A new approach to modelling interactions between oil price and exchange rate

    César Castro Rozo
    Department of Economics, Universidad Pública de Navarra

    Rebeca Jiménez Rodríguez
    Department of Economics, Universidad de Salamanca


II Workshop of the Spanish Macroeconomics Network (SMN)

Overview


  1. INTRODUCTION
    · Historical relationship between oil price (OP) and exchange rate (ER)
    · Literature review
    · Objective

  2. METHODOLOGY
    · Data
    · Preliminary analysis
    · Standard VAR model
    · Time varying parameter (TVP) VAR model

  3. RESULTS
    · Historical shock episodes vs. VAR and TVP-VAR shocks
    · Standard VAR IRFs
    · Time varying IRFs

  4. CONCLUSIONS




1. INTRODUCTION

1. INTRODUCTION: historical OP and ER relationship


1. INTRODUCTION: historical OP and ER relationship

5-Year rolling correlation

1. INTRODUCTION: literature review (theoretical)

The relatonship will depend on the oil dependence of the country. Then, the source of the shock will determine the sign of the effects through different transmission channels:
  1. \(\uparrow\) OP shock \(\rightarrow\) U.S. ER
    · Wealth channel (Oil importing): \(\quad\quad \ \downarrow\) U.S. ER \(\quad\) (-)
    · \(\downarrow\) trade balance (Oil importing): \(\quad\quad \ \downarrow\) U.S. ER \(\quad\) (-)
    · Petrodollar recycling (\(\uparrow D_{USD_{asset}}\)): \(\ \ \quad\uparrow\) U.S. ER \(\quad\) (+)

  2. \(\uparrow\) U.S. ER shock \(\rightarrow\) OP
    · Oil market: \(\quad\quad\quad \uparrow S_{oil} \ + \downarrow D_{oil} \rightarrow\quad\) \(\downarrow\) OP \(\ \quad\quad\) (-)
    · Financialization: \(\quad\quad\quad\quad\downarrow D_{oil}\rightarrow\quad\) \(\downarrow\) OP \(\ \quad\quad\) (-)

  3. US monetary policy shock
    \(\ \quad\quad\uparrow r \rightarrow\quad\) \(\uparrow\) U.S. ER \(\rightarrow\quad\downarrow D_{oil} \rightarrow\quad\) \(\downarrow\) OP \(\quad\quad\) (-)
    Links: O import;   VAR;   EER;   OP

1. INTRODUCTION: literature review (empirical)

From the point of view of the empirical literature, there is also a lack of consensus on the source and the subsequent direction of the effects:

  • ER shock\(\rightarrow\) OP:
    e.g., Trehan (1986), Zhang et al. (2008) and Coudert and Mignon (2016)

  • OP shock\(\rightarrow\) ER:
    e.g., Amano and Van Norden (1998), Lizardo and Mollick (2010) and Ferraro et al. (2015)

  • ER \(\leftrightarrow\) OP:
    e.g., Wang and Wu (2012) and Fratzscher et al., (2014)

1. INTRODUCTION: literature review (empirical)

  • The lack of consensus in the empirical literature seems to be related to:

    · Data frequency (e.g., monthly vs. quarterly data)
    · Period of analysis (e.g., 1974-2000 \(\neq\) 1974-2015)

  • The period of analysis could be related with the possible existence of structural breaks, but there is not a clear conclusion about such an existence in the related literature:

    · Chen and Chen (2007) do not find evidence for (G-7 countries, 1972-2005).
    · Fratzscher et al. (2014) show evidence in the early 2000s (euro area, 1995-2005)

1. INTRODUCTION: objective



Contribute to better understand the dynamic interaction over time between U.S. ER and OP analyzing the varying reaction of one variable to the shocks of the other.




