# A Two-Stage Estimator for Change Point in the Mean of Panel Data

Authors: Wenzhi Zhao; Yinqian Yang; Di Zhang
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, a two-stage consistency estimator for change point in the mean of panel data is given. Firstly, a single sequence is extracted, and the initial estimator and confidence interval of the change point are given by the least square method. Based on the confidence interval, a random interval containing change point with probability tending to 1 is constructed. Secondly, using all panel data falling into the random interval, the final estimator of change point is obtained by least square estimation. The asymptotic distribution is established. Simulation results show that our method can not only ensure the estimation accuracy but also greatly reduce time complexity.


---

## Body

## 1. Introduction

One of the tools to analyse large, high-dimensional datasets is the panel data model. This paper studies the problem of structural changes for panel data, in which there areN series (variables), and each series has T observations. It is assumed that there is only one change that has taken place in each series at an unknown common point, referred to as the common change point. Common change points in panel data are wide spread phenomena. For example, the outbreak of the epidemic may impact every country’s GDP. A tax policy change may alter each firm’s investment incentive. While it may be difficult to identify a change point with single series, it should be, naturally, much easier to locate the common change point using a number of series together. This paper explores the panel data approach to the estimation of change point.Joseph and Wolfson [1, 2] are the early researchers who laid the groundwork in change point for panel data. They proposed a random change point in which each series has its own change point; across N series, the change points are assumed to be independent and identically distributed (i.i.d.). They proved that the common distribution of the i.i.d. change points can be consistently estimated. This random change point model is extended to the autoregressive model proposed by Joseph et al. [3]. Joseph et al. [4] considered the Bayesian framework. Skates et al. [5] and Jackson and Sharples [6] studied the application-oriented Bayesian models. Bai [7] established the consistency of the estimated common change point in panel data by the least square method. Horváth and Hušková [8] and Shin and Hwang [9] used the CUSUM method to test change point in the mean of panel data. Bai [7] conducted ratio-type statistics to detect change point in panel data.Computer-based technology allows scientists to collect enormous datasets, and huge data demand new methodology. In massive data, how to quickly and accurately estimate the change point location has become a real problem to be solved. Cao and Xia [10] considered a fast estimation method for univariate sequence. This paper proposes a two-stage estimation method to locate change point in the mean of panel data, and the consistency of the estimator is proved.


## 2. Model and Assumptions

We assume that we studyN panels and we have T observations in each panel. We define our model as(1)Xit=μi1+εit,1≤t≤k0,μi2+εit,k0+1≤t≤T,t=1,2,…,T,i=1,2,…,N,where Eεit=0 for all i and t. In this model, each series has a change point at k0, where k0 is unknown and k0=τ0T for some τ0∈0,1. The prechange mean of Xit is μi1, and postchange mean is μi2. The difference λi=μi2−μi1 represents the magnitude of change in ith panel, which can be either random or nonrandom, and is assumed to be independent of error process εit. In this paper, we assume that each series has common change point at k0. Our purpose is to give consistent estimators with lower time complexity.For a givenk such that 1⩽k⩽T−1, define(2)X¯i1=1k∑t=1kXit,X¯i2=1T−k∑t=k+1TXit.So,X¯i1 and X¯i2 are estimators for μi1 and μi2, respectively. The classical least square estimator for k0 in Peštová and Pešta [11] is defined as(3)k^LS=argmin1≤k≤T−1SSRk,where(4)SSRk=∑i=1NSiTk,SiTk=∑t=1kXit−X¯i12+∑t=k+1TXit−X¯i22.This estimator is straightforward to compute, and the time complexity isONT2. When T and N are huge, it is not to easy to locate the change point. In this paper, a two-stage estimation method is proposed to reduce the time complexity.We adopt some assumptions in Bai [7].Assumption 1.
εit=∑j=0∞aijei,t−j,ei,t∼0,σie2 are i.i.d. over t; ∑jjaij≤M for all i. In addition, εit are independent over t. Let σi2=Eεit2=σie2∑jaij2.Assumption 2.
limN⟶∞∑i=11μi2−μi12=∞.Assumption 3.
T is larger than N such that loglogTN/T⟶0 as T and N go to infinity.Assumption 4.
μi2−μi1=N−1/2Δi, with limN⟶∞∑i=1Nμi2−μi12=limN⟶∞1/N∑i=1N△i2=λ, limN⟶∞∑i=1Nμi2−μi12σi2=limN⟶∞1/N∑i=1N△i2σi2=ϕ.


