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Forecasting the Price of Oil
Ron Alquist Lutz Kilian Robert J. Vigfusson
Bank of Canada University of Michigan Federal Reserve Board
CEPR
Prepared for the Handbook of Economic Forecasting
Graham Elliott and Allan Timmermann (eds.)
This presentation reflects the authors’ own views and should not be attributed to the Bank of Canada, the Federal Reserve System, or the Board of Governors of the Federal Reserve System.
Motivation
Potential to improve forecast accuracy of macroeconomic aggregates and macroeconomic policy responses
Forecasts of the prices of oil and its derivatives like gasoline or heating oil important for:
— Modeling purchases of energy-intensive durables
— Predicting carbon emissions and climate change
— Designing regulatory policies (fuel standards, gasoline taxes)
— Business decisions (e.g., airlines, auto manufacturers, utilities)
Key Oil Price Series
Since 1948: 1. US Producer Price Index (PPI) for crude oil
2. West Texas Intermediate (WTI) price of crude oil (almost identical with PPI series in pre-1974 era)
Since 1974: 1. Refiner’s acquisition cost (domestically produced crude oil)
2. Refiner’s acquisition cost (imported crude oil)
3. Refiner’s acquisition cost (composite) No single oil price series is perfect for all purposes.
Key Oil Price Series
Import price matters for standard theories of transmission of oil price shocks and is best proxy for global price of oil.
— Kilian (2009): Refiners’ acquisition cost (imports)
Retail energy price matters for theories of relative price shocks.
— Hamilton (2003): Crude oil PPI Retail gasoline price — Edelstein and Kilian (2009): Retail energy price Retail gasoline price
US domestic price series subject to regulation until early 1980s and unrepresentative of actual market price.
— Mork (1989): Refiners’ acquisition cost (composite)
1950 1955 1960 1965 19702
2.5
3
3.5
4
4.5
Nominal Price of Oil: 1948.1-1973.12
U.S
. D
olla
rs/B
arr
el
Actual WTI
Random Draw from Fitted Model
Pre-1974 Nominal Price of Oil
1950 1955 1960 1965 19702
2.5
3
3.5
4
4.5
Nominal Price of Oil: 1948.1-1973.12
U.S
. D
olla
rs/B
arr
el
WTI
1975 1980 1985 1990 1995 2000 2005 20100
20
40
60
80
100
120
140
U.S
. D
olla
rs/B
arr
el
Nominal Price of Oil: 1974.1-2010.4
WTI
RAC Domestic
RAC Imported
The Nominal Price of Crude Oil
1950 1955 1960 1965 1970-50
-40
-30
-20
-10
0
10
20
30
40
50
Real Price of Oil: 1948.2-1973.12
Perc
ent
Change
1975 1980 1985 1990 1995 2000 2005 2010-50
-40
-30
-20
-10
0
10
20
30
40
50
Perc
ent
Change
Real Price of Oil: 1974.2-2010.4
WTI WTI
RAC Domestic
RAC Imported
Percent Changes in the Real Price of Oil
Selective Survey of Topics in Chapter
1. Predictability in population
2. Forecasting the nominal price of oil
3. Forecasting the real price of oil
4. Joint forecasts of oil prices and US real GDP growth
5. Forecasting oil price volatility and quantifying oil price risks
6. Conclusions: How to forecast the price of oil?
1. Predictability in Population
Predictability in Population from Macroeconomic Aggregates to Price of Oil
It has become more widely accepted that price of oil is endogenous with respect to macroeconomic conditions.
— Hamilton (1983): Fails to reject null of no Granger causality from US macro aggregates to nominal oil prices pre-1973.
Thus, lagged macroeconomic aggregates should have predictive power for price of oil in population.
Predictability in population is a precondition for out-of-sample forecastability (Inoue and Kilian 2004).
