Time-varying relationship
between oil price and exchange rate

César Castro\(^*\)

*Department de Economics, Public University of Navarre

Rebeca Jiménez Rodríguez\(^*\)

*Department de Economics, University of Salamanca

November, 2018




Outline

Outline

  1. Motivation:
    Facts and literature (Theoretical, Empirical)
  2. Data
  3. Methodology
    • Shock episodes
    • Causality
    • VAR model
    • TVP-VAR model
  4. Results
    • VAR model
    • TVP-VAR model
  5. Concluding Remarks




1. Motivation

1. Motivation: Facts

Nominal oil price and US effective exchange rate

1. Motivation: Facts

Five-year rolling correlation between nominal oil price and US effective exchange rate

1. Motivation: Theoretical


Source of the shock: exchange rate (ER), oil price (OP)

  1. \(\uparrow\) OP shock \(\rightarrow\) ER:
    • Wealth channel (-): \(\quad\) \(\downarrow\) ER (Oil importing countries)
    • Deterioration of trade balance (-): \(\quad\) \(\downarrow\) ER (Oil importing countries)
    • Petrodollars recycling (+): \(\quad\uparrow D_{USD_{assets}} \rightarrow \quad\) \(\uparrow\) ER
  2. \(\uparrow\) ER shock \(\rightarrow\) OP:
    • Oil market (-): \(\quad \uparrow S_{oil} + \downarrow D_{oil} \rightarrow\quad\) \(\downarrow\) OP
    • Financialization (-): \(\quad\downarrow D_{oil}\rightarrow\quad\) \(\downarrow\) OP
  3. US monetary policy shock (-):
    \(\quad\uparrow r \rightarrow\quad\) \(\uparrow\) ER \(\rightarrow\quad\downarrow D_{oil} \rightarrow\quad\) \(\downarrow\) OP

1. Motivation: Literature


Source of the shock: exchange rate (ER), oil price (OP)

  • ER \(\rightarrow\) OP:
    Trehan (1986), Youse… and Wirjanto (2004), Breitenfellner and Cuaresma (2008), Zhang et al. (2008), Akram (2009), Chen et al. (2010), Beckmann and Czudaj (2013) and Coudert and Mignon (2016)
  • OP \(\rightarrow\) ER:
    Amano and Van Norden (1998), Chen and Chen (2007), Li zardo and Mollick (2010), Ferraro et al. (2015) and Habib et al. (2016)
  • ER \(\leftrightarrow\) OP:
    Wang and Wu (2012); Fratzscher et al., (2014)

1. Motivation: Empirical


The differences in the results in the literature:

  1. Data frequency
  2. Oil-dependence of the country
  3. Period of analysis
  4. Existence of structural breaks:
    • Chen and Chen (2007) do not find evidence of structural breaks for (G-7 countries, 1972-2005).
    • Fratzscher et al. (2014) show evidence of structural breaks in the early 2000s (euro area, 1995-2005)

1. Motivation: Empirical


What is the real relationship between OP and ER?

  1. ER: trade balance \(\rightarrow\quad\) macroeconomic variables
  2. USD ER: standard currency of international trade
  3. OP: basic input to production, indicator of economic activity worldwide




2. Data

2. Data


  • Monthly data
  • Common sample period: January 1974-December 2017
  • Total number of observations: 528
  • \(O_t\)
    Nominal oil price (WTI in USD per Barrel)
    Source: U.S. Energy Information Administration (EIA)
  • \(ER_t\)
    US nominal narrow index of effective exchange rate
    Source: Bank for International Settlements (BIS)




3. Methodology

3. Methodology: Shock episodes

Timing and duration of shocks

some text

3. Methodology: Granger causality

p-values for the linear Granger-causality test

some text

  • No evidence of causality in any direction
  • Exception: \(ER_t \rightarrow O_t\) when lag=1

3. Methodology: Granger causality

p-values for non-linear causality test (Diks and Panchenko, 2006)

some text

  • Evidence of causality from \(ER_t \rightarrow O_t\)

3. Methodology: VAR (time-invariant) model


\(y_t=a+\displaystyle\sum_{j=1}^pA_jy_{t-j}+u_t\)

