A Brief Review of Instrumented PCA

IPCA is particularly well suited for examining the relative importance of systematic mispricing and risk compensation, as it allows both alphas and betas to vary over time as functions of observable asset characteristics. IPCA is defined as a conditional latent factor model for returns of a cryptocurrency $ i $ at time $ t+1 $ Footnote ⁴:

(1) $$ {r}_{i,t+1}={\alpha}_{i,t}+{\beta}_{i,t}{f}_{t+1}+{\epsilon}_{i,t+1}, $$

where $ {\unicode{x1D53C}}_t\left[{\epsilon}_{i,t+1}\right]=0,{\unicode{x1D53C}}_t\left[{f}_{t+1}{\epsilon}_{i,t+1}\right]=0 $ and $ {f}_{t+1} $ is the vector of $ K $ latent factors extracted from cryptocurrency returns. Unlike standard factor models in which mispricing and factor loadings are static parameters, IPCA assumes that these evolve based on asset characteristics:

(2) $$ {\alpha}_{i,t}={\boldsymbol{z}}_{i,t}^{\prime }{\Gamma}_{\alpha }+{\nu}_{i,t}^{\alpha },\hskip2em {\beta}_{i,t}={\boldsymbol{z}}_{i,t}^{\prime }{\Gamma}_{\beta }+{\nu}_{i,t}^{\beta }, $$

where $ \Gamma =\left[{\Gamma}_{\alpha },{\Gamma}_{\beta}\right] $ are the loadings on the $ L\times 1 $ vector of asset characteristics $ {\boldsymbol{z}}_{i,t} $ . The scalar $ {\nu}_{i,t}^{\alpha } $ and the $ K\times 1 $ vector $ {\nu}_{i,t}^{\beta } $ are orthogonal to $ {\boldsymbol{z}}_{i,t} $ , allowing for the possibility that conditional alphas and betas may not be perfectly recoverable from observable characteristics.Footnote ⁵ Critically for our analysis, this framework allows us to assess whether a given characteristic contributes to systematic mispricing (if it significantly affects $ {\Gamma}_{\alpha } $ ) or risk compensation (if it significantly affects $ {\Gamma}_{\beta } $ ). By comparing model specifications with ( $ {\Gamma}_{\alpha}\ne 0 $ ) and without ( $ {\Gamma}_{\alpha }=0 $ ) the mispricing component, we can quantify the relative importance of these two explanations for cryptocurrency returns.

The model is estimated via an alternating least squares approach, which iterates the first-order conditions of $ {f}_{t+1} $ and $ \Gamma =\left[{\Gamma}_{\alpha },{\Gamma}_{\beta}\right]: $

(3) $$ {f}_{t+1}={\left({\Gamma}_{\beta}^{\prime }{Z}_t^{\prime }{Z}_t{\Gamma}_{\beta}\right)}^{-1}{\Gamma}_{\beta}^{\prime }{Z}_t^{\prime}\left({r}_{t+1}-{Z}_t{\Gamma}_{\alpha}\right)\hskip2em \forall t, $$

(4) $$ \mathrm{vec}\left(\Gamma \right)={\left(\sum \limits_{t=1}^{T-1}{Z}_t^{\prime }{Z}_t\otimes {\tilde{f}}_{t+1}{\tilde{f}}_{t+1}^{\prime}\right)}^{-1}\left(\sum \limits_{t=1}^{T-1}{\left[{Z}_t\otimes {\tilde{f}}_{t+1}^{\prime}\right]}^{\prime }{r}_{t+1}\right), $$

where $ {\tilde{f}}_{t+1}={\left[1,{f}_{t+1}^{\prime}\right]}^{\prime } $ and $ {Z}_t,{r}_{t+1} $ denote the stacked arrays of instruments and returns, respectively. To address the skewed cross-sectional distribution of some characteristics (such as market capitalization), we cross-sectionally rank, demean, and scale $ {\boldsymbol{z}}_{i,t} $ to be in the $ \left[-0.5,0.5\right] $ interval.Footnote ⁶

Kelly et al. (Reference Kelly, Pruitt and Su2019) show that latent factors in IPCA can be replaced with observable portfolios while maintaining the characteristic-based conditioning:

(5) $$ {r}_{t+1}={\boldsymbol{z}}_{i,t}^{\prime}\Gamma {\tilde{g}}_{t+1}+{\eta}_{t+1}=\mathrm{vec}{\left(\Gamma \right)}^{\prime}\left(\hskip0.1em {\boldsymbol{z}}_{i,t}\otimes {\tilde{g}}_{t+1}\right)+{\eta}_{i,t+1}, $$

where $ {\tilde{g}}_{t+1}={\left[1,{g}_{t+1}^{\prime}\right]}^{\prime } $ and $ {g}_{t+1} $ denotes the set of observable risk factors. We refer to this specification as an instrumented observable factor model and use it to verify that the importance of mispricing is robust to the choice of common factors.

A. Asset Pricing Performance

Following Kelly et al. (Reference Kelly, Pruitt and Su2019), we compute total and predictive $ {R}^2 $ as

(6) $$ {R}_{tot}^2=1-\frac{\sum_{i,t}{\left({r}_{i,t+1}-{\hat{\alpha}}_{i,t}-{{\hat{\beta}}^{\prime}}_{i,t}{\hat{f}}_{t+1}\right)}^2}{\sum_{i,t}{r}_{i,t+1}^2},\hskip2em {R}_{pred}^2=1-\frac{\sum_{i,t}{\left({r}_{i,t+1}-{\hat{\alpha}}_{i,t}-{{\hat{\beta}}^{\prime}}_{i,t}\hat{\lambda}\right)}^2}{\sum_{i,t}{r}_{i,t+1}^2}, $$

where $ {\hat{\alpha}}_{i,t}={\boldsymbol{z}}_{i,t}^{\prime }{\hat{\Gamma}}_{\alpha } $ , $ {\hat{\beta}}_{i,t}={\boldsymbol{z}}_{i,t}^{\prime }{\hat{\Gamma}}_{\beta } $ , and $ \hat{\lambda} $ is the vector of the unconditional time-series mean of the latent factors computed as $ {\hat{\lambda}}_k=\frac{1}{T}{\sum}_{t=1}^T{f}_{t,k}. $ The $ {R}_{tot}^2 $ indicates the ability of a model to describe the comovements of returns and $ {R}_{pred}^2 $ the proportion of predictable variation captured by the model.

We first implement an IPCA with eight latent factors and all characteristics in Table 1 as instruments. Next, we consider an instrumented observable factor model where alphas and betas are conditioned on the same characteristics, but latent factors are replaced with observable portfolios. We employ an 11-factor model that combines the market, size, momentum, and value factors from Liu et al. (Reference Liu, Tsyvinski and Wu2022), Cong et al. (Reference Cong, Karolyi, Tang and Zhao2021), and Liebi (Reference Liebi2022) with seven additional characteristic-managed portfolios, selected based on their incremental explanatory power.Footnote ⁷ Supplementary Material Appendix II provides descriptive statistics for the daily returns of all observable risk factors. Finally, we implement a static principal component analysis (PCA) to assess the contribution of time-varying parameters.

