STRUCTURAL EQUATION MODELING WITH LATENT VARIABLES FOR DIFFUSION PROCESSES BASED ON HIGH-FREQUENCY DATA SHOGO KUSANO1AND MASAYUKI UCHIDA12

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STRUCTURAL EQUATION MODELING WITH LATENT VARIABLES FOR

DIFFUSION PROCESSES BASED ON HIGH-FREQUENCY DATA

SHOGO KUSANO 1AND MASAYUKI UCHIDA 1,2

Abstract. We consider structural equation modeling (SEM) with latent variables for diﬀusion processes

based on high-frequency data. We derive the quasi-likelihood estimators for parameters in the SEM. The

goodness-of-ﬁt test based on the quasi-likelihood ratio is proposed. Furthermore, the asymptotic properties

of our proposed estimators are examined.

1. Introduction

We consider structural equation modeling (SEM) with latent variables for diﬀusion processes. The

stochastic process X1,t is deﬁned as the following factor model:

X1,t = Λx1,mξm,t +δm,t,(1.1)

where m∈Nis a model number, {X1,t}t≥0is a p1-dimensional observable vector process, {ξm,t}t≥0is a

k1,m-dimensional latent common factor vector process, {δm,t}t≥0is a p1-dimensional latent unique factor

vector process, Λx1,m ∈Rp1×k1,m is a constant loading matrix, p1is not zero, p1and k1,m are ﬁxed, and

k1,m ≤p1. The stochastic process X2,t is deﬁned by the factor model as follows:

X2,t = Λx2,mηm,t +εm,t,(1.2)

where {X2,t}t≥0is a p2-dimensional observable vector process, {ηm,t}t≥0is a k2,m-dimensional latent common

factor vector process, {εm,t}t≥0is a p2-dimensional latent unique factor vector process, Λx2,m ∈Rp2×k2,m is

a constant loading matrix, p2is not zero, p2and k2,m are ﬁxed, and k2,m ≤p2. Furthermore, the relationship

between ηm,t and ξm,t is expressed as follows:

ηm,t =B0,mηm,t + Γmξm,t +ζm,t,(1.3)

where {ζm,t}t≥0is a k2,m-dimensional latent unique factor vector process, B0,m ∈Rk2,m ×k2,m is a constant

loading matrix, whose diagonal elements are zero, and Γm∈Rk2,m×k1,m is a constant loading matrix. Assume

that {ξm,t}t≥0satisﬁes the following stochastic diﬀerential equation:







dξm,t =B1,m(ξm,t)dt+S1,mdW1,t (t∈[0, T ]),

ξm,0=c1,m,

1Graduate School of Engineering Science, Osaka University

2Center for Mathematical Modeling and Data Science (MMDS), Osaka University and JST CREST

Key words and phrases. Structural equation modeling; Asymptotic theory; High-frequency data; Stochastic diﬀerential

equation; Quasi-maximum likelihood estimation.

arXiv:2210.11677v1 [math.ST] 21 Oct 2022

2 S. KUSANO AND M. UCHIDA

where B1,m :Rk1,m →Rk1,m ,S1,m ∈Rk1,m×r1,c1,m ∈Rk1,m and W1,t is an r1-dimensional standard Wiener

process, {δm,t}t≥0is deﬁned as the following stochastic diﬀerential equation:







dδm,t =B2,m(δm,t)dt+S2,mdW2,t (t∈[0, T ]),

δm,0=c2,m,

where B2,m :Rp1→Rp1,S2,m ∈Rp1×r2,c2,m ∈Rp1and W2,t is an r2-dimensional standard Wiener process,