2. METHODOLOGY

2. METHODOLOGY: data


· Monthly data

· \(O_t\) : Nominal oil price (WTI in USD per Barrel)
Source: U.S. Energy Information Administration (EIA)

· \(EER_t\) US nominal narrow index of effective exchange rate
Source: Bank for International Settlements (BIS)

· Sample period: January 1974 - July 2019

· Total number of observations: 547


2. METHODOLOGY: preliminary analysis

Identification of OP and ER shock episodes
Based on accumulated rate of growth over time, similar to that taken by Blanchard and Galí (2010)

some text
Link: shocks

2. METHODOLOGY: preliminary analysis

Linear Granger-causality test some text

· No evidence of linear G-causality \(O_t \rightarrow EER_t\)
· Evidence of linear G-causality \(EER_t \rightarrow O_t\)

2. METHODOLOGY: preliminary analysis

Nonlinear G-causality test
Based on nonparametric test proposed by Diks and Panchenko (2006)

some text

· No evidence of nonlinear G-causality \(O_t \rightarrow EER_t\)
· Evidence of nonlinear G-causality \(EER_t \rightarrow O_t\)
Link: DP test

2. METHODOLOGY: 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

·
    \(A_1 ,A_2\) are (\(2 \times 2\)) matrices of parameters

·
    \(u_t\) (\(2\times1\)) \(\quad\) ; \(\quad u_t \sim \mathcal{N}(0,\) \(\Sigma_u\))

·
    Optimal lag length is 2, based on SIC


Link: CI

2. 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

·
    \(u_t\) (\(2 \times 1\)) \(\quad\quad\) \(u_t \sim \mathcal{N}(0,\) \(\Omega_t\))


2. METHODOLOGY: TVP-VAR model

The model can be rewritten as:

\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(u_t\)


    · \(I_2\) identity matrix (\(2 \times 2\))

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

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

    · \(u_t\) (\(2 \times 1\)) \(\quad\quad\) \(u_t \sim \mathcal{N}(0,\) \(\Omega_t\))

    · We consider the triangular reduction of the matrix \(\Omega_t\):

\(\quad B_t \Omega_t B^{\prime}_t = \Sigma_t\Sigma^{\prime}_t\)

2. METHODOLOGY: TVP-VAR model

\(\quad B_t \Omega_t B^{\prime}_t = \Sigma_t\Sigma^{\prime}_t\)

· \(B_t\) TV lower triangular matrix (\(2 \times 2\))
· \(\Sigma_t\) TV standard deviations of error term (\(2 \times 2\))

\(B_t = \begin{bmatrix} 1 & 0\\ b_{21,t} & 1\end{bmatrix} \quad\quad\quad \Sigma_t = \begin{bmatrix} \sigma_{1,t} & 0\\ 0 & \sigma_{2,t}\end{bmatrix}\)

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

Then, the model can be rewritten as:

\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)

There is one to one mapping with the starting specification:

\(y_t =\)\(a_t\) \(+\) \(A_{1,t}\) \(y_{t-1} +\) \(A_{2,t}\) \(y_{t-2} + u_t\)
Link: demonstration

2. METHODOLOGY: TVP-VAR model

\(y_t = (I_2 \otimes X_t)\) \(\alpha_t\) + \(B_t^{-1} \Sigma_t\) \(\varepsilon_t\)

  • In sum, there are three vectors of TVP:
    · \(\alpha_t=(a_t,\ A_{1,t}, \ A_{2,t})\), (\(10 \times 1\))
    · \(b_t=(b_{21,t})\) unrestricted TVP of \(B_t\), (\(1 \times 1\))
    · \(\sigma_t=(\sigma_{1,t}, \sigma_{2,t})\) diagonal TVP of the matrix \(\Sigma_t\), (\(2 \times 1\))

  • Dynamics for each of the vectors of TVP is specified as:
    · \(\alpha_t\) \(= \alpha_{t-1} + \eta_t^\alpha\) \(\quad\quad\quad\quad \ \eta_t^\alpha \sim \mathcal{N}(0,\) \(Q\)\()\)
    · \(b_t\) \(= b_{t-1} + \eta_t^b\)\(\quad\quad\quad\quad \ \ \eta_t^b \sim \mathcal{N}(0,\) \(S\)\()\)
    · \(log\) \(\sigma_t\) \(= log\, \sigma_{t-1} + \eta_t^\sigma \quad \ \ \ \eta_t^\sigma \sim \mathcal{N}(0,\) \(W\))
      where the error terms \(\eta_t^\alpha, \eta_t^b, \eta_t^\sigma\) are white noise processes

