Granger Causality in Mixed Frequency Vector Autoregressive Models

Transcription

1 .. Granger Causality in Mixed Frequency Vector Autoregressive Models Eric Ghysels, Jonathan B. Hill, & Kaiji Motegi University of North Carolina at Chapel Hill June 4, 23 Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 / 2

2 Motivation We develop the mixed frequency (MF) causality test based on Ghysels (23) MF-VAR model. Motivating empirical example: OIL CPI. Table : Causality from OIL to CPI HF is put if significant causality is detected in bivariate monthly VAR, while LF is put if significant causality is detected in quarterly VAR for OIL, CPI & GDP. p\h HF HF HF HF, LF HF HF HF HF 3 HF, LF HF HF HF HF 4 HF HF HF HF HF The MF causality test for OIL, CPI, and GDP detects significant causality from OIL to CPI for any h. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 2 / 2

3 Motivation Two advantages of MF causality tests:. Better recovery of underlying causality patterns. Example MF : Suppose that quarterly GDP causes monthly OIL, then does (underlying) monthly GDP cause monthly OIL? YES. Example LF : Suppose that quarterly GDP causes quarterly OIL, then does (underlying) monthly GDP cause monthly OIL? NOT NECESSARILY..2 Power improvement in both large sample and small sample. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 3 / 2

4 MF-VAR Causality Test x H (τ L, j) is monthly observations of a high frequency variable at the j-th month in quarter τ L. x L (τ L, j) is monthly observations of a low frequency variable at the j-th month in quarter τ L. Consider linear aggregation scheme: x H (τ L ) = 3 w j x H (τ L, j) and x L (τ L ) = j= 3 w j x L (τ L, j). j= Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 4 / 2

5 MF-VAR Causality Test Each of the ten scalars in the figure is observable in a HF process. Only scalars that are surrounded by solid lines are observable in a MF process, while only scalars that are surrounded by dashed lines are observable in a LF process. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 5 / 2

6 MF-VAR Causality Test Define: x H (τ L, ) X(τ L ) = x H (τ L, 2) x H (τ L, 3). x L (τ L ) Our model is VAR() with respect to X(τ L ): where X(τ L ) = AX(τ L ) + ɛ(τ L ) [ ] AHH A A HL. A LH A LL x L does not cause x H iff A HL = 3, while x H does not cause x L iff A LH = 3. These can be tested by linear Wald tests easily. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 6 / 2

7 Recovery of Underlying Causality Whether (non-)causality is preserved under temporal aggregation depends crucially on three factors: aggregation scheme (e.g. linear, flow, stock). VAR lag order p. presence of an auxiliary variable (e.g. when you are interested in causality between OIL and CPI, the presence of GDP may produce causality chains). Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 7 / 2

8 Recovery of Underlying Causality Theorem: Consider a linear aggregation scheme and bivariate HF-VAR(p) with p N { }. Then, the following two properties hold: (i) If x H does not cause x L in a MF process, then the non-causality is preserved in a LF process. (ii) If x L does not cause x H in a HF process, then the non-causality is preserved in a MF process. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 8 / 2

9 Recovery of Underlying Causality If we assume stock sampling with p =, then the corresponding MFand LF-VAR processes can be derived analytically and the following theorem holds. Theorem (Case I: High to Low): Non-causation in a HF process implies non-causation in a MF process, which is necessary and sufficient for non-causation in a LF process. Theorem (Case II: Low to High): Non-causation in a HF process is necessary and sufficient for non-causation in a MF process, which implies non-causation in a LF process. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 9 / 2

10 Power Improvement in Large Sample Our goal here is to show that the MF-VAR causality test has higher local asymptotic power than the LF-VAR causality test does. Case I: High to Low Assume that the true DGP is bivariate HF-VAR() with stock sampling: where X(τ L, k) = Φ(ν/ T )X(τ L, k ) + η(τ L, k), Φ(ν/ [ ] ρh T ) = ν/ T ρ L with ρ H, ρ L (, ). ν R is the Pitman local parameter. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 / 2

11 Power Improvement in Large Sample The coefficient of the corresponding MF-VAR() process is: (m ) ρ H A(ν/ T ) =... (m ) ρ m H m (m ) k= ρk H ρm k L (ν/. T ) ρ m L Suppose that we fit an MF-VAR() model with coefficient A. H : Rvec [A ] = m. H l : Rvec [A ] = (ν/ T )a, where a = [ (m ), m k= ρ k H ρm k L ]. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 / 2

