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In Search of the Black Swan: Analysis of the Statistical Evidence of Electoral Fraud in Venezuela

This study analyzes diverse hypotheses of electronic fraud in the Recall Referendum celebrated in Venezuela on August 15, 2004. We define fraud as the difference between the elector's intent, and the official vote tally. Our null hypothesis is that there was no fraud, and we attempt to search for evidence that will allow us to reject this hypothesis. We find no evidence that fraud was committed by applying numerical maximums to machines in some precincts. Equally, we discard any hypothesis that implies altering some machines and not others, at each electoral precinct, because the variation patterns between machines at each precinct are normal. However, the statistical evidence is compatible with the occurrence of fraud that has affected every machine in a single precinct, but differentially more in some precincts than others. We find that the deviation pattern between precincts, based on the relationship between the signatures collected to request the referendum in November 2003 (the so-called, Reafirmazo), and the YES votes on August 15, is positive and significantly correlated with the deviation pattern in the relationship between exit polls and votes in those same precincts. In other words, those precincts in which, according to the number of signatures, there are an unusually low number of YES votes (i.e., votes to impeach the president), is also where, according to the exit polls, the same thing occurs.


INTRODUCTION
This study presents a statistical evaluation of the results of the August 15, 2004 Recall Referendum on President Hugo Chávez's mandate. From the morning of August 16, 2004, when the CNE (Consejo Nacional Electoral) announced the results, opposition spokespersons expressed doubts about the validity of these results, and argued that an electronic fraud had been committed. These doubts had not been cleared up with the passing of time.
At the time, Súmate-a Venezuelan NGO that had organized the collection of signatures to request the referendum and monitored its execution-requested that we do a statistical analysis to verify if the available information is compatible with the hypothesis of fraud or if, on the contrary, it rejects this hypothesis. Súmate provided the data used in this study but gave us complete autonomy over the conduct of our research.
We were informed that the presumption of fraud is based on the following elements: (1) The adoption of a new automated voting system in spite of the fact that the opposition had requested a manual tally.
(2) The voting machines left a paper trail by printing ballots that allowed each elector to verify that the machine had counted his vote adequately. These ballots were collected in boxes. However, the CNE did not allow the boxes to be opened and counted. Instead, it performed a so-called "hot" audit of 1 percent of the machines on the evening of the election. Moreover, the CNE decided that the number of boxes to be opened would be chosen by a random number generator program run on its own computer.
(3) After a difficult negotiation, the CNE allowed the Organization of American States and the Carter Center to participate as observers in every phase of the process except for access to the central computer server that communicated with each machine in each voting precinct. No witness from the opposition was granted access to that room either. Only two people were allowed in that room until the results were ready.
(4) The adopted technology allowed-in fact required-bidirectional communication between the central servers and the voting machines. This bidirectional communication occurred. This is different from the information that was provided to opposition negotiators about the nature of the technology involved. (5) Contrary to what was initially stipulated, the voting machines communicated with the central server before printing the results in a document called Acta. This opens the possibility that the machines were instructed to print a result different from the one expressed by the voters.
(6) On August 15, 2004, different organizations, including Súmate, conducted exit polls in a number of precincts. To assure its quality, Súmate's poll was conducted with the assistance of the firm Penn, Shoen and Berland. Its results were radically different from official figures. The same thing occurred with the exit poll conducted by "Primero Justicia," a political party. The database of both surveys was given to us to conduct this study. (7) The "hot-audit" conducted at dawn on August 16, 2004 was not carried out to the satisfaction of either the opposition or the international observers. Only 78 of the 192 boxes stipulated were counted. The opposition only attended 28 counts, and the international observers were only present in less than 20. (8) As requested by the international observers, a second audit was conducted on August 18. The opposition did not participate in this audit because its conditions were not met; for example, the electoral materials were not delivered to a centralized location before choosing the boxes to be opened and there was no verification that the boxes selected had not been tampered with. Instead, the boxes were chosen 24 hours before they were opened, which in theory would give time for them to be altered. Notably, the CNE did not use the random number generator program proposed by the Carter Center, and instead insisted on using its own program run on its own computer and started with a seed defined by a pro-government member of the CNE. This raises doubts over whether the sample selected was truly a random one, or that the sample was unknown before the voting started.
All these facts raise the possibility of an electronic fraud in which the machines printed outcomes different from the real count. This could in theory have been done through software alterations, or through electronic communications with the computer hub.
Our main findings are the following. First, the paper finds that the sample used for the audit of August 18, which was observed by the OAS and the Carter Center, was not randomly chosen. In that sample, the relationship between the votes obtained by the opposition on August 15 and the signatures gathered requesting the Referendum in November 2003 was 10 percent higher than in the rest of the boxes. We calculate the probability of this taking place by pure chance at less than 1 percent. In fact, we create 1,000 samples of non-audited precincts to prove this.