2. METHODOLOGY: TVP-VAR model

It is assumed that the error terms (\(\varepsilon_t, \eta_t^\alpha, \eta_t^b, \eta_t^\sigma\)) are jointly normally distributed with the variance covariance matrix (V) matrix being:

\[\begin{equation}\def\r{\color{#8C001A}} V = Var \begin{pmatrix}\begin{bmatrix} \varepsilon_t\\ \eta_t^\alpha\\ \eta_t^b\\ \eta_t^\sigma \end{bmatrix}\end{pmatrix} \quad = \begin{bmatrix} I_2 & 0 & 0 & 0\\ 0 & \r Q & 0 & 0\\ 0 & 0 & \r S & 0\\ 0 & 0 & 0 & \r W \end{bmatrix} \end{equation}\]


· \(y_t\ (2 \times 1)\ \ \quad\rightarrow\quad I_2\)
· \(\alpha_t\), \((10 \times 1) \quad\rightarrow\quad\) \(Q\), \((10 \times 10) \quad\rightarrow\quad\) 55 free parameters
· \(b_t\), \((2 \times 2)\ \ \quad\rightarrow\quad\) \(S\), \((1 \times 1)\)
· \(log\)\(\sigma_t\), \((2 \times 2)\ \ \quad\rightarrow\quad\) \(W\), \((2 \times 2) \quad\rightarrow\quad\) 3 free parameters

2. METHODOLOGY: TVP-VAR model

Solution:
· Bayesian inference: Markov chain Monte Carlo (MCMC) algorithm based on a Gibbs sampler
· Joint posterior of 13 elements of the parameters \(\alpha^T, B^T, \Sigma^T\) and 59 elements of the hyperparameters \(Q, S, W\)
· Training sample (used to determine the numerical values in the prior based on the OLS estimate of a constant coefficient VAR model): 1974:01-1979:02 (60 months)
· The estimation uses data beginning in 1979:03 (to 2019:07, 485 obs.)
· Simulations: 50000 iterations
· Burn-in steps to initialize the sample: 5000


Link: Algorithm

2. METHODOLOGY: TVP-VAR model

The priors follow the same principles as in Primiceri (2005)

some text
Link: Priors

2. METHODOLOGY: TVP-VAR model

``The prior specification is arbitrary. Notably the multiplicative constants in the expressions of the table are chosen subjectively. The choices are made for tractability and the desire to reflect the lack of prior knowledge by the analyst.’’ (Kilian and Lütkepohl, 2018)

some text




3. RESULTS

3. RESULTS: shocks episodes vs. VAR and TVP-VAR shocks

Shocks
Sample period: 1979:03-2019:07 (485 months)

some text
Link: episodes

3. RESULTS: VAR IRFs

Responses to one unit shock

some text
Link: theory

3. RESULTS: TVP-VAR IRFs

Responses of \(EER_t\) to one unit \(O_t\) shock
Sample period: 1979:03-2019:07 (485 months)

3. RESULTS: TVP-VAR IRFs

Responses of \(O_t\) to one unit \(EER_t\) shock
Sample period: 1979:03-2019:07 (485 months)

3. RESULTS: TVP-VAR IRFs

Responses to one unit shock after 3, 6, 12, and 24 months
Sample period: 1979:03-2019:07 (485 months)

some text

3. RESULTS: TVP-VAR IRFs

Responses of \(EER_t\) to one unit \(O_t\) shock

some text

3. RESULTS: TVP-VAR IRFs

Responses of \(EER_t\) to one unit \(O_t\) shock
after 3, 6, 12, and 24 months
Sample period: 1979:03-2019:07 (485 months)

some text
Link: theory

3. RESULTS: TVP-VAR IRFs

Responses of \(O_t\) to one unit \(EER_t\) shock

some text

3. RESULTS: TVP-VAR IRFs

Responses of \(O_t\) to one unit \(EER_t\) shock
after 3, 6, 12, and 24 months
Sample period: 1979:03-2019:07 (485 months)

some text
Link: theory




4. CONCLUSIONS

4. CONCLUSIONS

\(O_t \rightarrow EER_t\)

  • The negative sign response of \(EER_t\) to \(O_t\) shocks in the short-run are highly similar over time (statistically significant over the period around the financial crisis), which is consistent with the wealth and the trade balance channels.

