Rough microstructure noise disappears on most trading days. Across 4,765 noisy ticker-days, the median daily estimate is exactly 0.000, and only 20.4% reject the null of no roughness at 5%. Christensen and Norlyk acknowledge the limitation in their abstract: rough noise, "while present, is not universal". They argue that its episodic appearance reconciles these daily results with the stronger and steadier roughness reported by Chong, Delerue and Li. The unresolved part of that argument lies in the 5-day rolling window.
Ticks appear, persist or reverse
Christensen and Norlyk construct the tick-by-tick price on an integer grid using a single integer-valued Lévy basis evaluated over two regions of space-time. A growing rectangle generates the permanent component, so a tick remains after landing there. The transitory component comes from a moving set shaped by a deterministic trawl function. A distortion arrives, persists for a random duration and is then corrected. The parameter b lies in (0,1) and divides the two components. In the scaling limit, the permanent Brownian component has variance rate b. Among their noisy ticker-days, b has a mean of 0.742 and a median of 0.775.
The scaling carries the paper. The Lévy measure's activity grows linearly in T, while the trawl function is capped at T. After rescaling, the price converges to a Bachelier semimartingale plus a Gaussian moving average whose kernel is inherited directly from the trawl function. Choosing the gamma kernel g(x) = x^alpha exp(-lambda x) yields an explicit trawl function and any roughness index alpha in (-1/2, 0], with 0 Brownian. Economically, roughness requires the trawl function to become unbounded at zero. Distortions with arbitrarily short lives then arrive in unbounded concentration and are cancelled almost immediately.
A bounded trawl cannot generate roughness.
The authors estimate the model by GMM, matching second moments of price increments over m = 100 lags. Prices are sampled every 0.1 seconds, weights are diagonal and the long-run covariance is truncated at 1,000 lags. Under the null of non-rough noise, alpha lies on the boundary. They therefore apply Andrews (2002): the estimator remains root-n, while its limit is a one-sided normal with a point mass at zero, and the test rejects below -1.645. Their simulation uses a Skellam basis with intensity 6.28 per second, just over six price changes a second. At a full trading day's length, bias is small. A true alpha of -0.400 produces -0.395 (Monte Carlo sd 0.094), while a true alpha of 0 produces -0.007 (sd 0.020).
The trading relevance comes through volatility estimation. Pre-averaging ceases to estimate integrated volatility consistently under rough noise, the result from Chong, Delerue and Li to which this work responds. The paper offers no return forecast.
Does daily measurement retain the roughness?
Earlier estimates use 5-day rolling windows. Over 2013 to 2022, their power-variation estimator finds a stable level corresponding to alpha of about -0.20. Christensen and Norlyk instead estimate one alpha for each ticker-day using 2024 TAQ transactions across all exchanges for the DJIA constituents.
The distribution bunches at the boundary. Median alpha is 0.000, its 5th percentile is -0.177, and estimates vary widely across days. The mean across 4,765 noisy ticker-days is -0.036. Only 20.4% reject non-roughness at 5%. Even among those rejections, the median estimate reaches only -0.08.
Their proposed reconciliation rests on the behavior of a 5-day rolling power-variation estimator. In their words, it will asymptotically be dominated by the roughest day in the window, allowing a few rough days to produce an apparently stable -0.20. The main text presents this as an argument rather than a measured result. We did not find an exercise that feeds their fitted daily alphas into a 5-day rolling estimator and checks whether it returns something near -0.20. That inexpensive check would move the reconciliation from plausible to demonstrated.
The authors give two qualifications. First, the test over-rejects in finite samples. At alpha = 0, size exceeds the nominal 5% for every simulated length and remains about 6% at T = 46,800 seconds, twice a trading day. Their application uses single days. The true share is probably below 20.4%, because the test rejects about 6% of the time at alpha = 0 against a nominal 5%.
Second, the microfoundation weakens in the rough case. Functional convergence in the Skorokhod topology is proved only at alpha = 0. When alpha lies in (-1/2, 0), the tightness condition fails, leaving finite-dimensional convergence as the link. Proposition 3.3 and Remark 3.7 state the limitation. Remark 3.7 argues that this failure is generic for any monotone trawl function that generates roughness.
The Roll spread largely explains the detection
The regressions (N = 4,559) place a rough-noise dummy on standard liquidity measures. Roll spread supplies the strongest joint prediction. Alone, its coefficient is 0.083 (t = 7.60, R-squared 4.59%). Jointly, it rises to 0.100 (t = 3.12), taking R-squared from 10.89% to 13.38%.
Average trade size is the strongest individual predictor, producing 6.43% R-squared with a coefficient of 0.276 (t = 4.39). It remains at 0.228 (t = 2.80) in the joint model excluding Roll spread. Once Roll spread enters, the coefficient falls to 0.036 (t = 0.43). Amihud illiquidity and trade count contribute nothing individually, with t statistics of -0.69 and 0.92 and R-squared values of 0.05% and 0.20%.
The definition of Roll spread changes how this result reads. It equals twice the square root of the negative part of the first-order autocovariance of intraday price changes, with zero assigned when that autocovariance is positive. The sample has already passed a Wald test, using Newey-West standard errors, that rejects non-negative first-order autocorrelation in one-second returns. This screen removes 36% of ticker-days. The dependent variable then comes from a GMM fit to the autocovariance structure of those same increments. The leading association is therefore close to definitional. Its magnitude is the relevant result: the reversal measure adds 2.5 points of R-squared beyond everything else. Small, given how closely it restates the left-hand side.
The collapse in trade size is more revealing. It suggests that temporary impact carries the trade-size channel, leaving little for information. The volatility result also deserves attention, provided its column is identified. In the joint model containing Roll spread, intraday realized volatility has a coefficient of -12.793 (t = -6.37), while lagged 21-day volatility enters at 0.584 (t = 3.23). Neither has an individual result. Intraday volatility is -3.611 (t = -0.80, R-squared 0.27%), and the lagged 21-day measure is 0.276 (t = 0.83, R-squared 0.21%). The pattern of rough noise appearing on quiet days within historically volatile names exists only in the joint fit.
Which tape determines the result?
The pre-test rejects zero first-order autocorrelation on 12% of ticker-days using single-exchange prices and 55% using all-exchange prices. Because the gap is so large, the authors abandon the standard single-venue cleaning rule. Venue consolidation raises rejection from 12% to 55%. They attribute the added dependence to cross-venue latency, since a single venue already contains bid-ask bounce. A two-exchange Hasbrouck (1995) calculation in the online appendix supports that explanation, though the authors do not split the 55% between latency and bounce. Noise estimates from a consolidated tape therefore pass their construction choice into the pre-averaging diagnostics.
Where the evidence ends
This diagnostic identifies a volatility-estimation regime using 30 mega-cap US names over one calendar year. It requires trade-level observations sampled every 0.1-second and, preferably, venue identifiers for reproducing the 12%-versus-55% comparison.
We could not run it.
Our equity data consists of daily and 1-minute OHLCV bars. One-second returns and tenth-of-a-second price paths cannot be recovered from the four prices in a minute bar. Venue-level trades are unavailable as well, putting the consolidation comparison entirely beyond reach.
The main text leaves out the exercise that would resolve its reconciliation. Take the 4,765 daily alphas, simulate their corresponding paths, apply a 5-day rolling power-variation estimator and show that it prints -0.20.