This paper:
Contribute to better understand the dynamic interaction between U.S. EER and OP
How the U.S. EER responses to OP shock may change over time?
Study the relationship between OP and U.S. EER changes over time by analyzing the varying reaction of one variable to the shocks of the other.
3. Methodology: Shock episodes
Timing and duration of shocks
3. Methodology: Shock episodes
Oil price shocks
3. Methodology: Shock episodes
U.S. EER shocks
3. Methodology: Granger causality
p-values for the linear Granger-causality test
- No evidence of Granger causality \(O_t \rightarrow EER_t\)
- Exception: \(EER_t \rightarrow O_t\) when lag=1
3. Methodology: Granger causality
p-values for non-linear G-causality test (Diks and Panchenko, 2006)
- Evidence of causality from \(EER_t \rightarrow O_t\)
3. Methodology: standard VAR model
\(y_t=\)\(a\)\(+\displaystyle\sum_{j=1}^2\) \(A_j\)\(y_{t-j}+u_t\)
\(y_t\) (\(2\times1\)) observed variables \((EER_t, O_t)^{\prime}\)
\(a\) (2x1) vector of parameters
Optimal lags = 2, based on SIC
Consistent with other studies (see Fratzscher et al., 2014; Ferraro et al., 2015)
\(A_1 ,A_2\) are (\(2 \times 2\)) matrices of parameters
\(u_t\) (\(2\times1\)) \(\quad\) ; \(\quad u_t \sim \mathcal{N}(0,\) \(\Sigma_u\))
3. Methodology: TVP-VAR model
Time varying parameter model (TVP) VAR model (Primiceri, 2005)
\(y_t =\)\(a_t\) \(+\displaystyle\sum_{j=1}^2\) \(A_{j,t}\) \(\ y_{t-j} + u_t\)
\(y_t\) (\(2 \times 1\)) observed variables \((EER_t, O_t)^{\prime}\)
\(a_t\) (\(2 \times 1\)) vector of TV-parameters
\(A_{1,t}, A_{2,t}\) (\(2 \times 2\)) matrices of TV-parameters
\(u_t\) (\(2 \times 1\)) \(\quad\quad\) \(u_t \sim \mathcal{N}(0,\) \(\Omega_t\))
3. Methodology: TVP-VAR model
\(y_t =\)\(a_t\) \(+\displaystyle\sum_{j=1}^2\) \(A_{j,t}\) \(\ y_{t-j} + u_t\)
\(y_t =\) \(a_t\) + \(A_{1,t}\) \(y_{t-1}\) + \(A_{2,t}\) \(y_{t-2}\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)
\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)
- \(I_2\) identity matrix (\(2 \times 2\))
- \(X_t=[1,\ y_{t-1}^{\prime},\ y_{t-2}^{\prime}]\) stacked vector of variables (\(1 \times 5\))
- \(\alpha_t=(a_t,\ A_{1,t}, \ A_{2,t})\) stacked vector of TV parameters (\(10 \times 1\))
- \(B_t\) TV lower triangular matrix (\(2 \times 2\))
- \(\Sigma_t\) TV standard deviations of error term (\(2 \times 2\))
3. Methodology: TVP-VAR model
\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)
Dynamics of the TV parameters
- \(\alpha_t\) \(= \alpha_{t-1} + \eta_t^\alpha\) \(\quad\quad\quad\quad \ \eta_t^\alpha \sim \mathcal{N}(0,\) \(Q\)\()\quad\ Past\) values of \(y_t\)
- \(b_t\) \(= b_{t-1} + \eta_t^b\)\(\quad\quad\quad\quad \ \ \eta_t^b \sim \mathcal{N}(0,\) \(S\)\() \ \ \quad\)Unrestricted \(B_t\)
- \(log\) \(\sigma_t\) \(= log\, \sigma_{t-1} + \eta_t^\sigma \quad \ \ \ \eta_t^\sigma \sim \mathcal{N}(0,\) \(W\)\()\quad\) \(\ \sigma_t\) \(= diag(\) \(\Sigma_t\)\()\)
- \(\eta_t^\alpha, \eta_t^b, \eta_t^\sigma\) are independent and white noise processes
3. Methodology: TVP-VAR model
Block diagonal matrix \(\rightarrow V\)
\[\begin{equation}
V = Var \begin{pmatrix}\begin{bmatrix}
\varepsilon_t\\
\eta_t^\alpha\\
\eta_t^b\\
\eta_t^\sigma
\end{bmatrix}\end{pmatrix} \quad =
\begin{bmatrix} I_2 & 0 & 0 & 0\\
0 & Q & 0 & 0\\
0 & 0 & S & 0\\
0 & 0 & 0 & W \end{bmatrix}
\end{equation}\]
\(y_t\ (2 \times 1)\ \ \quad\rightarrow\quad I_2\)
\(\alpha_t\) \((10 \times 1) \quad\rightarrow\quad\) \(Q\) \((10 \times 10) \quad\rightarrow\quad\) 55 free parameters
\(B_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(S\) \((1 \times 1)\)
\(\Sigma_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(W\) \((2 \times 2) \quad\rightarrow\quad\) 3 free parameters
3. Methodology: TVP-VAR model
The priors follow the same principles as in Primiceri (2005) and are summarized in the following Table:
3. Methodology: TVP-VAR model