  • However, the long-run responses of \(EER_t\) to \(O_t\) shocks are positive before the mid-2000s (statistically significant before the 1990s) which is in line with the petrodollar recycling argument.

  • The responses of \(EER_t\) to \(O_t\) shock differ over time and, consequently, the economic policy reaction may also differ.

4. CONCLUSIONS

\(EER_t \rightarrow O_t\)

  • The \(O_t\) declines after an appreciation of \(EER_t\), which is in conformity with the economic theory.

  • The responses are negative in the short- and long-run, although the reactions are more intensive from the beginning of the 1990s onwards.

  • The decline in oil price observed after an appreciation of \(EER_t\) is not the same over time and it may generate different adverse effects on investment over time.

4. CONCLUSIONS

Methodology

  • Unlike the (time-invariant) VAR model which considers that the responses of one variable to the shock of the other are equal across different period of time, the more realistic TVP-VAR model allows that these responses change over time without establishing specific breaks.




THANK YOU




APPENDIX

U.S. net imports of petroleum

some text

Link: nominal

A nonlinear causality test statistic

Granger test: \(\mathbb{H_0}: Y_{t+1}\mid (\textbf{X}_t, \textbf{Y}_t) \sim Y_{t+1}\mid \textbf{Y}_t\)

Considering a strictly stationary bivariate time series, the null hypothesis is a statement about the invariant distribution of the vector \(\textbf{W}_t=(\textbf{X}_t, \textbf{Y}_t, Z_t)\), where \(Z_t=Y_{t+1}\). Under \(\mathbb{H_0}\), the conditional distribution of \(Z\) given \((\textbf{X}, \textbf{Y})=(x, y)\) is the same as that of \(Z\) given \(\textbf{Y} = \textbf{y}\).

In terms of joint distributions: \(\quad \dfrac{f_{X,Y,Z}(\textbf{x,y},z)}{f_{Y}(\textbf{y})}=\dfrac{f_{X,Y}(\textbf{x,y})}{f_{Y}(\textbf{y})} \dfrac{f_{Y,Z}(\textbf{y},z)}{f_Y(\textbf{y})}\)

The proposed test statistic (Hiemstra and Jones, 1994) devise to measure the discrepancy between the left and right hand sides of this equation is given by:

\(Q_{T,W}(h)=\dfrac{{\hat{f}}_{X,Y,Z}(h)}{\hat{f}_{Y}(h)}-\dfrac{\hat{f}_{X,Y}(h)}{\hat{f}_{Y}(h)} \dfrac{\hat{f}_{Y,Z}(h)}{\hat{f}_Y(h)}\)

Under \(\mathbb{H_0}, \quad \sqrt{T}Q_{T,W}(h) \rightarrow \mathcal{N}(0,\sigma^2_W(h)), \quad\) as \(\quad T \rightarrow \infty\)

A nonlinear causality test statistic

Diks and Panchenko (2005, 2006) observe that, for a given nominal size, the actual rejection rate of \(Q_{T,W}\) may tend to one as \(T\) increases, i.e. the test statistic over- rejects the null hypothesis. Then, they rewrite:

\(\mathbb{H_0: E}\Bigg[\Bigg(\dfrac{{f}_{X,Y,Z}(\textbf{x},\textbf{y},z)}{f_{Y}(\textbf{y})}-\dfrac{f_{X,Y}(\textbf{x},\textbf{y})}{f_{Y}(\textbf{y})} \dfrac{f_{Y,Z}(\textbf{y}, z)}{f_Y(\textbf{y})}\Bigg)g(\textbf{x},\textbf{y},z)\Bigg]=0\)

Where, \(g(\textbf{x},\textbf{y},z)\) is a positive weight function set at \(g(\textbf{x},\textbf{y},z)= f^2_Y(\textbf{y})\)

