Signal Validation & Statistical Rigour

Research question. Do our alpha signals hold up under proper statistical correction, and can we recover Bollinger's edge by matching rebalance frequency to signal horizon?

Context

Notebook 02 identified two surviving factors, Momentum 12-1 and Bollinger, but flagged methodology problems:

1. Overlapping forward returns inflated t-statistics by ~4.6x. The reported t-stats assumed independent daily observations, but 21d forward returns share 20/21 days between consecutive dates.
2. Bollinger's IC-to-P&L gap: strongest IC yet worst backtest (Sharpe −0.93). Daily rebalancing generated turnover costs that removed the edge.
3. Regime dependence: Momentum's IC was far stronger post-2024 than pre-2024.

This notebook applies the validation module built in Phase 2.5, on a larger 50-stock universe, to answer: after correction, which signals are real, and can implementation close the gap?

Dropped factors

• XS Momentum: redundant (identical IC to raw Momentum, weaker backtest due to rank compression).
• RSI: insignificant IC at every horizon, high correlation with Bollinger (0.88), no robustness.

0. Setup

1. Data & Signal Generation

50 liquid US large-caps spanning all 11 GICS sectors. The larger cross-section increases statistical power (single-day IC SE ~0.14 vs ~0.28 with fewer stocks) and reduces sensitivity to any single name. Still survivorship-biased (current names selected with hindsight).

Fetching 33 new tickers: ['AVGO', 'ADBE', 'CRM', 'BAC', 'WFC', 'MS', 'BLK', 'PFE', 'ABT', 'LLY', 'TMO', 'TSLA', 'HD', 'NKE', 'MCD', 'KO', 'PEP', 'COST', 'CVX', 'COP', 'CAT', 'HON', 'UPS', 'RTX', 'MA', 'NFLX', 'DIS', 'NEE', 'DUK', 'AMT', 'PLD', 'LIN', 'SHW']
Done.

Universe: 47 tickers
Period: 2020-01-02 to 2026-05-15
Shape: 1601 days x 47 tickers
Missing: 0 NaNs

Momentum 12-1: first signal 2020-12-31, 63,403 values
Bollinger: first signal 2020-01-30, 74,354 values

2. Signal Diagnostics

Before any IC analysis, we characterize the signals themselves. Four questions:

• Autocorrelation: How sticky is the signal day-to-day? (quasi-static = daily rebalancing is churn)
• Turnover: How much do implied weights change daily? (low = rebalance less often)
• Dispersion: Does the signal differentiate stocks? (near-zero = no positions after neutralization)
• Lead-lag: Is the signal predictive or reactive? (negative-offset correlation = echoing past returns)

2a. Signal Autocorrelation

Momentum 12-1 lag-1 autocorrelation: 0.9903
Bollinger lag-1 autocorrelation: 0.8677

Observations: autocorrelation.

• Momentum (lag-1 = 0.990): extremely persistent, above the 0.95 quasi-static threshold through ~lag 6. A 252-day lookback shares 251/252 days between consecutive dates, so daily rebalancing would generate turnover for near-zero position change. This signal should be rebalanced infrequently.
• Bollinger (lag-1 = 0.868): decays faster, dropping below 0.50 by lag 5 and toward zero by lag 15. A 20-day window lets the signal flip within a few days as prices cross the bands. It can rebalance frequently, but as later sections show there is no predictive power to exploit at any frequency.
• Implication: Momentum should rebalance on the order of weeks to reduce cost drag. For Bollinger, the autocorrelation structure does not matter; frequency and smoothing only control transaction costs rather than how much alpha is captured, since there is no detectable alpha.

2b. Signal Turnover & Dispersion

Momentum 12-1 mean turnover: 0.0940
Momentum 12-1 mean dispersion: 0.3273
Bollinger mean turnover: 0.4369
Bollinger mean dispersion: 1.0955

Observations: turnover and dispersion.

