Improved calendar time approach for measuring long-run anomalies

Figures & data

Table 1. Specification of tests in random samples

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	1.2	2.8	2.4	2.8	1.2	2.4
Buy-and-hold abnormal return method	6.2^*	0.8	4.4^*	0.4	4.8^*	0.0
Fama–French three-factor model	4.4^*	0.8	2.0	0.8	3.2	1.6
Panel B: Value-weighted portfolios
Standardized calendar time approach	1.6	2.4	1.2	3.2	1.2	3.4
Fama–French three-factor model	3.6	1.2	2.0	2.0	0.8	2.8

Notes: This table presents the percentages of 1,000 random samples of 200 firms that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5 at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level. We, however, employ 25 size-BM reference portfolios to measure the BHARs and CTARs.

Table 2. Specification of tests in random samples

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardization only approach	1.3	8.3^*	2.0	3.2	3.4	2.8
Weighting only approach	5.6^*	0.4	2.0	3.0	1.1	4.2^*
Panel B: Value-weighted portfolios
Standardization only approach	0.2	4.0^*	1.6	2.7	4.1^*	2.3
Weighting only approach	3.8^*	2.6	1.2	3.1	0.3	5.8^*

Notes: This table reports the results from standardization only approach and weighting only approach. The standardization only approach does not weight the monthly portfolios, while the weighting only approach does not standardize the event firm anomalies. The numbers shown in this table present the percentages of 1,000 random samples of 200 firms that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level.

Table 3. Regression analysis of hot and cold markets

Periods	EW porfolios	VW portfolios
Hot period	0.12 (0.79)	0.08 (1.19)
Cold period	0.23 (-0.91)	0.14 (0.31)

Notes: This table presents the effect of hot and cold markets on CTAR obtained from SCTA over a three-year horizon. EW and VW portfolios indicate equally and value-weighted portfolios, respectively. The values in the parentheses report the t-statistics.

Table 4. Specification of tests in samples with large firms

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	0.4	4.8^*	0.8	2.0	1.6	2.0
Buy-and-hold abnormal return method	0.0	16.8^*	0.0	18.8^*	0.0	22.0^*
Fama–French three-factor model	0.8	4.0^*	1.6	2.8	2.8	2.0
Panel B: Value-weighted portfolios
Standardized calendar time approach	0.0	4.0^*	0.8	2.4	1.6	3.4
Fama–French three-factor model	0.8	4.8^*	0.8	2.8	2.8	2.4

Notes: This table presents the percentages of 1,000 samples of 200 large firms that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level.

Table 5. Specification of tests in samples with small firms

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	4.0^*	0.8	2.4	1.6	1.6	3.6
Buy-and-hold abnormal return method	24.6^*	0.0	9.2^*	0.0	9.2^*	0.0
Fama–French three-factor model	6.2^*	0.8	4.4^*	1.6	2.4	2.4
Panel B: Value-weighted portfolios
Standardized calendar time approach	4.6^*	2.4	3.2	2.4	0.4	2.8
Fama–French three-factor model	4.0^*	1.8	2.8	2.4	0.8	4.4^*

Notes: This table presents the percentages of 1,000 samples of 200 small firms that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level.

Table 6. Specification of tests in samples of firms with high BM value

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	2.0	3.2	1.6	2.0	0.0	6.0^*
Buy-and-hold abnormal return method	0.0	32.2^*	0.0	29.2^*	0.1	17.8^*
Fama–French three-factor model	0.4	7.2^*	0.2	5.6^*	0.8	3.6
Panel B: Value-weighted portfolios
Standardized calendar time approach	1.2	3.2	1.2	2.4	0.0	4.0^*
Fama–French three-factor model	0.0	10.1^*	0.4	5.4^*	0.8	5.6^*

Notes: This table presents the percentages of 1,000 samples of 200 firms with high BM value that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level.

Table 7. Specification of tests in samples of firms with low BM value

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	2.8	2.4	1.6	0.4	2.0	0.4
Buy-and-hold abnormal return method	14.0^*	0.0	10.4^*	0.2	8.2^*	0.0
Fama–French three-factor model	14.0^*	0.0	8.8^*	0.4	6.8^*	0.0
Panel B: Value-weighted portfolios
Standardized calendar time approach	1.6	2.4	2.4	0.8	1.2	0.4
Fama–French Three-Factor Model	11.0^*	0.0	6.0^*	0.8	4.0^*	0.0

Notes: This table presents the percentages of 1,000 samples of 200 firms with low BM value that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level.

Table 8. Specification of tests in samples with good pre-event returns: Case I

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	0.4	9.2^*	2.0	4.0^*	1.2	4.8^*
Buy-and-hold abnormal return method	0.0	6.8^*	14.0^*	0.2	21.4^*	1.6
Fama–French three-factor model	0.4	8.0^*	3.2	2.8	3.2	1.2
Panel B: Value-weighted portfolios
Standardized calendar time approach	0.0	12.0^*	1.2	3.6	0.4	4.8^*
Fama–French three-factor model	0.4	10.0^*	2.0	5.2^*	2.4	4.6^*

Notes: This table presents the percentages of 1,000 samples of 200 firms with good pre-event returns that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level. We, however, employ 25 size-BM reference portfolios to measure the BHARs and CTARs.