## 3. Two-Stage Estimator

### 3.1. The Initial Estimator

For any given1≤i0≤N, a univariate change point series with T observations Xi0t is selected from N panels. The initial estimator for k0 is defined as(5)l^=argmin1≤l≤T−1∑t=1lXi0t−X¯i012+∑t=l+1TXi0t−X¯i022=argmax1≤l≤T−1lT−lT21/2X¯i02−X¯i01.Actually, the initial change point estimatorl^ is an ordinary least square estimator for change point in mean of univariate series. Let γ^=l^/T and λi0=μi02−μi01. Then, according to Proposition 3 and Theorem 1 in Bai [12], the following conclusions can be drawn.(6)γ^−τ0=Op1Tλi02,(7)Tλi02γ^−τ0⟶d∑j=0∞ai0j2σi02argmaxυWυ−υ2,,where Wv is a two-sided Brownian motion on R.Denoteλ^i02=X¯i02−X¯i01; A^i0 is a consistent estimator for Dεi0t=∑j=0∞ai0j2σi02, and cα is the α quantile of argmaxυWυ−υ/2. Using (7), the 1−α% confidence interval for k0 is conducted as(8)l^−A^i0λ^i02⋅cα,l^+A^i0λ^i02⋅cα.GivenBT⟶∞T⟶∞ and BT/T⟶0, we enlarge confidence interval (8) to(9)l^−BT⋅A^i0λ^i02⋅cα,l^+BTA^i0λ^i02⋅cα.Defineh1=l^−BT⋅A^i0/λ^i02⋅cα, h2=l^+BT⋅A^i0/λ^i02⋅cα, and T0=h2−h1+1. So,(10)Pk0∈h1,h2⟶1,asT⟶∞.The time complexity for initial estimator isOT2.


### 3.2. The Finial Estimator

Using all samples falling into the random intervalh1,h2, the finial estimator is defined as(11)k^TS=argminh1≤k<h2SSR1k,where(12)SSR1k=∑i=1N∑t=h1kXit−X¯i12+∑t=k+1h2Xit−X¯i22.Denote(13)UNTk=1NTSSR1k=1NT∑i=1NSiTk,and then(14)k^TS=argmink∈h1,h2SSR1k=argmink∈h1,h2UNTk.It is easy to see that time complexity for finial estimator isONBT2. So, total time complexity for the two-stage estimation method is OT2+NBT2, which is smaller than ONT2 because BT/T⟶0. It implies that the two-stage estimation method can give the change point position faster. Furthermore, Theorems 1 and 2 in Section 4 ensure the accuracy of our estimator.In order to prove the properties of the two-stage estimator, we need the following lemmas described in Bai [7].Lemma (A.1).
Assume that model (1) and Assumption 1 hold, and we have(15)sup1≤k≤TUNTk−EUNTk=Op1NT.Lemma (A.2).
Assume that model (1) and Assumption 1 hold. For all k∈1,T, the expected value of UNTk satisfies(16)EUNTk−EUNTk0≥λCk−k0NT,where λ=∑i=1Nμi2−μi12 for some C>0.


## 3.1. The Initial Estimator

For any given1≤i0≤N, a univariate change point series with T observations Xi0t is selected from N panels. The initial estimator for k0 is defined as(5)l^=argmin1≤l≤T−1∑t=1lXi0t−X¯i012+∑t=l+1TXi0t−X¯i022=argmax1≤l≤T−1lT−lT21/2X¯i02−X¯i01.Actually, the initial change point estimatorl^ is an ordinary least square estimator for change point in mean of univariate series. Let γ^=l^/T and λi0=μi02−μi01. Then, according to Proposition 3 and Theorem 1 in Bai [12], the following conclusions can be drawn.(6)γ^−τ0=Op1Tλi02,(7)Tλi02γ^−τ0⟶d∑j=0∞ai0j2σi02argmaxυWυ−υ2,,where Wv is a two-sided Brownian motion on R.Denoteλ^i02=X¯i02−X¯i01; A^i0 is a consistent estimator for Dεi0t=∑j=0∞ai0j2σi02, and cα is the α quantile of argmaxυWυ−υ/2. Using (7), the 1−α% confidence interval for k0 is conducted as(8)l^−A^i0λ^i02⋅cα,l^+A^i0λ^i02⋅cα.GivenBT⟶∞T⟶∞ and BT/T⟶0, we enlarge confidence interval (8) to(9)l^−BT⋅A^i0λ^i02⋅cα,l^+BTA^i0λ^i02⋅cα.Defineh1=l^−BT⋅A^i0/λ^i02⋅cα, h2=l^+BT⋅A^i0/λ^i02⋅cα, and T0=h2−h1+1. So,(10)Pk0∈h1,h2⟶1,asT⟶∞.The time complexity for initial estimator isOT2.