• Turnover: Momentum's mean daily turnover (0.094) is 4.6x lower than Bollinger's (0.437). Momentum barely changes day-to-day (consistent with 0.99 autocorrelation), while Bollinger reshuffles nearly half the portfolio daily. This confirms NB02: Bollinger's cost drag at 10 bps is large under daily rebalancing (~4 bps/day, ~11% annualized).
• Momentum turnover spike (~0.18 in mid-2023): likely a regime shift, as several names crossed from negative to positive 12-month returns when the 2022 drawdown rolled out of the lookback window.
• Dispersion: Bollinger's cross-sectional dispersion (1.10) is 3.3x higher than Momentum's (0.33). Higher dispersion means stronger differentiation between stocks, which helps generate positions after dollar-neutral normalization. Momentum's lower dispersion clusters positions more tightly, reducing effective bet size.
• Momentum dispersion spikes (early 2021 ~0.8, mid-2023 ~0.45): periods where the cross-section of 12-month returns was unusually dispersed (post-COVID recovery, post-2022 recovery).

2c. Lead-Lag Profile

Observations: lead-lag profile.

• Momentum: all bars positive and nearly symmetric across reactive (negative offsets) and predictive (positive offsets) sides, around IC 0.015-0.020. This is consistent with a persistent factor: momentum stocks tend to keep outperforming regardless of the measurement window. The symmetry means the signal captures a slow-moving characteristic rather than being purely predictive or purely reactive.
• Bollinger: strong negative IC on the reactive side (offsets −15 to 0), peaking at offset 0 with IC ~ −0.40; the predictive side (offsets +1 to +15) is near zero. This is mean-reversion mechanics by construction: Bollinger z-scores are high when price is above the band (after a run-up), so the signal is mechanically anti-correlated with recent returns. The near-zero predictive side indicates Bollinger has no detectable forward-looking edge at any short-term horizon.
• The lead-lag profile separates mechanical correlation from genuine prediction. Bollinger's reactive side is a construction artefact (a z-score from recent prices correlates with those recent returns); the predictive side is what matters, and it is flat.

3. Corrected IC Analysis

Standard t-statistics assume independent daily IC observations. With overlapping forward returns (a 21d horizon shares 20/21 days between consecutive dates), this overstates significance by ~√h. We apply three corrections:

1. Newey-West HAC t-statistic: adjusts standard errors for autocorrelation (bandwidth = horizon − 1).
2. Non-overlapping IC: samples every h-th day for independent observations.
3. Block bootstrap 95% CI: preserves serial dependence to give honest confidence intervals.

IC Summary — Naive vs Newey-West Corrected

	Factor	Horizon	Mean IC	ICIR	t (naive)	t (NW)	Inflation	95% CI
0	Momentum 12-1	1d	0.0195	0.064	2.35	2.35	1.0x	[+0.0030, +0.0354]
1	Momentum 12-1	5d	0.0226	0.077	2.82	1.57	1.8x	[-0.0054, +0.0508]
2	Momentum 12-1	10d	0.0237	0.085	3.12	1.29	2.4x	[-0.0122, +0.0593]
3	Momentum 12-1	21d	0.0307	0.122	4.43	1.28	3.5x	[-0.0174, +0.0771]
4	Bollinger	1d	-0.0061	-0.023	-0.91	-0.91	1.0x	[-0.0194, +0.0068]
5	Bollinger	5d	-0.0052	-0.021	-0.84	-0.50	1.7x	[-0.0259, +0.0148]
6	Bollinger	10d	0.0007	0.003	0.12	0.06	2.1x	[-0.0240, +0.0264]
7	Bollinger	21d	0.0094	0.041	1.61	0.63	2.5x	[-0.0180, +0.0399]

Observations: corrected IC.

• Inflation scales with horizon as expected. At 1d there is no overlap, so naive = NW (1.0x). At 21d the inflation is 3.5x for Momentum, close to the theoretical √21 ≈ 4.6x; the gap from theory is because the IC series is not perfectly autocorrelated.
• Momentum's naive t = 4.43 at 21d falls to 1.28 after NW correction, below the 95% threshold, and the bootstrap CI [−0.017, +0.077] straddles zero. The signal that looked strong in NB02 is marginal after correction.
• Bollinger is insignificant at every horizon after correction. Best case (21d): t(NW) = 0.63, CI [−0.018, +0.040]. The daily IC is slightly negative (−0.006), consistent with the lead-lag finding.
• Momentum's mean IC rises with horizon (0.020 at 1d to 0.031 at 21d), consistent with a slow-moving signal, but statistical confidence falls because the NW correction grows harsher as overlap increases.
• Reality check: these are crude, well-known factors on a survivorship-biased universe. Marginal significance is the expected outcome; the NB02 results were inflated by overlapping returns treated as independent observations.

