## wpiea2020047-print-pdf — 2.1 Measuring global activity

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### Measuring global activity: rationale and new measure
- Rationale and critique of existing proxies:
  - No unique indicator of global activity is directly available; proxies require additional calculations.
  - Kilian (2009b) proposes using the real cost of shipping as a proxy; subsequently adopted in several studies.
  - Hamilton (2018) proposes an alternative based on monthly industrial production for the OECD plus 6 other major economies.
  - Validity of industrial production as a proxy depends on the industrial sector representing a substantive share of total production.
    - Figure 1a: services share has grown and currently accounts for at least half of total production activity in most advanced and emerging countries.
    - Figure 1b: decline in the share of oil consumption in total production over time.
- New measure proposed:
  - Quarterly global real GDP (PPP, US dollars) for the OECD plus Brazil, China, India, Indonesia, and South Africa.
  - Quarterly real GDP time series for these 5 countries became available only during the 1990s; appendix A.1 describes methodology used to infer quarterly observations for earlier periods.
  - Resulting estimate of aggregate real production for the OECD plus 5 is plotted in Figure A.1.

### Transitory and permanent shocks: shock structure and variables
- Model treats four endogenous variables as driven by five exogenous shocks:
  - nonstationary oil supply: Ψ_s
  - stationary oil supply shock: φ_s
  - stationary speculative shock: φ_sp
  - nonstationary demand shock: Ψ_d
  - stationary demand shock: φ_d
- Cointegration test results (appendix B, Tables A.1–A.3):
  - Oil production, inventories, and prices have unit roots and share the same nonstationarity component; 1980Q3 identified as a break date.
  - Global activity is not cointegrated with oil sector specific variables.
- Modeling implication:
  - Oil production, inventories, and prices share common trend Ψ_s.
  - Global output trend is explained by an independent variable Ψ_d.
- Stationary variable definitions:
  - ˆx_t = x_t − Ψ_d_t
  - ˆz_t = z_t − Ψ_s_t
  - ˆi_t = i_t − Ψ_s_t
  - ˆp_t = p_t − Ψ_s_t
- Law of motion:
  - Endogenous vector (4 variables) depends on lagged endogenous terms and structural shocks.
  - ∆Ψ_s_t ≡ Ψ_s_t − Ψ_s_{t−1}; ∆Ψ_d_t ≡ Ψ_d_t − Ψ_d_{t−1}.
- Structural shocks follow univariate AR(1) laws with one dummy variable to account for trend changes identified in cointegration analysis.

### Identification restrictions: cointegration-based assumptions and sign restrictions
- Cointegration-based assumptions:
  - Oil production, inventories, and prices are cointegrated with Ψ_s_t.
  - Global output is cointegrated with Ψ_d_t.
- Sign restrictions (impact only; weak inequalities allowed) using Uhlig (2005):
  - Transitory oil-supply shock:
    - B_{1,2} ≥ 0, B_{2,2} ≥ 0, B_{4,2} ≤ 0
    - Interpreted: oil production and real oil price move in opposite directions, and it increases global activity.
  - Transitory speculative shock:
    - B_{1,3} ≤ 0, B_{2,3} ≥ 0, B_{4,3} ≥ 0
    - Interpreted: generates a positive response of oil production and its price, and reduces global activity.
  - Transitory demand shock:
    - B_{1,5} ≥ 0, B_{2,5} ≥ 0, B_{4,5} ≥ 0
    - Interpreted: induces positive correlations between oil production, its price, and inventories.
  - Normalizations (without loss of generality):
    - B_{1,5} = B_{2,2} = B_{3,3} = 1

### Priors and estimation
- Prior specifications (Minnesota-flavored):
  - Diagonal elements of A_1:
    - A_1(j,j) for j = 1 to 4 ~ Normal(Mean 0.8, Std. Dev. 0.15)
  - Other A_i (i = 2,...,L):
    - A_i(j,k) ~ Normal(Mean 0, Std. Dev. 0.25)
  - B matrix priors:
    - B_{1,2} and −B_{1,3} ~ Gamma(Mean 1, Std. Dev. 1)
    - B_{2,3} and −B_{1,5} ~ Gamma(Mean 1, Std. Dev. 1)
    - −B_{4,1}, −B_{4,2}, B_{4,3}, and B_{4,5} ~ Gamma(Mean 1, Std. Dev. 1)
    - Other elements of B ~ Normal(Mean 0, Std. Dev. 1)
  - κ_{1,1} ~ Normal(Mean 0, Std. Dev. 1)
  - Prior serial correlations:
    - ρ(j,j) for j = 1 to 4 ~ Normal(Mean 0.3, Std. Dev. 0.15)
  - Prior standard deviations of structural shocks:
    - ω(j,j) for j = 1 to 4 ~ Normal(Mean 1, Std. Dev. 0.25)
- Model selection and MCMC:
  - Lag length identification via posterior odds for L = 1 to 6:
    - Posterior odds largely prefer L = 2, which implies a Log data density of 1766.8 (example alternative: 1649.6 with L = 4).
  - MCMC sampler:
    - Metropolis-Hastings sampler with Monte-Carlo Markov chain of 500,000 draws after burning initial 20 percent draws.
    - Posterior means and error bands around impulse responses constructed from a random subsample of the MCMC chain of length 100,000 with replacement.

### Empirical results — posterior of contemporaneous coefficients and impulse responses
- Posterior distributions of matrix B:
  - Data informative: posteriors noticeably different from priors.
  - Global activity immediate-impact parameters (first row) under sign restrictions:
    - Posterior averages for constrained parameters around 0.13 (significantly lower than prior mean of 1).
    - Global activity is rather irresponsive to permanent oil market specific shocks.
    - Posterior mean of B_{1,4} = −0.36 reflects short-term positive reaction of global activity to a permanent demand shock when variable defined as ˆx_t = x_t − Ψ_t.
  - Immediate reaction of oil availability:
    - Constrained parameters for speculative and transitory global demand shocks have posterior means 0.27 and 0.43, respectively.
    - Contemporaneous effect of permanent oil market shock shows negative posterior mean for B_{2,1}; impulse responses build over time to a positive steady state.
  - Posterior means of third and fourth row elements of B move toward higher absolute values, reflecting large volatility of oil inventories and real prices.
- Impulse-response key findings (responses to a one percent unexpected increase in each structural shock ε_t):
  - Supply shocks (transitory vs permanent):
    - Qualitatively similar effects; differ in timing and amplitude.
    - Transitory supply shocks:
      - Larger contemporaneous effect on real oil price and oil production than permanent supply shocks.
      - Real price reaction dissipates after 10 quarters.
      - Oil availability remains significantly above long-term level for about five years.
    - Permanent supply shocks:
      - Global output eventually rises to a higher medium-run level; response not rising significantly above zero until about eight quarters after the shock; maximum impact of 0.4 percent.
      - Cumulative global output response following a permanent innovation is significantly larger than following a transitory shock.
  - Inventories:
    - Strong contemporaneous reaction and then slowly decline to below zero.
    - After a transitory oil supply shock: both global output and inventories increase on impact, but inventories show a persistent positive response while global output does not.
  - Speculative shock:
    - Persistent negative reaction of global output, statistically significant for about 5 years.
    - Global output attains lowest value of −0.26 percent after eight quarters.
    - Oil production responds positively to speculative shock.
    - Short-term oil price reaction to speculative shock resembles that of a negative permanent supply shock.
  - Demand shocks (global activity side):
    - Shocks imply moderate and not significant positive change in oil prices which revert quickly.
    - A permanent disruption to global revenue causes inventories to increase on impact by almost 0.7 percent.
    - Permanent global activity shock yields a negative and persistent response of oil production (novel empirical finding); possible rationales:
      - Innovations and adoption of alternative energy sources embedded in permanent component of global activity.
      - Decline in share of oil-dependent economic activities (industrial sector) reflected in global activity trend.
    - Authors plan to further investigate by contrasting transmission when industrial production is used as measure of global oil demand.

