Simulation Study of Causality Change with Infinite
Order Vector Autoregressive Processes
Ryo Kinoshita∗, Kosuke Oya† and Mototsugu Shintani‡
Abstract — The causality relationship between the time series is aﬀected by the change in parameters of model. However,the change in parameters does not tell us the magnitude of causality change. In this study, we explore the measure ofcausality change between the time series and propose the test statistic whether there is any significance change in the causal
relationship using frequency domain causality measure. To avoid the misspecification of the model, we employ infinite
vector autoregressive models and the sieve approximation with increasing lag orders with sample size. The properties of themeasure and test statistic are examined through the Monte Carlo simulation.

Keyword: Causality; autoregressive sieve estimation; structural change.
1Introduction

Economic time series often has structural change that incurred by some event such as policy change
or financial crisis. When there is a structural change, the time series is characterized by diﬀerent
parameters of models for the sample periods before and after the change point. However it is not clear
what kind of change in the causal relationship are carried by the change in the model parameters. This
paper proposes a measure of change in the causal relationship and a statistical testing procedure forthe significance of the change. The causality measure function is defined in the frequency domain as
proposed in Geweke (1982) and Hosoya (1991). The frequency wise causality measure can be regarded
as the decomposition of well known Granger causality measure that based on the variance ratio of theone step ahead prediction errors between two diﬀerent models.
The inference procedure for the frequency wise causality measure proposed in the previous studies
is only for the finite order vector autoregressive (VAR) model. However, such finite order VAR models
are often insuﬃcient to estimate the spectral densities of true generation process. In such case, thecausality measure can not be estimated correctly so that it brings a wrong implication for the analysis.
Instead of the finite order VAR model, we consider the VAR model with larger lag order which
serves to avoid such diﬃculties. Lewis and Reinsel (1985) introduce an infinite order VAR modeland establish the consistency and asymptotic normality for the estimated VAR coeﬃcients. In thispaper, the statistical inference for the causality measure function is introduced under infinite order
VAR model. The parameters of model are estimated by ordinary least square (OLS) estimation. The
causality measure functions are estimated as the nonlinear function of the estimated model parameters.
The causality change can be defined as the diﬀerence of two causality measures for the periods beforeand after the change point. The asymptotic variance of the causality change which is required for thehypothesis testing is derived with the standard delta method for finite order VAR model. However, for
the infinite order VAR model, we propose a bootstrap procedure to obtain the variance of the estimatedchange in the causal measure. The finite sample performance of the test with the bootstrap variance isinvestigated by Monte Carlo simulations.
Next section, we give a brief explanation and definitions for VAR model and the related causality
measure. The asymptotic distributionof the change in the causality measures for the diﬀerent sample
periods is also given. The estimation procedure of the variance of the change in the causality measures∗ Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan, Email: r-kinoshita@econ.osaka-u.ac.jp, Tel: +81-6-6850-5265† Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan‡ RCAST, Tokyo University, Meguro-ku, Tokyo, 153-8904, Japan, and Department of Economics, Vanderbilt University, Nashiville, TN 37235, USA

is presented in section 3. We conduct a series of Monte Carlo experiments to see the properties of thetest statistic in section 4. Section 5 gives some discussion.
22.1Methodology
Causality measure for VAR Model

Let Z(t) = [X(t) Y(t)]′ , we consider the following VAR model
A(L)Z(t) = ϵ(t), ϵ(t) ∼ N(0, Ω),(2.1)

where Ap (L) = I − A1 L − A2 L2 − · · · − Ap Lp and A(L) = limp→∞ Ap (L). It is assumed that the zeros ofdet(A(L)) are inside of the unit circle. The moving average (MA) representation of (2.1) isZ(t) =A(L)−1 ϵ(t)= Ψ(L)u(t), u(t) ∼ N(0, I),
where Ψ(L) = A(L)−1 Ω1/2 and Ω1/2 is the cholesky decomposition of Ω. Let parameter vector θ ={vec(A1 )′ , vec(A2 )′ . . . , vec(Ap )′ , vech(Ω)′ }′ , the causality measure Y(t) to X(t) at frequency λ is given by{}||Ψ12 (e−iλ )||MY→X (λ; θ) = log 1 +.
||Ψ11 (e−iλ )||(2.2)