4. Non-Overlapping IC

An alternative to Newey-West: sample every h-th day so return windows are completely disjoint. Fewer observations (~65 at the 21d horizon vs ~1,300 daily), but each is genuinely independent.

Daily vs Non-Overlapping IC Comparison

	Factor	Horizon	Daily Mean IC	Daily N	Daily t	Non-Overlap Mean IC	Non-Overlap N	Non-Overlap t
0	Momentum 12-1	1d	0.0195	1348	2.35	0.0195	1348	2.35
1	Momentum 12-1	5d	0.0226	1344	2.82	0.0195	269	1.14
2	Momentum 12-1	10d	0.0237	1339	3.12	0.0205	134	0.89
3	Momentum 12-1	21d	0.0307	1328	4.43	0.0266	64	0.78
4	Bollinger	1d	-0.0061	1581	-0.91	-0.0061	1581	-0.91
5	Bollinger	5d	-0.0052	1577	-0.84	-0.0130	316	-0.96
6	Bollinger	10d	0.0007	1572	0.12	-0.0061	158	-0.33
7	Bollinger	21d	0.0094	1561	1.61	0.0062	75	0.21

Observations: non-overlapping IC.

• At 1d, daily = non-overlapping (N = 1,348 both ways, t = 2.35). No overlap means no correction needed, a useful sanity check.
• Momentum 21d: N drops from 1,328 to 64 independent observations. Mean IC barely changes (0.031 → 0.027), but t collapses from 4.43 to 0.78. With ~64 draws there is not enough power to detect a signal this weak.
• Bollinger 21d: mean IC drops from 0.009 to 0.006, t = 0.21, indistinguishable from noise.
• Non-overlapping agrees with NW directionally (neither signal is significant at 21d) but is more pessimistic due to the sample reduction. The effective sample size at 21d is ~1,328/21 ≈ 63, almost matching the non-overlapping N = 64, which confirms the NW inflation estimate is well-calibrated.
• Tradeoff: NW keeps all observations and adjusts standard errors (more power, model-dependent); non-overlapping uses independent observations only (less power, assumption-free). When both agree, the conclusion is trustworthy.

5. IC Decay with Bootstrap Confidence Intervals

The IC decay curve shows mean IC at each forward horizon, but without error bars it is impossible to tell signal from noise. Block bootstrap 95% CIs show which lags are significant and where the peak is.

Momentum 12-1: IC peaks at lag 15d with mean IC = 0.0327
Bollinger: IC peaks at lag 29d with mean IC = 0.0129

Observations: IC decay.

• Momentum peaks at lag 15d (mean IC 0.033). Built from a 252d lookback, it captures a slow-moving characteristic, but the predictive sweet spot is ~3 weeks. Beyond 15d, IC holds roughly flat through lag 30. The bootstrap CIs from Section 3 show this IC is not significant after NW correction (t = 1.28 at 21d), so the peak should be read cautiously.
• Bollinger peaks at lag 29d (mean IC 0.013), but the CI band contains zero at every lag (consistent with t(NW) = 0.63 at 21d). The apparent peak is estimation noise drifting upward at the boundary of the test range.
• Natural rebalance frequencies: Momentum 15d, Bollinger 29d. These feed the backtest in Section 6, though for Bollinger the natural frequency is just the lag where noise was highest.
• The decay shape is a useful diagnostic. Momentum has a shallow hump suggesting some horizon structure, even if marginal; Bollinger's curve is flat noise, with the upward drift an artefact of sampling variability at longer lags.

6. Backtest: Daily vs Natural Frequency

The key test: does matching rebalance frequency to the IC decay peak recover Bollinger's edge, and does momentum benefit from less frequent rebalancing (lower costs)?