### Quantitative impulse-response highlights (preserved exactly)
- Maximum impact of global output after a permanent supply shock: 0.4 percent.
- Lowest value of global output after a speculative shock: −0.26 percent (after eight quarters).
- Inventories increase on impact by almost 0.7 percent after a permanent global demand disruption.
- Duration of notable responses:
  - Real price dissipates after 10 quarters for transitory supply shock.
  - Global output response to speculative shock remains significant for about 5 years.
  - Oil availability elevated for about five years after a transitory supply shock.
  - Global output reaches notable positive response around eight quarters after a permanent supply shock.

### Variance decomposition — short- and long-run drivers of fluctuations
- Short- and long-run drivers:
  - Transitory shocks to oil demand are the most important driving force of global activity fluctuations in the short-term — more than twice the permanent shock — while oil market innovations yield about 14 percent of the variable dynamics.
  - Oil production variability mainly explained by unanticipated transitory supply shocks.
  - Shocks to global activities explain about 14 percent of oil production dynamics in the short run and remain virtually unchanged in the long run, a contribution almost equivalent to speculative innovations.
  - Large fraction of short-run volatility of oil inventories attributable to speculative shocks, with declining share over the horizon reaching 40 percent in the long-run.
  - Three most important drivers of oil prices:
    - permanent oil-supply shocks,
    - transitory oil-supply shocks,
    - speculative shocks (explaining between 32 and 34 percent of the volatility over different horizons).
  - Permanent and transitory oil-supply shocks together account for about two-thirds of oil price fluctuations in both short- and long-term horizons.
- Key numbers from Table 2 (selected horizons and variables — preserved exactly):
  - Horizon 1:
    - Global activity: ∆Ψs t = 2.89; φs t = 4.98; φs p t = 5.79; ∆Ψd t = 26.27; φd t = 60.07
    - Oil availability: ∆Ψs t = 10.83; φs t = 62.46; φs p t = 12.73; ∆Ψd t = 9.16; φd t = 4.82
    - Inventories: ∆Ψs t = 24.22; φs t = 16.85; φs p t = 7.13; ∆Ψd t = 8.89; φd t = 2.91
    - Oil prices: ∆Ψs t = 28.37; φs t = 37.28; φs p t = 34.02; ∆Ψd t = 0.20; φd t = 0.12
  - Horizon 4:
    - Global activity: ∆Ψs t = 6.23; φs t = 6.83; φs p t = 7.09; ∆Ψd t = 27.89; φd t = 51.96
    - Oil prices: ∆Ψs t = 28.73; φs t = 37.67; φs p t = 33.21; ∆Ψd t = 0.24; φd t = 0.16
  - Horizon 12:
    - Global activity: ∆Ψs t = 7.58; φs t = 7.52; φs p t = 6.92; ∆Ψd t = 27.61; φd t = 50.37
    - Oil prices: ∆Ψs t = 29.41; φs t = 37.25; φs p t = 32.92; ∆Ψd t = 0.25; φd t = 0.17
  - Horizon ∞:
    - Global activity: ∆Ψs t = 8.04; φs t = 7.81; φs p t = 7.03; ∆Ψd t = 27.42; φd t = 49.70
    - Oil prices: ∆Ψs t = 29.82; φs t = 37.04; φs p t = 32.68; ∆Ψd t = 0.27; φd t = 0.20
- Robustness on trend attribution:
  - Assuming ηi = ηp = 1 mildly increases contribution of permanent demand shocks in long-term volatility of oil prices by about 1 percent.
  - Estimating ηi and ηp yields posterior averages of −1.34 and 3.01, respectively; with these estimates the share of oil price volatility explained by permanent demand innovations increases to about 9 percent.

### Conditional correlations
- Observed unconditional correlation between growth rates of global GDP (∆xt) and oil production (∆zt) is 0.13.
- Conditional correlations (iteratively assuming each structural shock is the only shock) — preserved exactly:
  - Corr(∆xt, ∆zt):
    - Unconditional: 0.12
    - Conditional on ∆Ψs t: 0.10
    - Conditional on φs t: 0.56
    - Conditional on φs p t: -0.71
    - Conditional on ∆Ψd t: -0.54
    - Conditional on φd t: 0.79
  - Corr(∆xt, ∆it):
    - Observed: -0.12
    - Unconditional: -0.10
    - Conditional on φs p t: 0.30
    - Conditional on ∆Ψd t: -0.73
    - Conditional on φd t: 0.52
  - Corr(∆xt, ∆pt):
    - Observed: -0.15
    - Unconditional: -0.17
    - Conditional on φs p t: -0.71
    - Conditional on ∆Ψd t: -0.70
    - Conditional on φd t: 0.37
- Interpretation:
  - Negative conditional comovement following permanent demand innovations explains about:
    - 27 percent of volatility of global GDP,
    - 10 percent of volatility of oil production.
  - Speculative shocks also yield negative correlations between global GDP and oil production, but variance decomposition shows a relatively weak role for speculative shocks in explaining dynamics of global GDP and oil production.
  - Numbers in square brackets in Table 3 represent the 16th and 86th percentile Hall confidence bands.

### Historical decomposition of oil prices (selected episode attributions)
- General overview (1974–2018):
  - Until late 1980s speculative shocks contributed considerably to large swings in oil prices; their impact declined in subsequent decades.
  - Transitory and permanent supply innovations played important roles across many episodes, with differing magnitudes and persistence.
  - Distinguishing transitory vs. permanent supply shocks informed by inventories and global activity trajectories (positive medium-term for persistent supply shock; negative for transitory).
- Selected episodes (model attribution highlights):
  - Iranian revolution (1978):
    - Speculative shocks contributed to price increases later; permanent supply shocks responsible for bulk of price volatility; transitory supply shock (OPEC response) also impacted prices.
  - Collapse of OPEC (1986):
    - Positive transitory and demand shocks occurred concurrently; transitory shock contributed slightly more to collapse in prices; speculative shock muted.
  - Gulf War (1990):
    - Supply shock perceived as transitory; permanent shocks barely affected prices; speculative innovations moderately affected prices and heavily contributed to decline in subsequent periods.
  - Venezuelan Crisis and Iraq War (2002–03):
    - Sharp temporary decline in oil supply accounts for price upswing; speculative innovations prominent for price softening afterward; permanent supply trend muted.
  - Global Financial Crisis (2008):
    - Bulk of decline in oil prices attributed to negative oil-specific permanent supply shocks.
    - Global activity shocks minor as quarterly real growth rates remained positive in 2008 and 2009 except in 2008Q3.
    - Negative shocks on inventories and transitory supply shocks equally contributed to price dip; combined impact almost matched permanent supply shock.
  - Oil Price Plunge (2014):
    - Decline from 2014Q2 amid large increase in oil production and inventories rules out speculative shocks.
    - Supply factors explain most decline since mid-2014.
    - Reported evidence: oil “break-even” price declined by roughly 50 per cent between 2013 and 2016 supports oil market–specific shocks as main drivers.