Note that the causality measure function MY→X (λ; θ) is the function of VAR parameters, θ.
2.2

VAR model with structural break

We assume that a data is generated by infinite order VAR model with structural break which describedas follows. In this paper, it is assumed that a break point is known and defined as a constant fractionof sample size. Let T be the sample size and T1 be the break point, T1 = [cT] where [x] means Gausssymbol. The VAR model with structural break is as follows
A(k) (L)Z(t) = ϵ(t), ϵ(t) ∼ N(0, Ωk )(k)

(2.3)
(k)

where Ap (L) = I − A1,k L − A2,k L2 − · · · − Ap,k Lp and A(k) (L) = limp→∞ Ap (L) and k = 1 if t < T1 and k = 2 ift ≥ T1 . For the notational simplicity, we express the lag order of the model as p although which depends
on k. We approximate (2.3) by VAR models with increasing lag order p which satisfies p3 /T → 0 as
Ap (L)Z(t) = ϵ∗ (t), ϵ∗ (t) ∼ N(0, Ω∗k ).
(k)

We define parameter vectors of VAR models before and after break as θk = {vec(A1,k )′ , vec(A2,k )′ , . . . ,vec(Ap,k )′ , vech(Ω∗k )′ }′ and k = 1, 2.
2.3

Inference for the change in the causality measure

We introduce a test statistic for the causality change. The parameters are estimated by OLS and those aredenoted by θ̂k = {vec(Â1,k )′ , vec(Â2,k )′ , . . . , vec(Âp,k )′ , vech(Ω̂∗k )′ }′ . The causality measure functions areestimated by casting estimated parameters into (2.2). Since θ̂1 and θ̂2 are asymptotically independent sothat MY→X (λ; θ̂1 ) and MY→X (λ; θ̂2 ) are also asymptotically independent. Let m(λ; θ1 , θ2 ) = MY→X (λ; θ1 )−MY→X (λ; θ2 ). Assuming that true causality measure functions MY→X (λ; θ1 ) and MY→X (λ; θ2 ) are both
not zero, the asymptotic distribution of m(λ; θ̂1 , θ̂2 ) is given by√()
Tdm(λ; θ̂1 , θ̂2 ) → N m(λ; θ1 , θ2 ), V(λ, θ)pwhereV(λ, θ) = (1/c)V(λ,θ1 ) + (1/(1 − c))V(λ, θ2 ) and V(λ, θ1 ), V(λ, θ2 ) are the asymptotic variance of√√cT/p MY→X (λ; θ̂1 ), (1 − c)T/p MY→X (λ; θ̂2 ) respectively. Therefore the test statistic for null hypothesis
H0 : m(λ; θ1 , θ2 ) = 0 is given by√dm(λ; θ̂1 , θ̂2 )/ v1 + v2 → N(0, 1)
where v1 = p(cT)−1 V(λ, θ̂1 ) and v2 = p((1 − c)T)−1 V(λ, θ̂2 ).
3