Momentum 12-1: natural rebalance frequency = 15d
Bollinger: natural rebalance frequency = 29d

Backtest Sharpe by Rebalance Frequency (10 bps cost)

	Factor	Rebal Freq	Ann. Return (%)	Ann. Vol (%)	Sharpe	Natural
0	Momentum 12-1	1d	3.74	11.29	+0.332
1	Momentum 12-1	5d	3.62	11.32	+0.320
2	Momentum 12-1	10d	3.61	11.27	+0.321
3	Momentum 12-1	15d	4.81	11.33	+0.425	<<<
4	Momentum 12-1	21d	3.70	11.24	+0.330
5	Bollinger	1d	-14.98	9.48	-1.581
6	Bollinger	5d	-7.96	8.95	-0.890
7	Bollinger	10d	-5.55	8.32	-0.668
8	Bollinger	15d	-3.23	8.36	-0.386
9	Bollinger	21d	-0.34	8.85	-0.038

Observations: backtest frequency sweep.

• Momentum improves modestly with less frequent rebalancing. Sharpe goes from +0.33 (daily) to +0.42 (15d, the natural frequency). The 15d equity curve sits slightly above daily throughout, with the gap widening post-2024. This is almost entirely a cost saving: less turnover means fewer round-trips at 10 bps. The curve shape does not change, so the signal itself is no better-matched, only cheaper to trade.
• Bollinger goes from very negative to flat. Sharpe improves from −1.58 (daily) to −0.04 (21d), but less negative is not good. The daily equity curve declines steadily to 0.4x over 6 years (cost destruction); the 29d curve is roughly flat around 0.9x, so the signal itself contributes almost nothing after costs. The Sharpe improvement comes from trading less rather than from a better edge.
• The monotonic pattern is informative. For both factors, Sharpe improves monotonically as rebalance frequency falls. A signal with genuine predictive power at a specific horizon would show a peak there, worsening if rebalanced too infrequently (stale positions) or too frequently (cost drag). Here it is just: trade less, lose less to costs.

7. Backtest with Position Smoothing

Beyond rebalance frequency, position smoothing offers a second lever to reduce turnover. Instead of snapping to the new target on rebalance days, we blend: position = α × target + (1-α) × yesterday. This softens position transitions and reduces round-trip costs.

Best Bollinger combo: freq=21d, smooth=0.6, Sharpe=+0.195

Observations: position smoothing.

• The heatmap is a cost gradient. Every row improves left-to-right (more smoothing = less turnover = less cost) and every column improves top-to-bottom (less frequent rebalancing, same effect). The best combo (21d, smooth=0.6) reaches Sharpe +0.20, the only positive cell, but this is not signal recovery.
• The best-combo equity curve confirms it. The 21d/smooth=0.6 line ends at ~1.1x after 6 years (vs 0.4x for the daily baseline). The shape is nearly flat with a gentle upward drift, with no compounding-alpha signature, just linear cost reduction. The daily baseline's decline is mechanical: ~44% daily turnover × 10 bps × 252 days ≈ 11% annual cost drag, roughly matching the observed ~12% annual decline.
• Bottom line: frequency and smoothing control how much you pay to trade rather than how much alpha you capture. For Bollinger there is no detectable alpha to capture, so this only sets the rate of loss to friction. The signal would need a different construction (e.g. asymmetric entry/exit, conditional activation) to generate edge.

8. Robustness Tests

Four tests to challenge whether the corrected findings hold up:

1. Sub-period split: does IC persist in both halves (pre/post 2024)?
2. Leave-one-out: does removing NVDA or the most volatile stock destroy the signal?
3. Sector split: does the signal work within tech and non-tech separately?
4. Parameter sensitivity: is momentum IC stable across lookback windows?

8a. Sub-Period Stability

Sub-Period IC Stability (NW-corrected)

	Factor	Period	Mean IC	t (naive)	t (NW)	N dates
0	Momentum 12-1	Full	0.0307	4.43	1.28	1328
1	Momentum 12-1	Pre-2024	0.0052	0.57	0.17	754
2	Momentum 12-1	Post-2024	0.0643	6.11	1.83	574
3	Bollinger	Full	0.0094	1.61	0.63	1561
4	Bollinger	Pre-2024	0.0124	1.73	0.74	987
5	Bollinger	Post-2024	0.0043	0.43	0.15	574

Observations: sub-period stability.

