) i \(\text{diag}\left(\boldsymbol\sigma_2^2\right)\)
  • +
+

\[\begin{align} +\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex] +\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime} +\end{align}\]

+
+
+

identyfikacja modeli strukturalnych

+ +

identyfikacja przez heteroskedastyczność.

+

\[\begin{align} +\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex] +\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime} +\end{align}\]

+

identyfikacja.

    -
  • \(\mathbf{B}\) - \(N\times N\) structural matrix of contemporaneous relationships

  • -
  • \(u_t\) - \(N\)-vector of structural shocks at time \(t\)

    -

    Isolating these shocks allows us to identify dynamic effects of uncorrelated shocks on variables \(y_t\)

  • -
  • \(\epsilon_t\) - \(N\)-vector with error terms at time \(t\)

  • -
  • the rest as in the lecture on Bayesian VARs

  • +
  • \(\mathbf\Sigma_1\) i \(\mathbf\Sigma_2\) majÄ… \(N^2+N\) unikalnych elementów
  • +
  • wszystkie \(N^2\) elementy w \(\mathbf{B}_0\) mogÄ… być wyestymowane
  • +
  • oba wektory \(\boldsymbol\sigma_1^2\) i \(\boldsymbol\sigma_2^2\) mogÄ… być wyestymowane dziÄ™ki zaÅ‚ożeniu: \(E\left[\text{diag}\left(\boldsymbol\sigma_i^2\right)\right] = \mathbf{I}_N\)
+
+
+

identyfikacja modeli strukturalnych

+ +

Rozważ uogólnienie

+

\[\begin{align} +u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N, \text{diag}\left(\boldsymbol\sigma_t^2\right)\right)\\[1ex] +\mathbf\Sigma_t &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_t^2\right)\mathbf{B}_0^{-1\prime} +\end{align}\]

+

identyfikacja.

+
    +
  • identyfikacja macierzy \(\mathbf{B}_0\) co do znaków i kolejnoÅ›ci wierszy
  • +
  • szoki sÄ… identyfikowalne jeÅ›li wariancje warunkowe nie sÄ… proporcjonalne
  • +
  • wariancje warunkowe \(\boldsymbol\sigma_t^2\) mogÄ… być wyestymowane
  • +
+

modelowanie zmienności.

+

Wybierz model dla \(\boldsymbol\sigma_t^2\) o najlepszych właściwościach.

-
+
+

identyfikacja modeli strukturalnych

+ +
+
+

identyfikacja modeli strukturalnych

+ +

równanie strukturalne.

+

\[\begin{align} +\text{relacje strukturalne:}&&\mathbf{Q}\mathbf{B}\epsilon_t &= \mathbf{Q}u_t\\[1ex] +\text{efekty strukturalne:}&&\epsilon_t &= \mathbf{\Theta}_0\mathbf{Q}'\mathbf{Q} u_t\\[1ex] +\end{align}\]

+

identyfikacja co do macierzy obrotu.

+

\[\begin{align} +\mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{Q}'\mathbf{Q}\mathbf{B}^{-1\prime} = \mathbf{\Theta}_0\mathbf{Q}'\mathbf{Q}\mathbf{\Theta}_0^{\prime}\\[1ex] +\mathbf{Q}'\mathbf{Q} &= \mathbf{I}_N\\[1ex] +\end{align}\]

+
    +
  • funkcja wiarygodnoÅ›ci nie zależy od \(\mathbf{Q}\)
  • +
  • identyfikacja modely zawęża typ \(\mathbf{Q}\)
  • +
  • restrykcje zerowe zmieniajÄ… typ macierzy \(\mathbf{Q}\) do diagonalnej z elementami \(\pm1\)
  • +
+
+
+

identyfikacja modeli strukturalnych

+ +

restrykcje znaku.

+

\[\begin{align} +\text{relacje strukturalne:}&&\tilde{\mathbf{B}}\epsilon_t &= \tilde{u}_t\\[1ex] +\text{efekty strukturalne:}&&\epsilon_t &= \tilde{\mathbf{\Theta}}_0\tilde{u}_t\\[1ex] +\end{align}\]

+
    +
  • restrykcje na znak elementów w \(\tilde{\mathbf{B}}\) i/lub \(\tilde{\mathbf{\Theta}}_0\)
  • +
  • zawęża zbiór identyfikowalny: model jest identyfikowalny co do macierzy obrotu \(\mathbf{Q}\) koherentnej z restrykcjami
  • +
  • estymacja ma za cel przybliżenie zbioru identyfikowalnego
  • +
+

restrykcje znaku i zerowe.