### Iterative estimation — rolling windows and speculative shocks
- Rolling-window variance decomposition (fixed length 120 observations, quarterly):
  - Early samples: speculative innovations explain above 50 percent of oil price volatility until sample 1979Q1–2008Q4; then their contribution declines in later windows.
  - Recent samples: oil supply shocks (permanent and transitory) explain about 80 percent of overall volatility.
  - Innovations to permanent component of oil supply increased from virtually 0 in early samples to more than 60 percent in recent samples.
- Standard deviation parameter behavior over expanding windows:
  - Posterior mean of the permanent oil market shock varies considerably as sample covers more recent periods and is more precisely estimated.
  - Change in permanent shock’s posterior mean closely related to increased share of permanent shocks in variance decomposition of oil prices over time.
  - Posterior standard deviation means of remaining shocks broadly follow a milder decline.
  - Volatility of speculative shocks shows a small decline only towards end of sample; change in contribution of speculative shocks mainly driven by altered dynamic responses rather than variance decline.
- Oil price reaction to speculative shocks over time:
  - Impulse responses to an unexpected speculative shock evolve markedly.
  - Early samples: responses hump shaped, peak after two to three quarters.
  - Later samples: trajectories decline and remain below zero shortly after impulse.
  - On impact, oil price response drops from 12 percent to 2 percent in latest sample (1988Q1–2017Q4).
  - Interpretation: decreasing contribution of speculative shocks principally implied by changed oil market dynamics (impulse-response functions); little evidence of change in variance of speculative shock.

### Robustness check — alternative global activity measure (Baumeister and Hamilton (2018) industrial index)
- Purpose:
  - Compare results using alternative global activity measure (extended industrial production index) to assess whether negative conditional correlation between global activity and oil production to permanent demand shocks is due to technological innovations or change in industrial share.
- Key findings (median impulse-response comparisons using global GDP vs aggregated industrial index):
  - Changing global economic activity definition has little consequence on transitory shocks, except transitory demand shock where industrial index becomes less responsive.
  - Permanent supply shock has a positive and persistent impact on global industrial index; trajectory differs significantly from benchmark especially in short-term.
  - Oil production response to a permanent demand shock: impact response turns positive (as opposed to benchmark negative); trajectories significantly different over 5-year horizon and beyond.
  - Parallel robustness results:
    - Inventories decline and remain below zero for several years.
    - Oil prices persistently increase.
- Interpretation:
  - Negative response of oil production to shocks to trend of global economic activity can emerge from energy efficiency gains (lessened oil dependence) or decline in share of industrial activities.
  - Figure 8 suggests permanent shocks on global demand tend to capture changes in share of industrial sector rather than decreasing oil dependency, because oil production still exhibits positive persistent response under industrial index specification.
  - Does not preclude less pronounced long-term effect from innovation and energy diversification.

### Methodology summary and empirical conclusions
- Empirical approach distinguishes transitory and permanent shocks in global activity and oil market using:
  - SVAR techniques with sign restrictions,
  - Kalman filtering to infer permanent and transitory components,
  - Narrative identification of structural shocks.
- Advantages:
  - Accounts explicitly for sources contributing to non-stationarity.
  - Reflects identified cointegration relations.
  - Relies on standard Bayesian techniques and Kalman filter to assess relative importance of shock components over time.
- Empirical conclusions:
  - Supply side important determinant of real oil price in variance decomposition and major episodes.
  - On average, transitory and permanent shocks to oil supply played an equal role; trend shocks grew in influence on oil price volatility over time.
  - Speculative shocks significant in 1970s and 1980s; effect drastically declined in recent years due to weaker volatility of inventories and declining price responsiveness to speculative innovations.
- Suggested extension:
  - Modify identification of oil-sector-specific shocks to allow agents to learn about uncertain persistence of shocks; perceptions about persistence could be important for contributions of permanent vs transitory shocks to forecast error variance decomposition.

### Data sources and appendices (selected content)
- Global real activity construction (appendix A.1):
  - PPP-converted real quarterly GDP in U.S. dollars for all OECD members plus Brazil, China, India, Indonesia, Russia, South Africa.
  - Quarterly OECD and South Africa real GDP from OECD.Stat covering 1973Q1 until 2018Q1.
  - Quarterly nominal GDP availability start dates:
    - Brazil: 1996Q1
    - China: 1992Q1
    - India: 1996Q2
    - Indonesia: 1990Q1
  - Earlier-period observations constructed from annual World Bank Indicators local currency data, converted to PPP using implied PPP conversion rate from IMF World Economic Outlook database.
  - PPP index available starting from 1980; linear trend based on subsequent 10 years applied reversely to obtain index up to 1973.
  - Frequency conversion: Litterman frequency conversion method used with OECD quarterly GDP as indicator.
  - Final global activity variable: sum of seasonally adjusted OECD and country-specific GDP PPP in USD, divided by U.S. consumer price index to obtain real terms.
- Oil market variables:
  - World oil production: U.S. EIA series (thousands of barrels per day) monthly Jan 1973–Aug 2018; converted to quarterly.
  - OECD petroleum stocks: EIA monthly from 1988 and yearly from 1973; quarterly stocks from 1973Q1 constructed by scaling U.S. crude oil inventory data by ratio of OECD petroleum inventories over U.S. petroleum inventories.
  - Global oil availability: sum of global stock of previous quarter and production of current quarter (constructed as in Dvir and Rogoff (2014)).
  - Nominal oil prices: U.S. Crude Oil Imported Acquisition Cost by Refiners (Dollars per Barrel) from EIA, Jan 1974–Oct 2018.
  - Deflation: nominal oil prices and global GDP deflated by U.S. consumer price index for All Urban Consumers (All Items, Index 1982-1984=100, Quarterly, Seasonally Adjusted).
- Unit roots and cointegration (appendix B):
  - Implemented Clemente et al. (1998) unit root test with structural breaks (AO and IO models); conclusion: unit root and significant level shifts in all series at levels; first differences stationary; variables I(1).
  - Gregory and Hansen (1996) cointegration tests on oil variables indicate cointegration with break date 1980Q3; Johansen test finds at most one cointegrating equation.
  - Cointegration tests between oil variables and global GDP cannot reject null of no cointegration; break dates differ across models.
- State-space specification and identification (appendix C):
  - Endogenous levels: y_t ≡ [x_t z_t i_t p_t]′; transformed vector ̃y_t = [x_t − Ψ^d_t, z_t − Ψ^s_t, i_t − Ψ^s_t, p_t − Ψ^s_t]′; deviations ˆy_t = ̃y_t − E(̃y_t).
  - Autoregressive law: ˆy_t = sum_{j=1}^L A_j ˆy_{t−j} + B u_t.
  - u_t ≡ [ψ^s_t φ^s p_t φ^s_t ψ^d_t φ^d_t]′ with ψ^s_t ≡ ∆Ψ^s_t − E(∆Ψ^s_t), ψ^d_t ≡ ∆Ψ^d_t − E(∆Ψ^d_t).
  - Dynamics of u_t: u_{t+1} = κ d_{t+1} + ρ u_t + ω ε_{t+1}.
  - Observation equations express observed growth rates ∆x_t, ∆z_t, ∆i_t, ∆p_t as functions of trends, stationary components, and ψ shocks (Equation 7).
  - State-space representation used with Kalman filter; identification adapts Uribe (2018) to combine sign restrictions with Kalman filtering to handle more shocks than endogenous variables.