## 3.2. The Finial Estimator

Using all samples falling into the random intervalh1,h2, the finial estimator is defined as(11)k^TS=argminh1≤k<h2SSR1k,where(12)SSR1k=∑i=1N∑t=h1kXit−X¯i12+∑t=k+1h2Xit−X¯i22.Denote(13)UNTk=1NTSSR1k=1NT∑i=1NSiTk,and then(14)k^TS=argmink∈h1,h2SSR1k=argmink∈h1,h2UNTk.It is easy to see that time complexity for finial estimator isONBT2. So, total time complexity for the two-stage estimation method is OT2+NBT2, which is smaller than ONT2 because BT/T⟶0. It implies that the two-stage estimation method can give the change point position faster. Furthermore, Theorems 1 and 2 in Section 4 ensure the accuracy of our estimator.In order to prove the properties of the two-stage estimator, we need the following lemmas described in Bai [7].Lemma (A.1).
Assume that model (1) and Assumption 1 hold, and we have(15)sup1≤k≤TUNTk−EUNTk=Op1NT.Lemma (A.2).
Assume that model (1) and Assumption 1 hold. For all k∈1,T, the expected value of UNTk satisfies(16)EUNTk−EUNTk0≥λCk−k0NT,where λ=∑i=1Nμi2−μi12 for some C>0.


## 4. Theorem and Proof

The following properties can be obtained.Theorem 1.
Under Assumptions1–3, we have(17)limN,T⟶∞Pk^TS=k0=1.Proof.
Due to symmetry, it is sufficient to considerk≤k0. According to Lemmas (A.1) and (A.2), for any k∈h1,h2, since(18)UNTk−UNTk0=UNTk−EUNTk−UNTk0−EUNTk0+EUNTk−EUNTk0UNTk−UNTk0≥−2suph1≤k≤h2UNTk−EUNTk+EUNTk−EUNTk0≥−2suph1≤k≤h2UNTk−EUNTk+λCk−k0NT,and PUNTk^TS−UNTk0≤0⟶1, then we have that(19)k^TS−k0≤2λ−1C−1NTsuph1≤k≤h2UNTk−EUNTk=2λ−1C−1NT1/2Op1.
It can be concluded from Assumptions1 and 2 and (19) that(20)k^TS−k0T=2λCNTOp1=op1.
Thus,k^TS−k0=opT. So, for any ε>0, k^TS−k0≤εT for large T, with probability tending to 1. Because k0=Tτ0 and τ∈0,1, there exists δ>0 such that Pk^TS∈δT,1−δT⟶1 as T⟶∞. That is, Pk^TS∈D⟶1, where D=k:δT≤k≤1−δT,h1≤k≤h2.
Define the setDk0=D/k0 so that Dk0 excludes k0 from D. Then,(21)Pk^TS≠k0≤Pk^TS∉D+Pk^TS∈D,k^TS≠k0=Pk^TS∉D+Pk^TS∈Dk0.
By the definition ofk^TS, UNTk^TS≤UNTk0. So, a necessary condition for k^TS∈Dk0 is mink∈Dk0UNTkUNTk0≤0. Similar to Lemma A.3 of Bai [7], it can be proved that Pmink∈Dk0UNTk−UNTk0≤0⟶0, which implies that Pk^TS∈Dk0⟶0. Thus,(22)Pk^TS≠k0≤Pk^TS∉D+Pk^TS∈Dk0⟶0.
This completes the proof.Theorem 2.
Under Assumptions1, 3, and 4, as T,N⟶∞,(23)k^TS−k0⟶dargminℓℓλ+2ϕWℓ,where W0=0 and Wℓ=∑s=−ℓ+10Zs,ℓ=−1,−2,…, Wℓ=∑s=1ℓZs,ℓ=1,2,…, and Zs,s=…,−1,0,1,… are i.i.d. standard normal random variables.Proof.
Again, by symmetry, it is sufficient to considerk≤k0. For k≤k0 and k∈h1,h2,(24)X¯i1=μi1+1k∑t=h1kεit,X¯i2=μi1+T0−k0T0−kμi2−μi1+1T0−k∑t=k+1h2εit=k0−kT0−kμi1−μi2+μi2+1T0−k∑t=k+1h2εit.
Introduce(25)ε¯i1=1k∑t=h1kεit,ε¯i2=1T0−k∑t=k+1h2εit,and let(26)aik=T0−k0T0−kμi2−μi1,bik=k0−kT0−kμi1−μi2.
It follows that(27)X¯i1=μi1+ε¯i1,X¯i2=μi1+aik+ε¯i2=μi2+bik+ε¯i2.
By the definition ofSiTk, we get(28)SiTk=∑t=h1kεit−ε¯i12+∑t=k+1k0εit−aik−ε¯i22+∑t=k0+1h2εit−bik−ε¯i22=∑t=h1kεit−ε¯i12+∑t=k+1h2εit−ε¯i22+k0−kaik2+T0−kbik2−2aik∑t=k+1k0εit−ε¯i2−2bik∑t=k0+1h2εit−ε¯i2.
Notice that(29)∑t=h1kεit−ε¯i12+∑t=k+1h2εit−ε¯i22=∑t=h1h2εit2−kε¯i12−T0−kε¯i22,and thus(30)SSR1k=k0−k∑i=1Naik2+T−k0∑i=1Nbik2+∑i=1N∑t=h1h2εit2−∑i=1Nkε¯i12−∑i=1NT0−kε¯i22−∑i=1Nkε¯i12−∑i=1NT0−kε¯i22−2∑i=1Naik∑t=k+1k0εit−ε¯i2−2∑i=1Nbik∑t=k0+1h2εit−ε¯i2.
So,(31)SSR1k−SSR1k0=k0−k∑i=1Naik2+T−k0∑i=1Nbik2−∑i=1N1k∑t=h1kεit2−1k0∑t=h1k0εit2−∑i=1N1T0−k∑t=k+1h2εit2−1T0−k0∑t=k0+1h2εit2−2∑i=1Naik∑t=k+1k0εit−2∑i=1Nbik∑t=k0+1h2εit+2∑i=1Nk0−kaik+T0−k0bikε¯i2.
According to Theorem 4.2 of Bai [7], the first and the fifth terms are k0−kOp1 and all others are k0−kOp1.
BecauseT0−k0/T0−k⟶p1, we have(32)k0−k∑i=1Naik2=k0−k∑i=1NT0−k0T0−kμi2−μi12⟶pk0−kλ.
Similarly,(33)−2∑i=1Naik∑t=k+1k0εit=−2T0−k0T0−k∑t=k+1k0∑i=1Nμi2−μi1εit=−2T0−k0T0−k∑t=k+1k0N−1/2∑i=1NΔiεit=−2T0−k0T0−kϕN∑t=k+1k0N−1/2∑i=1Nωiεit,where(34)ωi=ΔiϕN=Δi1/N∑j=1NΔj2σ21/2.
Under Assumption4, ϕN=1/N∑j=1NΔj2σj2⟶ϕ and N−1/2∑i=1Nωiεit⟶Zt, where Zt∼N0,1. Thus, the limit of the fifth term of (31) is 2ϕ1/2∑t=k+1k0Zt.
In summary, fork≤k0,(35)SSR1k−SSR1k0⟶dk0−kλ+2ϕ1/2∑t=k+1k0Zt.
Similarly, fork>k0, we can prove that(36)SSR1k−SSR1k0⟶dk−k0λ+2ϕ1/2∑t=k0+1kZt.
Letℓ=k0−k, k^TS−k0⟶dargminℓℓλ+2ϕWℓ by functional central limit theorem.


