Cross-Sectional Pricing Tests

Methodology

We evaluate each factor parameter configuration using the Gibbons et al. (1989) test (GRS), which tests whether all intercepts (alphas) from time-series regressions of test portfolio returns on the factor model are jointly zero.

\[F_{\text{GRS}} = \frac{T - N - K}{N} \cdot \frac{\hat{\alpha}' \hat{\Sigma}^{-1} \hat{\alpha}}{1 + \hat{\mu}_f' \hat{\Sigma}_f^{-1} \hat{\mu}_f} \sim F(N, T - N - K)\]

A higher GRS p-value indicates the factor model better prices the cross-section (we fail to reject \(H_0\): all \(\alpha_i = 0\)).

Test Assets

Following Liu, Tsyvinski & Wu (2022, Tables III–VI), we form quintile (5-way) value-weighted portfolios sorted on 8 characteristics, yielding 40 test assets:

Characteristic	Label	Liu Table	Description
Market capitalisation	MCAP	III	Prior-week market cap
Price level	PRC	III	Prior-week closing price
Dollar volume	PRCVOL	V	Prior-week mean daily volume
1-week return	\(r_{1,0}\)	IV	Return over prior 1 week
2-week return	\(r_{2,0}\)	IV	Return over prior 2 weeks
3-week return	\(r_{3,0}\)	IV	Return over prior 3 weeks (baseline)
4-week return	\(r_{4,0}\)	IV	Return over prior 4 weeks
Skip-week return	\(r_{4,1}\)	IV	Return over 3 weeks, skipping most recent

All sort variables are lagged by one week relative to the portfolio return period to avoid look-ahead bias. Test portfolios are always value-weighted quintile sorts regardless of the factor model’s weighting scheme.

Factor Configurations Tested

Each factor model is characterised by 6 parameters. We test all 288 combinations:

Parameter	Values tested
Exclusions	All (stables + wrapped + derivs), Stablecoins only, None
Calendar	Sharp year (Liu et al., 2022), Monday–Monday
Weighting	Equal-weighted, Value-weighted
Size breakpoints	Median (2), Tercile (3), Quintile (5), Decile (10)
Momentum lookback	2, 3, 4 weeks
Delisting returns	Off, On (-100%)

The test portfolios (LHS) share the same exclusion, calendar, and delisting settings as the factor model (RHS) to ensure a consistent universe.

Results

Note

Results are generated by scripts/test_factor_pricing_liu.R and will be updated after each pipeline run. The table below shows the most recent results.

Top 20 Configurations

Rank	Exclusion	Calendar	Delist	Weight	Size N	Mom LB	GRS F	GRS p	Mean \|α\|	Avg R²
1	none	mon	On	VW	5	4	1.72	0.0045	0.00782	0.607
2	nostable	mon	On	VW	5	4	1.80	0.0021	0.00782	0.612
3	none	mon	On	VW	5	3	1.83	0.0016	0.00781	0.611
4	none	mon	On	VW	2	4	1.84	0.0015	0.00712	0.664
5	none	mon	On	VW	3	3	1.85	0.0015	0.00727	0.644
6	nostable	mon	Off	VW	10	4	1.86	0.0014	0.00711	0.531
7	none	mon	On	VW	3	4	1.86	0.0013	0.00743	0.643
8	nostable	mon	Off	EW	10	4	1.87	0.0012	0.00693	0.581
9	none	mon	On	VW	2	3	1.89	0.0010	0.00713	0.666
10	none	mon	On	VW	10	4	1.90	0.0009	0.00738	0.516
11	none	mon	On	VW	3	2	1.91	0.0008	0.00706	0.648
12	nostable	mon	On	VW	5	3	1.91	0.0008	0.00778	0.617
13	all	mon	On	VW	5	4	1.91	0.0008	0.00793	0.612
14	nostable	mon	On	VW	3	3	1.92	0.0008	0.00722	0.648
15	none	mon	Off	VW	10	4	1.92	0.0007	0.00716	0.522
16	none	mon	Off	EW	10	4	1.93	0.0007	0.00709	0.574
17	nostable	mon	On	VW	2	4	1.94	0.0006	0.00706	0.668
18	all	mon	Off	VW	10	4	1.95	0.0006	0.00732	0.532
19	nostable	mon	On	VW	2	3	1.95	0.0005	0.00707	0.670
20	all	mon	Off	EW	10	4	1.96	0.0005	0.00726	0.583

Marginal Effects by Parameter Dimension

Exclusion

Exclusion	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
none	0.000175	0.000010	0.00743	0.645	96
nostable	0.000118	0.000015	0.00729	0.649	96
all	0.000046	0.000004	0.00746	0.653	96

Calendar

Calendar	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
mon	0.000214	0.000037	0.00772	0.641	144
sharp	0.000012	0.000000	0.00707	0.657	144

Delist Return

Delist Return	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
TRUE	0.000160	0.000001	0.00738	0.666	144
FALSE	0.000066	0.000016	0.00741	0.633	144

Weighting

Weighting	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
vw	0.000184	0.000008	0.00750	0.637	144
ew	0.000042	0.000008	0.00729	0.662	144

Size Breakpoints

Size Breakpoints	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
5	0.000157	0.000007	0.00764	0.643	72
10	0.000125	0.000027	0.00715	0.602	72
3	0.000100	0.000008	0.00746	0.671	72
2	0.000071	0.000002	0.00732	0.681	72

Mom Lookback

Mom Lookback	Mean GRS p	Median GRS p	Mean \|α\|	Mean R²	n
4	0.000217	0.000020	0.00732	0.641	96
3	0.000089	0.000008	0.00737	0.648	96
2	0.000034	0.000002	0.00749	0.659	96

Best Configuration per Exclusion Level

Exclusion	Calendar	Delist	Weight	Size N	Mom LB	GRS F	GRS p	Mean \|α\|	Avg R²
all	mon	On	VW	5	4	1.91	0.000782	0.00793	0.612
none	mon	On	VW	5	4	1.72	0.004464	0.00782	0.607
nostable	mon	On	VW	5	4	1.80	0.002143	0.00782	0.612

Interpretation

The GRS test evaluates whether a given three-factor model (CMKT, CSMB, CMOM) can fully explain the cross-sectional return variation across the 40 test portfolios. Key metrics:

GRS F-statistic: lower is better (smaller unexplained alpha)
GRS p-value: higher is better (fail to reject H₀ that all alphas = 0)
Mean |α|: average absolute pricing error across test assets (lower is better)
Avg R²: average time-series explanatory power (higher is better)

References

Gibbons, M. R., Ross, S. A., & Shanken, J. (1989). A test of the efficiency of a given portfolio. Econometrica, 57(5), 1121–1152. https://doi.org/10.2307/1913625

Liu, Y., Tsyvinski, A., & Wu, X. (2022). Common risk factors in cryptocurrency. The Journal of Finance, 77(2), 1133–1177. https://doi.org/10.1111/jofi.13119