## 5 Methodologies
### Evaluation by Stylised Facts  
For this report, the performance of the proposed conditional RSQGAN is compared to the unconditional QGAN performance, keeping the topology and training hyperparameters constant. Denote \(r_{i,t}\) as the log-return for asset \(i\) over time period \((t-1,t)\) given by \(r_{i,t}:=\log S_{i,t}-\log S_{i,t-1}\), where \(S_{i,t}\) is the price of asset \(i=\{1,\ldots,N\}\) at time \(t=\{0,\ldots,T\}\). Table 1 summarises common univariate stylised facts (SFs) used and cited by [PERSON] et al. [11], [PERSON] et al. [12]. The original source papers for these SFs are in the right column. Loss metrics used to evaluate the RSQGAN and QGAN performance are based on the mean absolute difference of synthetic data versus the real data for various SF metrics. A summary of properties of SFs 1-6 was provided by [PERSON] et al. [11]. 1. **Linear unpredictability**[43] refers to the rapid decay of return autocorrelations even over small time lags, which reflects some evidence of market efficiency. This is captured by examining the autocorrelation function (ACF) of returns up to some maximum time lag. 2. **Heavy-tailed return distribution**[43; 53] refers to the phenomenon where the return distribution density fits a power, rather than exponential, law function. There is a tendency for very large positive returns for some stocks, which manifest as excess return kurtosis relative to a Gaussian distribution. 3. **Volatility clustering**[42] refers to the phenomenon where periods of high volatility (temporal standard deviation of returns) tend to be autocorrelated with periods of  
Figure 11: GGS is applied again locally to segment multivariate daily log-returns for only the study stocks WOW, BHP, and WBC. To form the study, the crisis regime was separated from normal regimes by forcing two breakpoints. Blue lines mark dates where structural breakpoints were optimally set. **Left**: historical stock prices for study stocks. **Right**: daily log-return scatter plots for study stocks. high volatility. This can be observed by fitting the ACF of absolute log-returns to a power-law function with low decay parameter \(\beta\). 4. **Leverage effect**[54; 55] refers to the observation that past price returns are negatively correlated with future volatility. It could be observed that the numerator of the lead-lag correlation expression \(L(k)\), given by \(\mathbb{E}\big{[}r_{t}|r_{t+k}|^{2}-r_{t}|r_{t}|^{2}\big{]}\), is negative if, for \(r_{t}<0\), \(|r_{t+k}|^{2}-|r_{t}|^{2}>0\), i.e., negative returns predict higher future volatility (and, vice-versa, that positive returns predict lower future volatility). 5. **Coarse-fine volatility correlation**[56; 57; 58] is a measure of the correlation of the volatility structure observed over a historical period \(\tau\) when looking at two different time resolutions. The lower-resolution volatility, coarse volatility, given by \(|\sum_{i=1}^{\tau}r_{t-i}|\), measures the absolute historical returns between time \((t-\tau,t)\). The fine volatility, measured by \(\sum_{i=1}^{\tau}|r_{t-i}|\), is a measure of the absolute daily return variations between time \((t-\tau,t)\). The coarse-fine volatility with \(\log k,\rho_{cf}^{\tau}(k)=Corr(|\sum_{i=1}^{\tau}r_{t+k-i}|,\sum_{i=1}^{\tau}| r_{t-i}|)\), measures the time dependency between fine and \(k\)-lagged coarse volatility. The lead lag correlation given by \(\Delta\rho_{cf}^{\tau}(k)=\rho_{cf}^{\tau}(k)-\rho_{cf}^{\tau}(-k)\) would indicate if a higher historical lead-lag correlation is predictive of a lower lead-lag correlation in the future--a kind of regime-switching behaviour as measured by coarse and fine volatility at total lag \(2k\). 6. **Gain/loss asymmetry**[59] is the phenomenon where positive total returns take a longer time to accrue than negative absolute returns. In other words, the speed at which return drawdowns occur tends to be faster than the speed at which return accumulation occurs.