Predictors of Nominal Oil Prices
US price changes: CPI inflation, %ΔM1 and %ΔM2
Commodity prices: CRB Industrial Raw Materials Index, CRB Metals Index
Other: 3-month T-bill rate, trade-weighted USD exchange rate
Commodity currencies: AUD, CAD, NZD, SAR (Chen, Rogoff,
and Rossi 2010).
Summary of Predictability Results
Strongest evidence of in-sample predictability for: — M1 — CRB indices — Currencies of some industrial commodity exporters (e.g., CAD)
Rejection of Granger non-causality at standard significance levels for WTI and RAC
Predictors of Real Oil Prices
Quarterly: US real GDP, world industrial production.
Monthly: CFNAI, US industrial production, OECD+6 industrial production, global real activity index (Kilian 2009).
Where applicable, Granger causality tests conducted on filtered series (e.g., US real GDP): — Linear — Hodrick-Prescott
— First difference
Summary of Predictability Results
Strongest evidence of in-sample predictability for linearly detrended series: — World industrial production — OECD+6 industrial production — Real activity index
Rejection of Granger non-causality at standard significance levels for real WTI and RAC
Why are linearly detrended global real activity measures good at predicting real price of oil?
US is not the world — Oil price determined in global market
GDP poor proxy for business-cycle driven fluctuations in oil demand because of large share of services — Industrial production is better indicator
Well-documented long swings in industrial commodity prices such as oil
2. Forecasting the Nominal Price of Oil
Do Oil Futures Prices Help Predict the Spot Price?
No-change (benchmark model)
𝑆 𝑡+ |𝑡 = 𝑆𝑡 = 1, 3, 6, 9, 12
Futures Price
𝑆 𝑡+ |𝑡 = 𝐹𝑡()
= 1, 3, 6, 9, 12
Futures Spread
𝑆 𝑡+ |𝑡 = 𝑆𝑡 1 + 𝛼 + 𝛽 ln(𝐹𝑡
/𝑆𝑡) , = 1, 3, 6, 9, 12
where 𝛼 and 𝛽 are recursive OLS estimates from
∆𝑠𝑡+ = 𝛼 + 𝛽 𝑓𝑡
− 𝑠𝑡 + 𝑢𝑡+
Forecast Accuracy of Futures Prices
Forecast Evaluation Period: 1991.1-2009.12
Monthly Forecasts
Small (≤ 6%) improvements in forecast accuracy
Not statistically significant
Daily Forecasts Short-horizon (1-12 months)
Similar to monthly forecasts, except at 12-month horizon.
Long-horizon (2-7 years)
No improvements in forecast accuracy
Alternative Monthly Forecasting Methods
Local trends and structural change
Recursive drift 𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝛼 = 1,… , 12
Rolling drift
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + ∆𝑠 𝑡()
= 1,… , 12
Random walk in growth
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + ∆𝑠𝑡 = 1,… , 12
Hotelling (1931)
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝑖𝑡 , /12
= 3, 6, 12
Alternative Monthly Forecasting Methods
CRB commodity prices 𝑆 𝑡+ |𝑡 = 𝑆𝑡 1 + ∆𝑝𝑡
𝑐𝑜𝑚 = 1, 3, 6, 9, 12
𝑆 𝑡+ |𝑡 = 𝑆𝑡 1 + ∆𝑝 𝑡 ,𝑐𝑜𝑚 = 1, 3, 6, 9, 12
where 𝑐𝑜𝑚 ∈ 𝑖𝑛𝑑,𝑚𝑒𝑡
Commodity currencies (Chen, Rogoff, and Rossi 2010)
𝑆 𝑡+ |𝑡 = 𝑆𝑡 1 + ∆𝑒𝑡𝑖
= 1, 3, 6, 9, 12
𝑆 𝑡+ |𝑡 = 𝑆𝑡 1 + ∆𝑒 𝑡 ,𝑖 = 1, 3, 6, 9, 12
where 𝑖 ∈ 𝐶𝑎𝑛𝑎𝑑𝑎,𝐴𝑢𝑠𝑡𝑟𝑎𝑙𝑖𝑎, 𝑆𝑜𝑢𝑡 𝐴𝑓𝑟𝑖𝑐𝑎
Summary of Forecasting Results
CRB Indices Up to 3-month horizon: Large (9-25%) and statistically significant improvements in forecast accuracy.