The comparison is based on the full-sample (in-sample) and recursive (out-of-sample) estimates. The out-of-sample performance is based on an expanding window estimation starting from Mar. 1, 2020. In each period $ t, $ we re-estimate the corresponding parameter $ {\hat{\Gamma}}_t=\left[{\hat{\Gamma}}_{\alpha, t},{\hat{\Gamma}}_{\beta, t}\right] $ using all the data through t (i.e., expanding window) and compute the realized factor return at $ t+1 $ as $ {\hat{f}}_{t+1}={\left({\hat{\Gamma^{\prime}}}_{\beta, t}{Z}_t^{\prime }{Z}_t{\hat{\Gamma}}_{\beta, t}\right)}^{-1}{\hat{\Gamma^{\prime}}}_{\beta, t}{Z}_t^{\prime}\left({r}_{t+1}-{Z}_t{\hat{\Gamma}}_{\alpha, t}\right) $ . Thus, the realized IPCA factors at $ t+1 $ require no information beyond time $ t $ . To test statistically the difference in asset pricing performance between models, we test the null hypothesis $ {\mathcal{H}}_0:E\left[\overline{\Delta}{L}_j\right]=0 $ , where $ \overline{\Delta}{L}_j\equiv \frac{1}{nT}{\sum}_{t=1}^T{\sum}_{i=1}^n\Delta {L}_{j,i,t} $ and $ \Delta {L}_{j,i,t}={\hat{e}}_{j,i,t}^2-{\hat{e}}_{bench,i,t}^2 $ is the squared error loss differential between the model $ j $ and a benchmark unrestricted IPCA. Supplementary Material Appendix III details this procedure.

Table 2 presents our findings for both daily and weekly returns.Footnote ⁸ The results show how systematic mispricing and time-varying risk compensation differentially affect the model’s ability to capture return comovements ( $ {R}_{tot}^2 $ ) versus predictable return variation ( $ {R}_{pred}^2 $ ).

Table 2Asset Pricing Performance

Table 2 Long description

The table is divided into two main sections: Panel A. In-Sample Estimation and Panel B. Out-of-Sample Estimation.

Columns include:

* Method: The model used (I P C A 8, Instrumented observable, or P C A 8).

* K: The number of factors (8 or 11).

* Daily Returns: Subdivided into R sub tot super 2 (%), Delta (%), R sub pred super 2 (%), and Delta (%).

* Weekly Returns: Subdivided into R sub tot super 2 (%), Delta (%), R sub pred super 2 (%), and Delta (%).

Panel A. In-Sample Estimation highlights:

* I P C A 8 (all characteristics) with K=8 shows a daily R sub tot super 2 of 15.55% and weekly of 31.92%.

* Instrumented observable with K=11 shows a daily R sub tot super 2 of 10.56% and weekly of 27.53%.

* P C A 8 and Gamma sub alpha = 0 with K=8 shows the highest daily R sub tot super 2 at 17.27% and weekly at 38.45%.

Panel B. Out-of-Sample Estimation highlights:

* I P C A 8 (all characteristics) with K=8 shows a daily R sub tot super 2 of 16.06% and weekly of 29.51%.

* Instrumented observable and Gamma sub alpha = 0 with K=11 shows the lowest daily R sub tot super 2 at 8.24%.

* P C A 8 and Gamma sub alpha = 0 with K=8 shows a daily R sub tot super 2 of 14.27% and weekly of 25.04%.

Negative Delta values are marked with asterisks to indicate statistical significance.

Allowing for systematic mispricing ( $ {\Gamma}_{\alpha}\ne 0 $ ) primarily enhances return predictability without affecting comovement patterns. Restricting mispricing to zero ( $ {\Gamma}_{\alpha }=0 $ ) leaves the total $ {R}^2 $ virtually unchanged (−0.43% for IPCA) but dramatically reduces predictive $ {R}^2 $ by 18.11% in-sample and 15.06% out-of-sample for daily returns. This pattern indicates that systematic mispricing represents a substantial source of predictable return variation that operates independently of the underlying factor structure. The importance of systematic mispricing extends beyond latent factors. For instrumented observable factors, eliminating mispricing reduces predictive $ {R}^2 $ by 68.99% (0.26% vs. 0.08%) and 58.52% (1.62% vs. 0.67%) for daily and weekly returns.

Time-varying risk compensation involves a fundamental trade-off between capturing comovements and generating predictable returns. The comparison between IPCA and static PCA reveals this tension clearly. Static PCA achieves a higher total $ {R}^2 $ (an increase of 11.03% for daily returns), indicating a superior ability to capture pure return comovements. However, PCA dramatically underperforms in terms of the predictive $ {R}^2 $ as we observe a reduction of 62.20% (0.26% vs. 0.10%) in predictive metrics. Furthermore, static PCA generates negative out-of-sample $ {R}^2 $ statistics.

The complementary nature of these components explains IPCA’s superior performance. Using latent factors enhances the model’s ability to capture return comovements compared to pre-specified portfolios (i.e., higher total $ {R}^2 $ ). Simultaneously, instrumenting alphas on characteristics substantially improves the model’s ability to generate predictable return variation (i.e., higher predictive $ {R}^2 $ ).

B. Economic Evaluation of Mispricing

We now investigate whether investors can profit from detecting systematic mispricing, as captured by the IPCA. To this end, we form a “pure-alpha” portfolio based on IPCA’s estimate of $ {\hat{\Gamma}}_{\alpha } $ . At the end of the day (week) $ t-1 $ , we estimate the model using the historical data and obtain parameter estimates $ {\hat{\Gamma}}_{\alpha, t-1} $ . We construct the portfolio with weights $ {w}_{t-1}={\boldsymbol{z}}_{t-1}{\left({\boldsymbol{z}}_{t-1}^{\prime }{\boldsymbol{z}}_{t-1}\right)}^{-1}{\hat{\Gamma}}_{t-1} $ , which combines the individual assets in proportion to their expected returns beyond the exposure to the latent factors. The portfolio construction starts in March 2020, which corresponds to the beginning of the out-of-sample period.

Table 3 reports the daily and weekly results. The pure-alpha portfolios generate highly significant risk-adjusted returns, with t-statistics consistently exceeding 7 (10) for daily (weekly) estimation across specifications. The Sharpe ratios exceed 0.7 per week in most cases. Most interestingly, the performance of the portfolios remains stable as we increase the number of latent factors to 8, which is consistent with significant systematic mispricing irrespective of return commonality. The robustness of these results across different risk adjustment models—from simple CAPM to the extended 11-factor specification—demonstrates that additional risk factors cannot explain away the economic value of systematic mispricing. This further supports the assumption that systematic mispricing represents a persistent and economically relevant feature in cryptocurrency markets rather than a statistical artifact.

