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	all	sharp	Off	EW	10	2	1.82	0.0018	0.00864	0.628
2	nostable	sharp	Off	EW	10	2	1.85	0.0014	0.00818	0.618
3	all	sharp	Off	VW	10	2	1.87	0.0012	0.00779	0.457
4	nostable	sharp	Off	VW	10	2	1.91	0.0008	0.00735	0.450
5	none	sharp	Off	EW	10	2	1.93	0.0007	0.00832	0.616
6	none	mon	Off	EW	10	4	1.94	0.0006	0.00648	0.587
7	all	sharp	Off	EW	10	3	1.95	0.0005	0.00896	0.620
8	nostable	sharp	Off	EW	10	3	1.97	0.0005	0.00863	0.615
9	none	mon	Off	VW	10	4	1.97	0.0005	0.00621	0.551
10	all	sharp	Off	VW	10	4	1.98	0.0004	0.00723	0.469
11	all	sharp	Off	VW	10	3	1.98	0.0004	0.00794	0.455
12	all	sharp	Off	EW	10	4	1.99	0.0004	0.00720	0.538
13	nostable	sharp	Off	EW	10	4	1.99	0.0004	0.00674	0.531
14	nostable	sharp	Off	VW	10	4	1.99	0.0004	0.00724	0.464
15	none	sharp	Off	VW	10	2	2.00	0.0003	0.00751	0.445
16	nostable	sharp	Off	VW	10	3	2.03	0.0003	0.00783	0.452
17	none	mon	Off	EW	10	2	2.03	0.0003	0.00661	0.601
18	nostable	mon	Off	EW	10	4	2.04	0.0002	0.00621	0.596
19	all	mon	Off	EW	10	4	2.08	0.0002	0.00675	0.598
20	nostable	mon	Off	VW	10	4	2.09	0.0001	0.00610	0.560

Marginal Effects by Parameter Dimension

Exclusion

Exclusion	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
all	0.000055	0.000000	0.00807	0.637	96
nostable	0.000046	0.000000	0.00787	0.633	96
none	0.000031	0.000000	0.00804	0.630	96

Calendar

Calendar	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
sharp	0.000070	0.000000	0.00902	0.624	144
mon	0.000018	0.000000	0.00696	0.642	144

Delist Return

Delist Return	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
FALSE	0.000088	0.000002	0.00816	0.610	144
TRUE	0.000001	0.000000	0.00782	0.656	144

Weighting

Weighting	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
ew	0.000053	0.000000	0.00801	0.655	144
vw	0.000036	0.000000	0.00798	0.611	144

Size Breakpoints

Size Breakpoints	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
10	0.000169	0.000006	0.00735	0.581	72
5	0.000007	0.000000	0.00781	0.612	72
3	0.000001	0.000000	0.00823	0.653	72
2	0.000000	0.000000	0.00857	0.687	72

Mom Lookback

Mom Lookback	Mean GRS p	Median GRS p	Mean |α|	Mean R²	n
2	0.000073	0.000000	0.00797	0.632	96
4	0.000040	0.000000	0.00787	0.631	96
3	0.000020	0.000000	0.00814	0.637	96

Best Configuration per Exclusion Level

Exclusion	Calendar	Delist	Weight	Size N	Mom LB	GRS F	GRS p	Mean |α|	Avg R²
all	sharp	Off	EW	10	2	1.82	0.001809	0.00864	0.628
none	sharp	Off	EW	10	2	1.93	0.000672	0.00832	0.616
nostable	sharp	Off	EW	10	2	1.85	0.001369	0.00818	0.618

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