Commodity Currencies Up to 3-month horizon: Small (7-13%) but statistically significant improvements in forecast accuracy for AUD and CAD.
Survey Forecasts
Monthly oil price forecasts from Consensus Economics, Inc.
𝑆 𝑡+|𝑡 = 𝑆𝑡 ,𝐶𝐹 = 3, 12
Quarterly oil price forecasts from Energy Information Administration (EIA)
𝑆 𝑡+|𝑡 = 𝑆𝑡 ,𝐸𝐼𝐴 = 3, 12
Monthly Michigan Survey of Consumers (MSC) forecasts of the price of gasoline
𝑃 𝑡+|𝑡𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒
= 𝑃𝑡 ,𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒 ,𝑀𝑆𝐶
= 60
Survey Forecasts Relative to No-Change Forecast = 3 = 12 = 60
MSPE Ratio
Success Ratio
MSPE Ratio
Success Ratio
MSPE Ratio
Success Ratio
𝑆 𝑡+|𝑡 = 𝑆𝑡 ,𝐶𝐹
1.519 0.447 0.944 0.539 - - 𝑆 𝑡+|𝑡 = 𝑆𝑡 ,
𝐸𝐼𝐴 0.918 0.417 0.973 0.562 - -
𝑃 𝑡+|𝑡𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒
= 𝑃𝑡 ,𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒 ,𝑀𝑆𝐶
- - - - 0.765 0.9071
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝜋𝑡 ,𝑀𝑆𝐶 - - 1.047 0.5661 - -
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝜋𝑡 ,𝑆𝑃𝐹 - - 1.016 0.5791 0.855 0.8111
NOTES: Boldface indicates statistical significance at the 10% level. 1 No significance test possible due to lack of variation in success ratio.
1994 1996 1998 2000 2002 2004 2006 2008 20100
100
200
300
400
500
600
1994 1996 1998 2000 2002 2004 2006 2008 20100
100
200
300
400
500
1994 1996 1998 2000 2002 2004 2006 2008 20100
100
200
300
400
500
600
Expected price 5 years from now
Current price
Expected real price 5 years from now
Current price
Expected price 5 years from now
Actual price 5 years from now
Household Expectations of U.S. Retail Gasoline Prices (Cents/Gallon) 1992.11-2010.1
Survey Forecasts Relative to No-Change Forecast = 3 = 12 = 60
MSPE Ratio
Success Ratio
MSPE Ratio
Success Ratio
MSPE Ratio
Success Ratio
𝑆 𝑡+|𝑡 = 𝑆𝑡 ,𝐶𝐹
1.519 0.447 0.944 0.539 - - 𝑆 𝑡+|𝑡 = 𝑆𝑡 ,
𝐸𝐼𝐴 0.918 0.417 0.973 0.562 - -
𝑃 𝑡+|𝑡𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒
= 𝑃𝑡 ,𝑔𝑎𝑠𝑜𝑙𝑖𝑛𝑒 ,𝑀𝑆𝐶
- - - - 0.765 0.9071
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝜋𝑡 ,𝑀𝑆𝐶 - - 1.047 0.5661 - -
𝑆 𝑡+|𝑡 = 𝑆𝑡 1 + 𝜋𝑡 ,𝑆𝑃𝐹 - - 1.016 0.5791 0.855 0.8111
NOTES: Boldface indicates statistical significance at the 10% level. 1 No significance test possible due to lack of variation in success ratio.