Table 3Pure-Alpha Portfolios

Table 3 Long description

The table consists of 10 columns and two main data panels. The columns are labeled: I P C A Factors, Mean (%), Std. Dev. (%), S R, alpha sub C A P M (%), t sub C A P M, alpha sub F 4 (%), t sub F 4, alpha sub F 11 (%), and t sub F 11.

Panel A: Daily Returns

* K = 1: Mean 1.412, Std. Dev. 1.951, S R 0.724, alpha sub C A P M 1.411, t sub C A P M 7.450, alpha sub F 4 1.282, t sub F 4 6.863, alpha sub F 11 1.064, t sub F 11 5.811.

* K = 2: Mean 1.009, Std. Dev. 1.883, S R 0.536, alpha sub C A P M 1.007, t sub C A P M 4.832, alpha sub F 4 0.929, t sub F 4 4.642, alpha sub F 11 0.767, t sub F 11 3.986.

* K = 3: Mean 1.064, Std. Dev. 1.772, S R 0.601, alpha sub C A P M 1.063, t sub C A P M 5.625, alpha sub F 4 0.983, t sub F 4 5.420, alpha sub F 11 0.809, t sub F 11 4.679.

* K = 4: Mean 1.041, Std. Dev. 1.423, S R 0.732, alpha sub C A P M 1.041, t sub C A P M 7.620, alpha sub F 4 0.969, t sub F 4 7.456, alpha sub F 11 0.856, t sub F 11 6.807.

* K = 5: Mean 0.949, Std. Dev. 1.206, S R 0.787, alpha sub C A P M 0.949, t sub C A P M 8.192, alpha sub F 4 0.889, t sub F 4 7.881, alpha sub F 11 0.786, t sub F 11 7.086.

* K = 6: Mean 0.772, Std. Dev. 1.017, S R 0.759, alpha sub C A P M 0.771, t sub C A P M 7.385, alpha sub F 4 0.733, t sub F 4 7.197, alpha sub F 11 0.653, t sub F 11 6.526.

* K = 7: Mean 0.573, Std. Dev. 0.800, S R 0.717, alpha sub C A P M 0.573, t sub C A P M 7.426, alpha sub F 4 0.541, t sub F 4 7.347, alpha sub F 11 0.482, t sub F 11 6.496.

* K = 8: Mean 0.491, Std. Dev. 0.722, S R 0.680, alpha sub C A P M 0.490, t sub C A P M 7.289, alpha sub F 4 0.469, t sub F 4 7.379, alpha sub F 11 0.414, t sub F 11 6.460.

Panel B: Weekly Returns

* K = 1: Mean 1.092, Std. Dev. 1.485, S R 0.736, alpha sub C A P M 1.086, t sub C A P M 12.791, alpha sub F 4 1.083, t sub F 4 12.977, alpha sub F 11 1.072, t sub F 11 15.456.

* K = 2: Mean 1.023, Std. Dev. 1.332, S R 0.768, alpha sub C A P M 1.021, t sub C A P M 8.918, alpha sub F 4 1.019, t sub F 4 8.682, alpha sub F 11 1.046, t sub F 11 10.308.

* K = 3: Mean 1.002, Std. Dev. 1.343, S R 0.746, alpha sub C A P M 1.002, t sub C A P M 7.603, alpha sub F 4 1.002, t sub F 4 7.465, alpha sub F 11 1.026, t sub F 11 9.163.

* K = 4: Mean 1.003, Std. Dev. 1.318, S R 0.761, alpha sub C A P M 1.002, t sub C A P M 8.987, alpha sub F 4 1.002, t sub F 4 8.711, alpha sub F 11 1.027, t sub F 11 10.708.

* K = 5: Mean 0.953, Std. Dev. 1.229, S R 0.775, alpha sub C A P M 0.953, t sub C A P M 9.098, alpha sub F 4 0.951, t sub F 4 8.710, alpha sub F 11 0.973, t sub F 11 10.673.

* K = 6: Mean 0.875, Std. Dev. 1.085, S R 0.807, alpha sub C A P M 0.879, t sub C A P M 9.371, alpha sub F 4 0.873, t sub F 4 9.073, alpha sub F 11 0.883, t sub F 11 11.589.

* K = 7: Mean 0.765, Std. Dev. 1.043, S R 0.734, alpha sub C A P M 0.768, t sub C A P M 8.074, alpha sub F 4 0.764, t sub F 4 7.730, alpha sub F 11 0.771, t sub F 11 10.407.

* K = 8: Mean 0.746, Std. Dev. 0.977, S R 0.764, alpha sub C A P M 0.748, t sub C A P M 10.363, alpha sub F 4 0.741, t sub F 4 9.854, alpha sub F 11 0.755, t sub F 11 13.964.

C. Which Characteristics Matter for Alphas and Betas?

We now investigate which groups of characteristics drive systematic mispricing and risk compensation. In our analysis, we identify the distinct sources of two components of cryptocurrency returns and quantify their significance over time. We implement bootstrap simulations to test the significance of groups of characteristics for conditional alphas ( $ {\alpha}_{i,t} $ ) and betas ( $ {\beta}_{i,t} $ ). For systematic mispricing, we test $ {\mathcal{H}}_0:{\Gamma}_{\alpha}^g=0 $ , where $ {\Gamma}_{\alpha}^g $ is the subvector corresponding to a particular characteristic group. We compute a Wald-type test statistic $ {W}_{\alpha}^g={\hat{\Gamma}}_{\alpha}^{g^{\prime }}{\hat{\Gamma}}_{\alpha}^g $ and obtain p-values using wild bootstrap simulations following Kelly et al. (Reference Kelly, Pruitt and Su2019).Footnote ⁹ For risk compensation, we implement analogous tests for factor loadings ( $ {\Gamma}_{\beta}^g=0 $ ).

1. Systematic Mispricing Drivers

Table 4 reveals distinct patterns for daily (Panel A) and weekly (Panel B) returns. For daily returns, two groups of characteristics are important for systematic mispricing: speculative demand maintains strong significance across models (p-values below 0.05), and liquidity shows robust significance throughout. All other characteristic groups lose significance as more factors are included. For weekly returns, reversal characteristics become highly significant across all specifications, although speculative demand and liquidity remain important. This suggests that mispricing operates through behavioral channels (speculative demand) and microstructure frictions (liquidity and reversal).

Table 4Characteristics and Systematic Mispricing

Table 4 Long description

The table is divided into two main sections: Panel A for Daily Returns and Panel B for Weekly Returns. The columns include Groups, L (number of characteristics), and the Number of Factors ranging from 1 to 8.

Panel A (Daily Returns) includes:

* All characteristics (L=35): Values range from 0.00 to 0.03.

* Market, size, and momentum (L=5): Values increase from 0.00 to 0.95.