+
    +
  • restrykcje zerowe i na znak elementów w \(\tilde{\mathbf{B}}\) i/lub \(\tilde{\mathbf{\Theta}}_0\)
  • +
  • zawęża zbiór identyfikowalny
  • +
+
+
+

identyfikacja modeli strukturalnych

+ +

restrykcje narracyjne.

+
    +
  • restrykcje na znak elementów lub wielkoÅ›ci \(u_t\) lub \(\tilde{u}_t\)
  • +
  • restrykcje w oparciu na narracjÄ™, teoriÄ™ ekonomii i konsensus naukowy
  • +
  • należy dostosować metody estymacji
  • +
  • zawęża zbiór identyfikowalny
  • +
+

nowa cecha.

+
    +
  • restrykcje zerowe, narracyjne i na znak w jednym modelu w paczce bsvarSIGNs
  • +
+
+
+

modelowanie rozkładu i zmienności

+
+
+

niescentrowana zmienność stochastyczna

+ +

+

\[\begin{align} +&\\ +\text{wariancja warunkowa:}&&\sigma_{n.t}^2 &= \exp\left\{\omega_n h_{n.t}\right\}\\ +\text{w skali log:}&&h_{n.t} &= \rho_n h_{n.t-1} + v_{n.t}\\ +\text{innowacje zmienności:}&&v_{n.t}&\sim N\left(0,1\right)\\ +\end{align}\]

+
    +
  • Å›wietna zdolność do prognozowania
  • +
  • normalizajca \(\sigma_{n.t}^2 = 1\)
  • +
  • verify_identification() przez ocene restrykcji \(H_0:\omega_n = 0\)
  • +
+
+
+

scentrowana zmienność stochastyczna

+ +

+

\[\begin{align} +&\\ +\text{wariancja warunkowa:}&&\sigma_{n.t}^2 &= \exp\left\{ \tilde{h}_{n.t}\right\}\\ +\text{w skali log:}&&\tilde{h}_{n.t} &= \rho_n \tilde{h}_{n.t-1} + \tilde{v}_{n.t}\\ +\text{innowacje zmienności:}&&\tilde{v}_{n.t}&\sim N\left(0,\omega_n^2\right)\\ +\end{align}\]

+
    +
  • Å›wietna zdolność do prognozowania
  • +
+
+
+

zmienność stochastyczna: rozkłady a priori

+ +
+
+

proces Markowa dla zmienności.

+ +

+

\[\begin{align} +&\\ +\text{szoki strukturalne:}&&\mathbf{u}_t\mid s_t \sim N\left( \mathbf{0}_N, \text{diag}\left(\boldsymbol{\sigma}_{s_t}^2\right) \right)\\ +\text{a priori:}&& M^{-1}\left(\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2\right) \sim Dirichlet(\underline{a}\boldsymbol\imath')\\ +\text{proces Markowa:}&& s_t\sim \text{Markov}(\mathbf{P},\boldsymbol\pi_0) +\end{align}\]

+
    +
  • modelowanie proces Markowa dla zmiennoÅ›ci
  • +
  • zapewnia identyfikacjÄ™
  • +
  • poprawa zdolnoÅ›ci do prognozowania
  • +
  • verify_identification() przez ocenÄ™ restrykcji \(H_0:\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2 = 1\)
  • +
+
+
+

mieszanka rozkładów normalnych.

+ +

+

\[\begin{align} +&\\ +\text{szoki strukturalne:}&&\mathbf{u}_t\mid s_t \sim N\left( \mathbf{0}_N, \text{diag}\left(\boldsymbol{\sigma}_{s_t}^2\right) \right)\\ +\text{a priori:}&& M^{-1}\left(\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2\right) \sim Dirichlet(\underline{a}\boldsymbol\imath')\\ +\text{prawdopodobieństwo:}&& Pr[s_t]=\boldsymbol\pi_0 +\end{align}\]

+
    +
  • modelowanie mieszanki rozkÅ‚adów normalnych
  • +
  • zapewnia identyfikacjÄ™
  • +
  • verify_identification() przez ocenÄ™ restrykcji \(H_0:\boldsymbol{\sigma}_{1}^2, \dots, \boldsymbol{\sigma}_{M}^2 = 1\)
  • +
+
+
+

rozkład t-studenta.

+ +

+

\[\begin{align} +&&&\\ +\text{szoki strukturalne:}&&\mathbf{u}_t\mid\mathbf{x}_t &\sim t\left( \mathbf{0}_N, \mathbf{I}_N, \nu \right) +\end{align}\]

+
    +
  • \(\nu\) - szacowane z danych stopnie swobody
  • +
  • grube ogony zapewniajÄ… identyfikacjÄ™
  • +
  • poprawa zdolnoÅ›ci do prognozowania
  • +
  • verify_identification() przez ocenÄ™ restrykcji \(H_0:\nu \rightarrow\infty\)
  • +
+
+
+

+

+
+
+

Australian Monetary Policy Analysis

+

\[ \]

+

Download reproduction script

+
+
+

Australian Monetary Policy Analysis

+ +

System of four variables.