Estimation of the variance of the estimator

The asymptotic variance V(λ, θ) is the complicated nonlinear function of the parameters. To estimate
the asymptotic variance, we employ the bootstrap procedure. The bootstrap variance of estimator iscalculated as the sample variance of repeated samples of the following procedure.
1. Bootstrap residuals ϵ̂(t)b are calculated using OLS residuals ϵ̂(t) before break (t < T1 ) and afterbreak (t ≥ T1 ) respectively. In this step, OLS residuals are resampled as a vector.
2. Using ϵ̂(t)b , the bootstrap data Z(t)b are generated as
Âp,k (L)Z(t)b = ϵ̂(t)b ,where k = 1 if t < T1 and k = 2 if t ≥ T1 .
3. Estimating parameters with Z(t)b by OLS, we get bootstrap samples of the parameter estimates
which are denoted by θ̂1,b and θ̂2,b .
4. Compute the causality measure functions M(λ; θ̂1,b ) and M(λ; θ̂2,b ).
With bootstrap variance of M(λ; θ̂1 ) and M(λ; θ̂2 ) which are denoted by v1,B and v2,B , we construct afollowing test statistic,√dm(λ; θ̂1 , θ̂2 )/ v1,B + v2,B → N(0, 1).
4Simulation

We demonstrate a test of causality changes for a infinite order VAR model with Gegenbauer polynomial
as follows,
X(t) = (0.3 + b(L))Y(t − 1) + e1 (t)Y(t) = e2 (t)∏nwhere b(L) = j=1 [α(1 − 2 cos λ j L − L2 )] and [e1 (t) e2 (t)]′ ∼ N(0, I). We set λ j = 0.4 + 0.5(j/n). The breakpoint is set to a half of sample size (T1 = 0.5T) and T=2000, 4000, 8000. The left panel of Figure 1 shows
the case of n = 5 and α = 0 before break so that MY→X (λ; θ1 ) is constant at each frequency and α = 0.3
after break. This case implies that m(λ; θ1 , θ2 ) = 0 for any λ ∈ [0.5π, π) as n → ∞. It is noted thatm(λ; θ1 , θ2 ) is not exactly equal to zero except for λ = 0.5π, 0.6π, 0.7π and 0.9π since we set n = 5. The
right panel of Figure 1 shows rejection rates for the causality change at each frequency. The nominal
size for tests is 5% which is depicted by dashed line. The actual size is smaller than the nominal size atλ ∈ [0.5π, π) when the sample size is T = 2000 and it gets close to the nominal size as the sample size
becomes large. The power is high enough at low frequencies as we expected.

0.7

1before breakafter break

T=2000
T=4000T=8000

0.6
0.8
0.5
0.6

0.4
0.3

0.4

0.2
0.2
0.1
0

0
0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

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Figure 1: True causality measures (left panel) and Rejection rates for the causality change (right panel)
5Discussions and summary

In this paper, a statistical inference for causality change is proposed with infinite order VAR model. The
proposed method is an extension of Kinoshita and Oya (2014) which derive an inference for finite order
VAR model. It is shown that the asymptotic distribution of the causality change is normal, however
its asymptotic variance is complicated function of the parameters. A bootstrap procedure is proposed
to estimate the asymptotic variance. Monte Carlo simulations shows that there is under bias in test
size and it gets close to the nominal size as the sample size becomes large. Although we assume thatthe lag order is known in our simulation, it is unknown in empirical applications. We have several
candidates for the lag selection such that a linear function of T1/4 , AIC and the general to specificapproach proposed by Kuersteiner (2005). An inference with large lag order is asymptotically valid,
however there may be a size distortion when sample size is finite or the true lag order is finite. It isnecessary to try to find a well balanced lag selection procedure even for the infinite order VAR models.
6

Acknowledgment

This work was partly supported by JSPS KAKENHI Grant Number 16K17103 and 25245033.
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Journal of the American Statistical Society. 77, 303-314.
[2] Hosoya, Y.(1991). The decomposition and measurement of the interdependency between secondorder stationary processes. Probability Theory, 88, 429-444.
[3] Kinoshita, R. and Oya, K.(2014). Measurement of causality change between Multiple time series.
Journal of the Japan Statistical Society Japanese Issue, 44, 19-40.
[4] Kuersteiner, G. M.(2005). Automatic inference for infinite order vector autoregressions. EconometricTheory, 21, 85-115.
[5] Lewis, R. and Reinsel, G. C.(1985). Prediction of multivariate time series by autoregressive modelfitting. Journal of multivariate analysis, 16, 393-411.