*Source: wpiea2020047-print-pdf — 2.1 Measuring global activity*

### 2.1    Measuring global activity

### 2.1    Measuring global activity

### Rationale and critique of existing proxies
- A unique indicator of global activity is not directly available; additional calculations are generally required.
- Kilian (2009b) proposes using the real cost of shipping as a proxy for global economic activity; this indicator was subsequently adopted in several studies (e.g., Antolín-Díaz and Rubio-Ramírez, 2018; Gambetti and Moretti, 2017; Juvenal and Petrella, 2015; Ratti and Vespignani, 2015).
- Hamilton (2018) argues that the Kilian (2009b) index can mischaracterize particular downturns and proposes an alternative measure based on an estimate of monthly industrial production for the OECD plus 6 other major economies.
- The validity of industrial production as a proxy for aggregate output depends on the industrial sector representing a substantive share of total production across countries.
  - Figure 1a shows the share of services has grown considerably and currently accounts for at least half of total production activity in most advanced and emerging countries, challenging the use of industrial production as a "good" proxy.
  - Figure 1b shows a decline in the share of oil consumption in total production over time, consistent with technological innovations and/or structural shifts toward services.

### New measure proposed
- To capture structural changes in economic and oil activities, the authors refrain from using proxies that exclusively mimic industrial activity.
- They propose a new measure of global real GDP (at purchasing power parity) in US dollars on a quarterly basis for the OECD plus 5 other major countries: Brazil, China, India, Indonesia, and South Africa.
  - Quarterly real GDP time series for these 5 countries became available only during the 1990s; appendix A.1 describes the methodology used to infer quarterly observations for earlier periods.
  - The resulting estimate of aggregate real production for the OECD plus 5 is plotted in Figure A.1.

---

### 2.2    Transitory and permanent shocks

### Shock structure and variables
- The model treats four endogenous variables as driven by five exogenous shocks:
  - nonstationary oil supply: Ψ_s
  - stationary oil supply shock: φ_s
  - stationary speculative shock: φ_sp
  - nonstationary demand shock: Ψ_d
  - stationary demand shock: φ_d
- Cointegration test results (appendix B, Tables A.1–A.3):
  - Oil production, inventories, and prices have unit roots and share the same nonstationarity component; 1980Q3 is identified as a break date.
  - Global activity is not cointegrated with oil sector specific variables.
- Modeling implication:
  - Oil production, inventories, and prices are cointegrated and share a common trend Ψ_s.
  - Global output trend is explained by an independent variable Ψ_d.
- Stationary variables are defined as:
  - ˆx_t = x_t − Ψ_d_t
  - ˆz_t = z_t − Ψ_s_t
  - ˆi_t = i_t − Ψ_s_t
  - ˆp_t = p_t − Ψ_s_t
- Law of motion (autoregressive form with L lags):
  - Endogenous vector (4 variables) depends on lagged endogenous terms (A_i matrices) and structural shocks (B matrix times vector of 5 shocks).
  - ∆Ψ_s_t ≡ Ψ_s_t − Ψ_s_{t−1}, ∆Ψ_d_t ≡ Ψ_d_t − Ψ_d_{t−1}.
- Structural shocks follow univariate AR(1) laws with one dummy variable to account for trend changes in the oil market variables identified in cointegration analysis.

---

### 2.3    Identification restrictions

### Cointegration-based assumptions
- Four identification assumptions from cointegration tests:
  - Oil production, inventories, and prices are cointegrated with Ψ_s_t.
  - Global output is cointegrated with Ψ_d_t.

### Sign restrictions (impact only; weak inequalities allowed)
- Methodology: Uhlig (2005) sign restrictions on contemporaneous (impact) responses; normalizations applied.
- Transitory oil-supply shock:
  - B_{1,2} ≥ 0, B_{2,2} ≥ 0, B_{4,2} ≤ 0
  - Interpreted as: oil production and real oil price move in opposite directions, and it increases global activity.
- Transitory speculative shock:
  - B_{1,3} ≤ 0, B_{2,3} ≥ 0, B_{4,3} ≥ 0
  - Interpreted as: generates a positive response of oil production and its price, and reduces global activity.
- Transitory demand shock:
  - B_{1,5} ≥ 0, B_{2,5} ≥ 0, B_{4,5} ≥ 0
  - Interpreted as: induces positive correlations between oil production, its price, and inventories.
- Normalizations (without loss of generality):
  - B_{1,5} = B_{2,2} = B_{3,3} = 1

---

### 2.4    Priors and estimation

### Prior specifications (Minnesota-flavored)
- Diagonal elements of A_1:
  - A_1(j,j) for j = 1 to 4 ~ Normal with Mean 0.8 and Std. Dev. 0.15
- Other A_i (i = 2,...,L):
  - A_i(j,k) ~ Normal with Mean 0 and Std. Dev. 0.25
- B matrix:
  - B_{1,2} and −B_{1,3} ~ Gamma with Mean 1 and Std. Dev. 1
  - B_{2,3} and −B_{1,5} ~ Gamma with Mean 1 and Std. Dev. 1
  - −B_{4,1}, −B_{4,2}, B_{4,3}, and B_{4,5} ~ Gamma with Mean 1 and Std. Dev. 1
  - Other elements of B ~ Normal with Mean 0 and Std. Dev. 1
- κ_{1,1} ~ Normal with Mean 0 and Std. Dev. 1
- Prior serial correlations:
  - ρ(j,j) for j = 1 to 4 ~ Normal with Mean 0.3 and Std. Dev. 0.15
- Prior standard deviations of structural shocks:
  - ω(j,j) for j = 1 to 4 ~ Normal with Mean 1 and Std. Dev. 0.25

### Model selection and MCMC
- Lag length identification via posterior odds comparisons for L = 1 to 6:
  - Posterior odds largely prefer the model with L = 2, which implies a Log data density of 1766.8, considerably higher than alternatives (e.g., 1649.6 with L = 4).
- MCMC sampler:
  - Metropolis-Hastings sampler with a Monte-Carlo Markov chain of 500,000 draws after burning the initial 20 percent draws.
  - Posterior means and error bands around impulse responses constructed from a random subsample of the MCMC chain of length 100,000 with replacement.