+

Based on Turnip (2017)

+

\[\begin{align} +y_t = \begin{bmatrix} \Delta rgdp_t & \pi_t & cr_t & \Delta rtwi_t \end{bmatrix}' +\end{align}\]

+

A lower-triangular identification pattern.

+

\[\begin{align} +\begin{bmatrix} +B_{11}&0&0&0\\ +B_{21}&B_{22}&0&0\\ +B_{31}&B_{32}&B_{33}&0\\ +B_{41}&B_{42}&B_{43}&B_{44} +\end{bmatrix} +\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} \end{align}\]

+
    +
  • In the extended model, the monetary policy shock is not identified
  • +
  • Use identification via heteroskedasticity to identify it
  • +
+
+
+

Four-Variable Monetary System

+
+
# Gross domestic product (GDP); Chain volume
+rgdp_dwnld      = readrba::read_rba(series_id = "GGDPCVGDP")
+rgdp_tmp        = xts::xts(rgdp_dwnld$value, rgdp_dwnld$date, tclass = 'yearqtr')
+drgdp           = na.omit(400 * diff(log(rgdp_tmp)))
+drgdp           = xts::to.quarterly(drgdp, OHLC = FALSE)
+
+# Consumer price index; All groups; Quarterly change (in per cent)
+picpi_dwnld     = readrba::read_rba(series_id = "GCPIAGSAQP")
+pi              = 4 * xts::xts(picpi_dwnld$value, picpi_dwnld$date, tclass = 'yearqtr')
+pi              = xts::to.quarterly(pi, OHLC = FALSE)
+
+# Interbank Overnight Cash Rate
+cr_dwnld        = readrba::read_rba(series_id = "FIRMMCRID")   # Cash Rate Target
+cr_tmp          = xts::xts(cr_dwnld$value, cr_dwnld$date)
+cr              = xts::to.quarterly(cr_tmp, OHLC = FALSE)
+
+# Real Trade-Weighted Index
+rtwi_dwnld      = readrba::read_rba(series_id = "FRERTWI")
+rtwi_tmp        = xts::xts(rtwi_dwnld$value, rtwi_dwnld$date, tclass = 'yearqtr')
+rtwi            = 100 * na.omit(diff(log(rtwi_tmp)))
+drtwi            = xts::to.quarterly(rtwi, OHLC = FALSE)
+
+y               = na.omit(merge(drgdp, pi, cr, drtwi))
+plot(y, main = "Australian monetary system",
+     legend.loc = "bottomleft", col = c("#FF00FF","#990099","#ededed","#330033"))
+
+ + +
+

Four-Variable Monetary System

+
+
+
+

+
+
+
+
+

Model Estimation

+ +

Lower-triangular model with zero prior mean for \(\mathbf{A}\).

+
+
# estimation - lower-triangular model
+############################################################
+library(bsvars)
+set.seed(123)
+
+spec = specify_bsvar$new(
+  as.matrix(y), 
+  p = 4, 
+  stationary = rep(TRUE, 4)
+)
+spec |>
+  estimate(S = 1000) |>
+  estimate(S = 5000) -> post
+
+
**************************************************|
+bsvars: Bayesian Structural Vector Autoregressions|
+**************************************************|
+ Gibbs sampler for the SVAR model                 |
+**************************************************|
+ Progress of the MCMC simulation for 1000 draws
+    Every draw is saved via MCMC thinning
+ Press Esc to interrupt the computations
+**************************************************|
+**************************************************|
+bsvars: Bayesian Structural Vector Autoregressions|
+**************************************************|
+ Gibbs sampler for the SVAR model                 |
+**************************************************|
+ Progress of the MCMC simulation for 5000 draws
+    Every draw is saved via MCMC thinning
+ Press Esc to interrupt the computations
+**************************************************|
+
+
+
+
+

Compute impulse responses

+ +
+
post |> 
+  compute_impulse_responses(horizon = 20) |> 
+  plot()
+ +
+
+
+

Compute forecast error variance decompositions

+ +
+
post |> 
+  compute_variance_decompositions(horizon = 20) |> 
+  plot()
+ +
+
+
+

Compute structural shocks

+ +
+
post |> 
+  compute_structural_shocks() |> 
+  plot()
+ +
+
+
+

Compute fitted values

+ +
+
post |> 
+  compute_fitted_values() |> 
+  plot()
+ +
+
+
+

Compute forecasts

+ +
+
post |> 
+  forecast(horizon = 8) |> 
+  plot(data_in_plot = 0.3)
+ +
+
+

-

+

@@ -630,6 +1509,44 @@

] }); + + + + + +