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Causality Along Subspaces
Majid Al-Sadoon
University of Cambridge
Causality Along Subspaces Majid Al-Sadoon University of Cambridge - - PowerPoint PPT Presentation
Causality Along Subspaces Majid Al-Sadoon University of Cambridge - - PowerPoint PPT Presentation
Causality Along Subspaces Majid Al-Sadoon University of Cambridge Royal Economic Society Fifth PhD Presentation Meeting, 16/01/2010 Outline Introduction Subspace Causality Subspace Causality Test Sample of Results Other Applications
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Abstract
This paper extends existing notions of causality due to Dufour & Renault (1998) and Bruneau & Jondeau (1999) in two directions:
- 1. Causality along subspaces.
- 2. Causality in the long run.
These two extensions allow us to show the following:
- 1. The appropriate test for causality in multivariate systems requires
testing the rank of coefficient matrices rather than zero restrictions.
- 2. ρ–mixing and cointegration are instances of long run subspace
non–causality.
- 3. Non–controllability and rational expectations are instances of
non–causality at every horizon.
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Subspace Causality
◮ In a nutshell: Y Granger causes X (both multivariate) if Y helps
forecast X.
◮ Dufour et al. (2006) test for causality at horizon h by estimating the
regression equation, X(t + h) Y (t + h) Z(t + h) =
lag polynomial
-
πXX(L) πXY (L) πXZ(L) πY X(L) πY Y (L) πY Z(L) πZX(L) πZY (L) πZZ(L) X(t) Y (t) Z(t) + UX(t) UY (t) UZ(t) , and testing , H0 : πXY (L) = 0. If we reject H0 then we write Y →h X and if not we write Y h X.
◮ But suppose we reject H0, have we got the full picture of the
dependence structure?
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Granger Non–causality Along Subspaces – Case I
✲ ✻
- ✠
- ✒
X1 X3 X2 U C ❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍ ❍ ❍❍❍❍❍❍❍❍❍❍ ❍ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ Y X|U
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Granger Non–causality Along Subspaces – Case II
✲ ✻
- ✠
❅ ❅ ❅ ❅ ■ Y1 Y3 Y2 D V ✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟ ✟ ✟✟✟✟✟✟✟✟✟✟ ✟ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ Y |V X
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Monetary Policy
◮ The Bernanke & Mihov (1998) data set consists of monthly logged
and differenced data on:
- 1. Real GDP (GDP),
- 2. The GDP deflator (P),
- 3. Non–borrowed reserves (NBR),
- 4. The federal funds rate, (r),
for the period January 1965 to December 1996.
◮ We would like to study the causal effect of monetary policy
(NBR, r) on (GDP, P).
◮ If monetary policy fails to cause GDP growth then forecasts which
include monetary policy as predictors will be the same as forecasts which exclude monetary policy as predictors.
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Difference in Forecasts with and without The Federal Funds Rate as a Predictor
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Rotating Policy Space
◮ Suppose that in place of the given monetary policy instruments, we
were to construct two different instruments, I1(θ) = cos(θ)NBR + sin(θ)r I2(θ) = − sin(θ)NBR + cos(θ)r
◮ Such a transformation amounts to rotating (NBR, r) space by θ
degrees.
◮ If we test H0 : I2 h GDP This will allow us to see in which
direction monetary policy has its strongest and weakest predictive power for GDP.
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Testing H0 : I2 h GDP
Figure: Horizontal lines are the asymptotic 10% and 5% critical values.
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Subspace Causality Test
To capture this structure all we have to do is to estimate the same equation as before, X(t + h) Y (t + h) Z(t + h) = πXX(L) πXY (L) πXZ(L) πY X(L) πY Y (L) πY Z(L) πZX(L) πZY (L) πZZ(L) X(t) Y (t) Z(t) + UX(t) UY (t) UZ(t) ,
- 1. If we want to find U then we test rank restrictions on,
[πXY 1 · · · πXY p]
- 2. If we want to find V then we test rank restrictions on,
πXY 1 . . . πXY p
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Sample of Results
h 10 11 12 13 14 15 16 r h (GDP, P ) ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ r h (GDP, P )|U U −0.0231 0.9997 −0.0408 0.9992 0.0154 0.9999 −0.0115 0.9999 −0.0320 0.9995 −0.0365 0.9993 −0.0703 0.9975
- ⋆⋆ indicates significance at 5%, ⋆ indicates significance at 10%.
◮ The effect of the Federal Funds rate on output growth and inflation
at horizons 10–16 months is primarily along a subspace. ✲ ✻ ✘ ✘ ✘ ✾ ✘✘✘✘✘✘✘✘ ✿ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❖ ❈ ❈ ❈ ❈ ❲
Not predicted by r Predicted by r GDP P
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Sample of Results
h 4 5 6 7 8 9 10 (NBR, r) h GDP ⋆⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ (NBR, r)|V h GDP V 0.0325 0.9995 0.0716 0.9974 0.1829 0.9831 0.2054 0.9787 0.2204 0.9754 0.2155 0.9765 0.2087 0.9780
- ⋆⋆ indicates significance at 5%, ⋆ indicates significance at 10%.
◮ Monetary policy has an effect on output growth only along a
subspace in policy space for horizons 4–10. ✲ ✻ ❳ ❳ ❳ ② ❳❳❳❳❳❳❳❳ ③ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✗ ✄ ✄ ✄ ✄ ✎
No effect of Monetary Policy on GDP The effective dimension of Monetary Policy NBR r
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Other Applications
◮ Testing dynamic models (every such model has implicit predictability
properties, e.g. DSGE forward components).
◮ Testing for controllability in quadratic–loss optimal policy problems. ◮ Model reduction (reducing a VAR to its bare causal “bones”). ◮ Forecasting (may sharpen forecasts if we focus on the most highly
correlated directions).
◮ A new interpretation of VAR coefficients.
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