# Change Point Estimation in Panel Data without Boundary Issue

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## Abstract

**:**

## 1. Introduction and Main Aims

#### 1.1. Current State of the Art

#### 1.2. Motivation in Non-Life Insurance

#### 1.3. Structure of the Paper

## 2. Abrupt Change in Panel Data

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Change Point Estimator

#### 3.1. Consistency

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Theorem 1**(Change point estimator consistency)

**.**

## 4. Simulation Study

## 5. Theoretical Usage in Hypothesis Testing

#### 5.1. Estimation of Correlation Structure

#### 5.2. Bootstrapping

**Theorem 2**(Bootstrap justification)

**.**

- (i)
- under ${H}_{0}$,$${\mathcal{S}}_{N,T}\underset{N\to \infty}{\overset{\mathcal{D}}{\to}}\mathcal{L};$$
- (ii)
- under additional Assumptions 3, 4 and under ${H}_{0}$, as well as under ${H}_{1}$,$${\mathcal{S}}_{N,T}^{*}|\mathbb{Y}\underset{N\to \infty}{\overset{\mathcal{D}}{\to}}{\mathcal{L}}^{*}\phantom{\rule{1.em}{0ex}}in\text{}probability\text{}\mathsf{P};$$
- (iii)
- under additional Assumptions 3, 4 and under ${H}_{0}$, $\mathcal{L}$ and ${\mathcal{L}}^{*}$ coincide.

## 6. Practical Application in Non-Life Insurance

## 7. Results and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of Theorem 1.**

**Proof**

**of Theorem 2.**

**Λ**is

## References

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**Figure 1.**Histograms of the estimated change points ${\widehat{\tau}}_{N}$ for various structures and distributions of the panel disturbances ($\tau =8$, $T=10$, $N=20$, $\sigma =0.2$; all of the panels are subject to a break of size ${\delta}_{i}\sim U[0,2]$).

**Figure 2.**Histograms of the estimated change points ${\widehat{\tau}}_{N}$ for various values of the change point τ ($T=10$, $N=20$, $\sigma =0.2$, $75\%$ of the panels are subject to a break of size ${\delta}_{i}\sim U[0,2]$ panel disturbances from AR(1) with $N(0,1)$ innovations).

**Figure 3.**Histograms of the estimated change points ${\widehat{\tau}}_{N}$ for various values of N ($\tau =9$, $T=10$, $\sigma =0.2$, $50\%$ of the panels are subject to a break of size ${\delta}_{i}\sim U[0,2]$, panel disturbances from AR(1) with ${t}_{5}$ innovations).

**Figure 4.**Histograms of the estimated change points ${\widehat{\tau}}_{N}$ for various values of σ ($\tau =1$, $T=10$, $N=10$, all of the panels are subject to a break of size ${\delta}_{i}\sim U[0,2]$, panel disturbances from GARCH(1,1) with $N(0,1)$ innovations).

**Figure 5.**Histograms of the estimated change points ${\widehat{\tau}}_{N}$ when various portion of the panels are subject to a break of size ${\delta}_{i}\sim U[0,2]$ ($\tau =5$, $T=10$, $N=20$, $\sigma =0.2$, panel disturbances from GARCH(1,1) with ${t}_{5}$ innovations).

**Figure 6.**Development of the yearly total claim amounts normalized by the earned premium together with the estimated change point ${\widehat{\tau}}_{146}=9$ (corresponding to year 1996).

**Figure 7.**Development of the yearly total claim amounts normalized by the earned premium together for the second half of the original observation period with the estimated change point ${\widehat{\tau}}_{146}=4$ (corresponding to year 1996).

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**MDPI and ACS Style**

Peštová, B.; Pešta, M.
Change Point Estimation in Panel Data without Boundary Issue. *Risks* **2017**, *5*, 7.
https://doi.org/10.3390/risks5010007

**AMA Style**

Peštová B, Pešta M.
Change Point Estimation in Panel Data without Boundary Issue. *Risks*. 2017; 5(1):7.
https://doi.org/10.3390/risks5010007

**Chicago/Turabian Style**

Peštová, Barbora, and Michal Pešta.
2017. "Change Point Estimation in Panel Data without Boundary Issue" *Risks* 5, no. 1: 7.
https://doi.org/10.3390/risks5010007