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Research article

Time-varying impact of U.S. financial conditions on China's inflation: a perspective of different types of events

  • Received: 31 May 2021 Accepted: 22 September 2021 Published: 29 September 2021
  • JEL Codes: E31, E44

  • In recent years, the frequency adjustment of U.S. monetary policy has a dynamic and global impact on other countries' economy. Based on the financial conditions index (FCI), the paper employs the time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) and spillover index respectively to investigate the time-varying impact of U.S. financial conditions (UFCI) on China's inflation (CINF) and its impact mechanisms. Some results are achieved as follows: first, the impacts of UFCI on CINF vary greatly over time both in the dimension of action duration and time point. Second, the effects of UFCI on CINF directly relate to different types of major events, and they are heterogeneous in action duration, degree, direction as well as the trend and range of fluctuations. In addition, UFCI can work on CINF through trade flow and China's financial market, and the China's financial market plays a main conductive role, and its conductive effect changes over time.

    Citation: Yanhong Feng, Shuanglian Chen, Wang Xuan, Tan Yong. Time-varying impact of U.S. financial conditions on China's inflation: a perspective of different types of events[J]. Quantitative Finance and Economics, 2021, 5(4): 604-622. doi: 10.3934/QFE.2021027

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  • In recent years, the frequency adjustment of U.S. monetary policy has a dynamic and global impact on other countries' economy. Based on the financial conditions index (FCI), the paper employs the time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) and spillover index respectively to investigate the time-varying impact of U.S. financial conditions (UFCI) on China's inflation (CINF) and its impact mechanisms. Some results are achieved as follows: first, the impacts of UFCI on CINF vary greatly over time both in the dimension of action duration and time point. Second, the effects of UFCI on CINF directly relate to different types of major events, and they are heterogeneous in action duration, degree, direction as well as the trend and range of fluctuations. In addition, UFCI can work on CINF through trade flow and China's financial market, and the China's financial market plays a main conductive role, and its conductive effect changes over time.



    HIV spreads through either cell-free viral infection or direct transmission from infected to healthy cells (cell-to-cell infection) [1]. It is reported that more than 50% of viral infection is caused by the cell-to-cell infection [2]. Cell-to-cell infection can occur when infected cells encounter healthy cells and form viral synapse [3]. Recently, applying reaction-diffusion equations to model viral dynamics with cell-to-cell infection have been received attentions (see, e.g., [4,5,6]). Ren et al. [5] proposed the following reaction-diffusion equation model with cell-to-cell infection:

    T(t,x)t=(d1(x)T)+λ(x)β1(x)TTβ2(x)TVd(x)T, t>0, xΩ,T(t,x)t=(d2(x)T)+β1(x)TT+β2(x)TVr(x)T, t>0, xΩ,V(t,x)t=(d3(x)V)+N(x)Te(x)V, t>0, xΩ,T(0,x)=T0(x)>0, T(0,x)=T0(x)0, V(0,x)=V0(x)0, xΩ. (1.1)

    In the model (1.1), T(t,x), T(t,x) and V(t,x) denote densities of healthy cells, infected cells and virus at time t and location x, respectively. The detailed biological meanings of parameters for the model (1.1) can be found in [5]. The well-posedness of the classical solutions for the model (1.1) have been studied. The model (1.1) admits a basic reproduction number R0, which is defined by the spectral radius of the next generation operator [5]. The model (1.1) defines a solution semiflow Ψ(t), which has a global attractor. The model (1.1) admits a unique virus-free steady state E0=(T0(x),0,0), which is globally attractive if R0<1. If R0>1, the model (1.1) admits at least one infection steady state and virus is uniformly persistent [5].

    It is a challenging problem to consider the global stability of E0 in the critical case of R0=1. In [6], Wang et al. studied global stability analysis in the critical case by establishing Lyapunov functions. Unfortunately, the method can not be applied for the model consisting of two or more equations with diffusion terms, which was left it as an open problem.

    Adopting the idea in [7,8,9], the present study is devoted to solving this open problem and shows that E0 is globally asymptotically stable when R0=1 for the model (1.2). For simplicity, in the following, we assume that the diffusion rates d1(x), d2(x) and d3(x) are positive constants. That is, we consider the following model

    T(t,x)t=d1ΔT+λ(x)β1(x)TTβ2(x)TVd(x)T, t>0, xΩ,T(t,x)t=d2ΔT+β1(x)TT+β2(x)TVr(x)T, t>0, xΩ,V(t,x)t=d3ΔV+N(x)Te(x)V, t>0, xΩ,T(0,x)=T0(x)>0, T(0,x)=T0(x)0, V(0,x)=V0(x)0, xΩ, (1.2)

    with the boundary conditions:

    T(t,x)ν=T(t,x)ν=V(t,x)ν=0, t>0, xΩ, (1.3)

    where Ω is the spatial domain and ν is the outward normal to Ω. We assume that all the location-dependent parameters are continuous, strictly positive and uniformly bounded functions on ¯Ω.

