1 Abstract

This paper examines whether recent cryptocurrency returns contain useful linear information about subsequent daily returns, and whether the remaining forecast errors exhibit economically important risk structure. The sample contains 2,191 aligned daily simple returns for Bitcoin (BTC), BNB and Cardano (ADA), covering 02 May 2020 to 01 May 2026.

Autoregressive models from AR(0) to AR(20) are estimated by maximum likelihood. Model selection balances parsimony, stationarity and residual whiteness. The resulting specifications are BTC AR(9), BNB AR(18) and ADA AR(9). The models remove most short-to-medium-lag linear autocorrelation, but the residual analysis shows that conditional-mean predictability is small relative to risk. Residuals are heavy-tailed, non-normal and volatility-clustered, while cross-asset correlations limit the diversification available from holding the three cryptocurrencies together.

Central finding. The defining statistical feature is not strong predictability in the mean. It is large, time-varying and heavy-tailed uncertainty around a comparatively weak autoregressive signal.

2 Data and return construction

The analysis uses daily closing prices for BTC, BNB and ADA. The allocated price window is 1 May 2020 to 1 May 2026, inclusive. Raw histories extend beyond this window so that coverage can be verified before filtering.

For asset \(j\), the simple daily return is

\[ r_{j,t}=\frac{P_{j,t}}{P_{j,t-1}}-1, \]

where \(P_{j,t}\) is the closing price on day \(t\). The first return is dated 2 May 2020 because the 1 May closing price is required as its denominator. Cryptocurrency markets operate every day, so the expected sequence contains calendar days rather than business days.

Data quality and coverage checks
Asset Raw rows Raw start Raw end Missing closes Duplicate dates Retained prices Daily returns
BTC 2398 2020-01-10 2026-08-03 0 0 2192 2191
BNB 2398 2020-01-10 2026-08-03 0 0 2192 2191
ADA 2398 2020-01-10 2026-08-03 0 0 2192 2191

The raw histories contain no missing closing prices or duplicate dates. Filtering leaves 2,192 prices and 2,191 daily returns for each asset, with complete alignment across the three series.

Descriptive statistics for daily simple returns
Asset Mean return Daily SD Minimum Median Maximum Skewness Excess kurtosis Annualised volatility
BTC 0.145% 3.026% -15.975% 0.041% 18.746% 0.121 3.742 57.8%
BNB 0.249% 4.263% -33.266% 0.130% 69.760% 2.438 40.348 81.4%
ADA 0.201% 5.178% -26.009% -0.080% 71.328% 1.732 19.399 98.9%
Daily simple returns. Volatility is visibly clustered rather than constant through time.

Daily simple returns. Volatility is visibly clustered rather than constant through time.

3 Autoregressive modelling framework

For each asset, the conditional mean is represented by an AR(\(p\)) process:

\[ r_t-\mu=\phi_1(r_{t-1}-\mu)+\cdots+\phi_p(r_{t-p}-\mu)+\varepsilon_t, \]

where \(\mu\) is the unconditional mean, \(\phi_i\) measures linear dependence at lag \(i\), and \(\varepsilon_t\) is the innovation or one-step prediction error. R’s arima() function uses this centred-mean parameterisation when include.mean = TRUE for an undifferenced ARMA model.

Candidate orders \(p=0,\ldots,20\) are estimated using maximum likelihood. AR(0) is included as a white-noise benchmark. A model enters the final candidate set when:

  1. estimation converges;
  2. every autoregressive root lies outside the unit circle, implying stationarity; and
  3. Ljung–Box tests at lags 20 and 30 do not reject residual whiteness at the 5% level.

Among eligible candidates, the model with the smallest Bayesian Information Criterion (BIC) is selected. BIC penalises unnecessary parameters more strongly as sample size increases, while the residual condition prevents selection of a parsimonious model that leaves material linear dependence unmodelled.

BIC by AR order. Coloured points satisfy the stationarity and residual-whiteness criteria; red points are selected.

BIC by AR order. Coloured points satisfy the stationarity and residual-whiteness criteria; red points are selected.