3. Forecasting the Real Price of Oil
Forecasting Models of the Real Price of Oil
AR, ARMA, ARIMA
Kilian-Murphy (2010) VAR 1. Percent change in world oil production 2. Real activity index 3. Change in inventories 4. Real price of oil
Consider two specifications — Unrestricted VAR
— Bayesian VAR (Giannone, Lenza, Primiceri 2010)
Summary of Forecasting Results
AR(I)MA Models Statistically significant improvements (≤ 17%) in forecast accuracy at 1-3 month horizon.
VAR Models
UVAR: Statistically significant improvements in forecast accuracy (≤ 19%) at 1-6 month horizon.
BVAR: Results similar to those from unrestricted VAR — Shrinkage improves forecast accuracy in more heavily
parameterized model with 24 lags.
2010 2011-100
-50
0
50
100
Forecast Adjustment Based on U.S. Oil Production Stimulus Scenario
Perc
ent
2010 2011-100
-50
0
50
100
Forecast Adjustment Based on 2007-08 World Recovery Scenario
Perc
ent
2010 2011-100
-50
0
50
100
Forecast Adjustment Based on Iran 1979 Speculation Scenario
Perc
ent
Forecasting Scenarios for Real Price of Oil based on Kilian-Murphy SVAR Conditional Forecast Expressed Relative to Baseline Forecast
4. Joint Forecasts of Oil Prices and US Real GDP Growth
Joint Forecasting Models
A key reason price of oil considered important is its perceived predictive power for US real GDP.
Examining this predictive power requires joint forecasting model of price of oil and domestic real activity.
Two Models: — VAR models — Nonlinear dynamic forecasting models
Bivariate VAR Models
Unrestricted VAR
∆𝑟𝑡 = 𝛼1 + 𝐵11,𝑖∆𝑟𝑡−𝑖4𝑖=1 + 𝐵12,𝑖∆𝑦𝑡−𝑖
4𝑖=1 + 𝑒1,𝑡
∆𝑦𝑡 = 𝛼2 + 𝐵21,𝑖∆𝑦𝑡−𝑖4𝑖=1 + 𝐵22,𝑖∆𝑟𝑡−𝑖
4𝑖=1 + 𝑒2,𝑡
where 𝑟𝑡 is the real price of oil; and 𝑦𝑡 is US real GDP. Restricted VAR Assume oil price is exogenous (𝐵12,𝑖 = 0 ∀ 𝑖)
Summary Only small (1-8%) improvements in forecast accuracy at 3-8
quarter horizon relative to AR(4) benchmark.
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010-10
-5
0
5
10
Perc
ent
at
Annual R
ate
s
(a) Four Quarters Ahead
Realizations
AR Forecast
Linear VAR Forecast
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010-10
-5
0
5
10
Perc
ent
at
Annual R
ate
s
(b) One Quarter Ahead
Realizations
AR Forecast
Linear VAR Forecast
Autoregressive Forecasts of Cumulative Real GDP Growth based on the Real Price of Oil
Possible Explanations for Limited Success of Linear Forecasting Models
Reflects inability to forecast more accurately real price of oil
— No. Can rule this out.
Underlying predictive relationship is weak
Predictive relationship between oil prices and domestic macroeconomic aggregates is time-varying:
1. Variation in the expenditure share of energy (Edelstein and Kilian 2009)
2. Variation in the extent of price regulation (Ramey and Vine 2010)
3. Variation due to changes in composition of underlying oil demand and oil supply shocks (Kilian 2009)
Linear forecasting models inherently misspecified
Nonlinear Dynamic Forecasting Models
Relationship between oil prices and US real GDP may be asymmetric in that only oil-price increases matter (Mork 1989; Hamilton 1996, 2003).