    Let Y=C(¯Ω,R3) with the supremum norm Y, Y+=C(¯Ω,R3+). Then (Y,Y+) is an ordered Banach space. Let T be the semigroup for the system:

    T(t,x)t=d2ΔT+β1(x)T0(x)T+β2(x)T0(x)Vr(x)T,V(t,x)t=d3ΔV+N(x)Te(x)V,

    where T0(x) is the solution of the elliptic problem d1ΔT+λ(x)d(x)T=0 under the boundary conditions (1.3). Then T has the generator

    ˜A=(d2Δ+β1(x)T0(x)r(x)β2(x)T0(x)N(x)d3Δe(x)).

    Let us define the exponential growth bound of T as

    ¯ω=¯ω(T):=limt+lnTt,

    and define the spectral bound of ˜A by

    s(˜A):=sup{Reλ,λσ(˜A)}.

    Theorem 2.1. If R0=1, E0 of the model (1.2) is globally asymptotically stable.

    Proof. We first show the local asymptotic stability of E0 of the model (1.2). Suppose ζ>0 and let v0=(T0,T0,V0) with v0E0ζ. Define m1(t,x)=T(t,x)T0(x)1 and p(t)=maxx¯Ω{m1(t,x),0}. According to d1ΔT0(x)+λ(x)d(x)T0(x)=0, we have

    m1td1Δm12d1T0(x)m1T0(x)+λ(x)T0(x)m1=β1(x)TTT0(x)β2(x)TVT0(x).

    Let ˜T1(t) be the positive semigroup generated by

    d1Δ+2d1T0(x)T0(x)λ(x)T0(x)

    associated with (1.3) (see Theorem 4.4.3 in [10]). From Theorem 4.4.3 in [10], we can find q>0 such that ˜T1(t)M1eqt for some M1>0. Hence, one gets

    m1(,t)=˜T1(t)m100˜T1(ts)[β1()T(,s)T(,s)T0()+β2()T(,s)V(,s)T0()]ds,

    where m10=T0T0(x)1. In view of the positivity of ˜T1(t), it follows that

    p(t)=maxx¯Ω{ω1(t,x),0}=maxx¯Ω{˜T1(t)m100˜T1(ts)[β1()T(,s)T(,s)T0()+β2()T(,s)V(,s)T0()]ds,0}maxx¯Ω{˜T1(t)m10,0}˜T1(t)m10M1eqtT0T0(x)1ζM1eqtTm,

    where Tm=minx¯Ω{T0(x)}. Note that (T,V) satisfies

    T(t,x)t=d2ΔT+β1(x)T0(x)T+β2(x)T0(x)Vr(x)T   +β1(x)T0(x)(TT0(x)1)T+β2(x)T0(x)(TT0(x)1)V,V(t,x)t=d3ΔV+N(x)Te(x)V.

    It then follows that

    (T(,t)V(,t))=T(t)(T0V0)
    +0T(ts)(β1()T0()(T(,s)T0()1)T(,s)+β2()T0()(T(,s)T0()1)V(,s)0)ds.

    From Theorem 3.5 in [11], we have that s(˜A)=sup{Reλ,λσ(˜A)} has the same sign as R01. If R0=1, then s(˜A)=0. Then we easily verify all the conditions of Proposition 4.15 in [12]. It follows from R0=1 and Proposition 4.15 in [12] that we can find M1>0 such that T(t)M1 for t0, where M1 can be chosen as large as needed in the sequel. Since p(s)ζM1eqsTm, one gets

    max{T(,t),V(,t)}M1max{T0,V0}     +M1(β1+β2)T00p(s)max{T(s),V(s)}dsM1ζ+M2ζ0eqsmax{T(s),V(s)}ds,

    where

    M2=M21(β1+β2)T0Tm.

    By using Gronwall's inequality, we get

    max{T(,t),V(,t)}M1ζe0ζM2eqsdsM1ζeζM2q.

    Then Ttd1ΔT>λ(x)d(x)TM1ζeζM2q(β1(x)+β2(x))T. Let ˆu1 be the solution of the system:

    ˆu1(t,x)t=d1Δˆu1+λ(x)d(x)ˆu1M1ζeζM2q(β1(x)+β2(x))ˆu1, xΩ, t>0,ˆu1(t,x)ν=0, xΩ, t>0,ˆu1(x,0)=T0, x¯Ω. (2.1)

    Then T(t,x)ˆu1(t,x) for x¯Ω and t0. Let Tζ(x) be the positive steady state of the model (2.1) and ˆm(t,x)=ˆu1(t,x)Tζ(x). Then ˆm(t,x) satisfies

    ˆm(t,x)t=d1Δˆm[d(x)+M1ζeζM2q(β1(x)+β2(x))]ˆm, xΩ, t>0,ˆm(t,x)ν=0, xΩ, t>0,ˆm(x,0)=T0Tζ(x), x¯Ω.

    For sufficiently large M1, from Theorem 4.4.3 in [10], we have F1(t)M1e¯α0t, where ¯α0<0 is a constant and F1(t):C(¯Ω,R)C(¯Ω,R) is the C0 semigroup of d1Δd() subject to (1.3) [5]. Hence, we have

    ˆm(,t)=F1(t)(T0Tζ(x))0F1(ts)M1ζeζM2q(β1()+β2())ˆm(,s)ds,ˆm(,t)M1T0Tζ(x)e¯α0t+0M21e¯α0(ts)ζeζM2q(β1()+β2())ˆm(,s)ds.