{εm,t}t≥0satisﬁes the following stochastic diﬀerential equation:







dεm,t =B3,m(εm,t)dt+S3,mdW3,t (t∈[0, T ]),

εm,0=c3,m,

where B3,m :Rp2→Rp2,S3,m ∈Rp2×r3,c3,m ∈Rp2and W3,t is an r3-dimensional standard Wiener process,

and {ζm,t}t≥0is deﬁned by the stochastic diﬀerential equation as follows:







dζm,t =B4,m(ζm,t)dt+S4,mdW4,t (t∈[0, T ]),

ζm,0=c4,m,

where B4,m :Rk2,m →Rk2,m ,S4,m ∈Rk2,m×r4,c4,m ∈Rk2,m and W4,t is an r4-dimensional standard Wiener

process. We assume that W1,t,W2,t,W3,t and W4,t are independent. Set Xt= (X>

1,t, X>

2,t)>.{Xtn

i}n

i=1 are

discrete observations, where tn

i=ihnand T=nhn, and p1,p2,k1,m and k2,m are independent of n.

SEM is a method that describes the relationships between latent variables that cannot be observed. SEM

has been used in various ﬁelds, e.g., behavioral science, economics, engineering, and medical science. For

example, in psychology, SEM is used to investigate the relationships between intelligence and motivation.

Note that intelligence and motivation are latent variables. J¨oreskog [16] proposed this method by combining

path analysis and conﬁrmatory factor analysis. For path analysis and conﬁrmatory factor analysis, see, e.g.,

Mueller [23]. Several models have been proposed to formulate SEM. In this paper, we consider the model

deﬁned by (1.1), (1.2) and (1.3), which is called the LInear Structural RELations (LISREL) model ( J¨oreskog

[17]). The LISREL model is one of the most well-known models in SEM and can be expressed complex

relationships between latent variables. For more information on the LISREL model, see, e.g., Everitt [10].

Note that SEM is a conﬁrmatory analysis method rather than an exploratory analysis method. SEM is used

to specify the model from a theoretical viewpoint of each research ﬁeld before conducting the analysis. This is

the diﬀerence between conﬁrmatory analysis methods and exploratory analysis methods such as exploratory

factor analysis. In behavioral science, factor analysis for time series data has been actively studied; see,

e.g., Molenaar [22] and Pena and box [25]. Moreover, Czi´aky [8] proposed SEM for time series data called

dynamic structural equation model with latent variables (DSEM). Asparouhov et.al. [3] studied the more

general DSEM model.

Recently, we can easily obtain high-frequency data such as stock price data and life-log data (blood

pressure and EEG, etc.) thanks to the development of measuring devices, and statistical inference for

stochastic diﬀerential equations based on high-frequency data has been developed. For parametric estimation

of diﬀusion processes based on high-frequency data, see for example, Yoshida [28], Genon-Catalot and Jacod

[11], Kessler [18], Uchida and Yoshida [27] and references therein. In ﬁnancial econometrics, the factor

model for high-frequency data has been extensively researched. In this ﬁeld, parameters and the number of

SEM FOR DIFFUSION PROCESSES BASED ON HIGH-FREQUENCY DATA 3

factors are estimated by using principal component analysis for high-frequency data (A¨ıt-Sahalia and Xiu

[2]) when the factor is latent; see, e.g., A¨ıt-Sahalia and Xiu [1]. However, these studies are based on high

dimensionality. For a low-dimensional model, the estimator does not have consistency; see Bai [4]. On the

other hand, Kusano and Uchida [20] proposed classical factor analysis for diﬀusion processes. Their method

works well for a low-dimensional model. However, to the best of our knowledge, there have been few studies

of SEM for high-frequency data. Oud and Jansen [24] and Driver et.al. [9] considered SEM for stochastic

diﬀerential equations. Note that their model diﬀers from the model in this paper. In the ﬁeld of causal

inference, Hansen and Sokol [13] studied SEM for stochastic diﬀerential equations. However, their model is

the path analysis model, so that their method cannot describe the relationships between latent variables.

Note that these studies do not assume that the data is sampled at high-frequency. On the other hand, we

propose SEM for diﬀusion processes based on high-frequency data.