* Reversal (L=9): Values increase from 0.00 to 0.25.

* On-chain activity (L=8): Values range from 0.04 to 0.63.

* Trading activity (L=9): Values range from 0.03 to 0.98.

* Liquidity (L=9): All values are 0.00.

* Speculative demand (L=9): Values range from 0.00 to 0.04.

* Volatility and downside risk (L=9): Values increase from 0.00 to 0.82.

* Social media activity (L=9): Values range from 0.00 to 0.10.

* Equity market exposure (L=8): Values range from 0.22 to 0.89.

Panel B (Weekly Returns) includes:

* All characteristics (L=34): All values are 0.00.

* Market, size, and momentum (L=5): Values range from 0.00 to 0.69.

* Reversal (L=8): All values are 0.00.

* On-chain activity (L=8): Values range from 0.07 to 0.54.

* Trading activity (L=9): Values range from 0.47 to 0.82.

* Liquidity (L=9): Values range from 0.01 to 0.16.

* Speculative demand (L=9): Values range from 0.01 to 0.12.

* Volatility and downside risk (L=9): Values range from 0.12 to 0.44.

* Social media activity (L=9): Values range from 0.07 to 0.47.

* Equity market exposure (L=8): Values range from 0.25 to 0.97.

To examine the time-varying nature of systematic mispricing, we estimate the 8-factor IPCA using a 2-year rolling window and implement bootstrap tests for each period. Figure 1 shows test statistics along with bootstrap percentiles for daily returns. Mispricing is a persistent phenomenon throughout our sample. Speculative demand exhibits the strongest and most consistent significance, intensifying during the 2021 cryptocurrency boom and remaining elevated through 2022. Liquidity effects are pronounced during the early period and the COVID-19 pandemic, when trading frictions were most severe.

Figure 1Bootstrap Statistics for Daily Alphas over Time

Figure 1 illustrates the daily Wald-type test statistics (black line) and different percentiles of bootstrap statistics (gray areas) for the conditional alphas from an 8-factor IPCA model estimated on daily returns in a 2-year rolling window.

Figure 1 Long description

A four-panel line graph display. All panels share a horizontal x-axis representing time from 2020 to 2023. The y-axis represents statistical values. A legend in the top-left panel identifies a black line as Rolling Gamma prime sub alpha Gamma sub alpha and shaded gray regions as bootstrap percentiles: 99th percentile (lightest), 95th, 90th, 75th, and Median (darkest).

* Graph A. All Characteristics: The y-axis ranges from 0 to 40. The black line shows a steady increase from 10 in 2020 to a peak near 30 in late 2021, followed by a sharp drop to 10 in late 2022.

* Graph B. Liquidity: The y-axis ranges from 0 to 15. The black line fluctuates between 2 and 5 until mid-2022, where it drops significantly toward 0.

* Graph C. Speculative Demand: The y-axis ranges from 0 to 15. The black line remains low near 2 until 2021, spikes to a sharp peak of 13 in late 2021, and then declines back to 2 by 2023.

* Graph D. Volatility and Downside Risk: The y-axis ranges from 0 to 15. The black line stays near 0 throughout 2020, rises gradually to a peak of 5 in late 2021, and fluctuates between 2 and 6 through 2022.

Weekly results (Figure 2) show the persistent statistical significance of conditional alphas over time. Consistent with unconditional bootstrap tests, reversal characteristics drive systematic mispricing throughout the whole sample, whereas volatility characteristics gain prominence during the 2021–2022 market cycle. Supplementary Material Appendix IV reports the significance of other groups for daily (Supplementary Material Figure A1) and weekly (Supplementary Material Figure A2) returns. These results show that other characteristics do not influence mispricing over time.

Figure 2Bootstrap Statistics for Weekly Alphas over Time

Figure 2 illustrates the weekly Wald-type test statistics (black line) and different percentiles of bootstrap statistics (gray areas) for the conditional alphas from an 8-factor IPCA model estimated on weekly returns in a 2-year rolling window.

Figure 2 Long description

A four-panel grid of line graphs. Each graph shares a horizontal x-axis representing time from late 2019 to early 2023. The y-axis represents the test statistic value.

Common Legend:

* Shaded gray areas represent bootstrap percentiles: 99th percentile (lightest gray), 95th, 90th, 75th, and Median (darkest gray).

* Solid black line represents Rolling Gamma prime sub alpha Gamma sub alpha.

Panel Breakdown:

* Top-Left: Graph A. All Characteristics. Y-axis ranges from 0 to 200. The black line fluctuates between 75 and 180, consistently remaining above the 99th percentile shaded region.

* Top-Right: Graph B. Reversal. Y-axis ranges from 0 to 60. The black line fluctuates between 25 and 60, staying significantly above the shaded bootstrap regions which peak around 30.

* Bottom-Left: Graph C. Speculative Demand. Y-axis ranges from 0 to 60. The black line is lower, often dipping into the shaded regions, particularly between 2021 and 2022 where it falls below the median. It spikes toward the 95th percentile in early 2023.

* Bottom-Right: Graph D. Volatility and Downside Risk. Y-axis ranges from 0 to 60. The black line shows high volatility, frequently crossing through the shaded bootstrap percentiles, which themselves show large vertical spikes reaching up to 60 in 2022.

We note that the evidence of significant mispricing in cryptocurrency returns, reflecting behavioral biases (speculative demand) and microstructural frictions (liquidity risk), presents an interesting comparison with findings from equity markets, which show that demand for lottery-like assets leads to overpricing of illiquid assets (Kumar (Reference Kumar2009), Bali, Cakici, and Whitelaw (Reference Bali, Cakici and Whitelaw2011)).

2. Risk Compensation Drivers

Table 5 reports that the determinants of conditional betas differ substantially from those of alphas. Core characteristics (market, size, and momentum) are consistently significant across factor specifications. For daily returns, equity market exposure becomes strongly significant in larger factor models (p-value = 0.00 with $ K=7,8 $ ), while speculative demand and volatility characteristics also gain significance. For weekly returns, core characteristics maintain strong significance, alongside reversal and trading activity.

Table 5Characteristics and Risk Compensation

Table 5 Long description

The table is divided into two main sections: Panel A for Daily Returns and Panel B for Weekly Returns. The columns are labeled Groups, L, and No. of Factors (numbered 1 through 8).

Panel A. Daily Returns:

* Market, size, and momentum: L=5; Factors 1-8: 0.00, 0.08, 0.05, 0.01, 0.02, 0.04, 0.05, 0.04.

* Reversal: L=9; Factors 1-8: 0.01, 0.21, 0.62, 0.22, 0.19, 0.38, 0.54, 0.75.

* On-chain activity: L=8; Factors 1-8: 0.06, 0.29, 0.61, 0.80, 0.93, 0.99, 0.96, 0.99.

* Trading activity: L=9; Factors 1-8: 0.19, 0.67, 0.47, 0.79, 0.91, 0.63, 0.35, 0.42.