Solution:
Simulation algorithm for the joint posterior of \(\alpha^T, B^T, \Sigma^T, Q, S, W\)
Bayesian inference: Markov chain Monte Carlo (MCMC) algorithm is based on a Gibbs sampler
Training sample used for determining prior parameters (least squares): 60
Simulations: 50000 iterations
Burn-in steps to initialize the sample: 5000
4. Results: VAR (time-invariant) model
Response of one variable to one unit shock of the other variable in the (time invariant) VAR model for the whole period
4. Results: TVP-VAR model
Standard deviations
4. Results: TVP-VAR model
Impulses \(O_t \rightarrow\) Responses \(EER_t\)TV responses to one standard deviation shock
4. Results: TVP-VAR model
Impulses \(EER_t \rightarrow\) Responses \(O_t\) TV responses to one standard deviation shock
4. Results: TVP-VAR model
Responses to one unit shock after 3, 6, 12, and 24 months
4. Results: TVP-VAR model
Responses of \(EER_t\) to one unit \(O_t\) shock
- \(O_t \rightarrow EER_t\): Wealth channel (-), trade balance (-), petrodollars recycling (+)
4. Results: TVP-VAR model
Responses of \(O_t\) to one unit \(EER_t\) shock
- \(EER_t \rightarrow O_t\): Oil market (-), Financialization (-)
4. Results: TVP-VAR model
Responses of \(U.S. EER_t\) to one unit \(O_t\) shock
4. Results: TVP-VAR model
Responses of \(O_t\) to one unit \(EER_t\) shock
Triangular reduction
To facilitate structural analysis, \(\Omega_t\) is parameterized as:
\(B_t\) \(\Omega_t\) \(B^{\prime}_t = \Sigma_t\) \(\Sigma^{\prime}_t\)
- \(B_t = \begin{bmatrix} 1 & 0\\ b_{21,t} & 1\end{bmatrix} \quad\quad\quad \Sigma_t = \begin{bmatrix} \sigma_{1,t} & 0\\ 0 & \sigma_{2,t}\end{bmatrix}\)
- \(u_t = B_t^{-1}\ \Sigma_t\ \varepsilon_t \quad\quad V(\varepsilon)=I_2\)
Triangular reduction
To facilitate structural analysis, the error covariance matrix \(\Omega_t\) is parameterized as:
\(B_t \Omega_t B_t^{\prime} = \Sigma_t \Sigma_t^{\prime}\)
If \(\ u_t = B_t^{-1} \Sigma_t \varepsilon_t\)
\(\Omega_t \ = E[u_t u_t^{\prime}] \ = E[B_t^{-1} \Sigma_t \varepsilon_t \ (B_t^{-1} \Sigma_t \varepsilon_t)^{\prime}] = E[B_t^{-1} \Sigma_t \ \varepsilon_t \varepsilon_t^{\prime} \ \Sigma_t^\prime B_t^{-1 \prime}]\)
If \(\ V(\varepsilon_t) = E[\varepsilon_t \varepsilon_t^\prime] = I_2\)
\(\Omega_t = B_t^{-1} \Sigma_t \ I_2 \ \Sigma_t^{\prime} {B_t^{\prime}}^{-1}\)
- \(B_t \Omega_t B_t^{\prime} = \Sigma_t \Sigma_t^{\prime}\)
Priors
Choice of \(k_Q\) and \(k_S\) in the prior inverse-Wishart (\(\mathcal{IW}\)) distributions for the hyperparameters
\(X \sim \mathcal{IW}(\bar{X}, \nu)\)
\(V(X)=\bar{X}/\nu\)
- \(\bar{Q}=k_Q^2 \times \nu \times \hat{V}(\hat{\alpha}_{OLS}) \quad\ \ \nu=pQ\)
- \(\bar{S}_1=k_S^2 \times \nu \times \hat{V}(\hat{B}_{OLS}) \quad \nu=pS_1\)
- Lower \(k_Q\) reduces the degree of time variation of shocks to \(past\) values of \(y_t \ (\downarrow k_Q \rightarrow \ \downarrow Q \rightarrow \ \downarrow\alpha_t)\)
- Larger \(k_S\) increases the degree of time variation of shocks to \(current\) values of \(y_t \ (\uparrow k_S \rightarrow \ \uparrow S_1 \rightarrow \ \uparrow B_t)\)
MCMC Algorithm
\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)
\(\begin{array}{c|c} \ \alpha^T & \ B^T & & \\ \downarrow & \downarrow & \\ Q & S & W \end{array}\) \(\begin{matrix} \quad \nearrow \ \ s^T \searrow \\ =\theta \quad \rightarrow \quad\Sigma^T \\ \quad\nwarrow \quad \hookleftarrow \end{matrix}\)
Initialize \(B^T,\ \Sigma^T,\ s^T, \ Q,\ S,\ W\)
\(p(\alpha^T/\ \theta^{-\alpha T}, \Sigma^T)\) \(\rightarrow \alpha^T\) \(\ \Rightarrow p(Q\ /\ \alpha^T)\) \(\rightarrow Q\)
\(p(B^T/\ \theta^{-S}, \Sigma^T) \ \Rightarrow\) \(B^T \leftarrow\)\(\ S \leftarrow\) \(p(S\ /\ \theta^{-S}, \Sigma^T)\)
\(s^T \leftarrow\) \(p(s^T/\ \Sigma^T, \theta) \ \Rightarrow\) \(\ \Sigma^T \leftarrow\) \(p(\Sigma^T/\ \theta, s^T)\ \Rightarrow\) \(\ W \leftarrow\) \(p(W\ /\ \Sigma^T)\)