In this paper, we assume that the volatilities for diﬀusion processes and loading matrices are not time-

variant but constant to simplify the discussion. We leave for future work the discussion on the model where

the volatilities for diﬀusion processes and loading matrices are time-varying. Furthermore, we do not discuss

a high-dimensional case. Bai [5] studied the asymptotic properties of factor analysis based on the maximum

likelihood estimation for a high-dimension model. We expect that our quasi-likelihood method will also work

well for a high-dimension model. The investigation is future work.

The paper is organized as follows. In Section 2, notation and assumptions are introduced. In Section 3,

we study SEM for diﬀusion processes in the ergodic and non-ergodic cases. First, the asymptotic properties

of the realized covariance are examined. Next, we derive the quasi-likelihood estimators for parameters in

the SEM. It is shown that the estimators have good asymptotic properties. Furthermore, we propose the

goodness-of-ﬁt test based on the quasi-likelihood ratio and investigate the asymptotic properties. In Section

4, we give examples and simulation studies to investigate the asymptotic performance of the results described

in Section 3. Section 5 is devoted to the proofs of theorems given in Section 3.

2. Notation and assumptions

For any vector v,|v|=√tr vv>and v(i)is the i-th element of v, where >denotes the transpose. For

any matrix A, kAk=√tr AA>, and Aij is the (i, j)-th element of A.Ipdenotes the identity matrix of

size p. Deﬁne Op×qas the p×qzero matrix. For any symmetric matrix A∈Rp×p, vec A, vech Aand Dp

denote the vectorization of A, the half-vectorization of Aand the p2×¯pduplication matrix, respectively.

Here, vec A=Dpvech Aand ¯p=p(p+ 1)/2; see, e.g., Harville [14]. For any matrix A, the Moore-Penrose

inverse of Ais denoted by A+. If Ais a positive deﬁnite matrix, we write A > 0. For any positive

sequence un,R: [0,∞)×Rd→Ris deﬁned as |R(un, x)| ≤ unC(1 + |x|)Cfor some C > 0. Let Fn

σ(W1,s, W2,s, W3,s, W4,s, s ≤tn

i) for i= 1,···n. Let Ck

↑(Rd) be the space of all functions fsatisfying the

following conditions:

(i) fis continuously diﬀerentiable with respect to x∈Rdup to order k.

(ii) fand all its derivatives are of polynomial growth in x∈Rd, i.e., gis of polynomial growth in x∈Rd

if g(x) = R(1, x).

4 S. KUSANO AND M. UCHIDA

Np(µ, Σ) represents the p-dimensional normal random variable with mean µ∈Rpand covariance matrix

Σ∈Rp×p. Let χ2

rbe the random variable which has the chi-squared distribution with rdegrees of freedom.

χ2

r(α) denotes an upper αpoint of the chi-squared distribution with rdegrees of freedom, where 0 ≤α≤1.

The symbols P

−→ and d

−→ express convergence in probability and convergence in distribution, respectively.

Let Σξξ,m =S1,mS>

1,m, Σδδ,m =S2,mS>

2,m, Σεε,m =S3,mS>

3,m, Σζζ,m =S4,mS>

4,m and Ψm=Ik2,m −B0,m.

Furthermore, we make the following assumptions.

[A1] (a) There exists a constant C > 0 such that for any x, y ∈Rk1,m ,

|B1,m(x)−B1,m(y)| ≤ C|x−y|.

(b) For all `≥0, sup

E|ξm,t|`<∞.

↑(Rk1,m ).

[A2] The diﬀusion process ξm,t is ergodic with its invariant measure πξm: For any πξm-integrable function

g, it holds that

TZT

g(ξm,t)dt P

−→ Zg(x)πξm(dx)

as T−→ ∞.

[B1] (a) There exists a constant C > 0 such that for any x, y ∈Rp1,

|B2,m(x)−B2,m(y)| ≤ C|x−y|.

(b) For all `≥0, sup

E|δm,t|`<∞.

↑(Rp1).

[B2] Σδδ,m >0.