• Momentum is regime-dependent. Pre-2024: mean IC 0.005, t(NW) 0.17, indistinguishable from zero. Post-2024: mean IC 0.064, t(NW) 1.83, the strongest corrected t-stat in the notebook. The full-sample IC (0.031) is dominated by the post-2024 period. This is a concern: the signal may be a one-regime effect rather than a persistent anomaly.
• Bollinger shows the opposite pattern, slightly stronger pre-2024 (IC 0.012, t(NW) 0.74) than post-2024 (IC 0.004, t(NW) 0.15). Neither period is significant, so this is noise either way.
• The pre-2024 period includes COVID and the 2022 bear market, both extreme momentum regimes (crash, recovery, drawdown). Momentum's near-zero IC there suggests it struggled with the whipsaw: the 12-month lookback was long enough to include the crash, contaminating the ranking. Post-2024 was a calmer directional bull market.

8b. Leave-One-Out

Most volatile (excl. NVDA): TSLA

Leave-One-Out IC (NW-corrected, at 21d horizon)

	Condition	Factor	Mean IC	t (NW)	N tickers
0	Full (50)	Momentum 12-1	0.0307	1.28	47
1	Full (50)	Bollinger	0.0094	0.63	47
2	No NVDA	Momentum 12-1	0.0274	1.18	46
3	No NVDA	Bollinger	0.0093	0.65	46
4	No TSLA	Momentum 12-1	0.0343	1.39	46
5	No TSLA	Bollinger	0.0121	0.80	46

Observations: leave-one-out.

• Neither signal is driven by a single name. Removing NVDA barely changes Momentum (IC 0.031 → 0.027, t(NW) 1.28 → 1.18) or Bollinger (0.009 → 0.009, t(NW) 0.63 → 0.65). Removing TSLA (most volatile ex-NVDA) slightly improves both (Momentum t(NW) rises to 1.39), suggesting TSLA's volatility adds noise to the cross-sectional correlation.
• This is reassuring for portfolio construction. With 47 stocks, no single name dominates the IC. This is an advantage of the larger universe over the 14-stock NB02 setup, where single-name effects were more pronounced.

8c. Sector Split

IC by Sector (NW-corrected, at 21d horizon) — sectors with N >= 5 only

	Sector	N tickers	Factor	Mean IC	t (NW)
0	Tech	8	Momentum 12-1	0.0760	1.88
1	Tech	8	Bollinger	0.0080	0.32
2	Financials	6	Momentum 12-1	0.0217	0.47
3	Financials	6	Bollinger	0.0398	1.55
4	Healthcare	6	Momentum 12-1	0.0546	1.30
5	Healthcare	6	Bollinger	0.0182	0.70
6	Cons. Disc.	5	Momentum 12-1	0.0284	0.58
7	Cons. Disc.	5	Bollinger	0.0004	0.01
8	Cons. Staples	5	Momentum 12-1	0.1179	2.36
9	Cons. Staples	5	Bollinger	0.0087	0.25
10	Other	10	Momentum 12-1	-0.0388	-1.04
11	Other	10	Bollinger	-0.0023	-0.10

Observations: sector split.

• Momentum works in some sectors but not others. Strongest in Consumer Staples (IC 0.118, t(NW) 2.36, the only sector-level significance) and Tech (IC 0.076, t(NW) 1.88); weakest in Other (IC −0.039, t(NW) −1.04, slightly negative). Momentum captures slow-moving sector trends (staples trending steadily, tech during the AI run-up) but fails in heterogeneous groups.
• Bollinger is insignificant in every sector. Best case is Financials (IC 0.040, t(NW) 1.55), still below the bar.
• Small cross-sections weaken IC. Sectors with 5-6 stocks have noisy rank correlations, where a single outlier can dominate. The Tech (8 stocks) and Other (10 stocks) results are more reliable than Cons. Disc. (5 stocks). Read the Consumer Staples result with caution: 5 defensive names in a bull market will show strong momentum almost by construction.

8d. Momentum Parameter Sensitivity

Momentum IC by Lookback Window (NW-corrected)

	Mean IC		t (NW)
Horizon	10d	21d	10d	21d
Window
126d	0.002	-0.008	0.108	-0.336
189d	0.005	-0.001	0.252	-0.043
252d	0.024	0.031	1.287	1.281
378d	0.034	0.047	1.621	1.552
504d	0.020	0.024	0.917	0.790
63d	-0.009	-0.004	-0.517	-0.171

Observations: parameter sensitivity.

