eric ghysels1 jonathan b. hill1 kaiji motegi2

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Simple Granger Causality Tests for Mixed Frequency Data

Eric Ghysels1 Jonathan B. Hill1 Kaiji Motegi2

1University of North Carolina at Chapel Hill2Waseda University

Dec. 11, 2015Tohoku University

Ghysels, Hill & Motegi (UNC & Waseda) Simple GC Tests for MF Data Dec. 11, 2015 1 / 21

Introduction

Consider a bivariate stationary time series model of a highfrequency variable xH and a low frequency variable xL.

We have m observations of xH in each low frequency period τL.

. . .

Example: xH = unemployment rate (announced monthly).xL = gross domestic product (announced quarterly).

How to define and test for Granger causality from xH to xL?


Introduction

Ghysels, Hill, and Motegi’s (2015, J. of Econometrics) mixedfrequency Granger causality tests are a type of Wald tests:

xL(τL) =

q∑k=1

αkxL(τL − k)

+ β1xH(τL − 1,m+ 1− 1) + · · ·+ βhxH(τL − 1,m+ 1− h)︸︷︷︸You need so many lags when m is large!

+u(τL).

How can we avoid poor statistical inference due toparameter proliferation?

We propose a new test called the max test.

The max test achieves higher power in finite sample than theWald test.


Table of Contents

...1 Introduction

...2 Methodology – Formulating the Max Test

...3 Monte Carlo Simulations

...4 Empirical Applications – Weekly Yield Spread vs. Quarterly GDP

...5 Conclusions


Methodology: Max Tests

Model 1 xL(τL) =

q∑k=1

αk,1xL(τL − k) + β1xH(τL − 1,m+ 1− 1) + u1(τL).

Model 2 xL(τL) =

q∑k=1

αk,2xL(τL − k) + β2xH(τL − 1,m+ 1− 2) + u2(τL).

......

......

Model h xL(τL) =

q∑k=1

αk,hxL(τL − k) + βhxH(τL − 1,m+ 1− h) + uh(τL).

Fit OLS for each model. Under H0 : xH 9 xL, all of β̂1, . . . , β̂h converges to 0

in probability. This implies that max{β̂21 , . . . , β̂

2h}

p→ 0. Using this property, we

propose the max test statistic:

T = max{√

TLwTL,1β̂21 , . . . ,

√TLwTL,hβ̂

2h},

where {wTL,1, . . . , wTL,h} is a weighting scheme (typically equal weights).


Methodology: Main Theorems

.Theorem (Limit Distribution under H0 : xH 9 xL)..

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Under H0 : xH 9 xL,

T d→ max{N 21 , . . . ,N 2

h}, N ≡ [N1, . . . ,Nh]′ ∼ N(0,V ).

V is a block matrix which consists of covariance matricesbetween all regressors in Models 1, 2, . . . , h.

V̂p→ V can be constructed from sample.

.Theorem (Consistency)..

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Consistency T p→ ∞ is ensured, assuming lag length h is larger thanor equal to the true lag length.


Monte Carlo Simulations: Two Purposes

MF Max Test MF Wald Test

LF Max Test LF Wald Test


Monte Carlo Simulations: DGP

The DGP is Ghysels’ (2015, J. of Econometrics) structuralMF-VAR(1):

1 0 . . . . . . . . . 0

−0.8 1. . .

. . .. . . 0

0 −0.8. . .

. . .. . .

......

.... . .

. . .. . .

...0 0 . . . −0.8 1 00 0 . . . 0 0 1

xH(τL, 1)

...xH(τL, 12)xL(τL)

=

0 0 . . . 0.8 0.40 0 . . . 0 −0.2...

.... . .

......

0 0 . . . 0 −0.03b12 b11 . . . b1 0.2

xH(τL − 1, 1)

...xH(τL − 1, 12)xL(τL − 1)

+

ηH(τL, 1)

...ηH(τL, 12)ηL(τL)

.


Monte Carlo Simulations

Week vs. quarter mixture (m = 12 approximately).

Sample size is TL = 80 quarters, a reasonable sample size.

b = [b1, . . . , b12]′ is a key parameter representing causal patterns

of xH on xL.

...1 Decaying Causality: b = [0.3, −0.15, 0.1, . . . , −0.025]′.– Decaying impact with +/− signs.

...2 Lagged Causality: b = [0, 0, . . . , 0, 0.3]′.– Delayed impact reflecting seasonality or laggedinformation transmission.

...3 Sporadic Causality: (b3, b7, b10) = (0.2, 0.05, −0.3) and allothers are 0.

– No clear patterns in terms of signs and decayingstructures.


Monte Carlo Simulations

We implement the max test based on 12 parsimonious models:

xL(τL) =2∑

k=1

αk,1xL(τL − k) + β1xH(τL − 1, 12 + 1− 1) + u1(τL),

xL(τL) =2∑

k=1

αk,2xL(τL − k) + β2xH(τL − 1, 12 + 1− 2) + u2(τL),

......