This result opens the possibility that the fraud was committed only in a subset of the 4,580 automated precincts, say 3,000, and that the audit was successful because it directed the search to the 1,580 unaltered precincts. This sheds new light on the fact that the Electoral Council did not accept the use of the random number generator proposed by the Carter Center and under these conditions one can infer why the Carter Center could not identify the fraud with the audit they observed. In other words, before the voting process starts the random seed might be known, and therefore, the computer only changes the machines that ex-ante knows that have no chance of being audited. The machines audited look like a random sample regarding regions, social characteristics, etc. except for the fact that they were not affected by the fraud.
In addition, we develop a statistical technique to identify whether there are signs of fraud in the data. To do so, we depart from previous work on the subject that was based on finding patterns in the number of votes per machine or precinct. Instead, we look for two independent variables that are imperfect correlates of the intention of voters. Fraud is nothing other than a deviation between the voters' intention and the actual count. Since each variable used is correlated with the intention, but not with the fraud, we can develop a test as to whether fraud is present. In other words, each of our two independent measures of the intention to vote predicts the actual number of votes imperfectly. If there is no fraud, the errors these two measures generate would not be correlated, as they each would make mistakes for different reasons. However, if there is fraud, the variables would make larger mistakes where the fraud was bigger and hence the errors would be positively correlated. The paper shows these errors to be highly correlated and the probability that this is pure chance is again less than 1 percent.
The first variable we use is the number of registered voters in each precinct that signed the recall petition in November, 2003. This clearly shows intent to vote yes in a future election but it does so imperfectly. Our second measure is the exit poll conducted by Penn, Schoen and Berland and complemented with an independent exit poll conducted by Primero Justicia. This is also an imperfect measure as it depends on potential biases in the sample, differences in the skill of the interviewer, etc. But this source of error should not be correlated at the precinct level with the one that affects the signatures. Therefore, it is very telling that in the precincts where the Penn, Schoen and Berland exit poll makes bigger mistakes is also where the number of petitioners suggests that the Yes votes would be higher.
This evidence is troubling because it resonates with three facts about the conduct of the election. First of all, contrary to the agreed procedure, the voting machines were ordered to communicate with the election computer server before printing the results. Second, contrary to what had been stated publicly, the technology utilized to connect the machines with the computer hub allowed two-way communication and this communication actually took place. This raises the possibility that the hub could have informed the machines what numbers to print, instead of the other way around. Finally, after an arduous negotiation, the Electoral Council allowed the OAS and the Carter Center to observe all aspects of the election process except for the central computer hub, a place where they also prohibited the presence of any witnesses from the opposition. At the time, this appeared to be an insignificant detail. Now it looks much more meaningful.
The structure of the paper is as follows. First, we describe the evidence coming from the exit polls. We show that the difference between the exit polls and the actual vote is not likely to have been caused by a sampling error, due for example, to an overrepresentation of anti-Chavez precincts, but instead to a generalized but variable difference, precinct by precinct.
Second, we discuss some of the previous evidence of fraud and its validity. We address the popular so-called "topes" hypothesis. According to this theory, machines were ordered not to surpass a certain maximum number of Yes votes. If this was the case, there should be an unusually large number of repeated Yes totals in each precinct and the repeated number should also be the maximum Yes vote total in the precinct. We show that the frequency with which the repeated number is also the maximum Yes vote of the precinct is consistent with a random event-which means that it does not constitute evidence of fraud in our view. We then move on to study whether the variance of results at the precinct level is unusual. This would be the case if some but  not all machines were manipulated at the precinct level. We find the variance at the precinct level to be if anything smaller than would be expected by pure chance. Again, we do not find evidence of fraud in this dimension. In the end, the objective of this section is to take a balanced view to the discovery of fraud. The next section develops our test for fraud using our two independent but correlated measures of voters' intent. We then move on to test whether the sample used for the audit of August 18 was random. The final section concludes.

Exit Polls Versus Votes: Analysis of the Differences
The first evidence of potential irregularities in the election count derives from the exit polls conducted independently by Súmate and Primero Justicia (PJ). As shown in Table 1, according to the CNE, 41.1 percent of voters voted YES to impeach the president. On the other hand, in the Súmate and PJ surveys, the weighted projections were 62.0 and 61.6 percent, respectively, a difference of more than 20 points.
We check whether this difference is due to the fact that the sample chosen by Súmate and Primero Justicia was not representative of the electoral universe. In other words, we check whether the problem arises because of an over-representation of precincts in favor of the YES vote in relation to those in favor of the NO. We show that this is not the source of the problem. As shown in Table 1  in the sample chosen by both organizations, the result reported by them differs from the official tally by more than 17 percentage points. Hence, the difference in the results is not principally due to the sample composition but to a systematic difference across the sample where the exit polls were conducted.
