# On Deep-Fake Stock Prices and Why Investor Behavior Might Not Matter
## 4 Results
Since an exhaustive evaluation of all computer-generated data series is not tenable, we have generated a small sub-sample of simulation data and identified several of them that exhibit patterns similar to those of real data. Of a sub-sample of 600 generated data series, we estimate that roughly 7% of them, or 42 data series, look extremely realistic. A side-by-side cursory analysis of real stock prices and ABM-generated series reveal how strikingly similar they look. Upon visual inspection, it is almost impossible to tell the difference among them. There is no discernible visual clue to help telling apart real from generated data (Figure 5). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Stock** & **Type of Observation** & **Period Covered** \\ \hline NASDAQ & Daily closing price & 12 September 2018 to 16 September 2022 \\ \hline TESLA & Daily closing price & 1 February 2017 to 1 February 2021 \\ \hline AIR CANADA & Daily closing price & 1 February 2017 to 1 February 2021 \\ \hline CARNIVAL & Daily closing price & 17 September 2018 to 17 September 2022 \\ \hline RENAULT & Daily closing price & 17 September 2018 to 17 September 2022 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Five common stocks used as a comparison benchmark. Figure 3: Three data series generated with P2(t) by varying the initial conditions. We run unit root tests for all data series, real and simulated. The Augmented Dickey-Fuller test coefficients are not statistically significant in virtually all instances, hence, as expected, we fail to reject the null hypothesis of mean stationarity for both real and simulated data. We also perform a Wald-Wolfowitz runs test to determine if indeed the data is occurring at random and if there are any patterns. The null hypothesis assumes the data is generated by a random process. In the case of real data, we fail to reject the null hypothesis in all five instances (we only report the results for three of them). The results are split in the case of realistically looking simulated price series. In several instances, the results fail to reject the null hypothesis, whereas in others they appear to  
Figure 4: Two data series generated with P\({}_{3}\)(t) by varying the initial conditions. Figure 5: A side-by-side comparison of visual patterns: real market data vs. simulated. Not all first-order differences have similar distribution characteristics. The ones who look just like real data have first-order distributions that are undistinguishable from those of real data. Also as expected, none of these distributions are normal. Tests for normality have been performed, but they are not reported here. We have decided to run pairwise non-parametric tests instead, and failed to reject the null hypothesis in all but six instances. We do not report all results, however. For example, first-order differences of Renault's stock price are compared to those of ABM3_28_DL_4 RIGHT_MINUS_5 RIGHT_Z. The Wilcoxon signed-rank test (H0: the median difference is zero) returns a z = 1.41216 and a two-tailed \(p\)-value of 0.157904, thus failing to reject the null hypothesis that the two distributions share the same location. The visual inspection of return time series shows the familiar pattern of volatility clustering and suggests the series are not variance stationary (Figure 8). To determine whether Auto-Regressive Conditional Heteroskedasticity (ARCH) is present, we run the McLeod-Li test with a default of twenty lags. The null hypothesis is that of no heteroskedasticity. In the case of real time series, the tests reject the null at all lags except for Renault, where we reject the null only for the lags 8 to 13 (Figure 9). In the case of simulated time series, we find that all 42 series that bear close resemblance to real data exhibit the same pattern of heteroskedasticity. The return distributions show the familiar leptokurtic shape for both real market and simulated data (Figure 10). As expected, return distributions are not normal. We run normality tests, but we do not report the results here.