    Let K=M21ζeζM2q(β1+β2). By employing Gronwall's inequality, one gets

    ˆu1(,t)Tζ(x)=ˆm(,t)M1T0Tζ(x)eKt+¯α0t.

    Choosing ζ>0 sufficiently small such that K<¯α02, then

    ˆu1(,t)Tζ(x)M1T0Tζ(x)e¯α0t/2,

    and

    T(,t)T0(x)ˆu1(,t)ˆU(x)=ˆu1(,t)Uπ(x)+Uπ(x)ˆU(x)ζM1(M1+1)Tζ(x)T0(x).

    Since p(t)ζM1Tm, one gets T(,t)T0(x)ζM1T0(x)Tm, and hence

    T(,t)T0(x)=max{ζM1+(M1+1)Tζ(x)T0(x),ζM1T0(x)Tm}.

    From

    limζ0Tζ(x)=T0(x),

    by choosing ζ small enough, for t>0, there holds T(,t)T0(x), T(,t), V(,t)ε, which implies the local asymptotic stability of E0.

    By Theorem 1 in [5], the solution semiflow Ψ(t):Y+Y+ of the model (1.2) has a global attractor Π. In the following, we prove the global attractivity of E0. Define

    Y1={(˜T,~T,˜V)Y+:~T=˜V=0}.

    Claim 1. For v0=(T0,T0,V0)Π, the omega limit set ω(v0)Y1.

    Since Ttd1ΔT+λ(x)d(x)T, T is a subsolution of the problem

    ˆT(t,x)t=d1ΔˆT+λ(x)d(x)ˆT, xΩ, t>0,ˆT(t,x)ν=0, xΩ, t>0,ˆT(x,0)=T0(x), x¯Ω. (2.2)

    It is well known that model (2.2) has a unique positive steady state T0(x), which is globally attractive. This together with the comparison theorem implies that

    lim supt+T(t,x)lim supt+ˆT(t,x)=T0(x),

    uniformly for xΩ. Since v0=(T0,T0,V0)Π, we know T0T0. If T0=V0=0, the claim easily holds. We assume that either T00 or V00. Thus one gets T(t,x)>0 and V(t,x)>0 for x¯Ω and t>0. Then T(t,x) satisfies

    T(t,x)t<d1ΔT+λ(x)d(x)T(t,x),xΩ, t>0,T(t,x)ν=0,xΩ, t>0,T(x,0)T0(x), xΩ.

    The comparison principle yields T(t,x)<T0(x) for x¯Ω and t>0. Following [7], we introduce

    h(t,v0):=inf{˜hR:T(,t)˜hϕ2,V(,t)˜hϕ3}.

    Then h(t,v0)>0 for t>0. We show that h(t,v0) is strictly decreasing. To this end, we fix t0>0, and let ¯T(,t)=h(t0,v0)ϕ2 and ¯V(,t)=h(t0,v0)ϕ3 for tt0. Due to T(,t)<T0(x), one gets

    ¯T(t,x)t>d2Δ¯T+β1(x)T¯T+β2(x)T¯Vr(x)¯T,¯V(t,x)t=d3Δ¯V+N(x)¯Te(x)¯V,¯T(x,t0)T(x,t0), ¯V(x,t0)V(x,t0), xΩ. (2.3)

    Hence (¯T(t,x),¯V(t,x))(T(t,x),V(t,x)) for x¯Ω and tt0. From the model (2.3), one gets h(t0,v0)ϕ2(x)=¯T(t,x)>T(t,x) for x¯Ω and t>t0. Similarly, we get h(t0,v0)ϕ3(x)=¯V(t,x)>V(t,x) for x¯Ω and t>t0. Since t0>0 is arbitrary, h(t,v0) is strictly decreasing. Let h=limt+h(t,v0). Then we have h=0. Let Q=(Q1,Q2,Q3)ω(v0). Then there is {tk} with tk+ such that Ψ(tk)v0Q. We get h(t,Q)=h for t0 due to limt+Ψ(t+tk)v0=Ψ(t)limt+Ψ(tk)v0=Ψ(t)Q. If Q20 and Q30, we repeat the above discussions to illustrate that h(t,Q) is strictly decreasing, which contradicts to h(t,Q)=h. Thus, we have Q2=Q3=0.

    Claim 2. Π={E0}.

    Since {E0} is globally attractive in Y1, {E0} is the only compact invariant subset of the model (1.2). From the invariance of ω(v0) and ω(v0)Y1, one gets ω(v0)={E0}. By Lemma 3.11 in [9], we get Π={E0}.

    The local asymptotic stability and global attractivity yield the global asymptotic stability of E0.

    The research is supported by the NNSF of China (11901360) to W. Wang and supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China (2019030196) to X. Lai. We also want to thank the anonymous referees for their careful reading that helped us to improve the manuscript.

    The authors decare no conflicts of interest.



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