• Short lookbacks (63d, 126d) produce near-zero signal. IC is near zero or negative, t(NW) well below 1. Short-term momentum (1-6 month) does not work cross-sectionally here, consistent with short-term reversal dominating at the weekly/monthly horizon.
• The sweet spot is 252d-378d. The 12-month window (252d) has t(NW) ~1.28; extending to 18 months (378d) improves slightly to t(NW) ~1.55-1.62 with higher mean IC (0.034 at 10d, 0.047 at 21d). The effect here is closer to a long-term trend signal than classic 12-month momentum.
• 504d (2 years) falls off. IC drops to 0.020-0.024, t(NW) below 1; too long a lookback smooths over the trend and includes stale information.
• The 252d choice is reasonable but not optimal. The 378d window would have been slightly stronger, but the difference is modest and could be overfitting to this sample. The signal is robust to ±50% window changes around 252d, which matters more than the exact optimum.

9. Updated Decision Table

Incorporating all corrections and robustness checks: Newey-West t-stats, natural-frequency backtests, sub-period stability, leave-one-out, and sector analysis.

Updated Decision Table — Corrected Statistics

	Natural Freq	Mean IC (21d)	t (NW)	Sharpe (daily)	Sharpe (natural)	Lag-1 Autocorr	Decision	Reason
Factor
Momentum 12-1	15d	0.0307	1.28	+0.332	+0.425	0.9903	Discard	t(NW) < 2, regime-dependent, no sector consistency
Bollinger	29d	0.0094	0.63	-1.581	-0.145	0.8677	Discard	IC insignificant at all horizons, no alpha after cost control

10. Conclusion

Both signals are discarded. After applying proper statistical corrections (Newey-West HAC t-statistics, non-overlapping IC, block bootstrap confidence intervals), neither Momentum 12-1 nor Bollinger z-score survives as a tradeable alpha source.

What we found

	Momentum 12-1	Bollinger
Corrected t(NW) at 21d	1.28	0.63
Bootstrap 95% CI	Straddles zero	Straddles zero
Sub-period stability	Regime-dependent (pre-2024 t=0.17, post-2024 t=1.83)	Insignificant in both halves
Sector consistency	Works in staples/tech, fails elsewhere	Insignificant in every sector
Best Sharpe after optimization	+0.42 (15d rebal)	+0.20 (21d rebal, smooth=0.6)
Source of improvement	Reduced trading cost	Reduced trading cost

Why NB02's results were misleading

1. Overlapping returns inflated t-statistics by ~3.5x. Momentum's naive t = 4.43 at 21d falls to 1.28 after NW correction. The statistical case did not survive once we accounted for the 20/21-day overlap between consecutive observations.
2. Bollinger's IC-to-P&L gap reflected an absence of signal rather than an implementation problem. The high IC at longer horizons (NB02) was noise that happened to look structured. Position smoothing and frequency matching improved Sharpe only by reducing trading cost, rather than by capturing predictive power.
3. Momentum's post-2024 strength is a single-regime effect. A calm, directional bull market is where naive momentum thrives; the pre-2024 period (COVID crash, 2022 bear, whipsaw recovery) showed IC indistinguishable from zero.

What we learned (methodology)

• NW correction is necessary for overlapping forward returns. Without it, every slow-moving signal looks significant.
• IC decay + bootstrap CIs separate real horizon structure from estimation noise. A peak without a CI is meaningless.
• Better Sharpe from less trading is not signal recovery. The monotonic Sharpe improvement with lower frequency is a cost gradient rather than an alpha signature; a real signal would show a peak at its natural horizon rather than improving monotonically toward trading as little as possible.
• Sub-period and sector splits are cheap robustness checks that catch regime dependence early.

Next steps

These signals served as a test bed for the validation framework. The toolkit (corrected IC, bootstrap CIs, IC decay, lead-lag profiles, robustness battery) is now tested and ready for signals that might work. Phase 3 focuses on sourcing more sophisticated alphas where this framework can distinguish real edges from noise.

9b. Tearsheets: Best Parameters (In-Sample Backtest)

Full performance tearsheets for each signal at its best configuration in this notebook. These are in-sample backtest results rather than live, out-of-sample, or paper-traded performance. The validation in Sections 3-8 shows neither signal is statistically significant after correction; the metrics below describe what happened historically rather than what to expect going forward.

• Momentum 12-1: 15d rebalance, no smoothing (Sharpe +0.42).
• Bollinger: 21d rebalance, smooth=0.6 (Sharpe +0.20).