  • \(y_t\) (\(2\times1\)) observed variables \((ER_t, O_t)^{\prime}\)
  • \(a\) (2x1) vector of parameters
  • \(A_1, \dots ,A_p\) are (\(2 \times 2\)) matrices of parameters
  • \(u_t\) (\(2\times1\)) ; \(u_t \sim \mathcal{N}(0,\) \(\Sigma_u\))
  • Optimal lags = 2, based on SIC
    Consistent with other studies (see Fratzscher et al., 2014; Ferraro et al., 2015)

3. Methodology: TVP-VAR model


\(y_t = a(t) + \displaystyle\sum_{j=1}^2 A_j(t)\ y_{t-j} + u_t\)


  • \(y_t\) (\(2 \times 1\)) observed variables \((ER_t, O_t)^{\prime}\)

  • \(a(t)\) (\(2 \times 1\)) 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


To facilitate structural analysis, \(\Omega(t)\) is parameterized as:


\(B(t)\) \(\Omega(t)\) \(B^{\prime}(t) = \Sigma(t)\) \(\Sigma^{\prime}(t)\)


  • \(B(t) = \begin{bmatrix} 1 & 0\\ b_{21}(t) & 1\end{bmatrix} \quad\quad\quad \Sigma(t) = \begin{bmatrix} \sigma_1(t) & 0\\ 0 & \sigma_2(t)\end{bmatrix}\)


  • \(u_t = B(t)^{-1}\ \Sigma_t\ \varepsilon_t \quad\quad V(\varepsilon)=I_2\)

3. Methodology: TVP-VAR model


\(y_t =\) \(a(t)\) + \(A_j(t)\) \(y_{t-j}\) + \(B(t)^{-1} \Sigma(t)\) \(\varepsilon_t\)

\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B(t)^{-1} \Sigma(t)\) \(\varepsilon_t\)


  • \(I_2\) is the identity matrix (\(2 \times 2\))

  • \(X_t=[1, y_{t-1}^{\prime}, y_{t-2}^{\prime}]\) is the stacked vector of variables (\(1 \times 5\))

  • \(\alpha_t=(a_t\ A_{1,t} \ A_{2,t})\) is the stacked vector of TV parameters (\(10 \times 1\))

  • \(B_t\) TV 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


  • \(\alpha_t\) \(= \alpha_{t-1} + \eta_t^\alpha\)\(\quad\quad\)(Shocks to the \(past\) values of \(y_t\))
  • \(b_t\) \(= b_{t-1} + \eta_t^b\)\(\quad\)(Unrestricted elements of \(B_t\): Shocks to the \(\quad\quad\quad\quad\quad\quad\quad\quad \ \ \ \ \ \ current\) values of \(y_t\))
  • \(log\) \(\sigma_t\) \(= log\, \sigma_{t-1} + \eta_t^\sigma \quad\quad\) \(\sigma_t\) \(= diag(\) \(\Sigma_t\)\()\)

    The error terms \(\eta_t^\alpha, \eta_t^b, \eta_t^\sigma\) are independent and white noise processes

3. Methodology: TVP-VAR model


\[\begin{equation} V = Var \begin{pmatrix}\begin{bmatrix} \varepsilon_t\\ \eta_t^\alpha\\ \eta_t^b\\ \eta_t^\sigma \end{bmatrix}\end{pmatrix} \quad = \begin{bmatrix} I_2 & 0 & 0 & 0\\ 0 & Q & 0 & 0\\ 0 & 0 & S & 0\\ 0 & 0 & 0 & W \end{bmatrix} \end{equation}\]
  • \(y_t\ (2 \times 1)\ \ \quad\rightarrow\quad I_2\)

  • \(\alpha_t\) \((10 \times 1) \quad\rightarrow\quad\) \(Q\) \((10 \times 10) \quad\rightarrow\quad\) 55 free paremeters

  • \(B_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(S\) \((1 \times 1)\)

  • \(\Sigma_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(W\) \((2 \times 2) \quad\rightarrow\quad\) 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:

some text

3. Methodology: TVP-VAR model


  • Simulation algorithm for the joint posterior of \(\alpha(1),\dots,\alpha(T), b(1)\dotsb(2), \sigma(1)\dots\sigma(T), Q, S, W\)

  • 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

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

some text

4. Results: TVP-VAR model

Standard deviations

some text

4. Results: TVP-VAR model

Impulses \(O_t \rightarrow\) Responses \(ER_t\) TV responses to one standard deviation shock