Unrestricted multivariate nonlinear forecasting model
∆𝑟𝑡 = 𝛼1 + 𝐵11,𝑖∆𝑟𝑡−𝑖4𝑖=1 + 𝐵12,𝑖∆𝑦𝑡−𝑖
4𝑖=1 + 𝑒1,𝑡
∆𝑦𝑡 = 𝛼2 + 𝐵21,𝑖∆𝑦𝑡−𝑖4𝑖=1 + 𝐵22,𝑖∆𝑟𝑡−𝑖
4𝑖=1 + 𝛿𝑖𝑟 𝑡−𝑖
4𝑖=1 + 𝑒2,𝑡
where 𝑟 𝑡 ∈ ∆𝑟𝑡𝑛𝑒𝑡 ,+,3𝑦𝑟
,∆𝑟𝑡𝑛𝑒𝑡 ,+,1𝑦𝑟
,∆𝑟𝑡+ is censored oil-price
increase variable (e.g., ∆𝑟𝑡𝑛𝑒𝑡 ,+,3𝑦𝑟
= max{0, 𝑟𝑡 − 𝑟𝑡∗} and 𝑟𝑡
∗ is the
maximum log real price of oil during the past 3 years).
Summary of Nonlinear Forecasting Results
Can forecast US GDP growth using model with:
— Exogenous oil prices (𝐵12,𝑖 = 0 ∀ 𝑖) and
— No feedback from lagged oil prices to GDP (𝐵22,𝑖 = 0 ∀ 𝑖)
— Models that combine restrictions 𝐵12,𝑖 = 0 ∀ 𝑖 and
𝐵22,𝑖 = 0 ∀ 𝑖
3-year net nominal and real oil-price increases achieve forecast-accuracy improvements for US real GDP growth at 4-quarter horizon.
1992 1994 1996 1998 2000 2002 2004 2006 2008 2010-10
-5
0
5
10
Perc
ent
(Annual R
ate
s)
4-Quarter Ahead Recursive Forecasts of Cumulative Real GDP Growth
Actual
Nonlinear Forecast Based on 3-Year Net Oil Price Increase
1992 1994 1996 1998 2000 2002 2004 2006 20080
0.5
1
1.5
2
4-Quarter Ahead Recursive MSPE Ratio Relative to AR(4) Benchmark
MS
PE
Ratio
Nonlinear Forecasts of Cumulative Real GDP Growth: Forecasting Success?
5. Forecasting Oil Price Volatility and Quantifying Oil Price Risks
0 20 40 60 80 100 120 140 160 180 2000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
2009.12 Dollars/Barrel
12-Month Ahead Predictive Density of the Real WTI Price as of 2009.12 Based on No-Change Forecast
2002 2003 2004 2005 2006 2007 2008 2009 20100
10
20
30
40
1-Month Implied Volatility
Perc
ent
2002 2003 2004 2005 2006 2007 2008 2009 20100
10
20
30
40
Perc
ent
Realized Volatility
2002 2003 2004 2005 2006 2007 2008 2009 20100
10
20
30
40
Recursive GARCH Volatility
Perc
ent
Alternative Measures of Nominal Oil Price Volatility
Limitations of Volatility Measures
Models of delayed investment decisions require long-run, real oil price volatility, not short-run nominal volatility.
Volatility is not risk. — Consumers are not concerned with real oil-price volatility associated with price decreases.
Risks depend on the predictive distribution of the variable of interest and on user’s loss function (Machina and Rothschild 1987).
Need to distinguish between upside and downside risk.
