📈 economics

Spot Regressions with Candlesticks

This paper introduces a new estimation and inference framework for spot regressions that leverages high-frequency candlestick data (including high and low prices) to produce more accurate spot beta estimates and more powerful hypothesis tests than conventional return-based methods, as demonstrated by its application to Bitcoin's market neutrality.

Original authors: Yasin Simsek

Published 2026-05-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yasin Simsek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Seeing the Whole Candle, Not Just the Wick

Imagine you are trying to guess how much a stock price is moving up and down (its "risk") compared to the whole market. Traditionally, economists have looked at this by checking the price at the start of a time period (like 9:30 AM) and the price at the end (like 4:00 PM). They draw a line between these two points and call it the "return."

Think of this like looking at a candlestick only by its wick (the start and end points). You know where it started and where it finished, but you have no idea if the flame (the price) went way up to the ceiling or crashed to the floor in between.

This paper, written by Yasin Simsek, introduces a new way to look at the data. Instead of just the start and end, it uses the entire candlestick: the Open, the High, the Low, and the Close. This gives a much fuller picture of the price's journey during that specific minute.

The Problem: The "Zoom-In" Dilemma

The author wants to measure risk right now (called "spot" risk), not just on average over a whole year. To do this, you have to zoom in very close, looking at tiny windows of time (like 10 minutes).

The Old Way: If you zoom in too much using only start/end prices, your data is "noisy" and unreliable. It's like trying to guess the weather by looking at the sky for only 10 seconds; you might miss a sudden storm.
The Trade-off: Usually, to get a more accurate answer, you need to look at a longer time period. But if you look too long, you miss the current changes. The author wanted to solve this: How do we get a super-accurate answer for a tiny window of time?

The Solution: The "Candlestick" Super-Tool

The author built a new mathematical tool that uses the High and Low prices inside those tiny windows.

The Analogy:
Imagine you are trying to measure the size of a room.

The Old Method: You measure the distance from the front door to the back wall.
The New Method: You measure the front-to-back distance, plus the distance from the left wall to the right wall, plus the height of the ceiling.

By using all these extra measurements (the High and Low prices), the author's tool can calculate the risk much more precisely, even when looking at a very small window of time.

How It Works: The "Weighted Average"

The author didn't just throw all the numbers together. He created a special recipe (a set of weights) to mix the data.

He tested millions of different ways to combine the "Open-Close" price, the "High-Open" price, and the "Low-Open" price.
He found the perfect combination that minimizes the "error" (the risk of being wrong).
The Surprise: The recipe turned out to rely heavily on the "Open-Close" price and the "Asymmetry" (how lopsided the candle is), but it barely used the "Range" (the total distance from High to Low) because that specific piece of data was too confusing for the math to handle accurately in this context.

The Result: A Sharper Lens

The paper proves two main things through computer simulations:

More Accurate: The new method is much less "jittery" than the old method. If you only have a few minutes of data (a small sample), the new method gives you a much clearer picture of the true risk.
Better Detection: When testing if Bitcoin is truly "neutral" (meaning it doesn't move with the stock market), the new method is much better at spotting the truth. It is less likely to miss a real connection (a "false negative").

The Real-World Test: Is Bitcoin "Digital Gold"?

To prove the tool works, the author applied it to Bitcoin (specifically the IBIT ETF) and the S&P 500 (SPY) using 1-minute data from 2024.

The Question: Is Bitcoin a safe haven that stays calm when the stock market crashes? (This is the "Digital Gold" theory).
The Finding: The new tool found that Bitcoin is not always neutral. In fact, during times of high stress (like August and September 2024), Bitcoin often moved with the stock market, showing a positive risk connection.
Why it matters: The old methods might have missed these short-term connections because they were too "blurry." The new candlestick method saw the connection clearly, suggesting that Bitcoin might not be as safe a hedge as people think during volatile times.

Summary

This paper gives investors and researchers a better magnifying glass. By using the full "candlestick" (Open, High, Low, Close) instead of just the start and end points, they can measure risk more accurately in real-time. This helps us understand that assets like Bitcoin might be riskier and more connected to the stock market than previously thought, especially during chaotic market moments.

1. Problem Statement

The paper addresses the challenge of estimating spot regression parameters (specifically spot beta, $\beta_t$ ) in high-frequency financial data.

The Trade-off: Traditional spot regression relies on local windows of high-frequency returns (open-to-close). There is a fundamental bias-variance trade-off: using very short windows (small $k$ ) provides high localization but suffers from high estimation variance due to limited data. Using larger windows reduces variance but loses the ability to capture rapid time-variation in risk exposures.
Information Loss: Conventional estimators discard valuable information contained in the High and Low prices within a sampling interval, utilizing only the Open and Close prices.
Inference Gap: While literature exists for spot volatility estimation using candlesticks (Open, High, Low, Close), there is a lack of feasible inference procedures (hypothesis testing) for spot betas using this richer data structure, particularly in multivariate settings where limiting distributions are non-pivotal.