To illustrate this problem more clearly, in Figure 1 we show the percentage of votes and the survey results for the 340 precincts surveyed by both groups. If the surveys were perfect, the points would align in a ray from the origin with a 45 degree slope (drawn in the graph). In other words, where the YES option received respectively 10 percent, 50 percent or 80 percent, the surveys would show the same result. If the points in the graph are above the 45 degree line, it means that the poll overestimates the result in that precinct. If the points are below, the poll underestimates it.
As can be seen, the bulk of the 342 precincts polled are above the 45 degree line. Moreover, the graph indicates that the differences between the votes and the surveys are very variable among precincts. The distances to the 45 degree line are largest in places where the YES option garnered between 20 and 40 percent.
This analysis has the following implications. First, it indicates that the difference between the surveys and the votes is not due, in any important way, to problems in the selection of the precincts to be included in the survey. Second, the analysis implies that the difference may be due to one of the two reasons, or to a combination of both. It may be due to a generalized failure in both surveys in each precinct, or to a quite generalized and nonlinear manipulation of the results. It will be a challenge of the statistical work to distinguish between these two hypotheses and investigate which is the right one.

The Caps or "Topes" Hypothesis
The fraud hypothesis most discussed in Venezuela has been based on the idea that numerical caps were imposed on the amount of YES votes that could be allowed in a precinct and that the overflow of YES votes would be switched into NO votes. In this section we evaluate this hypothesis.
To analyze the feasibility of this hypothesis we examine how many times the number of YES and NO votes are repeated at the precinct level in the CNE's database, which contains 19,062 automated machines (see Table 2).
The repetition of the YES count occurs with a frequency of 9.8 percent while that of the NO occurs with a frequency of 7.7 percent. We do not test whether this frequency is unusually high or low. 1 However, the relatively high frequency is at least in part due to the fact that the number of electors as well as the voting percentage tends to be very similar among machines in the same precinct. The fact that the repeated YES totals occur with a slightly higher frequency than the NO is at least in part due to the fact that YES has a lower percentage of votes. Let us illustrate this point with an example. Suppose the preference for the YES vote in a single precinct is approximately 40 percent and the number of voters at each machine is 100. A 5 percent variation would imply 2 votes, so the expected result in each machine could be between 38 and 42. The result could be in some of the five numbers in-1 Jonathan Taylor from Stanford University has argued that it is unusually high. See http://www-stat.stanford. edu/~jtaylo/venezuela/.  cluded in that interval. On the contrary, the same percent variation for the NO would yield a variation between 57 and 63 votes, which gives seven possible numbers. Since the amount of possible numbers is higher for the NO than for the Yes, it is logical the latter would repeat less frequently. More importantly, the cap hypothesis implies that the number that repeats itself is also the maximum from the precinct and that the difference is assigned to the NO. For this, it is necessary that the repeated number also be the maximum YES vote in the precinct. We study this hypothesis in Table 3. The table includes all precincts in which repeated numbers are observed and classifies them by the number of voting machines in the precinct. Column one shows the number of machines per precinct. Column 2 shows the number of repeated numbers that are not the maximum of the precinct, as required by the theory. The third column shows the number of repeats that are the maximum, while the final column adds the two.
If the repeated number was randomly distributed, it would occur with a frequency equal to 1/(Number of machines -1). For example, in the case of precincts with two machines, the repeated number is simultaneously the maximum and the minimum, for there is only one number. In the case of three machines, the probability that the repeated number is the maximum is 50 percent.
As we see in Table 3, 66 is not very far from being half of 124. In the case of five machines, 54 is not far from being one-fourth of 198. We conclude that if there was fraud, this was not done through the imposition of numerical caps to the YES votes in the machines of a precinct.

Variance Analysis of the Within-Precinct Results
The caps hypothesis, if true, would also affect the percentage difference in the results of the machines belonging to the same precinct. This is due to the fact that the amount of voters per machine varies due to differences in the abstention rate or in the number of electors assigned to each machine. This variation would show in the number of NO votes, and therefore would create a source of variation in the results across machines of the same precinct. This hypothesis and any other hypothesis that is based on the idea of altering some machine more than others at the precinct level can be tested.