12 Power Improvement in Large Sample W [H ] d χ 2 m under H. W [H l ] d χ 2 m (κ MF ) under H l, where χ2 m(κ MF ) is the noncentral chi-squared distribution with degrees of freedom m and noncentrality parameter κ MF. κ MF is given by: κ MF = ν 2 a [ RΣ (Â )R ] a, where Σ (Â ) is the covariance matrix for OLS estimator, which is analytically available in this simple case. The local asymptotic power of the MF high-to-low causality test, P, is given by: P = F [ F ( α) ], where F is the c.d.f. of χ 2 m, while F be the c.d.f. of χ 2 m(κ MF ). Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 2 / 2

13 Power Improvement in Large Sample The corresponding LF-VAR process is characterized by: A(ν/ [ ρ m ] T ) = H m k= ρk H ρm k L (ν/ T ) ρ m. L In this case, F is the c.d.f. of χ 2, while F is the c.d.f. of χ 2 (κ LF ). We are interested in the power ratio (i.e., P MF /P LF ). Case II: Low to High Assume that the true DGP is bivariate HF-VAR() with: Φ(ν/ [ ρh ν/ ] T T ) =. ρ L Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 3 / 2

14 Power Improvement in Large Sample The corresponding MF-VAR process is characterized by: (m ) ρ H A(ν/ k= ρk H ρ k L (ν/ T ) T ) =... (m ) ρ m m H k= ρk H ρm k L (ν/ T ). (m ) ρ m L The corresponding LF-VAR process is characterized by: A(ν/ [ ρ m m T ) = H k= ρk H ρm k L (ν/ ] T ) ρ m. L Up to these changes, P MF /P LF can be computed analogously. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 4 / 2

15 Power Improvement in Large Sample MF/LF MF/LF MF/LF MF/LF m 5.5 ν 5.5 m 5.5 ν.5 5 m 5.5 ν.5 5 m 5.5 ν MF/LF.8.6 MF/LF.8.6 MF/LF MF/LF m 5.5 ν m 5.5 ν.5 5 m 5.5 ν.5 5 m 5.5 ν.5 Note: The z-axis of each figure has P MF /P LF. Note that the scale and grid of each z-axis is different. The x-axis has ν [.5,.5], while the y-axis has m {3,..., 2}. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 5 / 2

16 Power Improvement in Large Sample The power ratio in Panel B is decreasing in m for (ρ H, ρ L ) = (.25,.25) and increasing in m for (ρ H, ρ L ) = (.75,.75). To interpret this fact, let ρ H = ρ L = ρ and consider a key quantity in the upper-right block of A, m ρm k = mρ m f(m). k= ρk H L P MF /P LF increases by switching from m to m + if f(m) is large. The figure below plots f(m) for ρ {.25,.75} (Sampling frequency m) Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 6 / 2

17 Power Improvement in Finite Sample Assume that the true DGP is bivariate HF-VAR() with stock sampling. Prepare two kinds of Φ depending on causality directions: (a) Unidirectional causality from x H to x L : [ ] φhh φ HL = φ LH φ LL [ ]. (b) Unidirectional causality from x L to x H : [ ] [ ] φhh φ HL.4.2 =. φ LH φ LL.4 Aggregate generated HF data into MF or LF data, and apply VAR causality tests. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 7 / 2

18 Power Improvement in Finite Sample Rejection frequencies at the 5% level. Newey and West s (994) automatic bandwidth selection is used. Dufour, Pelletier, and Renault s (26) bootstrap is used for T L = in order to avoid size distortions. Size Sample Size T L = Case (a) Case (b) [m = 2] [m = 3] [m = 2] [m = 3] MF: LF: Power MF:.229 LF: Size Sample Size T L = 5 Case (a) Case (b) [m = 2] [m = 3] [m = 2] [m = 3] MF: LF: Power MF:.9 LF: Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 8 / 2

19 Empirical Application MF-VAR() causality test for monthly CPI, monthly OIL, and quarterly GDP. LF-VAR(4) causality test for quarterly CPI, quarterly OIL, and quarterly GDP. (%) CPI (LHS) -4. GDP (LHS) OIL (RHS) (%) Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 9 / 2

20 Empirical Application MF ( LF ) is put if the mixed (low) frequency causality test rejects the null hypothesis of non-causality. h CPI OIL MF MF, LF LF MF, LF MF CPI GDP - - MF MF MF OIL GDP - LF LF LF LF OIL CPI MF MF MF MF MF GDP CPI GDP OIL MF MF CPI OIL GDP Solid arrows correspond to the MF case, while dashed arrows correspond to the LF case. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 2 / 2

21 Conclusions MF approach recovers underlying causality better than LF approach does. MF causality tests are more powerful than LF causality tests especially for low-to-high causality. MF and LF causality tests produce different empirical evidence. Ghysels, Hill, & Motegi (UNC) Granger Causality in MF-VAR Models June 4, 23 2 / 2