* Liquidity: L=9; Factors 1-8: 0.18, 0.05, 0.23, 0.41, 0.61, 0.71, 0.67, 0.51.

* Speculative demand: L=9; Factors 1-8: 0.31, 0.15, 0.03, 0.02, 0.04, 0.05, 0.08, 0.01.

* Volatility and downside risk: L=9; Factors 1-8: 0.00, 0.59, 0.44, 0.21, 0.04, 0.02, 0.01, 0.01.

* Social media activity: L=9; Factors 1-8: 0.01, 0.27, 0.68, 0.82, 0.92, 0.94, 0.98, 1.00.

* Equity market exposure: L=8; Factors 1-8: 0.80, 0.83, 0.88, 1.00, 0.21, 0.04, 0.00, 0.00.

Panel B. Weekly Returns:

* Market, size, and momentum: L=5; Factors 1-8: 0.13, 0.02, 0.00, 0.01, 0.03, 0.01, 0.00, 0.00.

* Reversal: L=8; Factors 1-8: 0.01, 0.07, 0.13, 0.11, 0.09, 0.02, 0.05, 0.00.

* On-chain activity: L=8; Factors 1-8: 0.12, 0.82, 0.68, 0.08, 0.17, 0.24, 0.07, 0.00.

* Trading activity: L=9; Factors 1-8: 0.09, 0.28, 0.49, 0.29, 0.00, 0.06, 0.01, 0.02.

* Liquidity: L=9; Factors 1-8: 0.25, 0.38, 0.64, 0.68, 0.42, 0.25, 0.49, 0.65.

* Speculative demand: L=9; Factors 1-8: 0.40, 0.76, 0.53, 0.62, 0.40, 0.68, 0.25, 0.26.

* Volatility and downside risk: L=9; Factors 1-8: 0.05, 0.76, 0.76, 0.79, 0.95, 0.90, 0.84, 0.79.

* Social media activity: L=9; Factors 1-8: 0.04, 0.46, 0.62, 0.88, 0.85, 0.54, 0.65, 0.49.

* Equity market exposure: L=8; Factors 1-8: 0.65, 0.65, 0.41, 0.61, 0.62, 0.20, 0.44, 0.44.

Figure 3 shows how different characteristics affect factor loadings over time. Core characteristics and equity market exposure consistently demonstrate importance, particularly after 2020, suggesting a growing integration between cryptocurrency and equity markets. Volatility and downside risk also represent a key feature for risk compensation, whereas speculative demand shows relevance at the beginning and toward the end of the sample.

Figure 3Bootstrap Statistics for Daily Betas over Time

Figure 3 illustrates the daily Wald-type test statistics (black line) and different percentiles of bootstrap statistics (gray areas) for the conditional betas from an 8-factor IPCA model estimated on daily returns in a 2-year rolling window.

Figure 3 Long description

A multi-panel figure containing four line graphs labeled Graph A through Graph D. Each graph shares a horizontal x-axis representing time from early 2020 to mid-2023 and a vertical y-axis ranging from 0 to 2.

* Graph A. Market, Size, Momentum: The black line fluctuates between 1 and 1.5, generally staying above the shaded gray bootstrap regions until early 2022, where it dips toward 0.5 before recovering to 1.

* Graph B. Speculative Demand: The black line remains relatively stable between 1 and 1.5. The shaded bootstrap percentiles show a significant peak in height during 2021, briefly overlapping with the black line.

* Graph C. Volatility and Downside Risk: The black line shows a downward trend starting in late 2021, moving from 1.7 toward 1.1. The shaded bootstrap regions are densest at the start of 2020 and gradually thin out over time.

* Graph D. Equity Market Exposure: Unlike the other panels, the black line starts low (around 0.5) and climbs to a peak of 1.7 in mid-2021 before oscillating between 0.8 and 1.2. The shaded bootstrap regions remain consistently low, below 1.0, for the entire duration.

A legend in Graph D identifies the shaded layers from darkest to lightest as Median, 75th percentile, 90th percentile, 95th percentile, and 99th percentile. The solid black line is labeled as Rolling Gamma prime sub beta Gamma sub beta.

Figure 4 reports the bootstrap statistics for weekly betas over time. The main insights align with the daily results. Core characteristics have a significant influence on risk compensation, with exposure to the equity market also representing a considerable feature that drives conditional betas. Unlike daily results, volatility and downside risks are less relevant, whereas reversal and trading activity gain significant prominence in the dynamics of factor loadings.Footnote ¹⁰

Figure 4Bootstrap Statistics for Weekly Betas over Time

Figure 4 illustrates the weekly Wald-type test statistics (black line) and different percentiles of bootstrap statistics (gray areas) for the conditional betas from an 8-factor IPCA model estimated on weekly returns in a 2-year rolling window.

Figure 4 Long description

The image consists of four panels arranged in a two-by-two grid. Each panel shares a common X-axis representing years from 2020 to 2023 and a Y-axis ranging from 0 to 2.

* Graph A. Market, Size, Momentum: Located top-left. The black line representing Rolling Gamma prime sub beta Gamma sub beta starts high near 1.5, peaks around 1.8 in mid-2020, then trends downward to fluctuate between 1.0 and 1.4 through 2023. The gray bootstrap areas are densest below 1.0.

* Graph B. Reversal: Located top-right. The black line fluctuates between 0.5 and 0.8, showing a slight peak in late 2021. The bootstrap percentiles are concentrated lower than in Graph A, mostly staying below 0.5.

* Graph C. Trading Activity: Located bottom-left. The black line shows a steady upward trend, starting near 0.8 in 2020 and rising to approximately 1.4 by 2023. The bootstrap shaded regions follow a similar rising density.

* Graph D. Equity Market Exposure: Located bottom-right. The black line starts low near 0.3, climbs steadily to peak near 1.0 in early 2022, then fluctuates between 0.5 and 0.9.

A legend in the bottom-right panel identifies the shaded layers from darkest to lightest as Median, 75% percentile, 90% percentile, 95% percentile, and 99% percentile. The solid black line is labeled as Rolling Gamma prime sub beta Gamma sub beta.

The results overall suggest that while mispricing persists through behavioral and structural channels that resist arbitrage, risk compensation increasingly follows established asset pricing mechanisms. Furthermore, exposure to equity markets suggests that cryptocurrency systematic risk reflects broader market factors rather than solely crypto-specific risks.

We complement our analysis by performing two additional exercises. First, we further test the relevance of individual characteristics instead of groups for betas. This provides a more granular picture of which characteristics are most important within groups. We show that a few characteristics drive the significance of the most important groups. Regarding the strong role of equity characteristics in driving factor loadings, exposure to equity market returns (including their downside movements) matters for the risk compensation of cryptocurrencies.