In each precinct, voters are distributed to machines according to the last two digits in their identity card number (cédula de identidad). This allows each machine to be a random sample of the precinct's voters because the last digits in their identity card are not correlated with any variable relevant to the voting decision. This limits the possible distance between the results from two machines from the same precinct. To illustrate this, consider how opinion surveys are done in any country. A random sample is chosen-usually of a thousand or two thousand people-and the outcome is used to predict the results of millions of voters. In other words, a representative sample composed of a miniscule fraction of the electorate is used to predict the outcome of the whole. In the case of a precinct we are taking a much smaller and homogeneous universe than a country and we are dividing the population randomly according to the number of machines in the precinct. For example, in the case of a precinct with five machines, each machine represents approximately 20 percent of the total population of the precinct. In addition, in the case of this referendum, the options were limited to two: YES or NO. This imposes a condition for the standard deviation of the number of votes per machine. Suppose that in a machine, there were N votes cast and the probability that each vote is a YES is p. Then, the number of YES votes is a binomial random variable with expectation N p and with standard deviation equal to p(1 − p)N . To illustrate this, take the case in which p is the probability that an elector will vote YES in a given precinct, is equal to 50 percent and N is 400. In this case, the standard deviation would be 10 votes. The coefficient of variation (or the standard deviation of the percentage vote) would be 10 divided by 400, meaning, 2.5 percent. Given this, the typical deviation among machines in the same precinct must be compatible with this rule. If, for example, within a precinct the results of some machines were changed by 10 percent while the others were left unaltered, then we would see an increase in the deviation among all machines that would be four times the expected standard deviation of 2.5 percent. This would be abnormal.
One implication of this result is that the caps or "topes" theory would also violate the expected distribution of a binomial. If numerical caps were assigned to each machine in a precinct, the variation of the number of voters per machine would affect the number of NO votes and therefore alter the percentage results in a manner that would increase the dispersion of the results and cause these to violate the binomial rule.
To verify if the CNE vote data comply with the standard deviation predicted by probability theory, we calculate each machine's deviation with respect to the average of its precinct. Moreover, we divide this number by the standard deviation that would correspond to a precinct with the actual number of voters and machines. Figure 2 presents our results. It shows a histogram of the percent difference among machines of the same precinct with respect to the standard deviation expected by the binomial distribution. The curve reflects the expected theoretical distribution. The bars are the frequency calculated with the actual data. As can be seen, the coincidence is quite substantial. The graph indicates that only close to 1 percent of the machines have deviations higher than two times the expected standard deviation. This frequency is consistent with the theoretical distribution. In fact, if there is anything surprising about the graph, it is that the deviations of the results are if anything too small, as can be seen by the large concentration of results near zero variation.
This result has two possible interpretations. One is that there was no fraud. The other is that if fraud was committed, it must have been done by changing every machine in the precinct by a similar percentage. In fact, a fraud of this kind would not be detected with the analysis done so far for it would not alter the variance results among machines. Any hypothesis of fraud that does not comply with this condition would violate the restriction imposed on the deviation of the results by the binomial distribution.

A STATISTICAL STRATEGY TO DETECT THE PRESENCE OF FRAUD
To detect if the data are compatible with the presence of fraud we need to develop a model and fit it to the data. We define fraud as the difference between the voters' intent and what the electoral system registered about their decision. We will take as our null hypothesis the assumption that there was no fraud. We will then develop a test to see if the null hypothesis can be rejected.
The main challenge is that we cannot observe the voters' intent directly. The statistical strategy we adopted begins with finding two sets of independent variables that are correlated to the voters' intent, but not with the fraud. For our purposes, it is not too important that our variables do not predict the voters' intent perfectly. Even if they do so imperfectly, it may still give us a chance to reject the hypothesis of no fraud. Notice that the worse the quality of the data, the harder it will be to reject the null hypothesis, meaning that bad information makes it harder, not easier, to reject the hypothesis of no fraud.
To illustrate what we do, we start with a simplified presentation of our approach. In practice, things are a bit more complicated, but explaining the sources of complexity will be easier after the fundamental intuition is presented.
Let us take two variables that are correlated to the elector's intent: the number of signatures in favor of holding a recall referendum that were collected during the December 2003 (called Reafirmazo) drive and the proportion of YES responses in the exit polls. We use s i and e i to denote the number of signatures and the number of YES responses in the exit poll in the ith precinct, respectively. Each one of these variables is an imperfect measure of the voters' intent on August 15, 2004. The Reafirmazo was a public vote. Signatures were observed and identities known. The motivations to sign are different than voting YES in the referendum. For instance, some people that signed the petition may have changed their opinion in the intervening months. Others might have decided not to sign because it was not secret, but may have decided to vote later given its secrecy. Others may not have been registered in November and hence could not sign, but were registered by August and hence could vote. The lines in the August election were particularly long and slow and that may have reduced the number of voters, etc.