---

### 3    Empirical Results

### Posterior distributions of contemporaneous coefficients (matrix B)
- Data are informative: posterior distributions are noticeably different from priors.
- Global activity immediate-impact parameters (first row) under sign restrictions:
  - Posterior averages for constrained parameters around 0.13 (significantly lower than prior mean of 1).
  - Global activity is rather irresponsive to permanent oil market specific shocks.
  - Posterior mean of B_{1,4} = −0.36 reflects a short-term positive reaction of global activity to a permanent demand shock once the variable is defined as ˆx_t = x_t − Ψ_t (global activity reacts mildly to the permanent shock before reaching new steady state).
- Immediate reaction of oil availability:
  - Constrained parameters for speculative and transitory global demand shocks have posterior means 0.27 and 0.43, respectively.
  - Contemporaneous effect of the permanent oil market shock shows a negative posterior mean for B_{2,1}; impulse responses build over time to a positive steady state.
- Posterior means of third and fourth row elements of B move toward higher absolute values, reflecting large volatility of oil inventories and real prices.

### Impulse-response functions (Figure 3) — key findings
- Responses shown for a one percent unexpected increase in each structural shock (ε_t).
- Supply shocks (transitory vs permanent):
  - Qualitatively similar effects, differing in timing and amplitude.
  - Transitory supply shocks:
    - Larger contemporaneous effect on real oil price and oil production than permanent supply shocks.
    - Real price reaction dissipates after 10 quarters.
    - Oil availability remains significantly above long-term level for about five years.
  - Permanent supply shocks:
    - Global output eventually rises to a higher medium-run level, with the response not rising significantly above zero until about eight quarters after the shock; maximum impact of 0.4 percent.
    - Cumulative response of global output following a permanent innovation is significantly larger than following a transitory shock — highlighting the importance of distinguishing shock types.
- Inventories:
  - Tend to react strongly contemporaneously and then slowly decline to below zero in subsequent periods.
  - Following a transitory oil supply shock, both global output and inventories increase on impact, but inventories show a persistent positive response while global output does not.
- Speculative shock:
  - Persistent negative reaction of global output, statistically significant for about 5 years.
  - Global output attains a lowest value of −0.26 percent after eight quarters.
  - Oil production responds positively to speculative shock (consistent with the imposed contemporaneous sign restrictions and contrary to some previous findings).
  - Short-term oil price reaction to speculative shock resembles that of a negative permanent supply shock, indicating an important role for speculative innovations in oil price dynamics.
- Demand shocks (global activity side):
  - Shocks imply a moderate and not significant positive change in oil prices which revert quickly.
  - A permanent disruption to global revenue causes inventories to increase on impact by almost 0.7 percent.
  - Permanent global activity shock yields a negative and persistent response of oil production (puzzling outcome):
    - Authors interpret this as a novel empirical finding: a persistent innovation lowering dependency on oil (embedded in permanent component of global activity) can reduce oil production and build inventories while oil prices remain virtually stable.
    - Two possible rationales:
      - Innovations and adoption of alternative energy sources embedded in the permanent component of global activity.
      - Decline in share of oil-dependent economic activities (industrial sector) reflected in global activity trend.
    - The authors plan to further investigate this mechanism by contrasting transmission when industrial production is used as measure of global oil demand (discussed in Section 5).

### Quantitative impulse-response highlights
- Maximum impact of global output after a permanent supply shock: 0.4 percent.
- Lowest value of global output after a speculative shock: −0.26 percent (after eight quarters).
- Inventories increase on impact by almost 0.7 percent after a permanent global demand disruption.
- Duration of notable responses:
  - Real price dissipates after 10 quarters for transitory supply shock.
  - Global output response to speculative shock remains significant for about 5 years.
  - Oil availability elevated for about five years after a transitory supply shock.
  - Global output reaches notable positive response around eight quarters after a permanent supply shock.

*Source: wpiea2020047-print-pdf — 2.1 Measuring global activity*

### 3.2    Variance decomposition

### 3.2    Variance decomposition

### Short- and long-run drivers of fluctuations
- Transitory shocks to oil demand are the most important driving force of global activity fluctuations in the short-term — more than twice the permanent shock — while the oil market innovations yield about 14 percent of the variable dynamics.
- Oil production variability is mainly explained by unanticipated transitory supply shocks.
- Shocks to global activities explain about 14 percent of oil production dynamics in the short run and remain virtually unchanged in the long run, a contribution almost equivalent to that of speculative innovations.
- A large fraction of the short-run volatility of oil inventories is attributable to speculative shocks, with a declining share over the horizon reaching 40 percent in the long-run.
- The three most important drivers of oil prices are:
  - the permanent oil-supply shocks,
  - the transitory oil-supply shocks,
  - the speculative shocks (explaining between 32 and 34 percent of the volatility over different horizons).
- Permanent and transitory oil-supply shocks together account for about two-thirds of oil price fluctuations in both short- and long-term horizons.

### Key numbers from Table 2 (selected horizons and variables)
- Horizon 1:
  - Global activity: ∆Ψs t = 2.89; φs t = 4.98; φs p t = 5.79; ∆Ψd t = 26.27; φd t = 60.07
  - Oil availability: ∆Ψs t = 10.83; φs t = 62.46; φs p t = 12.73; ∆Ψd t = 9.16; φd t = 4.82
  - Inventories: ∆Ψs t = 24.22; φs t = 16.85; φs p t = 7.13; ∆Ψd t = 8.89; φd t = 2.91
  - Oil prices: ∆Ψs t = 28.37; φs t = 37.28; φs p t = 34.02; ∆Ψd t = 0.20; φd t = 0.12
- Horizon 4:
  - Global activity: ∆Ψs t = 6.23; φs t = 6.83; φs p t = 7.09; ∆Ψd t = 27.89; φd t = 51.96
  - Oil prices: ∆Ψs t = 28.73; φs t = 37.67; φs p t = 33.21; ∆Ψd t = 0.24; φd t = 0.16
- Horizon 12:
  - Global activity: ∆Ψs t = 7.58; φs t = 7.52; φs p t = 6.92; ∆Ψd t = 27.61; φd t = 50.37
  - Oil prices: ∆Ψs t = 29.41; φs t = 37.25; φs p t = 32.92; ∆Ψd t = 0.25; φd t = 0.17
- Horizon ∞:
  - Global activity: ∆Ψs t = 8.04; φs t = 7.81; φs p t = 7.03; ∆Ψd t = 27.42; φd t = 49.70
  - Oil prices: ∆Ψs t = 29.82; φs t = 37.04; φs p t = 32.68; ∆Ψd t = 0.27; φd t = 0.20

(Note: Table 2 entries are preserved exactly as reported, including numeric precision and variable labels.)

### Robustness on trend attribution
- Assuming ηi = ηp = 1 mildly increases the contribution of permanent demand shocks in the long-term volatility of oil prices by about 1 percent.
- Estimating ηi and ηp yields posterior averages of −1.34 and 3.01, respectively; with these estimates the share of oil price volatility explained by permanent demand innovations increases to about 9 percent (still modest).