Second, we augment the bootstrap significance tests by measuring the relative contribution of different groups of characteristics to the sum of squared alpha and beta parameters. We delegate the details of this exercise to the Supplementary Material and discuss the main results here. Focusing on contributions to daily alphas, speculative demand tends to account for the large share throughout the sample, with its impact increasing from mid-2020 to the end of 2022, consistent with the strong statistical significance reported in Figure 1. Similarly, the economic impact of liquidity characteristics is more substantial during the early sample period, consistent with their statistical importance. We observe a similar degree of association between the statistical and economic relevance of variables for weekly alphas. Turning to betas, we find that the relative contributions of various groups exhibit similar patterns for daily and weekly factor loadings. Furthermore, the economic impact of groups of characteristics on daily and weekly betas is also associated with their statistical significance.

D. Asset Quality and Model Performance

To gain additional insight into the impact of modeling mispricing and risk compensation on asset pricing performance, we compute the $ {R}^2 $ statistics for coins grouped by different characteristics. Each day, we sort the cryptocurrencies into quartiles based on various variables, one at a time. For each quartile, we compute the total and predictive $ {R}^2 $ for the 8-factor IPCA, 8-factor PCA, and a dynamic observable 11-factor model instrumented by all characteristics. We compare these different approaches to better understand where various modeling mechanisms (time-varying coefficients and latent factors) are most relevant. Following Kelly et al. (Reference Kelly, Pruitt and Su2019), we do not re-estimate models for different subsamples, as this would mechanically improve fit. Instead, we keep factors and parameters fixed at their full-sample estimates and recalculate $ {R}^2 $ statistics within each subsample.

Table 6 reports daily results for quartiles with the lowest and highest values of a given characteristic. Focusing on the total $ {R}^2 $ , IPCA maintains substantial advantages over instrumented observable factors for volatile and illiquid assets, with IPCA outperforming by 62%–63% for high-volatility cryptocurrencies. For larger, more liquid assets, instrumented observable factors often align with IPCA performance. Static PCA shows mixed performance relative to IPCA. For lower-quality assets, it typically underperforms IPCA in $ {R}_{pred}^2 $ by 75%–90%, but often outperforms in $ {R}_{tot}^2 $ by 10%–20%. This suggests that while static PCA can capture realized return variation, it struggles with prediction for assets where mispricing effects are most pronounced.

Table 6Asset Quality and Asset Pricing Performance

Table 6 Long description

The table is divided into two main sections: Panel A and Panel B.

Header Structure:

The columns are grouped into three main models: I P C A (Low and High), Instrumented Observable Factors (Low, High, Delta L percent, Delta H percent), and Static P C A (Low, High, Delta L percent, Delta H percent).

Panel A: R sub tot super 2 (percent)

This panel lists 15 asset characteristics including capm beta, size, new_add, act_add, bm, dollar vol, bidask, ivol, illiq, VaR (5 percent), max 30, reddit_act, reddit_post, twitter_f, and Equity capm beta.

- For capm beta, I P C A values are 12.11 (Low) and 16.61 (High). Instrumented Observable Factors are 5.46 (Low) and 10.77 (High) with deltas of minus 55 and minus 35. Static P C A values are 14.68 (Low) and 18.58 (High) with deltas of 21 and 12.

- For ivol, I P C A values are 36.94 (Low) and 12.02 (High). Instrumented Observable Factors are 37.93 (Low) and 4.40 (High) with deltas of 3 and minus 63. Static P C A values are 41.33 (Low) and 14.17 (High) with deltas of 12 and 18.

Panel B: R sub pred super 2 (percent)

This panel uses the same 15 asset characteristics to measure predictive performance.

- For capm beta, I P C A values are 0.38 (Low) and 0.31 (High). Instrumented Observable Factors are 0.10 (Low) and 0.08 (High) with deltas of minus 73 for both. Static P C A values are 0.16 (Low) and 0.15 (High) with deltas of minus 58 and minus 51.

- For bm (book-to-market), I P C A values are 0.03 (Low) and 0.48 (High). Instrumented Observable Factors are 0.09 (Low) and 0.13 (High) with deltas of 223 and minus 73. Static P C A values are 0.14 (Low) and 0.14 (High) with deltas of 388 and minus 72.

Turning to the predictive $ {R}^2, $ IPCA demonstrates its strongest relative performance among lower-quality assets. For instance, IPCA achieves an $ {R}_{pred}^2 $ of 0.42% among cryptocurrencies with the highest idiosyncratic volatility compared to 0.10% for observable factors, representing a 76% underperformance by the conditional observable factor model that ignores mispricing. Similar patterns emerge for illiquid assets with the highest bid–ask spreads, where IPCA generates 0.58% $ {R}_{pred}^2 $ versus 0.12% for observable factors (a 79% underperformance). The pattern extends to speculative demand (max), where observable factors achieve only 0.09% compared to IPCA’s 0.40%. In contrast, IPCA’s $ {R}_{pred}^2 $ is negative or near-zero for large, more liquid, less volatile assets with more social media and on-chain activity.

These results suggest that the impact of mispricing on the $ {R}_{pred}^2 $ is not straightforward. The time-varying alphas make a positive contribution to the predictive performance for smaller and illiquid cryptocurrencies. This likely happens because the mispricing of these cryptocurrencies is more significant and time-varying. However, frequent changes in alphas are detrimental to the return prediction of larger and liquid cryptocurrencies, as their mispricing is likely less significant. Since the cross section tends to be skewed toward smaller and illiquid cryptocurrencies, the IPCA produces, on average, the higher $ {R}_{pred}^2 $ statistics when allowing for systematic mispricing.

Overall, Table 6 provides strong empirical validation that the advantages from time-varying alphas are systematically concentrated where economic theory predicts mispricing should be most prevalent—among assets with high arbitrage costs and limited liquidity. The weekly frequency (as reported in Supplementary Material Table A5) amplifies the distinction between asset quality segments, suggesting that the benefits of modeling systematic mispricing and time-varying exposures are particularly pronounced over coarser frequencies, especially for assets where arbitrage constraints are most binding.

E. Additional Checks

A. Latent Factors and Characteristic-Managed Portfolios

Following Ludvigson and Ng (Reference Ludvigson and Ng2009), we first examine the correlation between latent factors and managed portfolios. Graph A of Figure 5 shows the marginal $ {R}^2 $ , which is the $ {R}^2 $ statistic from univariate regressions of each characteristic-managed portfolio on each latent factor.Footnote ¹² The first latent factor (F1) is primarily associated with volatility measures, showing the highest correlations with rvol, rskew, and std_vol. The second factor (F2) captures the exposure to the equity market and liquidity risk, with the strongest correlations observed for equity capm $ \beta $ , to (turnover), and illiq.

Figure 5Characteristic-Managed Portfolios and IPCA Latent Factors

Graph A of Figure 5 shows the marginal $ {R}^2, $ which are $ {R}^2 $ statistics from univariate regressions of each of the 35 characteristic-managed portfolios on each latent factor. Graph B shows the regression coefficients of a series of multivariate regressions in which all latent factors are projected onto each characteristic-managed portfolio.