......

xL(τL) =

2∑k=1

αk,12xL(τL − k) + β12xH(τL − 1, 12 + 1− 12) + u12(τL).

For comparison, we implement the Wald test based on onelarge model:

xL(τL) =2∑

k=1

αkxL(τL − k) +12∑j=1

βjxH(τL − 1, 12 + 1− j) + u(τL).


Monte Carlo Simulations: Low Frequency Tests

Implement flow sampling xH(τL) = (1/12)∑12

j=1 xH(τL, j) orstock sampling xH(τL) = xH(τL, 12).We implement the max test based on 3 parsimonious models:

xL(τL) =2∑

k=1

αk,1xL(τL − k) + β1xH(τL − 1) + u1(τL),

xL(τL) =2∑

k=1

αk,2xL(τL − k) + β2xH(τL − 2) + u2(τL),

xL(τL) =

2∑k=1

αk,3xL(τL − k) + β3xH(τL − 3) + u3(τL).

We implement the Wald test:

xL(τL) =2∑

k=1

αkxL(τL − k) +3∑

k=1

βkxH(τL − k) + u(τL).


Simulation Results

Rejection Frequencies (Empirical Size)

MF LF (flow) LF (stock)Max Wald Max Wald Max Wald.055 .043 .054 .039 .058 .038

All tests are correctly sized, so we can compare powermeaningfully.

The max test has accurate size due to parsimoniousspecifications.

The Wald test has accurate size only after bootstrapping (heavycomputation).


Simulation Results: MF Max Test vs. MF Wald Test

Rejection Frequencies (Empirical Power)

MF Max MF WaldDecaying Causality .642 .480Lagged Causality .931 .731Sporadic Causality .803 .740

MF max test is more powerful than MF Wald test for all causalpatterns.


Simulation Results: MF Tests vs. LF Tests

Rejection Frequencies (Empirical Power)

MF LF (flow) LF (stock)Max Wald Max Wald Max Wald

Decaying Causality .642 .480 .198 .159 .748 .697Lagged Causality .931 .731 .539 .527 .804 .749Sporadic Causality .803 .740 .107 .067 .332 .325

Stock-sampling tests are more powerful than MF tests forDecaying Causality.

Stock-sampling tests are roughly as powerful as MF tests forLagged Causality.

MF tests are much more powerful than LF tests for SporadicCausality.


Simulation Results: Summary

MF max test is more powerful than MF Wald test for all causalpatterns.

If there exist relatively simple causal patterns, LF tests mayoutperform MF tests due to parsimony (cfr. Decaying Causalityand Lagged Causality).

MF tests are more robust against complex causal patterns thanLF tests (cfr. Sporadic Causality).


Empirical Applications

We test for Granger causality from weekly yield spread toquarterly GDP growth in the U.S. (1962 - 2013).

-15

-10

-5

0

5

10

15

20

25(%)

10Y

FF

GDP

10Y - FF


Empirical Applications

Background Knowledge

Historically, negative yield spread used to be a well-knownpredictor of immediate economic recession (cfr. Stock andWatson, 2003, JEL).

More recently, there is a possibility that such a relationshipbecame less apparent.

Methodology

To see how causal patterns from spread to GDP changed overtime, we conduct rolling window analysis with 20-year rollingwindows.


Empirical Results

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(d) LF Wald Test

p-values for Causality from Weekly Yield Spread to Quarterly GDP GrowthGhysels, Hill & Motegi (UNC & Waseda) Simple GC Tests for MF Data Dec. 11, 2015 18 / 21

Empirical Results

MF Wald test says spread was not a valid predictor until recently.– Counter-Intuitive.

MF max test and LF tests say spread used to be a validpredictor of GDP.

– Intuitive.

MF max test produces longer periods of significant causalitythan LF tests.

– High power.


Conclusions

Mixed frequency Granger causality tests often involve manyparameters.

We have proposed the max test, which has higher power thanthe existing Wald test in small sample with many parameters.

In the empirical application on weekly yield spread and quarterlyGDP, the max test produces more intuitive results than theWald test.

The max test has a broad range of applications in hypothesistesting.


References

Ghysels, E. (2015). ”Macroeconomics and the Reality of MixedFrequency Data”. Journal of Econometrics, forthcoming.

Ghysels, E., J. B. Hill, & K. Motegi (2015). ”Testing forGranger Causality with Mixed Frequency Data”. Journal ofEconometrics, forthcoming.

Stock, J. H. & M. W. Watson (2003). ”Forecasting Output andInflation: The Role of Asset Prices”. Journal of EconomicLiterature, 41, 788-829.


eric ghysels1 jonathan b. hill1 kaiji motegi2

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