Determining oil price drivers with Dynamic Model Averaging - - PowerPoint PPT Presentation
Determining oil price drivers with Dynamic Model Averaging Krzysztof Drachal Faculty of Economic Sciences, University of Warsaw Research funded by the Polish National Science Centre grant under the contract number DEC-2015/19/N/HS4/00205. K.
Krzysztof Drachal
Faculty of Economic Sciences, University of Warsaw
Research funded by the Polish National Science Centre grant under the contract number DEC-2015/19/N/HS4/00205.
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supply and demand forces stock markets global economic activity exchange rates interest rate(s) speculative pressures (. . . ) A lot of, moreover, the impact can be time-varying
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What people sometimes do? OLS: yt = xtθ + ǫt ⇓ TVP-OLS: yt = xtθt + ǫt
Consider all possible OLS models which can be constructed out of m independent variables, i.e., 2m OLS models, and select some subset of models from them (or even take all of them!). OLS: yt = xk
t θk + ǫk t
⇓ TVP-OLS: yt = xk
t θk t + ǫk t
Final forecast is produced as some weighted average.
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Raftery, A.E., Karny, M., Ettler, P., (2010). Online prediction under model uncertainty via dynamic model averaging: Application to a cold rolling mill. Technometrics 52, 52–66. Let there be m independent variables. Then, K = 2m different regression models can be constructed. The state space model is given by yt = xk
t θk t + ǫk t
, (1) θk
t = θk t−1 + δk t
. (2) k = {1, . . . , K} θk
t are parameters.
ǫk
t ∼ N(0,Vk t ) and δk∼ N(0,Wk t ).
Initially, values of V k
0 and W k 0 have to be set.
Further, V k
t is estimated by a recursive method of moments estimator, and W k t by the
Kalman filter updating. During this, a forgetting factor λ ∈ (0, 1] has to be set.
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Initialize the process with π0|0,k := 1
K , i.e., all models are initially equally
”good” (non-informative prior). Then, πt|t−1,k = (πt−1|t−1,k)α + c K
i=1(πt−1|t−1,i)α + c
, (3) πt|t,k = πt|t−1,kfk(yt|y0, y1, . . . , yt−1) K
i=1 πt|t−1,ifi(yt|y0, y1, . . . , yt−1)
. (4) α ∈ (0, 1] is a fixed forgetting factor. fk(yt|y0, y1, . . . , yt−1) is the predictive density of the k-th model at yt given the data from the previous periods. πt|t,k are called posteriori inclusion probabilities. A small constant is specified, c := K · 10−3, in order to avoid reducing the probabilities to 0 due to numerical approximations.
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DMA forecast
ˆ yt =
K
πt|t−1,k · ˆ yk
t
, (5) where ˆ yk
t is the forecast produced by the k-th model.
Notice that weights πt|t−1,k are continuously updated, basing on the models’ performances. Economic part: For every t sum up πt|t−1,k for models containing a given independent variable. Time-paths of these (summed as above) weights can be used to describe the importance of a given variables as a driver.
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WTI – WTI spot price in USD per barrel MSCI – MSCI World Index EM – EM (Emerging Markets) MSCI Index TB3MS – U.S. 3-month treasury bill secondary market rate in % CSP – Crude steel production in thousand tonnes TWEXM – Trade weighted U.S. dollar index (Mar, 1973 = 100) PROD – U.S. product supplied for crude oil and petroleum products in thousands
CONS – Total consumption of petroleum products in OECD in quad BTU IMP – Average U.S. imports of crude oil in thousands of barrels per day INV – U.S. total stocks of crude oil and petroleum products in thousands of barrels VXO – Implied volatility of S&P 100 Monthly data beginning on Jan, 1990 and ending on Dec, 2016. Log-differences, except TB3MS and VXO (i.e., rates) which were taken in ordinary differences. Also, data were normalized (i.e., rescaled to fit between 0 and 1, which is desirable from computational point of view).
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Mean Squared Error λ \α 1 0.99 0.95 0.90 1 0.0148 0.0147 0.0154 0.0163 0.99 0.0146 0.0146 0.0152 0.0161 0.95 0.0144 0.0143 0.0149 0.0159 0.90 0.0143 0.0142 0.0147 0.0157 naive MSE = 0.0198 TVP OLS = 0.0155 Also, Mean Absolute Error preferres DMA. Also, DMA according to minimization of MSE and MAE is preferred over Dynamic Model Selection.
(In Dynamic Model Selection the final forecast is produced not as a weighted average, but comes from the model which has the highest weight in a given period.)
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developed stock markets emerging stock markets
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interest rates global economic activity
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supply forces demand forces
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exchange rates speculative pressures
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demand forces market stress
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https://CRAN.R-project.org/package=fDMA
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