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

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.

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, lag polynomial � �� �         X ( t + h ) π XX ( L ) π XY ( L ) π XZ ( L ) X ( t ) U X ( t )  =  , Y ( t + h ) π Y X ( L ) π Y Y ( L ) π Y Z ( L ) Y ( t )  + U Y ( t )      Z ( t + h ) π ZX ( L ) π ZY ( L ) π ZZ ( L ) Z ( t ) U Z ( t ) and testing , H 0 : π XY ( L ) = 0 . If we reject H 0 then we write Y → h X and if not we write Y � h X . ◮ But suppose we reject H 0 , have we got the full picture of the dependence structure?

Granger Non–causality Along Subspaces – Case I Y � X | U ❍❍❍❍❍❍❍❍❍❍ ❏ ❏ ❏ ❍❍❍❍❍❍❍❍❍❍ U ❏ ❏ ❏ � ✒ ❏ ❍❍❍❍❍❍❍❍❍❍ ❏ � ❏ ❏ ❏ ❍ ❏ � ❏ ❏ ❏ ❏ ❍❍❍❍❍❍❍❍❍❍ ❏ � ❏ ❏ ❏ ❍ ❏ ❏ ❏ ❏ ❏ C ❏ ❍❍❍❍❍❍❍❍❍❍ ❏ ❏ ❏ ❏ ❍ ❏ X 2 ❏ ❏ ❏ ❏ ✻ ❏ ❍ ❏ ❏ ✲ ❏ ❍ ❏ � X 1 � ✠ X 3

Granger Non–causality Along Subspaces – Case II ✟ ✟✟✟✟✟✟✟✟✟✟ ✡ ✡ ✡ ✟ ✟✟✟✟✟✟✟✟✟✟ V ✡ D ✡ ❅ ■ ✡ ✡ ✡ ✡ ❅ ✟ ✟✟✟✟✟✟✟✟✟✟ ✡ ✡ ✡ ✡ ❅ ✡ ✡ ✡ ✡ ✡ ✟ ✟✟✟✟✟✟✟✟✟✟ ❅ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✟ ✟✟✟✟✟✟✟✟✟✟ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ Y 2 ✡ ✡ Y | V � X ✻ ✲ � Y 1 ✠ � Y 3

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.

Difference in Forecasts with and without The Federal Funds Rate as a Predictor

Rotating Policy Space ◮ Suppose that in place of the given monetary policy instruments, we were to construct two different instruments, I 1 ( θ ) = cos( θ ) NBR + sin( θ ) r I 2 ( θ ) = − sin( θ ) NBR + cos( θ ) r ◮ Such a transformation amounts to rotating ( NBR, r ) space by θ degrees. ◮ If we test H 0 : I 2 � h GDP This will allow us to see in which direction monetary policy has its strongest and weakest predictive power for GDP .

Testing H 0 : I 2 � h GDP Figure: Horizontal lines are the asymptotic 10% and 5% critical values.

Subspace Causality Test To capture this structure all we have to do is to estimate the same equation as before,         X ( t + h ) π XX ( L ) π XY ( L ) π XZ ( L ) X ( t ) U X ( t )  =  +  , Y ( t + h ) π Y X ( L ) π Y Y ( L ) π Y Z ( L ) Y ( t ) U Y ( t )      Z ( t + h ) π ZX ( L ) π ZY ( L ) π ZZ ( L ) Z ( t ) U Z ( t ) 1. If we want to find U then we test rank restrictions on, [ π XY 1 · · · π XY p ] 2. If we want to find V then we test rank restrictions on,   π XY 1 .  .  .   π XY p

Sample of Results 10 11 12 13 14 15 16 h r � h ( GDP, P ) ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ r � h ( GDP, P ) |U � − 0 . 0231 �� − 0 . 0408 �� 0 . 0154 �� − 0 . 0115 �� − 0 . 0320 �� − 0 . 0365 �� − 0 . 0703 � U 0 . 9997 0 . 9992 0 . 9999 0 . 9999 0 . 9995 0 . 9993 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. P ✻ Not predicted by r ❖ ❈ ❈ ❈ ❈ ❈ ❈ ✘✘✘✘✘✘✘✘ ✿ ❈ ❈ ✘ ✘ Predicted by r ✘ ✾ ❈ ❈ ❈ ❈ ❲ ✲ GDP

Sample of Results h 4 5 6 7 8 9 10 ( NBR, r ) � h GDP ⋆⋆ ⋆ ⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆⋆ ( NBR, r ) |V � h GDP � 0 . 0325 �� 0 . 0716 �� 0 . 1829 �� 0 . 2054 �� 0 . 2204 �� 0 . 2155 �� 0 . 2087 � V 0 . 9995 0 . 9974 0 . 9831 0 . 9787 0 . 9754 0 . 9765 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. r ✻ No effect of Monetary Policy on GDP ✗ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ❳ ② ❳ ❳ ✄ ❳❳❳❳❳❳❳❳ The effective dimension of Monetary Policy ✄ ✄ ③ ✄ ✎ ✄ ✲ NBR

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.

Bernanke, B. S. & Mihov, I. (1998). Measuring monetary policy. The Quarterly Journal of Economics , 113 (3), 869–902. Bruneau, C. & Jondeau, E. (1999). Long-run causality, with an application to international links between long-term interest rates. Oxford Bulletin of Economics and Statistics , 61 (4), 545–568. Dufour, J.-M., Pelletier, D., & Renault, E. (2006). Short run and long run causality in time series: inference. Journal of Econometrics , 127 (2), 337–362. Dufour, J.-M. & Renault, E. (1998). Short run and long run causality in time series: Theory. Econometrica , 66 (5), 1099–1125.