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Table 5 Bai and Perron (1998, 2003) and Elliot and Mueller (2004)’ Test Statistics for Breaks and Model Selection: 01/01/2019–06/30/2020

From: COVID-19 and instability of stock market performance: evidence from the U.S.

SupF (1)

SupF (2)

SupF (3)

UDmaxa

WDmax

SupFT(1|0)

SupFT(2|1)

SupFT(3|2)

\(\widehat{J}\)-Stat

BIC

LWZ

SM

Panel A: \(\pi =20\mathrm{\%},m=3\mathrm{ breaks}\)

 Model I: \({R}_{t}^{S\&P 500}={\beta }_{0}^{S\&P 500}+{\beta }_{1}^{S\&P 500}{DS}_{t-1}^{US}+{\beta }_{2}^{S\&P 500}{R}_{t-1}^{S\&P 500}+{\varepsilon }_{t}\)

  8.6434**

5.5298

4.2090

8.6434**

8.6434**

7.4733*

3.4279

0.0000

− 16.9766**

1

0

1

 Model II: \({R}_{t}^{DJIA}={\beta }_{0}^{DJIA}+{\beta }_{1}^{DJIA}{{DS}_{t-1}^{US}+\beta }_{2}^{DJIA}{R}_{t-1}^{DJIA}+{\varepsilon }_{t}\)

  7.4514*

5.1430

3.6911

7.4514*

7.4514*

6.9766*

5.0930

0.2501

− 13.8949*

1

0

1

Panel B:\(\pi =25\mathrm{\%},m=2\mathrm{ breaks}\)

 Model I: \({R}_{t}^{S\&P 500}={\beta }_{0}^{S\&P 500}+{\beta }_{1}^{S\&P 500}{DS}_{t-1}^{US}+{\beta }_{2}^{S\&P 500}{R}_{t-1}^{S\&P 500}+{\varepsilon }_{t}\)

  7.9936*

5.1009

/

7.9936*

7.9936*

7.5632*

3.4504

–

− 14.1311*

1

0

1

 Model II: \({R}_{t}^{DJIA}={\beta }_{0}^{DJIA}+{\beta }_{1}^{DJIA}{{DS}_{t-1}^{US}+\beta }_{2}^{DJIA}{R}_{t-1}^{DJIA}+{\varepsilon }_{t}\)

  7.3832*

5.1663

/

7.3832*

7.3832*

6.9766*

5.1307

–

− 13.8012*

1

0

1

  1. Table 5 reports the results of various test for structural breaks in both S&P 500 and DJIA return models and those for the number of true breaks over the period January 1st, 2019 to June 30th, 2020. The SupF statistics are used to test the hypothesis of no structural break against the one-sided (upper-tail) alternative of m = k (k is a predefined number). The double maximum statistics and the J-test statistic are used to test hypothesis of no structural break against the one-sided (upper-tail) alternative of an unknown number of breaks. The \({Sup\mathrm{F}}_{\mathrm{T}}(l+1|l)\) statistics have the null hypothesis of l breaks against the one-sided (upper-tail) alternative of l+1 breaks. The Bayesian information criterion (BIC) (Yao 1988), a modified Schwarz criterion (LWZ) (Liu et al. 1997) and the sequential method (SM) (Bai and Perron 1998) are used to test the number of true breaks. The \({Sup\mathrm{F}}_{\mathrm{T}}(l+1|l)\) test in the sequential context is different from the sequential method because the first l breaks are not necessarily the global minimizers of the sum of squared residuals. The statistics allow for the possibility of heteroscedasticity and serial correlation in the errors. The heteroscedasticity and autocorrelation consistent covariance matrix is constructed following Andrews (1991) and Andrews and Monahan (1992) using a quadratic kernel with automatic bandwidth selection based on a first order Vector Auto-regression (VAR (1)) approximation. The autocorrelation from the residuals is removed also using a VAR (1)
  2. * and **indicate significance at the 10% and 5% levels, respectively