A general panel break test based on the self-normalization method

Abstract

We propose new break tests for parameters such as mean, variance, quantile and others of panel data sets, in a general setup based on the self-normalization method. The self-normalization tests show much better size than existing tests, resolving their over-size problem for panels with serial dependence, cross-sectional dependence, conditional heteroscedasticity and/or N relative larger than T, which is demonstrated theoretically by a nuisance parameter free limiting null distribution and experimentally by very stable finite sample sizes. The proposed test is also implemented much more easily than the existing tests in that the proposed test needs no bandwidth selection for the long-run variance estimation and is computed very simply. Applications of the self-normalization test to the financial stock return and realized volatility indicate more toward absence of breaks of mean and/or variance than the existing tests which neglect cross-sectional correlation and other features apparent in the data sets.

Appendix

1.1 A.1 Proofs

Proof of Theorem 3.2

The result is straightforwardly derived by combining delta method and the invariance principle of ${\hat{\Theta }}_{1,T}$ in Assumption 3.1. $\square$

Proof of Theorem 3.3

We get the desired result by showing

$$\begin{aligned}&\sqrt{T} ({\hat{\xi }}_{1,[Tz]} - {\hat{\xi }}_{[Tz]+1,T}) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega (\frac{1}{z} + \frac{1}{1-z}) B^0(z), \end{aligned}$$

(12)

$$\begin{aligned}&\frac{1}{T} \sum _{s=1}^{[Tz]} \{\sqrt{T} \frac{s}{T} ({\hat{\xi }}_{1,s} -{\hat{\xi }}_{1,[Tz]})\}^2 {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega ^2 \int _0^z [B^0(u) - \frac{u}{z} B^0(1)]^2 du, \end{aligned}$$

(13)

$$\begin{aligned}&\frac{1}{T} \sum _{s=[Tz]+1}^T \{\sqrt{T}(1-\frac{s}{T})({\hat{\xi }}_{s,T}\nonumber \\&\quad - {\hat{\xi }}_{[Tz]+1,T})\}^2 {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega ^2\int _z^1 [B(1)-B(u) - \frac{1-u}{1-z} (B(1) - B(z))]^2) du. \end{aligned}$$

(14)

We first show (12). By Assumption 3.1, we have

$$\begin{aligned} -\sqrt{T}(1-z)({\hat{\xi }}_{[Tz]+1,T} - {\hat{\xi }}_{1,T}) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega B^0(z). \end{aligned}$$

(15)

Accordingly, by applying the continuous mapping theorem, we obtain (12) from

$$\begin{aligned} \begin{aligned}&\sqrt{T}({\hat{\xi }}_{1,[Tz]} - {\hat{\xi }}_{[Tz]+1,T}) = \sqrt{T}({\hat{\xi }}_{1,[Tz]}-{\hat{\xi }}_{1,T}\\&\quad +{\hat{\xi }}_{1,T}-{\hat{\xi }}_{[Tz]+1,T} ) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega \{\frac{1}{z} B^0(z) + \frac{1}{(1-z)} B^0(z)\}. \end{aligned} \end{aligned}$$

Similarly, the results (13), (14) are attained by combining Lemma 3.2, (15),

$$\begin{aligned}&\frac{1}{T} \sum _{s=1}^{[Tz]} \{\sqrt{T}\frac{s}{T} ({\hat{\xi }}_{1,s} - {\hat{\xi }}_{1,[Tz]})\}^2 = \frac{1}{T} \sum _{s=1}^{[Tz]} \{ \sqrt{T} \frac{s}{T} ({\hat{\xi }}_{1,s} - {\hat{\xi }}_{1,T}\\&\quad + {\hat{\xi }}_{1,T} - {\hat{\xi }}_{1,[Tz]}) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \int _0^z [B^0(u) - \frac{u}{z}B^0(z)]^2 du\end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\frac{1}{T} \sum _{s=[Tz]+1}^T \{ \sqrt{T}(1-\frac{s}{T})({\hat{\xi }}_{s,T} - {\hat{\xi }}_{[Tz]+1,T})\}^2\\&\quad = \frac{1}{T} \sum _{s=[Tz]+1}^T \{ \sqrt{T}(1-\frac{s}{T})({\hat{\xi }}_{s,T} - {\hat{\xi }}_{1,T} + {\hat{\xi }}_{1,T} - {\hat{\xi }}_{[Tz]+1,T}) \}^2 \\&\quad {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \int _{z}^1 [B(1)- B(u) - \frac{1-u}{1-z}(B(1)-B(z))]^2 du. \end{aligned} \end{aligned}$$