[B3] The diﬀusion process δm,t is ergodic with its invariant measure πδm: For any πδm-integrable function

g, it holds that

TZT

g(δm,t)dt P

−→ Zg(x)πδm(dx)

as T−→ ∞.

[C1] (a) There exists a constant C > 0 such that for any x, y ∈Rp2,

|B3,m(x)−B3,m(y)| ≤ C|x−y|.

(b) For all `≥0, sup

E|εm,t|`<∞.

↑(Rp2).

[C2] Σεε,m >0.

[C3] The diﬀusion process εm,t is ergodic with its invariant measure πεm: For any πεm-integrable function

g, it holds that

TZT

g(εm,t)dt P

−→ Zg(x)πεm(dx)

as T−→ ∞.

SEM FOR DIFFUSION PROCESSES BASED ON HIGH-FREQUENCY DATA 5

[D1] (a) There exists a constant C > 0 such that for any x, y ∈Rk2,m ,

|B4,m(x)−B4,m(y)| ≤ C|x−y|.

(b) For all `≥0, sup

E|ζm,t|`<∞.

↑(Rk2,m ).

[D2] The diﬀusion process ζm,t is ergodic with its invariant measure πζm: For any πζm-integrable function

g, it holds that

TZT

g(ζm,t)dt P

−→ Zg(x)πζm(dx)

as T−→ ∞.

[E] Ψmis non-singular.

[F] rank Λx1,m =k1,m.

Remark 1 [A1], [B1], [C1] and [D1] are the standard assumptions for ergodic diﬀusion processes. For

example, see Kessler [18]. [B2], [C2], [E] and [F] imply that Σm(θm) is non-singular. For details, see Lemma

3. Main theorems

3.1. Ergodic case. In the LISREL model, we will estimate Λx1,m, Λx2,m, Γm, Ψm, Σξξ,m, Σδδ,m, Σεε,m

and Σζζ,m. Note that some of these elements are assumed to be known in order to satisfy an identiﬁability

condition for parameter estimation. See Remark 4 for constraints on the parameter and the identiﬁability

condition. Set the parameter as θm∈Θm, where Θm⊂Rqmis a convex compact space. θmincludes only

unknown and non-duplicated elements of Λx1,m, Λx2,m, Γm, Ψm, Σξξ,m, Σδδ,m, Σεε,m and Σζζ,m. Deﬁne the

covariance structure as

Σm(θm) = ΣX1X1,m(θm) ΣX1X2,m(θm)

ΣX1X2,m(θm)>ΣX2X2,m(θm)!,(3.1)

where

ΣX1X1,m(θm) = Λx1,mΣξξ,mΛ>

x1,m + Σδδ,m,

ΣX1X2,m(θm) = Λx1,mΣξξ,mΓ>

mΨ−1>

mΛ>

x2,m,

ΣX2X2,m(θm) = Λx2,mΨ−1

m(ΓmΣξξ,mΓ>

m+ Σζζ,m)Ψ−1>

mΛ>

x2,m + Σεε,m.

To estimate (3.1), we use the realized covariance as follows:

QXX =1

i=1

(Xtn

i−Xtn

i−1)(Xtn

i−Xtn

i−1)>.

Let

Wm(θm)=2D+>

p(Σm(θm)⊗Σm(θm))D+

For the realized covariance, the following theorem holds.

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STRUCTURALEQUATIONMODELINGWITHLATENTVARIABLESFORDIFFUSIONPROCESSESBASEDONHIGH-FREQUENCYDATASHOGOKUSANO1ANDMASAYUKIUCHIDA1;2Abstract.Weconsiderstructuralequationmodeling(SEM)withlatentvariablesfordiusionprocessesbasedonhigh-frequencydata.Wederivethequasi-likelihoodestimatorsforparametersintheSEM.The...

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STRUCTURAL EQUATION MODELING WITH LATENT VARIABLES FOR DIFFUSION PROCESSES BASED ON HIGH-FREQUENCY DATA SHOGO KUSANO1AND MASAYUKI UCHIDA12

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