Equally, exit polls are an imperfect measure of the voter's intent. Pollsters may have, consciously or unconsciously, gathered a biased sample. People may have had more or less willingness to cooperate with the interview, etc. However, these errors are of a quite different nature from the errors generated by the relationship between signatures and votes and hence should not be correlated.
Suppose we have an imperfect measure of the voters' intent in each precinct and we build a graph relating this variable-say the signatures-and the voters' intent. As the signatures are an imperfect measure of the voters' intent, the graph will look like a cloud of dots around some basic relationship ( Figure 3). 2 Regression analysis can identify the line that relates the signature with the voters' intent.  The real relationship is 0.7, because that is how we built the data. The estimated relationship using the simulated data is 0.71 + / − 0.014, as is indicated by the graph.
We cannot observe the voters' actual intent but can only see the votes registered, and these, in theory, could be influenced by fraud. Suppose fraud takes place and it is directly proportional to the numbers of votes in that precinct. For example, let us suppose that fraud is committed by multiplying the total number of YES votes in a machine by 0.7 and the difference added to the NO votes. Figure 4 illustrates this case. In this case, the estimated slope is no longer 0.7 but 0.5. In addition, the pattern of errors-that is to say, the distance with respect to the regression line-looks similar. It reveals no evidence of fraud. If fraud were committed this way, we would be unable to detect it. In fact, a fraud that reduces a fixed percentage of YES votes across all machines would practically be impossible to detect by purely statistical methods without additional information; that is, it could only be detected using another source of information such as counting the paper ballots. Now, suppose the fraud was not committed in a proportional manner. For example, suppose it was committed in some precincts and not in others. Specifically, suppose fraud consists of eliminating 30 percent of the YES votes in precincts where signatures were less than 30 percent or more than 70 percent of the registered voters. In this case, the pattern of errors will have a peculiar shape, as shown in Figure 5. This peculiarity is not due to the imperfect nature of the number of signatures as predictor of votes, but is caused by the presence of fraud.
What happens if we now use a second measure of the voters' intended vote, for example the exit polls? This is also an imperfect measure of the voters' intended vote and as such when doing a regression analysis, this will generate errors. Nevertheless, if there is a nonproportional fraud, this will also generate an irregularity in the errors which will look similar, that is, will be correlated with the errors in the other relationship. A positive and significant correlation would identify nonproportional fraud.
Note that each measure-the one based on signatures and the one based on exit polls-is imperfect. Nevertheless, what makes each of them imperfect are factors different and independent from each other. The exit poll is not influenced by the turnout rate, as people are interviewed after they vote. The signatures do not depend on the ability or bias of the interviewer. People could have changed their minds between November and August, but people do not change their minds for the same reason between the act of voting and the exit interview. Signing is a public act and voting is secret, etc. Therefore, errors made by each measure may be larger or smaller but they should not be correlated. However, if there is nonproportional fraud, it will influence each of these measures in the same way. Hence, the errors made by both should be positively correlated. This is the essence of the method we used.

Theoretical Considerations
In this section we derive formally the technique we use. In particular, we show that for a variety of increasingly complex assumptions about the nature of the fraud the covariance between the errors of the instrumental variables regression is an appropriate test of the absence of fraud. 3 Assume that the fraud is defined as the difference between the votes for Si actually collected and an unobservable variable that is the intention of voting of the voters that showed up. We define the first one as ν i , the intention of voters as χ i , and the fraud as φ i : where i indexes precinct. As discussed earlier, two additional measures of the intention of voters are the number of YES responses in the exit poll in the precinct (e i ) and the number of signatures collected in the precinct (s i ). These measures, however, are imperfect but can be modeled as where we are assuming that the exit polls are possibly a biased estimate of the intention to vote: α can be smaller than 1. The signatures (s i 's), as well, could be a biased measure. Both equations have an error (ε i and η i ) that takes into account the fact that both the exit polls and the signatures are very imperfect measures of the voter's intentions-even the biased measured intentions. We assume that these errors are uncorrelated among themselves and with the fraud. 4 How can we detect fraud? Fraud can only affect the actual votes, not the exit polls, nor the signatures. In other words, fraud is a displacement of the distribution of votes that is not present in the other two measures. Statistically, this means that fraud could be detected by using the exit polls and the signatures as predictors of the voting process and analyzing the correlation structure of the residuals. Under the assumption that all residuals are uncorrelated-which makes sense given the definitions we have adopted-then the correlation of residuals is an indication of the magnitude of fraud.
The particular procedure used to detect fraud is the following: (1) Estimate the regression of ν i on e i plus controls and recover the residual. This residual has two components: the fraud and the errors in variables residual due to the fact that the exit polls are noisy.
(2) Estimate the regression of ν i on s i plus controls and recover the residual. This residual has two components: the fraud and the errors in variables residual due to the fact that the signatures are an imperfect measure of the intention of voters.