Figure 5 Long description

Graph A, titled Marginal R super 2 from Individual Factor Regressions, is a stacked bar chart. The Y-axis is labeled Cumulative R super 2 ranging from 0.0 to 0.8. The X-axis lists 35 Characteristic-Managed Portfolios including v w underscore m k t, capm beta, size, and various momentum and volatility metrics. Each bar is composed of up to eight segments representing latent factors F 1 through F 8, distinguished by color and pattern. F 8, shown in blue with a cross-hatch pattern, frequently appears at the top of the stacks, contributing significantly to portfolios like bidask and ivol.

Graph B, titled Standardized Regression Coefficients, is a heatmap. The Y-axis lists the same 35 portfolios, and the X-axis lists I P C A Factors F 1 through F 8. A color scale on the right indicates Standardized Coefficient values from negative 0.8 in dark blue to 0.6 in dark red. The heatmap shows strong positive coefficients (dark red) for F 8 in portfolios such as bidask, max 30, and ivol. Strong negative coefficients (dark blue) are visible for F 6 in the v w underscore m k t portfolio and F 2 in the twitter f portfolio.

The third factor (F3) correlates most strongly with down $ \beta $ and capm $ \beta $ . This echoes the sixth factor, which shows the highest correlations with down $ \beta $ , capm $ \beta $ , in addition to std_vol. The fourth factor (F4) emerges as the primary momentum factor, showing strong explanatory power for r30_1, r21_1, and r7_1. The fifth factor (F5) exhibits a distinctive pattern, correlating most strongly with equity downside risk (equity down $ \beta $ ). The seventh factor (F7) can be unambiguously identified as the value-weighted cryptocurrency market factor, accounting for 77.3% of the variation in the vw_mkt portfolio. Finally, the eighth factor (F8) is associated with trading frictions, exhibiting strong correlations with bidask, rvol, and speculative demand measures such as max 30.

Graph B of Figure 5 shows the results of a complementary regression analysis. We implement a multivariate regression in which all standardized latent factors are projected onto each standardized managed portfolio. Since the regression does not include an intercept, each coefficient can be interpreted as a partial correlation coefficient. The darker the color in the heatmap, the larger the partial correlation.

The results largely confirm the evidence from Graph A of Figure 5. Factor 1 (F1) exhibits strong positive correlations with volatility measures, and Factor 2 (F2) shows strong correlations with trading activity, in addition to a strong correlation with the equity market equity capm $ \beta $ . Factor 3 (F3) displays a strong positive correlation with down $ \beta $ but a negative correlation with capm $ \beta $ . Factor 4 (F4) confirms its role as the momentum factor, whereas Factor 5 (F5) shows an interesting dual pattern, with a strong negative correlation with equity down $ \beta $ but positive correlations with momentum measures. Factor 6 (F6) and Factor 7 (F7) exhibit strong correlations with broad market risk measures. In particular, the return on F7 is highly related to the return on the market portfolio vw_mkt. Finally, Factor 8 (F8) demonstrates strong positive correlations with liquidity frictions and extreme returns.

B. Correlation with Equity Risk Factors

In this section, we address a fundamental question that has been central to the debate among market participants and researchers: Do cryptocurrencies and traditional asset classes share common risk factors? The factor structure we have identified via IPCA suggests potential linkages with equity markets that warrant investigation. Notably, the fifth IPCA factor (F5) shows a distinctive pattern with equity-related characteristics, exhibiting the strongest correlation with equity down $ \beta $ in the marginal $ {R}^2 $ analysis and a strong negative partial correlation in the multivariate regression analysis. Similarly, the second factor (F2) demonstrates significant correlations with equity capm $ \beta $ , suggesting potential cross-market risk transmission.

1. IPCA Bootstrap Tests

To investigate these linkages more formally, we start by leveraging the flexibility of IPCA and consider an extended model that includes both latent cryptocurrency factors and observable equity factors:

(7) $$ {r}_{i,t+1}={\alpha}_{i,t}^{\prime }+{\beta}_{i,t}^{\prime }{f}_{t+1}+{\delta}_{i,t}^{\prime }{g}_{t+1}+{\epsilon}_{i,t+1}, $$

in which $ {\alpha}_{i,t}^{\prime },{\beta}_{i,t}^{\prime }, $ and $ {\delta}_{i,t}^{\prime } $ are time-varying coefficients instrumented with all characteristics. Here, $ {f}_{t+1} $ and $ {g}_{t+1} $ represent the latent cryptocurrency factors and the observable equity factors, respectively. The incremental explanatory power of equity factors can be tested using a Wald-like statistic for the null hypothesis $ {\mathrm{\mathscr{H}}}_0:{\Gamma}_{\delta }={\mathbf{0}}_{L\times M} $ (see Supplementary Material Appendix IV for details).Footnote ¹³. We consider the five equity factors of Fama and French (Reference Fama and French2015)—the market (MKT), size (SMB), value (HML), profitability (RMW), and investment (CMA)—and momentum (MOM) of Jegadeesh and Titman (Reference Jegadeesh and Titman1993).Footnote ¹⁴

Table 7 reports the p-values on testing the significance of $ {\delta}_{i,t}^{\prime } $ . The results on individual tests, which examine the individual significance of each equity factor separately, provide mixed evidence. The p-values for HML decline systematically from 0.60 in the single-factor model to 0.02 in the 8-factor specification. This finding aligns with recent evidence that value-like characteristics matter for cryptocurrency pricing (Cong et al. (Reference Cong, Karolyi, Tang and Zhao2021), Liebi (Reference Liebi2022)). The market factor (MKT) shows moderate evidence of correlation, with p-values improving from 0.79 to 0.50 as the number of latent factors increases, though this falls short of conventional significance levels. The momentum factor (MOM) displays marginal significance in higher-factor specifications, particularly in IPCA6 (p-value = 0.11) and IPCA7 (p-value = 0.07). In contrast, the size (SMB), profitability (RMW), and investment (CMA) factors show no significance across all specifications.

Table 7IPCA-Based Tests for Equity Factors

Table 7 Long description

The table presents results for I P C A (Instrumented Principal Component Analysis) across eight rows, labeled K equals 1 through K equals 8. The data is organized into two primary sections: Individual Tests and Joint Tests. Both sections include columns for the following equity factors: M K T, S M B, H M L, R M W, C M A, and M O M.

Key data points include:

* For K equals 1: Individual tests range from 0.49 to 0.98; Joint tests range from 0.21 to 0.71.

* For K equals 2: Individual tests range from 0.27 to 0.97; Joint tests range from 0.08 to 0.92.

* For K equals 3: Individual tests range from 0.15 to 0.94; Joint tests range from 0.08 to 0.96.

* For K equals 4: Individual tests range from 0.25 to 0.94; Joint tests range from 0.17 to 0.98.