4 Selected conditional-mean models

Selected autoregressive models
Asset Model BIC Ljung-Box p(20) Ljung-Box p(30) Minimum root modulus
BTC AR(9) -9,044.92 0.770 0.111 1.290
BNB AR(18) -7,537.21 0.634 0.121 1.108
ADA AR(9) -6,692.04 0.193 0.162 1.297

All minimum root moduli exceed one, supporting stationarity. The lag-20 and lag-30 Ljung–Box p-values exceed 0.05, so the selected models remove the principal short-to-medium-lag linear dependence under the stated diagnostic rule.

The fitted orders should not be confused with economic importance. A higher order means that more lag terms were required to whiten the residuals; it does not mean that returns are strongly predictable. Most estimated coefficients are small.

AR coefficients significant at the 5% level
Asset Lag Estimate Standard error p-value
BTC 8 -0.0473 0.0213 0.027
BTC 9 0.0567 0.0214 0.008
BNB 1 -0.0452 0.0213 0.034
BNB 2 0.0772 0.0213 <0.001
BNB 9 0.0503 0.0212 0.018
BNB 10 0.1276 0.0212 <0.001
BNB 13 0.0450 0.0214 0.035
BNB 18 0.0659 0.0213 0.002
ADA 9 0.0643 0.0213 0.003

BTC’s clearest effects occur at lags 8 and 9, with a negative lag-8 coefficient followed by a positive lag-9 coefficient. BNB displays the broadest lag structure; its largest coefficient occurs at lag 10. ADA’s only 5%-significant AR coefficient is positive at lag 9. These estimates are consistent with a mixture of weak short-run reversal and lagged continuation, rather than a simple universal momentum rule.

5 Residual distribution and model adequacy

Summary statistics for selected-model residuals
Asset Model N Mean Daily SD Minimum Q1 Median Q3 Maximum Skewness Excess kurtosis Annualised volatility
BTC AR(9) 2,191 0.00000% 3.013% -16.219% -1.430% -0.076% 1.306% 18.254% 0.107 3.667 57.6%
BNB AR(18) 2,191 -0.00005% 4.184% -32.151% -1.709% -0.098% 1.621% 61.627% 1.774 29.257 79.9%
ADA AR(9) 2,191 0.00002% 5.155% -25.234% -2.688% -0.313% 2.127% 71.908% 1.797 20.203 98.5%

Residual means are effectively zero, as expected for fitted models with a mean term. The residual standard-deviation ranking is unambiguous: BTC is lowest, BNB is intermediate and ADA is highest. The AR components reduce dispersion only modestly relative to the raw returns, indicating that the majority of daily variation remains innovation rather than forecastable conditional mean.

BNB and ADA show pronounced positive skewness because of rare, exceptionally large positive residuals. This should not be read as an absence of downside risk: their minimum residuals are also economically large. Excess kurtosis is positive for every asset and extreme for BNB and ADA, implying far more tail mass than a Gaussian model would assign.

Residual histograms and normal Q<U+2013>Q plots. Systematic departures from the reference lines indicate non-normal tails.

Residual histograms and normal Q<U+2013>Q plots. Systematic departures from the reference lines indicate non-normal tails.

Residual diagnostics
Asset LB p(10) LB p(20) LB p(30) LB p(50) Jarque-Bera p ARCH-LM(10) p
BTC 0.530 0.770 0.111 0.007 <0.001 <0.001
BNB 0.824 0.634 0.121 0.108 <0.001 <0.001
ADA 0.106 0.193 0.162 0.136 <0.001 <0.001

The Jarque–Bera tests reject normality for all three residual series. The ARCH-LM tests also reject constant conditional variance, showing that large squared residuals are serially related. In economic terms, risk arrives in clusters rather than at a stable daily rate.

BTC is the only asset that fails the lag-50 Ljung–Box check, with (p=0.007). Its AR(9) model is adequate under the lag-20 and lag-30 selection rule but leaves some longer-horizon linear structure. This is a reported limitation rather than a reason to keep adding lags without an out-of-sample test.

Thirty-day rolling annualised residual volatility. A value of 1 represents 100% annualised volatility.

Thirty-day rolling annualised residual volatility. A value of 1 represents 100% annualised volatility.