Notice that these two residuals are correlated: first, because both have fraud as an unobservable component, and second, because the right-side variables are correlated and there are errors in variables in the regression.
(3) Estimate the regression of ν i on e i plus controls using s i as an instrument. Recover the residual. Notice that in our model, because η i is uncorrelated with ε i and φ i , we can use s i as an instrument to correct for the error in variables.
(4) Using the same logic, estimate ν i on s i plus controls, and using e i as the instrument. Recover the residual. In this case, because the two coefficients are supposed to have solved the problem of error in variables the residuals can only be correlated if there is a common component-which in our case is the definition of fraud.
This procedure actually detects how important fraud is. The next section first explains why this procedure indeed is able to identify fraud. After that we also analyze the possibility that fraud is corre-lated with signatures-which is likely given what we have argued about the stochastic properties of the votes per machine and precinct. Finally, we present evidence.

OLS Estimation (No Correlation Between
Fraud and Intention to Vote) Solving for χ i in (2), substituting in (1) we find where For this model, the estimate of the slope coefficient, θ, by OLS is given by which is always smaller than 1/α-the true coefficient. This means that the residual from the regression (ζ 1,i ) is We can proceed in the same manner but now considering s i as opposed to e i . We solve for the intention of voting in equation (3). The slope coefficient is which is always smaller than 1 β . The residual is given by Notice that the two residuals are correlated. Under the assumption that ε i and η i are uncorrelated, and also uncorrelated with fraud, there are two components that create the correlation among these residuals: fraud, and the errors-in-variable bias: cov(ζ 1,i , ζ 2,i ) = var(φ i ) The first term is the variance coming from fraud, while the second term comes from the variance due to the error in variables that is present in both e i and s i . Notice that we are assuming that the errors in variables are independent. The covariance arises because the error in variables downward biases both coefficients (θ ols < 1 α and π ols < 1 β ) and because the exit polls and the signatures are correlated through the voter's intention.

Instrumental Variables
Under our assumptions, we have an easy solution to the error in variables in both regressions. Notice that η i and ε i are uncorrelated and that η i is uncorrelated with φ i . Additionally, e i and s i are correlated because both measure the same factor (χ i ). This means that s i can be used for instrumenting e i and e i for instrumenting s i . The outcome is as follows: The IV estimate is which means that the residual is Notice that now the errors-in-variables component has disappeared. Similarly, if we run the regression for votes on signatures and using the exit polls as instrument, we find: The IV estimate is which means that the residual is

11
The correlation between the residuals of the two IV regression is now cov(ζ 1,i , ζ 2,i ) = var(φ i ).
So, a simple test is to compare these two covariances, and determine if they are statistically different from zero. Furthermore, if the covariance of the IV residuals is different from zero, then we have an estimate of the importance of the fraud. 5

Estimation When There is Correlation Between Fraud and Intention to Vote
The previous exercise has assumed that fraud is uncorrelated with the signatures as a measure of the intent to vote, but as we have argued in the previous section, this is unlikely. In fact, most probably, fraud is correlated with the signatures because the government used that information in the design of the fraud. Let us repeat the previous exercise allowing for any covariance structure between fraud and the signatures. We assume that the number of votes in each precinct is ν i = χ i + φ i + ρs i , where ρ is the coefficient capturing the correlation between the fraud and the signatures.
The residual from running OLS of votes on exit polls is while the residual of running votes on signatures is Equations (15) and (16) are equivalent to equations (7) and (9). Notice that the two residuals are correlated as before, but now there are two additional terms: That arises from the correlation between fraud and the signatures.
On the other hand, the residual of the IV regression when votes are projected on the exit poll and 5 This procedure is in the spirit of the recent literature on identification through heteroskedasticity (Rigobon [6]). The classical reference on identification is Fisher [3].
signatures are used as instruments is while the residual of projecting votes on signatures and using the exit polls as instruments is One point worth noticing is that when exit polls are used as instruments, the results are identical to those in the previous subsection. In other words, whether fraud is correlated or not with the signatures, makes no difference in the validity and quality of exit polls as instruments. The fraud is not correlated with the exit polls or their innovations. Signatures, on the other hand, are correlated with fraud. This makes exit polls a good instrument for signatures, but signatures are not a good instrument for exit polls.
Let us compare the two covariances: the covariance for the OLS residuals with the covariance for the IV residuals. The OLS residuals have covariance equal to while the covariance for the IV estimates is First, notice that as before, if there is no fraud the covariance of the IV residuals should be zero. Furthermore, the covariance in equation (2) reflects the different forms of fraud. The first element (var(φ i )) is the component when fraud is random, while the last two terms capture fraud when it is correlated with the signatures.