---

### 3.3    Conditional correlations

### Main conditional and unconditional correlation findings
- Observed unconditional correlation between growth rates of global GDP (∆xt) and oil production (∆zt) is 0.13.
- Conditional correlations (iteratively assuming each structural shock is the only shock) reveal:
  - Corr(∆xt, ∆zt):
    - Unconditional: 0.12
    - Conditional on ∆Ψs t: 0.10
    - Conditional on φs t: 0.56
    - Conditional on φs p t: -0.71
    - Conditional on ∆Ψd t: -0.54
    - Conditional on φd t: 0.79
  - Corr(∆xt, ∆it):
    - Observed: -0.12
    - Unconditional: -0.10
    - Conditional on φs p t: 0.30
    - Conditional on ∆Ψd t: -0.73
    - Conditional on φd t: 0.52
  - Corr(∆xt, ∆pt):
    - Observed: -0.15
    - Unconditional: -0.17
    - Conditional on φs p t: -0.71
    - Conditional on ∆Ψd t: -0.70
    - Conditional on φd t: 0.37
- The negative conditional comovement following permanent demand innovations explains about:
  - 27 percent of the volatility of global GDP,
  - 10 percent of the volatility of oil production.
- Speculative shocks also yield negative correlations between global GDP and oil production, but variance decomposition shows a relatively weak role for speculative shocks in explaining dynamics of global GDP and oil production.

### Notes on interpretation
- Observed correlations computed directly from data; unconditional and conditional correlation estimates based on the benchmark empirical model.
- Numbers in square brackets in Table 3 represent the 16th and 86th percentile Hall confidence bands.

---

### 3.4    Historical decomposition of oil prices

### Overview of drivers across episodes (1974–2018)
- Until the late 1980s speculative shocks contributed considerably to large swings in oil prices; their impact declined in subsequent decades.
- Transitory and permanent supply innovations played important roles across many episodes, with differing magnitudes and persistence.
- Distinguishing transitory vs. permanent supply shocks is informed by:
  - oil production and inventories’ reactions showing different persistence depending on the shock,
  - sign of global activity trajectory in the medium term — positive for persistent supply shock, negative for transitory supply shock.

### Selected episode summaries with model attribution
- Iranian revolution (1978):
  - Speculative shocks contributed to the oil price increase later in the episode and dampened international prices at the outbreak.
  - Permanent supply shocks are responsible for the bulk of price volatility in that period.
  - Transitory supply shock (interpreted as OPEC response) simultaneously impacted prices.
- The Collapse of OPEC in 1986:
  - Model suggests positive transitory and demand shocks occurred concurrently, with the transitory shock contributing slightly more to the collapse in prices.
  - Speculative shock remained virtually muted during the episode.
- The Gulf War in 1990:
  - The supply shock was almost entirely perceived as transitory; permanent shocks barely affected prices then.
  - Speculative innovations moderately affected prices and heavily contributed to their decline in subsequent periods.
- The Venezuelan Crisis and Iraq War of 2002–03:
  - Decomposition captures a sharp temporary decline in oil supply accounting for about the total price upswing.
  - Speculative innovations became prominent drivers of price softening afterward.
  - Permanent shocks to supply trend remained muted.
- The Global Financial Crisis of 2008:
  - Bulk of the decline in oil prices attributed to negative oil-specific permanent supply shocks.
  - Global activity shocks played a minor role as quarterly real growth rates remained positive in 2008 and 2009 except in 2008Q3.
  - Negative shocks on inventories and transitory supply shocks equally contributed to the price dip; combined impact almost matched the permanent supply shock.
- The Oil Price Plunge of 2014:
  - Decline starting from 2014Q2 amid a large increase in oil production and inventories rules out speculative shocks as contributors.
  - Supply factors explain most of the decline in oil prices since mid-2014.
  - Reported evidence that the oil “break-even” price declined by roughly 50 per cent between 2013 and 2016 supports oil market–specific shocks as main drivers.

---

### 4    Iterative estimation

### Rolling-window variance decomposition of oil prices
- Estimation run on rolling-windows with fixed length 120 observations (quarterly data), starting from 1974Q1 and moving the starting date by one year.
- Figure 5 (ten-year rolling window results) findings:
  - In early samples, speculative innovations explain above 50 percent of oil price volatility until sample 1979Q1–2008Q4; then their contribution declines in later windows.
  - In more recent samples, oil supply shocks (permanent and transitory) explain about 80 percent of overall volatility.
  - Innovations to the permanent component of oil supply increased from virtually 0 in early samples to more than 60 percent in recent samples.

### Standard deviation parameter behavior over expanding windows
- Posterior means and 90 percent confidence intervals show:
  - Posterior mean of the permanent oil market shock varies considerably as sample covers more recent periods and is more precisely estimated.
  - This change in the permanent shock’s posterior mean is closely related to the increased share of permanent shocks in the variance decomposition of oil prices over time.
  - Posterior standard deviation means of the remaining shocks broadly follow a milder decline.
  - Volatility of speculative shocks shows a small decline only towards the end of the sample, suggesting propagation mechanisms largely drive the decline in the share of oil price fluctuations attributable to speculative shocks.

*Source: wpiea2020047-print-pdf - 3.2    Variance decomposition*

### 4.2    Oil price reaction to speculative shocks

### 4.2    Oil price reaction to speculative shocks

### Oil price response over time
- Impulse responses to an unexpected speculative shock evolve markedly over the sample, reflecting changes in the dynamic parameters of the autoregressive model.
- Early samples: responses are hump shaped with a peak reached after two to three quarters.
- Later samples: trajectories drastically decline and remain below zero shortly after the impulse.
- On impact, the oil price response drops from 12 percent to 2 percent in the latest sample (1988Q1–2017Q4).
- Interpretation:
  - The decreasing contribution of speculative shocks to oil market dynamics is principally implied by changed oil market dynamics (impulse-response functions).
  - Iterative estimation shows very little evidence of a change in the variance of the speculative shock; hence the change in contribution is driven mainly by altered dynamic responses.

### Robustness check — alternative measure of global activity (Baumeister and Hamilton (2018) industrial index)
- Purpose:
  - Compare empirical results using an alternative measure of global activity (an extended industrial production index in the OECD and 6 major other countries).
  - Assess whether the negative conditional correlation between global activity and oil production to permanent demand shocks is due to technological innovations (less dependence on oil) or a change in the share of industrial activity in aggregate output.
- Key empirical findings from Figure 8 (median impulse-response functions using global GDP vs. aggregated industrial index):
  - First: Changing the definition of global economic activity has little consequence on outcomes of transitory shocks, except for the transitory demand shock where the global activity index becomes less responsive.
  - Second: The permanent supply shock has a positive and persistent impact on the global industrial index and the trajectory is significantly different from the benchmark case especially in the short-term horizon.
  - Third: For oil production response to a permanent demand shock, the impact response turns out to be positive as opposed to the benchmark case; the two trajectories are significantly different over the 5-year horizon and beyond.
  - In parallel in this robustness specification:
    - Inventories decline and remain below zero for several years.
    - Oil prices persistently increase.
- Interpretation:
  - A negative response of oil production to shocks to the trend of global economic activity can emerge from continued energy efficiency gains in advanced economies (lessened oil dependence).
  - Alternatively, it can reflect a decline in the share of industrial activities in total output.
  - Figure 8 suggests permanent shocks on global demand tend to capture changes in the share of the industrial sector rather than decreasing dependency on oil, because oil production still exhibits a positive persistent response.
  - This does not preclude a less pronounced long-term effect from innovation and energy diversification.

