*2.3. Methodology—Monte Carlo Simulation and Forming Expectations*

To form expectations on the number of runs to expect in a sample, we rely on a Monte Carlo simulation. Consider the case of the DSEX that has 3119 returns. If we observe positive returns 53% of the time, we generate 3119 returns from the binomial distribution, with a success rate of 53%. We then count the number of runs found in our trial. The simulation then repeats this process 10,000 times to form a distribution of runs generated from all the trials.

Figure 1 illustrates the results of our 10,000 trial simulation given 3119 returns. Based on a success rate of 53%, 95% of the time we find that the number of runs is between 1497 and 1609. Thus, if the number of runs observed for the DSE Index is outside the 95% confidence interval, we reject the null hypothesis of statistical independence. If the number of runs exceeds 1609, that suggests negative serial correlation, whereas runs fewer than 1497 implies positive serial correlation.

**Figure 1.** Monte Carlo Simulation for Number of Runs Observed over 10,000 Trials. This graph assumes the DSE based parameters of 3119 returns and a success rate of 53%.

We perform a similar Monte Carlo simulation for the DJIA index with 3272 returns. Again assuming a 53% success rate, 95% of the time the number of runs is between 1573 and 1685. If the observed number of runs is outside this confidence interval, we reject the null hypothesis of independent returns.

### *2.4. Methodology—The Distribution of n Day Runs and Implications for the Short Selling Ban*

While the standard runs test considers statistical independence, it does not directly examine whether the tails of the runs distribution are larger than expected. If, as Diamond and Verrecchia (1987) and Beber and Pagano (2013) suggest, a short selling ban inhibits price discovery given bad news, then the returns distribution would be more negatively skewed than in an efficient market. This implies finding more runs than expected in the left hand tail of the distribution. Additionally, if markets overreact after good news, a short selling ban might impede a correction. If that were the case, we would observe longer, positive return runs than expected yielding a fatter right hand tail of the distribution.

To appreciate what the tails of the n day run distribution might look like, consider again the Monte Carlo simulation. For each trial, we can also collect the number of n day runs we observe. So for example, we can observe the number of two day, negative return runs (n = −2) in one trial. We can then generate the average number of −2 day runs for the 10,000 trials and calculate a 95% confidence interval for each n day run. We collect this information for all values of n starting with negative return runs of 10 days or longer (n = −10+) all the way up to positive return runs of 10 days or longer (n = 10+).

To give the flavor of what the tails of the distribution might look like, consider a probability of success equal to 50% for a sample of 4000 returns. Table 2 illustrates the tails of the distribution under these conditions. The second row shows the percentage of n day runs we observe, on average, for the 10,000 trials. From Table 2, negative and positive return runs of 10+ days each typically comprise 0.10% of all runs in a sample. Looking between |5| and |10+| day runs, we see that in total, this represents 6.16% of all runs (row 3, Table 2). Thus, the 5% tails of the distribution are for runs that are 5 days or longer.

In the following analysis, we examine the distribution of n day runs by splitting the sample into runs of 5 days or longer and runs between 1 and 4 days. Observing more runs than expected that are 5 days or longer suggests fat tails, and implies a short selling ban impedes price discovery leading to extended periods for markets to digest new information. If that result is obtained, it necessarily follows that shorter run periods (1–4 days) will have fewer runs than expected in an efficient market.


**Table 2.** Tails of the n day distribution (probability of success assumed equal to 50%).