Second, the difference between the two covariances is where the two terms are easily signed. We know that the error in variables, together with a negative ρ, implies that both OLS estimates are downward biased. We also know that a reasonable set of assumptions imply that signatures and exit polls are positively correlated. Hence, the first term is a multiplication of three positive elements. Additionally, we know that θ iv is closer to 1 α than θ ols . This means that the term in brackets is negative, and we have been analyzing only the case in which ρ is negative. Hence, the covariance of the OLS residuals has to be larger than the covariance of the IV residuals. Notice that if ρ were positive we could not have made this claim. And there would be circumstances in which the covariance actually goes up after instrumenting.

Results
In this section we estimate the covariance of the residuals from the OLS and instrumental variables. Our results are summarized as follows: first, we observe a reduction in the covariance, as would be consistent with the presence of ρ being negative (i.e., against the YES vote). Second, we find that cov IV is positive and statistically significant, which is informative of the fact that ρ is significantly different from zero and there is fraud.
It is important to remember that in this procedure we are allowing the exit polls and the signatures to be imperfect measures of the actual votes. Not only do we allow them to be noisy, but we also allow them to be biased. So, our results will NOT depend on the fact that the mean of the exit polls is different from the mean of the votes.
For the estimation, we included other explanatory variables in our analysis. These are: (i) the number of new voters in each precinct from the time the signatures were collected; and (ii) the turnout rate in each precinct.
The new voters were unable to take part in the Reafirmazo as they had not been previously registered to vote. The more new voters there are, the greater the number of votes there should be. Now then, the percentage of YES votes could increase or diminish according to the difference in political preferences of the new voters with respect to those registered previously. As in the previous case, the turnout rate obviously increases the number of votes and is able to do so in a differentiated manner between the YES and NO options.

The estimated equation is (all variables in logs)
: where ν i is the logarithm of the number of YES votes; s i is the logarithm of the number of signatures in each precinct; newvote i is the percentage of new voters; turnout i is the percentage of voters participating; and c 0 , c 1 , c 2 and c 3 are parameters to be estimated. Table 4 shows the results of our estimates for the 342 (voting) precincts for which we also have exit polls, using the most conventional method: the squared minimums. The estimate allows us to explain 97 percent of the variation in votes among (voting) precincts. It estimates parameters c 0 , c 1 , c 2 and c 3 with great precision. Specifically, c 0 is the constant, estimated at 0.306. Parameter c 1 is the elasticity between signatures and votes and is estimated at almost 1 (in reality it is 0.994). This implies that if a precinct has twice as many signatures as another, it obtains on average twice as many votes. Parameter c 2 is the elasticity of the YES votes to variations in the percentage of new voters. It is estimated at 0.46, which means that if the number of voters in a precinct increases by 100 percent, the YES votes would increase by 46 percent. Parameter c 3 is the elasticity of the number of YES votes compared to a change in the voters participating and is estimated at 0.306, which indicates that a 10 percent increase in the rate of voters participating would cause a 3.06 percent increase in the number of YES votes.
This equation does not indicate the actual ratio between the voters' intended vote and its explanatory variables, but between the latter and the votes published. As in Figure 4, the possible presence of fraud influences the estimated coefficients, biasing the slopes downward, and in part is found in the error term.
The second equation we estimated was the ratio between votes and exit polls also for the 342 precincts for which we have data. The equation we estimated is similar to equation (22) but now using the exit polls (e i ) as the proxy for voters' intentions: where e i is the number of YES votes which the poll for this precinct predicts given the number of voters that actually showed up to vote and the percentage of YES votes documented in the poll. The results appear in Table 5. Again, the equation explains a large part of the variance of the logarithm of votes (82%). The esti-mated elasticity of the voting intentions according to the polls is 0.97. These estimates could also be biased downward by the presence of fraud, but we are mostly interested in understanding what the implications are regarding the correlation of the residuals.  In Table 6 we present the covariances, their differences and their significance. The correlation is 24%, which is surprisingly high. This does not permit us to reject the fraud hypothesis. In other words, in places where the signatures are proportionally wrong in the sense of predicting more YES votes than those obtained, the exit polls also overestimate the YES votes. Since both measurements are independent, the implication is that what they have in common is fraud. This is the first result consistent with the fraud hypothesis. Formally, we can say that we cannot reject the hypothesis that fraud was committed. The presence of this correlation indicates that there is something in common between the errors committed by the exit poll and the errors committed by the signatures and this is consistent with a difference between the elector's voting intent and the official tally.