4. Results: TVP-VAR model

Impulses \(O_t \rightarrow\) Responses \(ER_t\) TV responses to one standard deviation shock

some text

4. Results: TVP-VAR model

Impulses \(ER_t \rightarrow\) Responses \(O_t\) TV responses to one standard deviation shock

4. Results: TVP-VAR model

Impulses \(ER_t \rightarrow\) Responses \(O_t\) TV responses to one standard deviation shock

some text




5. Concluding Remarks

5. Concluding Remarks

Responses after 3, 6, 12, and 24 months

some text

  • \(O_t \rightarrow ER_t\): Wealth channel (-), trade balance (-), petrodollars recycling (+)
  • \(ER_t \rightarrow O_t\): Oil market (-), Financialization (-)

5. Concluding Remarks


  • The negative sign response of \(ER_t\) to \(O_t\) shocks in the short-run are highly similar across different period of time.
  • This finding is consistent with most of economic theory, which establishes a depreciation after an increase in \(O_t\).
  • The mid- and long-run responses of \(ER_t\) to \(O_t\) shocks differ qualitative and quantitatively. These responses are positive before the mid-2000s, which in line with the petrodollar recycling argument.

5. Concluding Remarks


  • The \(O_t\) declines after an appreciation of \(ER_t\), which is in conformity with the economic theory.
  • While the negative short-run reaction has been similar across different period of time, the long-run reaction differs.
  • In fact, the pattern responses have been highly similar before the mid-1990s, date from which the responses start to be more intensive, with the highest impact being observed in the global financial crisis.

5. Concluding Remarks


  • These findings highlight the importance of considering the period of time in which the oil price shock occurs because the U.S. EER response may differ over time and, consequently, the economic policy reaction which is required to counteract to such a shock may also differ.
  • The decline in oil price observed after an appreciation of U.S. EER is not the same over time and it may generate different adverse effects on investment depending on the period of time the appreciation takes place. The knowledge of such effects may help financial investors to diversify their investments in order to optimize the risk-return profile of their portfolios.




Thank you




4. Appendix

Triangular reduction

To facilitate structural analysis, the error covariance matrix \(\Omega_t\) is parameterized as:

\(B_t \Omega_t B_t^{-1} = \Sigma_t \Sigma_t^{\prime}\)

Demonstration:

  • \(\Omega_t \ = V(u_t) = E[(u_t-0)(u_t-0)^{\prime}] = E[u_t u_t^{\prime}]\)

  • If \(\ u_t = B_t^{-1} \Sigma_t \varepsilon_t\)

  • \(\Omega_t \ = E[B_t^{-1} \Sigma_t \varepsilon_t \ (B_t^{-1} \Sigma_t \varepsilon_t)^{\prime}] = E[B_t^{-1} \Sigma_t \ \varepsilon_t \varepsilon_t^{\prime} \ \Sigma_t^\prime B_t^{-1 \prime}]\)

  • If \(\ V(\varepsilon_t) = E[\varepsilon_t \varepsilon_t^\prime] = I_2\)

  • \(\Omega_t = B_t^{-1} \Sigma_t \Sigma_t^{\prime} {B_t^{\prime}}^{-1}\)

  • \(B_t \Omega_t B_t^{-1} = \Sigma_t \Sigma_t^{\prime}\)

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


  • \(\alpha_t\) \((10 \times 1) \quad\rightarrow\quad\) \(Q\) \((10 \times 10)\)
  • \(B_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(S\) \((1 \times 1)\)
  • \(\Sigma_t\) \((2 \times 2)\ \ \quad\rightarrow\quad\) \(W\) \((2 \times 2)\)

    \(\theta= [ \alpha^T, B^T, V ]\)

  • 1 Initialize \(B^T, \Sigma^T, S^T\) and \(V\)
  • 2 Sample \(\alpha^T\) from \(p(\alpha^T|\theta^{-\alpha T}, \Sigma^T)\ \rightarrow\) sample \(Q\) from \(\ p(Q|\alpha^T)\)
  • 3 Sample \(B^T\) from \(p(B^T|\theta^{-S}, \Sigma^T) \ \rightarrow\) sample \(\ S\) from \(p(S|\theta^{-S}, \Sigma^T)\)
  • 4 Sample \(s^T\) from \(p(s^T|\Sigma^T, \theta) \ \rightarrow\) draw \(\ \Sigma^T\) from \(p(\Sigma^T|\theta, s^T)\ \rightarrow\) sample \(\ W\) from \(p(W|\Sigma^T)\)
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