Table 9. Specification of tests in samples with poor pre-event returns: Case I

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	14.0^*	0.4	0.4	2.8	1.2	1.6
Buy-and-hold abnormal return method	21.2^*	0.0	8.4^*	0.1	4.6^*	0.3
Fama–French three-factor model	16.8^*	0.0	2.6	4.2^*	2.1	2.0
Panel B: Value-weighted portfolios
Standardized calendar time approach	6.8^*	0.4	0.4	4.0^*	0.8	4.0^*
Fama–French three-factor model	9.6^*	0.4	0.6	2.4	0.4	2.8

Notes: This table presents the percentages of 1,000 samples of 200 firms with poor pre-event returns that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level).

* Empirical size is significantly different from the 5% significance level. We, however, employ 25 size-BM reference portfolios to measure the BHARs and CTARs.

Table 10. Specification of tests in samples with good pre-event returns: Case II

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	0.3	4.6^*	3.1	3.6	2.4	0.7
Buy-and-hold abnormal return method	0.2	4.2^*	3.8^*	2.3	3.0	1.1
Carhart four-factor model	0.0	5.6^*	2.3	2.4	0.9	3.1
Panel B: Value-weighted portfolios
Standardized calendar time approach	2.8	4.0^*	1.8	3.4	3.6	2.1
Carhart four-factor model	0.4	3.8^*	2.6	1.9	2.9	3.6

Notes: This table presents the percentages of 1,000 samples of 200 firms with good pre-event returns that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level). However, the difference between this table and Table 4A is that here we employ $5\times 5\times 5$ size-BM-past returns reference portfolios to measure the BHARs and CTARs. We also estimate the four-factor model of Carhart (1997) to include the momentum factor computed as the difference between returns of winners and losers. We, therefore, exclude the three-factor model while constructing Table 4C.

* Empirical size is significantly different from the 5% significance level.

Table 11. Specification of tests in samples with poor pre-event returns: Case II

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	3.7^*	1.6	2.8	2.0	3.5	1.3
Buy-and-hold abnormal return method	4.8^*	1.2	3.6	2.7	2.6	2.2
Carhart four-factor model	4.4^*	0.3	3.7^*	1.9	2.3	1.6
Panel B: Value-weighted portfolios
Standardized calendar time approach	4.1^*	2.1	2.6	0.3	3.4	2.0
Carhart four-factor model	3.8^*	0.2	1.1	2.3	3.6	2.7

Notes: This table presents the percentages of 1,000 samples of 200 firms with poor pre-event returns that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level). However, the difference between this table and Table 4B is that here we employ $5\times 5\times 5$ size-BM-past returns reference portfolios to measure the BHARs and CTARs. We also estimate the four-factor model of Carhart (1997) to include the momentum factor computed as the difference between the returns of winners and losers. We, therefore, exclude the three-factor model while constructing Table 4D.

* Empirical size is significantly different from the 5% significance level.

Table 12. Specification of tests in samples with overlapping returns

Method description	1 Year		3 Years		5 Years
	2.5	97.5	2.5	97.5	2.5	97.5
Panel A: Equally weighted portfolios
Standardized calendar time approach	3.2	2.0	2.9	3.6	2.4	3.2
Buy-and-hold abnormal return method	7.2^*	0.4	9.4^*	3.6	10.2^*	4.8^*
Fama–French three-factor model	1.2	1.6	2.3	2.4	2.8	2.1
Panel B: Value-weighted portfolios
Standardized calendar time approach	3.2	1.6	1.8	3.1	1.6	2.0
Fama–French three-factor model	2.8	0.8	1.6	3.4	2.2	3.3

Notes: This table presents the percentages of 1,000 samples of 200 firms with overlapping returns that reject the null hypothesis of no abnormal returns over one-year, three-year, and five-year holding periods at the theoretical significance level of 5% in favor of the alternative hypothesis of a significantly negative intercept (i.e. calculated $p$ value is less than 2.5% at the 5% significance level) or a significantly positive intercept (calculated $p$ value is greater than 97.5% at the 5% significance level). Selecting these samples is a two-step procedure. The first step involves a random selection of 100 firms from the population. In the second step, for each of these 100 firms, we randomly choose a second event month that is within $H-1$ periods of the original event month (either before or after), where H equals 12, 36, or 60. Hence, we have 200 firms with 200 event months, where the same firm appears in the sample twice and this generates the issue of overlapping returns. We repeat this procedure 1,000 times.

* Empirical size is significantly different from the 5% significance level.

Figure 1. Simulated power of different methods in random samples. Notes: This figure represents the percentages of 1,000 random samples of 200 firms that reject the null hypothesis of no abnormal returns over three-year holding period. We consider equally weighted portfolios to make a direct comparison with the BHAR approach. The horizontal axis indicates the induced level of abnormal returns (%), while the rejection rates are shown in the vertical axis.

Table 13. Power of alternative methods in random samples

Method description	Induced level of abnormal return (%) over 3 years
	–20	–15	–10	–5	0	5	10	15	20
Standardized calendar time approach	1.00	0.88	0.56	0.11	0.05	0.18	0.71	0.92	1.00
Buy-and-hold abnormal return method	0.94	0.71	0.46	0.11	0.05	0.17	0.59	0.80	0.97
Fama–French three-factor model	0.78	0.55	0.34	0.08	0.04	0.10	0.39	0.58	0.85

Notes: This table presents the percentages of 1,000 random samples of 200 firms that reject the null hypothesis of no abnormal returns over a three-year holding period. We add the levels of annual abnormal return indicated in the column heading. However, only equally weighted portfolios are considered to make a direct comparison with the BHAR approach.

Carhart, M. (1997). On persistence in mutual fund performance. Journal of Finance, 52, 57–82.

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