$\square$

Proof of Theorem 3.8

We get the result by showing

$$\begin{aligned} \sup _{z \in [0,1]} |L_{N,T} (z)| {{\mathop {\longrightarrow }\limits ^{\mathrm{p}}} }\infty . \end{aligned}$$

(16)

It sufficies to show, for $\tau /T \rightarrow z_0 \in (0,1)$,

$$\begin{aligned} \begin{aligned} \sup _{z \in [0,1]}&L_{N,T} (z) \ge L_{N,T} (z_0)\\ =&\frac{[\sqrt{T} z_0 (1-z_0) \{ \sum _{i=1}^N {\hat{\lambda }}_i^{-1}(W({\hat{F}}_{i,1,[Tz_0]}) - W({\hat{F}}_{i, [Tz_0]+1, T}))\}]^2}{\frac{1}{T} \sum _{s= 1}^{[Tz_0]} [\frac{s}{\sqrt{T}} \{ \sum _{i=1}^N {\hat{\lambda }}_i^{-1}( W({\hat{F}}_{i,1,s}) - W({\hat{F}}_{i,1,[Tz_0]}))\}]^2 + \frac{1}{T}\sum _{s= [Tz_0]+1}^T [\frac{T-s}{\sqrt{T}} \{\sum _{i=1}^N {\hat{\lambda }}_i^{-1}(W({\hat{F}}_{i,s,T}) - W({\hat{F}}_{i,[Tz_0]+1,T}))\}]^2}\\ =&\frac{D_1^2(z_0)}{D_2(z_0) + D_3(z_0)} {{\mathop {\longrightarrow }\limits ^{\mathrm{p}}} }\infty . \end{aligned} \end{aligned}$$

Therefore, (16) can be obtained by showing, as $T\rightarrow \infty$, for fixed N,

$$\begin{aligned}\text {(i)}~ D_1(z_0) \rightarrow \infty , ~\text {(ii)}~ D_2(z_0) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} d_2, ~\text {(iii)}~ D_3(z_0) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} d_3\end{aligned}$$

for some random variables $d_2$ and $d_3$.

We first show (i). From (5), we have

$$\begin{aligned} \begin{aligned}&D_1(z_0) = \sqrt{T} z_0(1-z_0) \{\sum _{i=1}^N {\hat{\lambda }}^{-1}_i (\frac{1}{[Tz_0]} \sum _{t=1}^{[Tz]} (W(F_{it}) + IF(Y_{it}; F_{it})) + R_{i,1,[Tz]} \\&\quad - \frac{1}{T-[Tz_0]} \sum _{t=[Tz_0]+1}^T (W(F_{it}) + IF(Y_{it}; F_{it})) + R_{i,[Tz_0]+1, T})\}. \end{aligned} \end{aligned}$$

(17)

Note that, combining Assumption 3.7 and the delta method, $IF(Y_{it}; F_{it})$ satisfies invariance principle

$$\begin{aligned}&\sum _{i=1}^N \frac{1}{\sqrt{T}} \sum _{t=1}^{[Tz]} IF(Y_{it}; F_{it}) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega _{IF} B(z),~~ \nonumber \\&\quad \sum _{i=1}^N \frac{1}{\sqrt{T}} \sum _{t=[Tz]+1}^{T} IF(Y_{it}; F_{it}) {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega _{IF}(B(1)-B(z)), \end{aligned}$$

(18)

and, by Assumption 3.7, $R_{i,1,[Tz]}$ and $R_{i,[Tz]+1, T}$ are uniformly negligible. Then, it is enough to show

$$\begin{aligned} D_{11}(z_0) = \sqrt{T}z_0(1-z_0) \{ \sum _{i=1}^N {\hat{\lambda }}_i^{-1} (\frac{1}{[Tz_0]} \sum _{t=1}^{[Tz_0]} W(F_{it}) - \frac{1}{T-[Tz_0]} \sum _{t=[Tz_0]+1}^T W(F_{it}))\} \rightarrow \infty . \end{aligned}$$

(19)