Oil Price Risks
Consider event of t hR
exceeding an upper threshold of 𝑅 (upside
risk) or falling below the lower threshold of 𝑅 (downside risk):
𝐷𝑅𝛼 ≡ − 𝑅 − 𝑅𝑡+ 𝛼𝑑𝐹 𝑅𝑡+
𝑅
−∞, 𝛼 ≥ 0
𝑈𝑅𝛽 ≡ 𝑅𝑡+ − 𝑅 𝛽𝑑𝐹 𝑅𝑡+ ∞
𝑅 , 𝛽 ≥ 0
Examples:
1. 𝛼 = 𝛽 = 0 ⇒ 𝐷𝑅0 = −Pr 𝑅𝑡+ < 𝑅
𝑈𝑅0 = Pr 𝑅𝑡+ > 𝑅
2. 𝛼 = 𝛽 = 1 ⇒ 𝐷𝑅1 = 𝐸 𝑅𝑡+ − 𝑅|𝑅𝑡+ < 𝑅 Pr 𝑅𝑡+ < 𝑅
𝑈𝑅1 = 𝐸 𝑅𝑡+ − 𝑅 |𝑅𝑡+ > 𝑅 Pr 𝑅𝑡+ > 𝑅 .
2002 2003 2004 2005 2006 2007 2008 2009 2010-1
-0.5
0
0.5
1
Target Probabilities
UR(0)>80
DR(0)<45
2002 2003 2004 2005 2006 2007 2008 2009 2010-30
-20
-10
0
10
20
30
40
50
60
Probability Weighted Expected Excess/Shortfall
UR(1)>80
DR(1)<45
12-Month-Ahead Upside and Downside Risks in the Real WTI Price Based on No-Change Forecast
6. Conclusions: How to Forecast the Price of Oil?
How to Forecast the Price of Oil?
Nominal price of oil
Futures prices generally poor predictors of spot prices
1-3 months: Adjust no-change forecast by recent price change in industrial raw materials
6-48 months: No-change forecast
60 months: Adjust no-change forecast by expected inflation
Real price of oil 1-6 months: Recursive VAR forecast
Beyond 6 months: No-change forecast
Background Slides
Predictability from Nominal U.S. Aggregates to the Nominal Price of Oil (p-values of the Wald test statistic for Granger Non-Causality)
Evaluation Period: 1973.1-2009.12 1975.2-2009.12
Monthly Predictors:
WTI WTI RAC Oil Imports
RAC Domestic Oil
RAC Composite
CPI
0.004 0.108 0.021 0.320 0.161
M1
0.181 0.039 0.010 0.000 0.000
M2
0.629 0.234 0.318 0.077 0.209
CRB Industrial Raw Materials Index
0.000 0.000 0.000 0.000 0.000
CRB Metals Index
0.000 0.002 0.005 0.000 0.003
3-Month T-Bill Rate
0.409 0.712 0.880 0.799 0.896
Trade-Weighted Exchange Rate
- 0.740 0.724 0.575 0.746
Predictability from Selected Bilateral Nominal Dollar Exchange Rates to the Nominal Price of Oil
(p-values of the Wald test statistic for Granger Non-Causality) Evaluation Period: 1973.1-
2009.12 1975.2-2009.12
Monthly
Predictors: WTI WTI RAC
Oil Imports RAC
Domestic Oil RAC
Composite
Australia
0.038 0.066 0.073 0.017 0.044
Canada
0.004 0.003 0.002 0.006 0.002
New Zealand
0.128 0.291 0.309 0.045 0.169
South Africa
0.017 0.020 0.052 0.021 0.037
Predictability from Selected Real Aggregates to the Real Price of Oil (p-values of the Wald test statistic for Granger Non-Causality)
Evaluation Period: 1973.I-2009.IV 1975.II-2009.IV
Quarterly Predictors:
WTI WTI RAC Oil Imports
RAC Domestic Oil
RAC Composite
U.S. Real GDP LT
0.353
0.852
0.676
0.397
0.561
HP 0.253 0.821 0.653 0.430 0.573 DIF 0.493 0.948 0.705 0.418 0.578
World Industrial Production
