Testing a Large Set of Zero Restrictions in Regression ...

Testing a Large Set of Zero Restrictions in Regression Models,with an Application to Mixed Frequency Granger Causality

Eric Ghysels1 Jonathan B. Hill1 Kaiji Motegi2

1University of North Carolina at Chapel Hill2Kobe University

June 30, 2017Workshop on Advances in Econometrics 2017 at Hakodate

Ghysels, Hill & Motegi (UNC & Kobe) Testing Many Zero Restrictions June 30, 2017 1 / 18

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 (2016) propose a Wald test for mixedfrequency Granger causality:

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 is more robust against parameter proliferation thanthe Wald test in terms of empirical size and power.


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 ↛ 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:

T̂ = 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 ↛ xL)

Under H0 : xH ↛ xL,

T̂ 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)

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


General Linear Regression Models

Max test can be applied to a general linear regression model:

yt = z′ta+ x′tb+ ϵt, H0 : b = 0h×1,

where h is large relative to sample size.

Split xt = [x1t, . . . , xht]′ and run parsimonious regression models:

yt = z′tαi + βixit + uit for i = 1, . . . , h.

Run the max test:

T̂ = max{√TLwTL,1β̂

21 , . . . ,

√TLwTL,hβ̂

2h}.


Proof of Consistency (Sketch)

Suppose that a true DGP is

yt = x′tb+ ϵt.

where xt = [x1t, . . . , xht]′ and b = [b1, . . . , bh]

′.

Fit parsimonious regression models

yt = βixit + uit for i = 1, . . . , h.

Write β∗i = plimβ̂i, and β∗ = [β∗

1 , . . . , β∗h]

′, then it follows that

β∗ = D × b,

where D is a positive definite matrix involving covariances of xt.


Proof of Consistency (Sketch)

Since β∗ = D × b, it follows that

β∗′β∗ = b′ (D′D)︸︷︷︸p.d.

b.

If b ̸= 0, then β∗′β∗ ̸= 0 and thus β∗ ̸= 0.

In words: If b ̸= 0, then at least one of {β̂1, . . . , β̂h} convergesin probability to a non-zero value.

Hence, consistency holds (b ̸= 0 ⇒ T̂ → ∞).


Monte Carlo Simulations

The DGP is Ghysels’ (2016) structural MF-VAR(1):

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

−0.2 1. . .

. . .. . . 0

0 −0.2. . .

. . .. . .

......

.... . .

. . .. . .

...0 0 . . . −0.2 1 00 0 . . . 0 0 1

xH(τL, 1)

...xH(τL, 12)xL(τL)

=

0 0 . . . 0.2 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)

,

or compactlyNX(τL) = M1X(τL − 1) + η(τL).



m = 12 (”month vs. year” or ”week vs. quarter”).

Sample size is TL = 160.

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.



We perform the max test based on 24 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,24xL(τL − k) + β24xH(τL − 1, 12 + 1− 24) + u24(τL).

For comparison, we perform the Wald test based on one largemodel:

xL(τL) =

2∑k=1

αkxL(τL − k) +

24∑j=1

βjxH(τL − 1, 12 + 1− j) + u(τL).


Rejection Frequencies

DGP: MF-VAR(1)Test: Max Wald

Non-Causality .041 .031Decaying Causality .692 .639Lagged Causality .685 .434Sporadic Causality .721 .644

Max test is asymptotic.

Wald test is bootstrapped in order to control for size.

Max and Wald tests have similar power under Decaying andSporadic Causality.

Max test has higher power than Wald test under LaggedCausality.


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

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

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 perform rolling window analysis with 20-year rollingwindows.


P-values over Rolling Window Subsamples

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1962:I -

1981:IV

1967:I -

1986:IV

1972:I -

1991:IV

1977:I -

1996:IV

1982:I -

2001:IV

1987:I -

2006:IV

1992:I -

2011:IV

a. MF Max Test

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1962:I -

1981:IV

1967:I -

1986:IV

1972:I -

1991:IV

1977:I -

1996:IV

1982:I -

2001:IV

1987:I -

2006:IV

1992:I -

2011:IV

b. MF Wald Test

Max test suggests spread used to be a valid predictor of GDP.– Intuitive.

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


Conclusions

Mixed frequency Granger causality tests often involve manyparameters.

We have proposed the max test, which is more robust againstparameter proliferation than the Wald test in terms of finitesample size and power.

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 generallinear regression models.


References

Ghysels, E. (2016). ”Macroeconomics and the Reality of MixedFrequency Data”. Journal of Econometrics, 193, 294-314.

Ghysels, E., J. B. Hill, & K. Motegi (2016). ”Testing forGranger Causality with Mixed Frequency Data”. Journal ofEconometrics, 192, 207-230.

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


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