* For K equals 5: Individual tests range from 0.16 to 0.88; Joint tests range from 0.05 to 0.91.

* For K equals 6: Individual tests range from 0.03 to 0.84; Joint tests range from 0.03 to 0.78.

* For K equals 7: Individual tests range from 0.06 to 0.73; Joint tests range from 0.01 to 0.80.

* For K equals 8: Individual tests range from 0.02 to 0.76; Joint tests range from 0.06 to 0.77.

Notably, the R M W factor shows a significant decrease in p-values in the Individual Tests as K increases, reaching 0.02 at K equals 8.

When including all equity factors simultaneously, the results reveal more substantial evidence for certain factors. Most notably, MOM achieves statistical significance in several specifications, with p-values of 0.05 in IPCA5, 0.01 in IPCA7, and 0.06 in IPCA8. The value factor (HML) maintains its significance in joint tests, particularly in IPCA6 and IPCA8, where p-values reach 0.03 and 0.06, respectively.

2. Factor-Spanning Regressions

In addition to the IPCA-based bootstrap tests, we conduct factor-spanning regressions that directly test whether IPCA factors can be replicated using linear combinations of equity risk factors. Table 8 reports the results where each IPCA latent factor is regressed on the six equity risk factors. Five out of eight IPCA factors exhibit negative adjusted $ {R}^2 $ values, indicating that most equity factors provide no meaningful explanatory power. However, a notable exception is the seventh factor (F7), which exhibits a substantial correlation with equity risk factors, achieving an adjusted $ {R}^2 $ of 9.1%. This finding is consistent with our earlier interpretation of F7 (see Figure 5), which showed the strongest correlation with the value-weighted cryptocurrency market portfolio ( $ {R}^2=77.3\% $ ).

Table 8Factor-Spanning Regressions

Table 8 Long description

The table consists of 8 columns labeled F 1 through F 8.

Row 1: alpha (%) values are 0.10, 1.60 (significant at 1 percent), 0.60 (significant at 1 percent), 2.50 (significant at 1 percent), 0.20, 0.70 (significant at 1 percent), 0.20, and 1.40 (significant at 1 percent).

Row 2: M K T coefficients are 0.30, minus 0.30, minus 0.40 (significant at 10 percent), 0.00, 0.10, minus 0.40 (significant at 1 percent), minus 1.20 (significant at 1 percent), and 0.00.

Row 3: S M B coefficients are minus 0.50, minus 0.50, minus 0.70 (significant at 10 percent), 0.50, 0.00, 0.00, minus 0.20, and 0.20.

Row 4: H M L coefficients are 0.40, 0.60, 0.70 (significant at 5 percent), minus 0.30, 0.10, minus 0.30, 0.40 (significant at 5 percent), and 0.10.

Row 5: R M W coefficients are minus 0.20, minus 0.90, 0.10, minus 0.10, minus 0.10, 0.20, 0.40, and 0.00.

Row 6: C M A coefficients are minus 0.80, minus 1.20, minus 0.70, 0.30, minus 0.40, 0.40, minus 0.40, and minus 0.20.

Row 7: M O M coefficients are minus 0.20, minus 0.40, 0.10, minus 0.20, 0.00, 0.00, 0.00, and 0.00.

Row 8: Adjusted R-squared (%) values are minus 0.20, 0.00, 0.40, minus 0.10, minus 0.30, 0.70, 9.10, and minus 0.20.

The regression intercepts provide additional evidence on the correlation between equity and cryptocurrency market returns. If equity risk factors could fully explain IPCA factors, the intercepts should be statistically indistinguishable from zero. This is indeed the case for the seventh IPCA latent factor (F7). The statistically insignificant intercept suggests that the equity market returns fully capture the presence of systematic components in the seventh IPCA cryptocurrency factor.

To provide additional insight into the temporal evolution of cryptocurrency-equity correlations, Figure 6 presents 2-year rolling-window estimates of the significance of market (MKT) and value (HML) factors for the two IPCA factors that showed the strongest correlations in our spanning regression analysis. The results provide important context for interpreting the static regression results in Table 8.

Figure 6Rolling-Window p-Values for Equity Factor Correlations

Graph A of Figure 6 shows the p-values from rolling 2-year window regressions of IPCA factors F6 and F7 on the market factor (MKT). Graph B shows the p-values from rolling 2-year window regressions of the same factors on the value factor (HML). The dashed horizontal lines indicate conventional significance thresholds of 5% (red) and 10% (orange).

Figure 6 Long description

A two-panel line graph. Both panels share a y-axis labeled p-value from 0.0 to 1.0 and an x-axis labeled Date from 2018 to 2023. Two horizontal dashed lines represent significance thresholds: a red line at 0.05 (5%) and an orange line at 0.10 (10%).

* Graph A. Rolling 2-Year Market p-Values (M K T): Shows factor F 6 (blue line) and F 7 (red line). F 6 fluctuates highly, mostly staying above 0.2, except for a sharp drop to near zero throughout 2021. F 7 remains below the 0.10 threshold for most of 2019 and from mid-2020 through 2023, indicating sustained significance.

* Graph B. Rolling 2-Year Market p-Values (H M L): Shows factor F 6 (blue line) and F 7 (red line). F 6 is significant (below 0.10) from late 2018 through 2021. F 7 shows high volatility, frequently spiking to 1.0, but drops below the significance thresholds in early 2021 and again in late 2022.

The rolling window analysis reveals a dramatic structural shift around March 2020. Panel A reports that F6 and F7 exhibited virtually no significant correlation with the equity market from 2018 to early 2020, with p-values consistently above 0.6. However, from March 2020 onward, both factors exhibit much stronger and more persistent correlations with equity markets, with p-values frequently dropping below the 5% significance threshold. F7 shows particularly strong significance during 2020–2021, with p-values near zero. Panel B reports intermittent but significant correlations between the value factor and IPCA factors, with distinct periods of high significance during 2019–2020 and 2021–2022.

The sharp increase in correlations after March 2020 aligns with accelerated institutional adoption and the integration of cryptocurrencies into traditional investment portfolios during the pandemic (Didisheim and Somoza (Reference Didisheim and Somoza2024)). Increasingly correlated trading could lead to cross-asset class correlations, even if the two markets are not fully integrated (Kyle (Reference Kyle1989)). This evolution supports the theoretical framework of Pástor and Veronesi (Reference Pástor and Veronesi2009), where increased investor exposure to innovative sectors reduces information asymmetries and strengthens risk spillovers between asset classes.

The convergence of evidence from our empirical approaches reveals a nuanced picture of cryptocurrency–equity factor relationships. While the bootstrap tests show mixed significance patterns, the spanning regressions provide more convincing evidence in favor of strong time-varying correlations between cryptocurrency and equity markets. These seemingly contradictory results are reconciled by recognizing that the modest correlations documented in our static tests mask substantial temporal variation in the underlying relationships.