LT 0.032 0.095 0.141 0.081 0.098 HP 0.511 0.766 0.800 0.665 0.704 DIF 0.544 0.722 0.772 0.668 0.691
Predictability from Selected Real Aggregates to the Real Price of Oil (p-values of the Wald test statistic for Granger Non-Causality)
Evaluation Period: 1973.1-2009.12 1976.2-2009.12
Monthly WTI WTI RAC
Oil Imports RAC
Domestic Oil RAC Composite
Predictors: 𝑝 = 12 𝑝 = 24 𝑝 = 12 𝑝 = 24 𝑝 = 12 𝑝 = 24 𝑝 = 12 𝑝 = 24 𝑝 = 12 𝑝 = 24
Chicago Fed National Activity
Index (CFNAI)
0.823 0.951
0.735 0.952
0.881 0.998
0.707 0.979
0.784 0.995
U.S. Industrial Production
LT 0.411 0.633 0.370 0.645 0.410 0.746 0.091 0.421 0.182 0.510 HP 0.327 0.689 0.357 0.784 0.415 0.878 0.110 0.549 0.194 0.668 DIF 0.533 0.859 0.458 0.866 0.473 0.909 0.114 0.490 0.222 0.699
OECD Industrial Production1
LT 0.028 0.001 0.009 0.033 0.023 0.199 0.021 0.187 0.018 0.230 HP 0.195 0.034 0.072 0.278 0.138 0.714 0.121 0.530 0.114 0.706 DIF 0.474 0.060 0.130 0.353 0.182 0.741 0.174 0.604 0.209 0.757
Global Real Activity Index
0.041 0.000
0.055 0.020
0.141 0.034
0.004 0.004
0.028 0.018
Forecast Accuracy Relative to Monthly No-Change Forecast Evaluation Period: January 1991- December 2009
𝐹𝑡()
𝑆𝑡 1 + 𝛼 + 𝛽 ln(𝐹𝑡()
/𝑆𝑡)
MSPE Ratio Success Ratio MSPE Ratio Success Ratio
1 0.988 0.465 1.001 0.539
3 0.998 0.465 1.044 0.531
6 0.991 0.509 1.051 0.535
9 0.978 0.548 1.042 0.583
12 0.941 0.557 1.240 0.537
NOTES: Boldface indicates statistical significance at the 10% level.
Forecast Accuracy Relative to Daily No-Change Forecast Evaluation Period: Since January 1986
𝐹𝑡()
MSPE Ratio Success Ratio
1 0.963 0.522
3 0.972 0.516
6 0.973 0.535
9 0.964 0.534
12 0.929 0.562
NOTES: There are 5968, 5926, 5861, 5744, and 5028 daily observations at horizons of 1 through 12 months, respectively. Boldface indicates statistical significance at Leamer’s (1978) critical value.
Forecast Accuracy Relative to Daily No-Change Forecast
NOTES: Boldface indicates statistical significance using Leamer’s (1978) critical value.
(in years) Starting date
Sample size
MSPE Ratio
Success Ratio
2 11/20/90
3283
1.159
0.515
3 05/29/91
515
1.168
0.518
4 11/01/95
194
1.212
0.294
5 11/03/97
154
1.280
0.247
6 11/03/97
134
1.158
0.276
7 11/21/97
22
1.237
0.500
Forecast Accuracy Relative to No-Change Forecast Evaluation Period: January 1991- December 2009
𝑆𝑡 1 + ∆𝑝 𝑡 ,𝐶𝑅𝐵 ,𝑖𝑛𝑑 𝑆𝑡 1 + ∆𝑝 𝑡 ,
𝐶𝑅𝐵 ,𝑚𝑒𝑡
MSPE Ratio Success Ratio MSPE Ratio Success Ratio
1 0.913 0.583 1.031 0.579
3 0.782 0.601 0.750 0.601
6 1.055 0.583 1.219 0.623
9 1.076 0.553 1.304 0.575
12 1.035 0.548 1.278 0.539
NOTES: Boldface indicates statistical significance at the 10% level.