6 Cross-asset dependence

Pearson correlations between daily returns
BTC BNB ADA
BTC 1.000 0.631 0.645
BNB 0.631 1.000 0.550
ADA 0.645 0.550 1.000

All pairwise correlations are positive and moderately large. BTC–ADA is the strongest pair at 0.645, followed by BTC–BNB at 0.631. BNB–ADA is lower but remains positive at 0.550. Holding all three assets therefore provides some idiosyncratic diversification, but it does not remove the shared cryptocurrency-market component.

7 Implications for quantitative strategy design

7.1 Conditional-mean signals

The fitted AR equations can produce one-day conditional-mean forecasts. The coefficient magnitudes and residual dispersion show, however, that the predictable component is small relative to the innovation. AR forecasts are better interpreted as a minor timing or position-sizing overlay than as a stand-alone directional strategy.

A practical signal should include a no-trade region. A forecast should trigger a position change only when its expected benefit is large enough to exceed estimated fees, bid–ask spread and slippage. Otherwise, statistical predictability can be converted into negative economic performance through turnover.

7.2 Volatility-responsive exposure

Fixed dollar positions would generate very different risk contributions across the three assets. For an equal risk budget, ADA would ordinarily receive the smallest notional position, BNB an intermediate position and BTC the largest. Because volatility changes markedly through time, these weights should respond to a forecast or recent estimate of conditional volatility rather than rely only on full-sample standard deviations.

The significant ARCH effects provide a direct statistical motivation for extending the mean equation to an AR–GARCH specification. A heavy-tailed innovation distribution, such as Student-\(t\), would address both volatility clustering and the observed departure from Gaussian tails.

7.3 Tail-risk measurement

Gaussian risk calculations would understate the probability of extreme residuals, especially for BNB and ADA. Historical simulation, stress testing and Expected Shortfall are more defensible starting points than relying only on a normal Value-at-Risk calculation. Position limits and drawdown controls remain necessary because a favourable average or positively skewed distribution does not rule out severe losses.

7.4 Portfolio construction

Portfolio weights should use the covariance matrix rather than equal dollar allocations. The assets differ substantially in residual volatility and are positively correlated. A covariance-aware or risk-budgeting approach would explicitly account for both facts. Broader diversification would require assets outside this three-coin set because common cryptocurrency-market movements remain material.

7.5 Validation protocol

No strategy should be assessed using the same sample that selected the model. A defensible evaluation would:

  1. divide the data chronologically into training and hold-out periods;
  2. re-estimate models using rolling or expanding windows;
  3. prevent future observations from influencing lag selection;
  4. deduct fees, spreads and slippage from every simulated trade;
  5. compare the AR strategy with simple benchmarks, including no trade; and
  6. report drawdown, turnover and Expected Shortfall alongside average return.

8 Limitations

The results describe in-sample linear dependence in one historical window. They do not establish stable causal relationships or guaranteed trading profits. Cryptocurrency market structure can change, making fixed coefficients unstable. The analysis also focuses on closing-price returns and does not include volume, liquidity, order-book conditions, macroeconomic variables or cross-asset lead–lag information.

The AR model addresses the conditional mean but assumes a simple innovation process during estimation. The diagnostics show that this is incomplete: variance is time-varying and tails are non-normal. Finally, statistical significance is not equivalent to economic significance. The small coefficients must be evaluated against implementation costs and forecast uncertainty.

9 Conclusion

BTC AR(9), BNB AR(18) and ADA AR(9) provide adequate in-sample descriptions of short-to-medium-lag linear dependence under the adopted BIC and residual-whiteness rule. The models are stationary and their residual means are close to zero. Nevertheless, the conditional-mean signal is weak relative to residual risk.

The most robust empirical findings concern the distribution and timing of risk. ADA has the highest residual volatility, followed by BNB and BTC. BNB and ADA are highly right-skewed and heavy-tailed; all three residual series reject normality and display volatility clustering. Positive cross-asset correlations further limit diversification within the selected cryptocurrency set.

Accordingly, the AR forecasts are most defensible as conservative tactical inputs embedded within a wider risk system. That system should use volatility-responsive sizing, heavy-tail risk measures, covariance-aware allocation, explicit cost thresholds and strict out-of-sample validation. An AR–GARCH model with heavy-tailed innovations is a natural next modelling step.

