Testing for shifts in mean with monotonic power against multiple structural changes

ABSTRACT

It is known that several widely used structural change tests have non-monotonic power because the long-run variance is poorly estimated under the alternative hypothesis. In this paper, we propose a modified long-run variance estimator to alleviate this problem. We theoretically show that the tests with our long-run variance estimator are consistent against large multiple structural changes. Simulation results show that the proposed test performs well in finite samples.

Keywords:

• Structural change
• non-monotonic power
• long-run variance

JEL classifications:

• C12
• C22

Acknowledgments

The author is grateful to Eiji Kurozumi, Yoshihiko Nishiyama, and the conference participants at Hitotsubashi University, Okayama University, and the University of Tokyo. All remaining errors are mine.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Methods to reduce the size distortion of the tests were proposed by Kejriwal [25] and Yamazaki and Kurozumi [26] in the case of a one-time break.

2 We may also use a Nadaraya-Watson estimator defined by (1) $\tilde{\theta }\left(\frac{t}{T}\right)=\frac{\sum _{s=1}^{\mathrm{T}}K\left(\frac{t-s}{Th}\right)y_s}{\sum _{s=1}^{\mathrm{T}}K\left(\frac{t-s}{Th}\right)}.$(1)

Note that the local linear estimator may be used.

3 The kernel and the bandwidth are selected so that they minimize the mean squared error of the long-run variance estimator (see [23]).

4 Assumption 3 (a)–(b) is the same as Assumption 3 in [27].

5 When $t=1,\dots ,0.05T$ and $t=T-0.05T+1,\dots ,T$, we use $y_{max\{1,t-0.05T\}},\dots ,y_{min\{T,t+0.05T\}}$ to obtain ${\hat{\sigma }}_t$, that is, ${\overline{y}}_t=\sum _{s=a_1\left(t\right)}^{a_2\left(t\right)}{y}_s/\left(a_2\right(t)-a_1(t)+1)$, ${\hat{\sigma }}_t^2=\sum _{s=a_1\left(t\right)}^{a_2\left(t\right)}(y_s-{\overline{y}}_t{)}^2/(a_2\left(t\right)-a_1\left(t\right))$ where $a_1\left(t\right)=max\{1,t-0.05T\}$ and $a_2\left(t\right)=min\{T,t+0.05T\}$.

6 Since ${\hat{\sigma }}_t$ tends to take large values when t is close to the structural break dates, the mean of ${\hat{\sigma }}_t$ tends to be greatly affected by the breaks, unlike the median. Therefore, we used the median of ${\hat{\sigma }}_t$ to select $L_T$.

7 The results for the cases with AR(2) and MA(1) errors are given in Tables S.1–S.3 in the Supplementary Material.

8 The results for $\phi =0.6$ are given in Figure S.1 in the Supplementary Material. In addition, the results for the case of c=1,2 for the $CUSU{M}^{\ast }$ test under DGP3, T=100 and $\phi =0.8$ are given in Figure A.2.

Additional information

Funding

The author gratefully acknowledges the financial support by the Japan Society for the Promotion of Science (JSPS) Research Fellowship for Young Scientists (KAKENHI Grant Number 16J07085).