Recursive Forecasts of Real Price of Oil from AR and ARMA Models U.S. Refiners’ Acquisition Cost for Imported Crude Oil
Evaluation period: 1991.12-2009.8 = 1 = 3 = 6 = 9 = 12 MSPE SR MSPE SR MSPE SR MSPE SR MSPE SR
AR(12) 0.849 0.599 0.921 0.552 0.969 0.522 1.034 0.441 1.022 0.517 AR(24) 0.898 0.576 0.978 0.557 1.008 0.565 1.056 0.446 1.058 0.453 AR(SIC) 0.826 0.613 0.936 0.557 1.015 0.488 1.039 0.515 1.007 0.532 AR(AIC) 0.842 0.613 0.940 0.562 0.983 0.483 1.013 0.500 0.989 0.527 ARMA(1,1) 0.837 0.580 0.932 0.514 0.982 0.493 1.006 0.510 0.992 0.527 ARI(11) 0.856 0.604 0.939 0.571 1.003 0.517 1.095 0.471 1.091 0.512 ARI(23) 0.898 0.561 0.978 0.538 1.009 0.546 1.068 0.500 1.068 0.508 ARI(SIC) 0.833 0.594 0.951 0.605 1.041 0.546 1.053 0.505 1.016 0.527 ARI(AIC) 0.849 0.604 0.958 0.605 1.008 0.556 1.042 0.500 1.015 0.527 ARIMA(0,1) 0.841 0.599 0.945 0.581 1.009 0.546 1.032 0.515 1.017 0.512
NOTES: Boldface means significance at the 10% level.
Recursive Forecasts of Real Price of Oil from Kilian-Murphy (2010) VAR Model
U.S. Refiners’ Acquisition Cost for Imported Crude Oil
Evaluation period: 1991.12-2009.8
UVAR BVAR 𝑝 MSPE Ratio SR MSPE Ratio
12 1 0.814 0.561 0.800 3 0.834 0.567 0.876 6 0.940 0.546 0.967 9 1.047 0.564 1.052 12 0.985 0.632 1.004
24 1 0.961 0.561 0.801 3 1.081 0.614 0.883 6 1.298 0.604 0.993 9 1.476 0.583 1.095 12 1.415 0.647 1.059
NOTES: Model includes the four oil market variables used in Kilian and Murphy (2010). BVAR based on Giannone, Lenza, and Primicieri (2010).
MSPE Ratios of VAR and VARX Models Relative to AR(4) Benchmark Cumulative U.S. Real GDP Growth Rates
Evaluation Period: 1990.Q1-2010.Q2
Real RAC Price of Imports Nominal RAC Price of Imports
Horizon Oil Price Endogenous
Oil Price Exogenous
Oil Price Endogenous
Oil Price Exogenous
1 1.10 1.10 1.11 1.11 2 1.04 1.04 1.05 1.05 3 0.99 0.98 1.00 0.99 4 0.97 0.96 0.98 0.97 5 0.96 0.95 0.96 0.96 6 0.95 0.94 0.95 0.95 7 0.92 0.92 0.92 0.92 8 0.92 0.92 0.92 0.92
MSPE Ratios for Alternative Specifications of Restricted Exogenous Model Cumulative U.S. Real GDP Growth Rate
Oil Price Series 1990.Q1-2010.Q2 1990.Q1-2007.Q4 Horizon Horizon 1h 4h 1h 4h
Real RAC imports 0.91 0.85 1.11 1.71 RAC composite 1.16 0.99 1.49 2.05 RAC domestic 1.23 0.89 1.55 1.73 WTI 1.03 0.70 1.23 1.22 PPI 1.24 1.09 1.63 2.28 Nominal RAC imports 1.01 0.74 1.22 1.37 RAC composite 1.26 0.82 1.58 1.54 RAC domestic 1.23 0.80 1.50 1.40 WTI 0.92 0.66 1.02 1.08 PPI 1.23 0.88 1.59 1.78