10 References

11 Technical appendix

11.1 Full coefficient estimates

Maximum-likelihood coefficient estimates
Asset AR order Term Estimate Standard error z statistic p-value Significant at 5%
BTC 9 ar1 -0.034989 0.021328 -1.641 0.101 FALSE
BTC 9 ar2 0.020288 0.021312 0.952 0.341 FALSE
BTC 9 ar3 -0.001696 0.021313 -0.080 0.937 FALSE
BTC 9 ar4 0.017802 0.021311 0.835 0.404 FALSE
BTC 9 ar5 0.020179 0.021311 0.947 0.344 FALSE
BTC 9 ar6 0.004563 0.021336 0.214 0.831 FALSE
BTC 9 ar7 0.001961 0.021332 0.092 0.927 FALSE
BTC 9 ar8 -0.047296 0.021330 -2.217 0.027 TRUE
BTC 9 ar9 0.056742 0.021375 2.655 0.008 TRUE
BTC 9 intercept 0.001451 0.000669 2.169 0.030 TRUE
BNB 18 ar1 -0.045161 0.021312 -2.119 0.034 TRUE
BNB 18 ar2 0.077220 0.021327 3.621 <0.001 TRUE
BNB 18 ar3 -0.011745 0.021385 -0.549 0.583 FALSE
BNB 18 ar4 0.025955 0.021381 1.214 0.225 FALSE
BNB 18 ar5 -0.015702 0.021377 -0.735 0.463 FALSE
BNB 18 ar6 0.009196 0.021361 0.431 0.667 FALSE
BNB 18 ar7 0.027639 0.021352 1.294 0.196 FALSE
BNB 18 ar8 -0.002015 0.021356 -0.094 0.925 FALSE
BNB 18 ar9 0.050256 0.021200 2.371 0.018 TRUE
BNB 18 ar10 0.127634 0.021199 6.021 <0.001 TRUE
BNB 18 ar11 -0.001988 0.021380 -0.093 0.926 FALSE
BNB 18 ar12 -0.021564 0.021371 -1.009 0.313 FALSE
BNB 18 ar13 0.044953 0.021371 2.103 0.035 TRUE
BNB 18 ar14 -0.016392 0.021393 -0.766 0.444 FALSE
BNB 18 ar15 -0.006419 0.021384 -0.300 0.764 FALSE
BNB 18 ar16 0.016783 0.021380 0.785 0.432 FALSE
BNB 18 ar17 -0.015626 0.021314 -0.733 0.463 FALSE
BNB 18 ar18 0.065895 0.021290 3.095 0.002 TRUE
BNB 18 intercept 0.002489 0.001291 1.928 0.054 FALSE
ADA 9 ar1 -0.037010 0.021315 -1.736 0.083 FALSE
ADA 9 ar2 0.031125 0.021320 1.460 0.144 FALSE
ADA 9 ar3 0.008014 0.021322 0.376 0.707 FALSE
ADA 9 ar4 0.013714 0.021318 0.643 0.520 FALSE
ADA 9 ar5 -0.025456 0.021309 -1.195 0.232 FALSE
ADA 9 ar6 0.007071 0.021310 0.332 0.740 FALSE
ADA 9 ar7 0.021158 0.021306 0.993 0.321 FALSE
ADA 9 ar8 -0.024494 0.021297 -1.150 0.250 FALSE
ADA 9 ar9 0.064275 0.021293 3.019 0.003 TRUE
ADA 9 intercept 0.002006 0.001169 1.715 0.086 FALSE

11.2 Reproducibility

The R Markdown document expects the following files in the same project:

Assignment1_Nathan_Foale/
├── Crypto_AR_Modelling_Paper.Rmd
├── Question1_AR_Analysis.R
└── raw/
    ├── BTC_raw.csv
    ├── BNB_raw.csv
    └── ADA_raw.csv

Open Crypto_AR_Modelling_Paper.Rmd in RStudio and select Knit. The analysis script uses only base R; rmarkdown and knitr are required to produce the HTML document.

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## Running under: macOS Sequoia 15.2
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## [1] C.UTF-8/C/C.UTF-8/C/C.UTF-8/C.UTF-8
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## time zone: Australia/Melbourne
## tzcode source: internal
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## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
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## loaded via a namespace (and not attached):
##  [1] digest_0.6.39   R6_2.6.1        fastmap_1.2.0   xfun_0.57      
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