### Methodology summary (as used across analysis)
- Empirical approach distinguishes between transitory and permanent shocks in global economic activity and the oil market.
- Methodological components:
  - Flexibility of SVAR techniques.
  - Narrative approach to identify structural shocks.
  - Kalman filtering to infer permanent and transitory components of oil demand and supply shocks.
- Advantages:
  - Explicitly accounts for sources contributing to non-stationarity of variables.
  - Reflects identified cointegration relations between endogenous variables.
  - Simple to use: relies on standard Bayesian techniques and the Kalman filter to assess the relative importance of shock components over time.

### Empirical conclusions and implications
- Supply side has been an important determinant of the real price of oil in both the variance decomposition and prominent episodes of oil fluctuations.
- In average, the transitory and permanent shocks to oil supply played an equal role; however, shocks on the trend witnessed a growing influence on the oil price volatility over time.
- Speculative shocks:
  - Exerted significant influence over major oil price swings in the 1970s and 1980s.
  - Their effect drastically declined in the most recent years, stemming from the weaker volatility of inventories and a declining price responsiveness to speculative innovations.
- Suggested extension:
  - Modify the identification scheme of oil-sector-specific shocks to allow agents to learn about the uncertain nature and persistence of underlying shocks.
  - Expectation: perceptions about the persistence of shocks could be important in explaining the contribution of permanent versus transitory shocks to the forecast error variance decomposition of endogenous variables affecting the global oil market.

*Source: wpiea2020047-print-pdf - 4.2    Oil price reaction to speculative shocks*

### References

### References and Appendices (Selected content)

### Major literature and methodological references
- Works addressing oil market dynamics, identification of shocks, and econometric techniques:
  - Al Rousan, S., R. Sbia, and B. K. O. Tas (2018). A dynamic network analysis of the world oil market: Analysis of OPEC and non-OPEC members. Energy Economics 75, 28–41.
  - Antolín-Díaz, J. and J. F. Rubio-Ramírez (2018). Narrative sign restrictions for SVARs. American Economic Review 108, 2802–2829.
  - Baumeister, C. J. and J. D. Hamilton (2018). Structural interpretation of vector autoregressions with incomplete identification: Revisiting the role of oil supply and demand shocks. NBER Working Paper No. 24167.
  - Blanchard, O. J. and J. Galí (2007, 2010). The macroeconomic effects of oil price shocks: Why are the 2000s so different from the 1970s? (NBER Chapters / University of Chicago Press).
  - Clemente, J., A. Montañés, and M. Reyes (1998). Testing for a unit root in variables with a double change in the mean. Economics Letters 59, 175–182.
  - Dickey, D. A. and W. A. Fuller (1979). Distribution of the estimators or autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.
  - Engle, R. F. and C. W. J. Granger (1987). Cointegration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.
  - Gregory, A. W. and B. E. Hansen (1996). Tests for cointegration in models with regime and trend shifts. Oxford Bulletin of Economics and Statistics 58, 555–560.
  - Kilian, L. (2008, 2009a, 2009b) and Kilian & Murphy (2014). Work on exogenous oil supply shocks, disentangling demand and supply shocks, and inventories/speculative trading.
  - Uribe, M. (2018). The neo-Fisher effect: Econometric evidence from empirical and optimizing models. NBER working paper 25089.
- Econometric testing and software:
  - Baum, C. F. (2004). A review of stata 8.1 and its time series capabilities. International Journal of Forecasting 20, 151–161.
  - Phillips, P. C. B. and P. Perron (1988). Testing for a unit root in time series regression. Biometrika 75, 335–346.
  - Kwiatkowski, D., P. C. B. Phillips, P. Schmidt, and Y. Shin (1992). Testing the null hypothesis of stationary against the alternative of a unit root. Journal of Econometrics 54, 159–178.
- Contributions on oil-market-specific phenomena:
  - Dvir, E. and K. Rogoff (2014). Demand effects and speculation in oil markets: Theory and evidence. Journal of International Money and Finance 42, 113–128.
  - Knittel, C. R. and R. S. Pindyck (2016). The simple economics of commodity price speculation. American Economic Journal: Macroeconomics 8, 85–110.
  - Lippi, F. and A. Nobili (2012). Oil and the macroeconomy: A quantitative structural analysis. Journal of the European Economic Association 10, 1059–1083.

### A. Data sources (A.1 Global real activity; A.2 Oil market variables)
- Global real activity construction:
  - Proposed PPP-converted real quarterly GDP in U.S. dollars for all OECD members plus six emerging countries: Brazil, China, India, Indonesia, Russia, South Africa.
  - Quarterly OECD and South Africa real GDP (current prices, current PPPs, seasonally adjusted) extracted from OECD.Stat covering 1973Q1 until 2018Q1.
  - Quarterly nominal GDP availability start dates for remaining countries:
    - Brazil: 1996Q1
    - China: 1992Q1
    - India: 1996Q2
    - Indonesia: 1990Q1
  - Earlier-period observations constructed from annual World Bank Indicators local currency data, converted to PPP using implied PPP conversion rate from the IMF World Economic Outlook database.
  - PPP index availability note: PPP index available starting from 1980; linear trend based on the 10 subsequent year applied reversely to obtain index up to 1973.
  - Frequency conversion: Litterman frequency conversion method used with OECD quarterly GDP as indicator.
  - Final global activity variable: sum of seasonally adjusted OECD and country-specific GDP PPP in USD, divided by the U.S. consumer price index to obtain real terms.
- Oil market variables:
  - World oil production: U.S. Energy Information Administration series (thousands of barrels per day) monthly from January 1973 to August 2018; February day-count differences accounted for; series transformed from monthly to quarterly.
  - OECD petroleum stocks: extracted from EIA; monthly from 1988 and yearly from 1973; quarterly stocks from 1973Q1 constructed by scaling U.S. crude oil inventory data by the ratio of OECD petroleum inventories over U.S. petroleum inventories (following Kilian and Murphy (2014), Dvir and Rogoff (2014), Baumeister and Hamilton (2018)).
  - Global oil availability: measured as sum of global stock of previous quarter and production of current quarter (constructed as in Dvir and Rogoff (2014)).
  - Nominal oil prices: U.S. Crude Oil Imported Acquisition Cost by Refiners (Dollars per Barrel) from EIA, ranging from January 1974 to October 2018; series selected for stability relative to alternatives like WTI.
  - Deflation: nominal oil prices and global GDP deflated by U.S. consumer price index for All Urban Consumers from Federal Reserve Economic Data (All Items, Index 1982-1984=100, Quarterly, Seasonally Adjusted).

