Removing Bid–Ask Bounce from Trade-Only Price Series

# Is there a way to derive a fair price from cryptocurrency trade (no quote) data that is free of bid-ask bounce?


# Is there a way to derive a fair price from cryptocurrency trade (no quote) data that is free of bid-ask bounce?












I'm working with Kraken historical ETH-USD trade data from 2017 onward, which includes:

- Timestamp (Datetime)

- Trade ID

- Trade price

- Trade volume

- Taker side (buy/sell)

- Order type (market or marketable limit)

My goal is to derive a fair (efficient) price series that's minimally affected by bid-ask bounce with the goal being to use it for execution backtesting and modeling/analysis. I am okay with downsampling to 1 and maybe 5 minute bars, but would like to keep the data as granular as possible.

Approaches tried:

- OHLCV bars (1, 5, 15m): Log returns on close prices still show autocorrelation (≤ -0.1) at lag 1–3.

- Spread estimation (Roll's, Corwin-Schultz): Often gives NaNs or zero/negative spreads — possibly due to model or implementation issues.

- Low-order linear ARMA on log returns: Limited success. I haven’t yet tried rolling ARMA.

I saw that this question somewhat addresses my issue, but it seems to me that Hasbrouck’s model requires simultaneous estimation of the efficient price and the spread?

Thanks for any input/references on this.

## Answer by dikovaxi (score 0)

https://quant.stackexchange.com/a/85818

Short version. Because your data has the taker side, you don't need Roll, Corwin–Schultz or a joint Hasbrouck-style estimation at all: the trade sign $q_t\in\{+1,-1\}$ is exactly the latent variable those models exist to infer. With the sign observed, the bounce is a known component and the sign-adjusted price

$$\hat m_t = p_t - c\,q_t,\qquad c=\frac{\operatorname{cov}(\Delta p_t,\Delta q_t)}{\operatorname{var}(\Delta q_t)}$$

removes it. That is the Huang–Stoll / Glosten–Harris regression $\Delta p_t = c\,\Delta q_t+\varepsilon_t$ — OLS with one regressor — and $c$ is the effective half-spread. Estimate it per day (or per hour if the spread moves intraday), then sample $\hat m_t$ at the bar close. Two things I would not do: don't build the return series from bar VWAP (averaging a random walk inside the bar produces positive autocorrelation, Working 1960; I measure +0.2 to +0.3 at lag 1 below), and don't use the "last taker-buy price = ask, last taker-sell price = bid" mid as your price series — it is the right tool for simulating fills, because it tells you the touch on each side, but as a mid it goes stale on one side and smooths returns.

A test on a market where the truth is available. Binance USD-M perpetuals publish both `aggTrades` (with taker side) and `bookTicker` (every top-of-book change) as daily files on data.binance.vision (bookTicker until March 2024), so trade-only estimators can be checked against the real quoted mid. Here is 2024-03-26 for XRPUSDT: tick = 1.56 bp, quoted spread = 1 tick 99% of the day, 130 trades/min, 97% of taker trades execute at the prevailing touch — a clean bounce regime, close to what you have. Lag-1 autocorrelation of log returns, and mean distance of the bar-close level from the true quoted mid:

|  | last trade | last-buy/last-sell mid | $p_t - c\,q_t$ | true quoted mid |
| ACF(1), 1 s bars | −0.173 | +0.095 | +0.034 | +0.044 |
| ACF(1), 5 s bars | −0.041 | +0.051 | +0.032 | +0.029 |
| ACF(1), 60 s bars | −0.045 | −0.036 | −0.035 | −0.036 |
| mean |level − true mid| | 0.78 bp | 0.23 bp | 0.15 bp | 0 |

$c$ estimated on the whole day is 0.47 tick = 0.73 bp, and the 0.78 bp error of the last-trade series is, as it should be, the half spread. The sign-adjusted price sits on the quoted mid within 0.15 bp and has the same autocorrelation as the mid at every horizon: the bounce (−0.17 at 1 s) is gone. Note also that by 60 s the bounce is invisible in every series on this venue — whatever negative autocorrelation remains at 1–5 min is present in the true mid too, i.e. it is genuine short-horizon mean reversion, not microstructure.

BTCUSDT on the same day teaches the opposite lesson. Quoted spread is 1 tick (0.014 bp), but the touch usually holds only a few thousandths of a BTC, only 40% of taker trades execute at the pre-trade touch (the rest sweep several levels within the same millisecond), and the regression half-spread is 3.35 ticks — 6.7× the quoted half-spread. For an execution backtest that effective spread is the number you want, and you get it from trades + signs; the quotes would have misled you.

Why Roll and Corwin–Schultz returned NaNs or nonsense. Roll assumes the negative autocovariance of price changes is caused by the spread. At the trade clock on BTCUSDT it returns 52 ticks (the autocovariance there comes from sweeps that revert, not from the quote). On 1–5-minute bars it simply estimates whatever mean reversion exists at that horizon: on XRPUSDT it gives 1.1 bp at 1 s (≈ the true 1.56), 3.4 bp at 1 min and 7.6 bp at 5 min — growing with the bar while the quoted spread is constant. Corwin–Schultz assumes the high–low range is spread plus diffusion volatility; on minute-scale crypto bars the range is dominated by volatility, the estimator goes negative and you get NaN. Nothing is wrong with your implementation; the estimators are being asked a question the data doesn't answer.

About lags 2–3. Bid–ask bounce is an MA(1) effect: it lives at lag 1 only. If −0.1 at lags 2 and 3 survives in $\hat m_t$, it is a transitory price component (Kraken in 2017 was thin and often slow relative to the larger venues, and large trades pushed price and reverted over minutes). The tool for that is a state-space model with a random-walk efficient price and a stationary transitory term — Hasbrouck (1993), Menkveld, Koopman & Lucas (2007). But note that the "simultaneous estimation of efficient price and spread" you were worried about disappears once $q_t$ is observed: the measurement equation is $p_t = m_t + c\,q_t + u_t$ with $q_t$ known, so the Kalman filter has three or four parameters and $c$ is essentially the OLS coefficient above.

Recipe

- $q_t=+1$ for a taker buy, $-1$ for a taker sell; $c$ by OLS of $\Delta p$ on $\Delta q$, per day or per hour.







```
q = np.where(df.taker_side == 'buy', 1, -1)
dp, dq = df.price.diff(), pd.Series(q, index=df.index).diff()
c = (dp * dq).sum() / (dq * dq).sum()            # effective half-spread
df['m_hat'] = df.price - c * q
df['ask_proxy'] = df.price.where(q == 1).ffill()
df['bid_proxy'] = df.price.where(q == -1).ffill()
bars = df.set_index('ts').resample('1min').last()  # last, not mean
```

References: Roll (1984) J. Finance; Glosten & Harris (1988) JFE; Huang & Stoll (1997) RFS; Hasbrouck (1993) RFS "Assessing the quality of a security market"; Menkveld, Koopman & Lucas (2007) JBES; Andersen, Bollerslev, Diebold & Labys (2000) "Great realizations"; Working (1960) Econometrica on autocorrelation induced by averaging.