We have

$$\begin{aligned} \begin{aligned} D_{11}(z_0) =&\sqrt{T}z_0(1-z_0)\{ \sum _{i=1}^N {\hat{\lambda }}_i^{-1} ( \frac{1}{[Tz_0]} \sum _{t=1}^{[Tz_0]} (W(F_i) +\delta _i I(t> \tau ))\\&\quad - \frac{1}{T-[Tz_0]} \sum _{t=[Tz_0]+1}^T (W(F_i) +\delta _i I(t > \tau ))) \}\\ =&\sqrt{T} z_0 (1-z_0) \{\sum _{i=1}^N \delta _iA_{Tz_0}/{\hat{\lambda }}_i\} , \end{aligned} \end{aligned}$$

where $A_{Tz } = \frac{\sum _{t=1}^{[Tz_0]} I(t> [Tz])}{[Tz_0]} - \frac{\sum _{t=[Tz_0]+1}^T I(t>[Tz])}{T-[Tz_0]},$ and, since $A_{Tz_0}{{\mathop {\longrightarrow }\limits ^{\mathrm{p}}} }-1$ and $\sum _{i=1}^N \delta _i/{\lambda }_i \ne 0$, we get (i).

Similarly to (17)-(19), (ii) is derived by observing

$$\begin{aligned} \begin{aligned} D_2(z_0)&= \frac{1}{T} \sum _{s=1}^{[Tz_0]} [ \frac{s}{\sqrt{T}} \{ \sum _{i=1}^N {\hat{\lambda }}_i^{-1} ( \frac{1}{s} \sum _{t=1}^s IF(Y_{it}; F_{it}) + R_{i,1,s}\\&\quad - \frac{1}{[Tz_0]} \sum _{t=1}^{[Tz_0]} IF(Y_{it}; F_{it}) + R_{i,1,[Tz_0]})\}]^2 {\mathop {\longrightarrow }\limits ^{\mathrm{d}}} \omega _{IF}^2 \int _0^{z_0} [B(u) - \frac{u}{z_0} B(z_0)]^2 du, \end{aligned} \end{aligned}$$

and (iii) is obtained similarly. $\square$

1.2 A.2 Bandwidth for long-run variance estimator and block length for moving block bootstrapping

We specify the bandwidth for the long-run variance estimator and the block length for the moving block bootstrapping, which are required by the existing tests considered in Sect. 5. The long run variance estimator for the tests $H_{opt}$, B, $B^*$ for panel mean break is considered with the optimal bandwidth of Andrews (1991) $l= 1.1447\{4T{\hat{\rho }}^2/(1-{\hat{\rho }}^2)^2\}^{1/3},$ where ${\hat{\rho }}$ is the estimated AR(1) coefficient in the AR(1) fitting to $\sum _{i=1}^N Y_{i,t},~t =1, \ldots , T$. For the test H, we consider the bandwidth $l = 3$, which were considered by Horvath and Huskova (2012). The tests U, W for the panel variance break use the bandwidth $l =3, T^{1/3}$, respectively, which were considered by Shi (2015)and Li et al. (2015).

The block length parameter $L_{MB}$ of the bootstrap tests $B^*$, $J^*$ is considered to be the optimal block length parameters of Politis and White (2004, Section 3.2),

$$\begin{aligned}L_{MB} = [2{\hat{G}}^2T/(\frac{4}{3}{\hat{g}}^2(0))]^{1/3}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}{\hat{G}} =&\sum _{ k=-M}^M \lambda (k/M) |k| {\hat{R}}(k),~~{\hat{g}}(r) = \sum _{ k=-M}^M \lambda (k/M) {\hat{R}}(k) cos(rk),~~ M = logT, \\ \lambda (y)&= 1~ \text { if } |y|<\frac{1}{2},~~\lambda (y)=2(1-|y|)~ \text { if }\frac{1}{2} \le |y| \le 1,~~\\ {\hat{R}}(k)&=\frac{1}{T}\sum _{t=1}^{T-|k|}(R_t-{\bar{R}}_T)(R_{t+|k|}-{\bar{R}}_T),\\ R_t =&\sum _{i=1}^N ({E}_{it} - {\bar{E}}_i),~{E}_{it}=(\frac{Y_{it}-{\bar{Y}}_{i}}{sd_{i}})^2,~~{\bar{E}}_i = \frac{1}{T}\sum _{t=1}^T E_{it} ~~{\bar{R}}_T = \frac{1}{T} \sum _{t=1}^T R_t, \end{aligned} \end{aligned}$$

and $sd_{i}^2$ is the sample variance of $Y_{i,t},t=1,\ldots ,T$.