### B. Unit roots and cointegration (B.1 Unit Roots; B.2 Cointegration)
- Unit root testing approach:
  - Regular unit root tests (ADF, PP, KPSS) may be misleading if structural breaks exist.
  - Implemented Clemente et al. (1998) unit root test with structural breaks (AO and IO models) to detect up to two structural breaks and capture sudden (AO) or gradual (IO) shifts.
  - Results:
    - Table A.1 reports Clemente-Montanes-Reyes test results indicating none of the 5 test statistics at level exceed approximate 5 percent critical value; t-statistics for both time breaks are very significant in both AO and IO models.
    - Conclusion: existence of unit root and significant level shifts in all series at levels; first differences are stationary and variables are integrated of order 1.
    - Note: Lag order k reported in parentheses in Table A.1. The 5 percent critical value for both models to reject the null of unit root is -5.490.
    - Table A.1 entries (examples):
      - x_t additive outliers t-stat −2.887(2) TB1 1995Q1 ∗∗∗ TB2 2008Q1 ∗∗∗ unit root yes; innovative outliers t-stat −1.388(5) TB1 1982Q2 ∗∗∗ TB2 2003Q1 ∗∗ unit root yes.
      - z_t additive outliers t-stat −2.830(4) TB1 1988Q4 ∗∗∗ TB2 2002Q4 ∗∗∗ unit root yes; innovative outliers t-stat −3.305(5) TB1 1987Q1 ∗∗∗ TB2 2003Q1 ∗∗∗ unit root yes.
      - i_t additive outliers t-stat −3.841(4) TB1 1978Q3 ∗∗∗ TB2 1990Q3 ∗∗∗ unit root yes; innovative outliers t-stat −4.467(4) TB1 1978Q4 ∗∗ TB2 1996Q1 ∗∗∗ unit root yes.
      - p_t additive outliers t-stat −4.957(1) TB1 1986Q4 ∗∗∗ TB2 2004Q4 ∗∗∗ unit root yes; innovative outliers t-stat −4.750(3) TB1 1985Q3 ∗∗∗ TB2 2003Q3 ∗∗∗ unit root yes.
    - Stars indicate significance of the time break at 1 and 5 percent levels.
- Cointegration testing using Gregory and Hansen (1996):
  - Motivation: standard cointegration tests (Engle-Granger, Johansen, Enders and Siklos) are biased in presence of structural break; Gregory and Hansen (1996) allows one endogenous structural break and provides ADF*, Z*t, Z*α statistics across four models:
    - Model 1: Level Shift (C)
    - Model 2: Level Shift with Trend (C/T)
    - Model 3: Regime Shift where intercept and slope change (C/S)
    - Model 4: Regime Shift where intercept, slope, and trend change (C/S/T)
  - Cointegration among oil variables (World availability, Global inventories, real oil prices):
    - Table A.2 results (all four models) show absolute values of ADF*, Z*t, Z*α greater than 1 percent critical values; reject null of no cointegration and accept alternative.
    - All four models suggest break date 1980Q3.
    - Table A.2 examples:
      - Model 1: ADF* −5.861 break 1980Q3; asymptotic critical values 1% −5.44, 5% −4.92, 10% −4.69.
      - Z*t −9.58 break 1980Q3; Z*α −109.72 break 1980Q3 (critical values listed).
      - Model 3 and 4 Z*α −122.77 break 1980Q3 (critical values listed).
    - Additional note: Johansen’s cointegration test also run, finding at most one cointegrating equation.
  - Cointegration between oil variables and global GDP:
    - Table A.3 results: absolute values of the three test statistics (in all four models) are lower than critical values even at 10 percent; cannot reject the null hypothesis of no cointegration.
    - Break dates differ among models (examples from Table A.3):
      - Model 1: ADF* −4.16 break 2004q2 (critical values 1% −5.77, 5% −5.28, 10% −5.02).
      - Model 3: ADF* −4.84 break 2003q1 (critical values 1% −6.51, 5% −6.00, 10% −5.75).
      - Model 4: Z*t −5.92 break 2003q1; Z*α −55.39 break 2003q1 (critical values 1% −90.84, 5% −78.87, 10% −72.75).
    - Conclusion: sources of non-stationarity differ between oil variables and global GDP.

### C. Estimated model: state-space specification and identification (Appendix C)
- Endogenous variables and transformed state:
  - Endogenous in levels: y_t ≡ [x_t z_t i_t p_t]′ where:
    - x_t = log real global output
    - z_t = log global oil production
    - i_t = log global oil inventories
    - p_t = log global real oil price
  - Transformed vector ̃y_t defined to account for trend shocks:
    - ̃y_t ≡ [x_t − Ψ^d_t, z_t − Ψ^s_t, i_t − Ψ^s_t, p_t − Ψ^s_t]′
  - Deviation from unconditional mean: ˆy_t ≡ ̃y_t − E(̃y_t)
- Autoregressive law of motion:
  - ˆy_t = sum_{j=1}^L A_j ˆy_{t−j} + B u_t  (Equation 5)
  - u_t ≡ [ψ^s_t φ^s p_t φ^s_t ψ^d_t φ^d_t]′ with definitions:
    - ψ^s_t ≡ ∆Ψ^s_t − E(∆Ψ^s_t)
    - ψ^d_t ≡ ∆Ψ^d_t − E(∆Ψ^d_t)
  - A_j are 4-by-4 matrices; B is 4-by-5; L is lag length of empirical model.
- Dynamics of u_t:
  - u_{t+1} = κ d_{t+1} + ρ u_t + ω ε_{t+1}  (Equation 6)
  - ρ is 5-by-5 diagonal with elements in [0,1)
  - ω is 5-by-5 diagonal of innovation variances
  - ε_{t+1} is 5-by-1 i.i.d. normal with variances equal to 1
- Observables and measurement equations:
  - Observed variables are growth rates expressed in percent per quarter: ∆x_t, ∆z_t, ∆i_t, ∆p_t.
  - They relate to unobserved states by:
    - ∆x_t = ∆Ψ^d + ˆx_t − ˆx_{t−1} + ψ^d_t
    - ∆z_t = ∆Ψ^s + ˆz_t − ˆz_{t−1} + ψ^s_t
    - ∆i_t = ∆Ψ^s + ˆi_t − ˆi_{t−1} + ψ^s_t
    - ∆p_t = ∆Ψ^s + ˆp_t − ˆp_{t−1} + ψ^s_t  (Equation 7)
  - o_t ≡ [∆x_t ∆z_t ∆i_t ∆p_t]′ denote observed quarter-t vector.
- State-space representation and identification approach:
  - State vector ζ_t = [ˆy_t ˆy_{t−1} ... ˆy_{t−L+1} u_t d_t]′.
  - State equations: ζ_{t+1} = C ζ_t + D ε_{t+1}; observation equation: o_t = G′ + H′ ζ_t.
  - C, D, G, H are functions of A_i, B, ρ, ω, E(∆Ψ^d_t), and E(∆Ψ^s_t).
  - Dimensions and definitions:
    - L = number of lags (=2)
    - S = number of shocks (=4)
    - V = number of endogenous variables (=4)
    - Specific matrix forms given (examples):
      - C matrix block structure shown (using ABρ /0 notation)
      - D = [Bω /0_{V(L−1),S} ω /0_{1,S}]′
      - G = [E(∆Ψ^d_t) E(∆Ψ^s_t) E(∆Ψ^s_t) E(∆Ψ^s_t)]
      - H′ = [M_ζ /0_{V,V(L−2)} M_u] with M_ζ and M_u provided explicitly:
        - M_ζ = 5×5 matrix with entries including values 1, 0, −10000, −1000, −100, 1000, −1
        - M_u = 4×5 matrix with specific 0/1 pattern
- Identification strategy:
  - Method adapts Uribe (2018) to identify transitory and permanent shocks by combining sign restrictions with a Kalman filter framework to handle unobserved variables (more shocks than endogenous variables) in an augmented SVAR setup.

*Italic: Content derived from the "References" and appended appendices (A–C) of the provided IMF content unit.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020047-print-pdf.pdf_