## 5. Monte Carlo Comparison

We compare two estimators (3) and (11) by Monte Carlo simulation on the same computer. The series is generated according to model (1), where εit∼i.i.d.N0,1, μi1∼i.i.d.U−10,10 and μi1−μi2∼N0.5,0.01. Experiments are carried out for N=80,100,120, T=1000,1200,1400, and k0=0.5T. We choose BT=lgT,lnT,log2T. Table 1 reports our simulation results based on 500 replications, where Mean, Sd, and Tc stand for the average estimator, standard deviation, and the operated time for the computer, respectively, while time is in seconds.Table 1
Simulation comparison of two-stage estimation and least square estimation.
N=80,T=1000N=100,T=1200N=120,T=1400MeanSdTcsMeanSdTcsMeanSdTcsTSlgT501.218 (26.411)0.926601.084 (21.981)1.180700.164 (20.535)1.468lnT499.998 (0.184)2.452600.012 (0.126)3.194700.006 (0.073)3.838log2T499.998 (0.118)3.834600.006 (0.134)5.074700.000 (0.063)5.946LS499.998 (0.195)7.640599.988 (0.141)11.378699.998 (0.077)24.856It can be seen from Table1 that with the increasing number of sequences of N and the sample size of T, the running times are getting longer and longer for both the two-stage method and least square method. When N and T are fixed, with the increase of BT, the two-stage estimator is closer to the true change point and the running time is increased slightly. The running time of the two-stage estimation method is much less than that of the least square method. This shows that in the case of massive data, the method in this paper can estimate the change point position faster.



---

*Source: 1455812-2021-09-18.xml*