2. Methodology

A. Theoretical Framework

Price Process: The author models asset prices as a bivariate Itô semimartingale:
$dX_t = b_t dt + \sigma_t dW_t$
where $X_t$ represents log-prices, $b_t$ is the drift, $\sigma_t$ is the stochastic volatility matrix, and $W_t$ is a Brownian motion.
Regression Representation: The second asset is modeled as a regression on the first (market portfolio):
$dX_{2,t} = b_{2,t} + \beta_t dX_{1,t} + \varsigma_t^{1/2} dW_{2,t}$
Here, $\beta_t$ is the spot beta, $\nu_t$ is market variance, and $\varsigma_t$ is idiosyncratic variance.
Candlestick Returns: Instead of just returns ( $r_i$ $r_{i}$ ), the author utilizes normalized High ( $h_i$ $h_{i}$ ) and Low ( $\ell_i$ $ℓ_{i}$ ) returns. These are transformed into three variables per interval:
1. Return: $r_i$ (Open-to-Close)
2. Range: $w_i = h_i - \ell_i$
3. Asymmetry: $a_i = h_i + \ell_i - r_i$

B. Estimation Framework

The core contribution is a new class of estimators for the spot covariance matrix ( $c_t$ ) that leverages all four prices.

General Estimator: The estimator is a weighted sum of quadratic forms of the candlestick variables over a local window of size $k$ :
$\hat{c}_{n,t}(\lambda) = \frac{1}{k} \sum_{i \in I_{n,t}} z_i \Lambda z_i^\top$
where $z_i = (r_i, a_i, w_i)^\top$ and $\Lambda$ is a matrix of weights determined by the vector $\lambda$ .
Risk Minimization: The weights $\lambda$ are chosen to minimize a quadratic risk function defined as the expected squared Frobenius norm of the relative estimation error:
$L(\lambda; c_t) = \|\sigma_t^{-1} \hat{c}_{n,t}(\lambda) \sigma_t^{-\top} - I\|^2$
Fixed- $k$ Asymptotics: Unlike traditional methods where $k \to \infty$ , this paper adopts a fixed- $k$ framework (Bollerslev et al., 2021). This acknowledges that in high-frequency event studies, the window size is often small and fixed.
Average Risk Optimization:
- In the multivariate setting, the limiting distribution of the estimator depends on the unknown correlation parameter $\rho_t$ , making the risk function non-pivotal and direct minimization infeasible.
- To solve this, the author minimizes the Average Risk $\bar{R}(\lambda)$ , defined as the integral of the asymptotic risk over the space of possible correlation matrices.
- This yields a closed-form solution for optimal weights $\lambda^*$ that is robust to the unknown correlation structure.
- Key Finding: The optimal weights assign significant weight to the Return ( $r_i$ ) and Asymmetry ( $a_i$ ) terms, while the Range ( $w_i$ ) receives near-zero weight because it loses sign information regarding correlation.

C. Inference Procedure

The paper develops a hypothesis test for $H_0: \beta_t = \beta_0$ .

Test Statistic: A studentized statistic is constructed using the candlestick-based estimates.
Critical Values:
- For $H_0: \beta_t = 0$ , the limiting distribution is pivotal and can be simulated directly.
- For $H_0: \beta_t = \beta_0 \neq 0$ , the distribution is non-pivotal. The author employs a sup-t test approach, constructing critical values as the envelope of conditional distributions across all possible correlation values. This ensures uniform asymptotic size control.

3. Key Results

A. Simulation Evidence (Monte Carlo)

Efficiency: The candlestick-based estimator significantly outperforms the standard return-based estimator.
- For small windows ( $k=5$ ), the candlestick estimator reduces asymptotic risk by approximately 65% compared to the return-based estimator.
- It performs nearly as well as an infeasible "oracle" estimator that knows the true correlation structure (only ~8-9% efficiency loss).
Power: The hypothesis test using candlestick data exhibits substantially higher power.
- At $k=10$ and $\alpha=5\%$ , the rejection rate for false nulls is roughly 25% higher than the return-based test.
- The power advantage is most pronounced when the true beta deviates moderately from the null hypothesis.

B. Empirical Application: Bitcoin Market Neutrality

Data: 1-minute candlestick data for the iShares Bitcoin Trust (IBIT) and the SPDR S&P 500 ETF (SPY) throughout 2024.
Objective: Test the "digital gold" narrative (i.e., is Bitcoin market neutral, $\beta_t = 0$ ?).
Findings:
- The null hypothesis of market neutrality is rejected approximately 35% of the time.
- Rejection rates spike to ~60% during August and September 2024, coinciding with high market volatility (e.g., Bank of Japan rate hike, weak US labor data).
- Implication: Bitcoin exhibits significant positive market beta during turbulent periods, challenging its role as a hedge or "digital gold" during crises.

4. Significance and Contributions

Methodological Innovation: Extends the range-based volatility literature (Garman-Klass, Parkinson) to multivariate spot regression. It provides the first feasible framework for estimating and testing spot betas using High/Low prices.
Practical Utility: Offers a solution to the bias-variance trade-off in high-frequency estimation. By utilizing candlestick data, researchers can achieve the precision of larger windows while maintaining the localization of smaller windows.
Feasible Inference: Solves the non-pivotal distribution problem in multivariate settings via the Average Risk minimization and sup-t test approach, providing a practical tool for hypothesis testing.
Data Accessibility: Advocates for the use of 1-minute or 5-minute candlestick data, which is widely available and less prone to microstructure noise than ultra-high-frequency tick data, making the method accessible to a broader range of researchers.
Empirical Insight: Provides granular evidence that Bitcoin's risk profile is time-varying and highly correlated with the equity market during stress periods, offering critical insights for portfolio risk management.

Conclusion

Yasin Simsek's paper establishes a robust, efficient, and practical framework for spot regression analysis. By integrating High and Low prices into the estimation process and developing a rigorous inference procedure, the paper demonstrates that candlestick data significantly enhances the precision of spot beta estimation and the power of statistical tests, particularly in small-sample, high-frequency contexts.