The proposed nonparametric Granger causality (bivariate) test statistic is given by

\(\Delta \equiv \mathbb{E} \big[\ f_{X,Y,Z}(\textbf{x}, \textbf{y}, z)f_Y(\textbf{y})-f_{X,Y}(\textbf{x}, \textbf{y})\ f_{Y,Z}(\textbf{y},z)\ \big]=0\)

Under \(\mathbb{H_0}\) the term within square brackets vanishes, so that the expectation is zero. The test statistic simplifies to

\(Q^*_{T,W}(h) = \dfrac{T-1}{T(T-2)}\displaystyle\sum_i \big[\ \hat{f}_{X,Y,Z}(X_i, Y_i, Z_i)\ \hat{f}_Y(Y_i)-\hat{f}_{X,Y}(X_i, Y_i)\ \hat{f}_{Y,Z}(Y_i,Z_i)\big]\)

Under \(\mathbb{H_0}, \quad \sqrt{T}\dfrac{Q^*_{T,W}(h)-\Delta}{S_{T,W}(h)} \rightarrow \mathcal{N}(0,1), \quad\) as \(\quad T \rightarrow \infty\)
Link: G-causality

Triangular reduction

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

\(B_t\) \(\Omega_t\) \(B^{\prime}_t = \Sigma_t\) \(\Sigma^{\prime}_t\)

\(B_t = \begin{bmatrix} 1 & 0\\ b_{21,t} & 1\end{bmatrix} \quad\quad\quad \Sigma_t = \begin{bmatrix} \sigma_{1,t} & 0\\ 0 & \sigma_{2,t}\end{bmatrix}\)

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


Demonstration:

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

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

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

· \(\Omega_t = B_t^{-1} \Sigma_t \ I_2 \ \Sigma_t^{\prime} {B_t^{\prime}}^{-1}\)

· \(B_t \Omega_t B_t^{\prime} = \Sigma_t \Sigma_t^{\prime}\)


Link: triangular reduction

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 \(B^T,\ \Sigma^T,\ s^T, \ Q,\ S,\ W\)

·
    \(p(\alpha^T/\ \theta^{-\alpha T}, \Sigma^T)\) \(\rightarrow \alpha^T\) \(\ \Rightarrow p(Q\ /\ \alpha^T)\) \(\rightarrow Q\)

·
    \(p(B^T/\ \theta^{-S}, \Sigma^T) \ \Rightarrow\) \(B^T \leftarrow\)\(\ S \leftarrow\) \(p(S\ /\ \theta^{-S}, \Sigma^T)\)

·
    \(s^T \leftarrow\) \(p(s^T/\ \Sigma^T, \theta) \ \Rightarrow\) \(\ \Sigma^T \leftarrow\) \(p(\Sigma^T/\ \theta, s^T)\ \Rightarrow\) \(\ W \leftarrow\) \(p(W\ /\ \Sigma^T)\)

·
    Go to 2


Link: solution

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 \(y_t\)
\((\uparrow k_S \rightarrow \ \uparrow S_1 \rightarrow \ \uparrow B_t)\)


Link: priors

Shocks episodes vs. VAR and TVP-VAR shocks

Shocks (series in real terms)
Sample period: 1979:03-2019:07 (485 months)

some text

Link: nominal

Cointegration

· Looking at the stochastic properties of the time series, the results of Augmented Dickey–Fuller test indicates that U.S. EER and oil price seem to be I(1). In addition, given that both variables seem to be I(1)

· We test for cointegration by using the Johansen cointegration test. The results of these cointegration tests indicate the lack of cointegration between the two variables.

· Authors such as Sims et al. (1990), Hamilton (1994) and Ramaswamy and Sløk (1998) state that it is possible to perform the study in levels allowing for implicit cointegrating relationships in the data if there are and still have consistent estimates of the parameters.

· Including misspecified cointegrating relationship in the VAR model in levels would give rise to biased estimates and estimating the VAR model in the first differences would lead to a loss of information in the levels (see Sims et al., 1990).



Link: VAR model