However, it is possible to argue that the observed correlation might be generated by two sources. One is the fact that our measurements of the voter's intent are very noisy or imperfect and that the errors in such variables might generate problems. The second is that we suppose fixed coefficients between signatures and votes or between exit polls and votes, and that these coefficients might be random. This opens the possibility that the correlation we are finding may have been generated by other factors and not by fraud.
To discard this possibility we estimate using the IV strategy. Table 7 shows the same equation as Table 4, but this time it uses the instrumental variables method where the exit polls is the instrument.
Note that the coefficient of the signatures now increases: from 0.994 in the estimate in Table 4 to 1,013 in Table 7. This is normal, as the existence of errors or noise in the data tends to lower the coefficients estimated with the method of Table 4. On cleaning or lowering the problem of errors in the data, higher coefficients are usually obtained.   Table 8 re-estimates the same equation as in Table 5 but using instrumental variables. This time, the coefficient of the exit poll (e i ) increases from 0.97 to 1.17. This is to be expected as the data of the exit polls, given their nature, are noisier than the signature data, which is why the method in Table 5 skews the coefficient more than in the case of the signatures.
On studying the relationship between the errors in these two equations, we obtain the data presented in Table 9. The analysis shows that even after using the method of instrumental variables to correct for problems of errors in variables, the correlation between errors generated using signatures and those generated using exit polls diminishes only from 0.24 to 0.17 and remains significantly different from zero. Table 9 summarizes our two main results: First, the correlation of the residuals in the OLS and in the IV strategies is statistically different from zero. Second, the OLS covariance and correlation are larger than in the IV. This means that we reject the hypothesis that there was no fraud, and we reject the hypothesis that fraud was random.
Our strategy has consisted in utilizing two sources of information related to the voters' intended vote but not to the possible fraud. If we use these sources or variables in estimating the votes imperfectly, then the residual will contain not only the imperfections of our sources but also a component associated with fraud. Our interpretation is that since the imperfections are independent from each other and the residuals are correlated, there must be a common factor tying them together, that is, fraud.

The Audit
Any hypothesis of fraud requires an explanation of why the audits that took place did not find any foul play. While the first audit carried out in the wee hours of the morning of August 16 failed, the audit conducted on August 18, if it was well carried out, should have settled the issue. The audit was based on opening 150 randomly selected ballot boxes, which contain the original paper ballots checked by the voters and which thus reflect their real intended vote. If these boxes were not tampered with and if they really are a random sampling of the universe of precincts, the audit should rule out any presumption of fraud. So, how could fraud have taken place, if the audit did not find it? It should be pointed out that any hypothesis of fraud which involves changing hundreds of ballot boxes would constitute a conspiracy involving a large number of participants and hence would be more likely to be revealed through disloyalty. One hypothesis is that fraud was not committed in all precincts but only in a fraction of them. To give an example, suppose that out of the 4,580 automated precincts used in the election, 3,000 precincts were altered but the rest were not. Let us further suppose that the unaltered 1,580 precincts were picked at random. This implies that they would represent a balanced sample of the country from a regional and social point of view. The same would be true of the 3,000 precincts in which the results supposedly had been altered. One reason to do things this way is that it was known (beforehand) that expost audits would be carried out and that a number of precincts would be checked. To accommodate this, they would have to be unaffected by fraud.
Note that if fraud is committed in some precincts and not in others, then it will not be perfectly proportional and the method used in the previous section would detect it. If the selection of the precincts left unaffected was done in this way, this creates an important complication but also opens up a great opportunity. The complication is that the selection of the boxes to be audited could not really be random ex-post. It is critical that the selection be made among the 1,580 untampered precincts and not among the 3,000 tampered ones. This is only possible if one has control over the random number generator that selects the boxes to be audited. In this sense, it has to be pointed out that the National Electoral Council refused to make use of the random numbers-generating program proposed by the Carter Center and insisted on the use of their own program installed in their own computer-which was the one counting the votes and connecting to the electronic machines.
The opportunity generated by this form of addressing the audit problem is that any sample taken of the 1,580 untampered precincts is a representative sample of the country in the social and regional sense. This makes it more difficult to know if the sample taken was really random, as it resembles the country in all the dimensions usually associated with representativeness, such as regional or social.
To solve this problem we must develop a methodology that allows us to test if the sample taken for the audit on August 18th really is a random sample. To understand the problem more clearly, let us call the tampered precincts "fat" and the untampered ones "thin." The sample taken for the audit must be a sample of only "thin" precincts, while the rest of the precincts are a mixture of "fat" and "thin." If we could "weigh" the audited precincts, we would be able to see that on average the un-audited precincts are "fatter." The problem is that we need to develop a methodology that can test whether the audited precincts weigh as much as the others do or whether on the contrary they have a statistically different body frame.
The method we suggest is as follows. There exists a theorem in statistics that states that if a ratio applies to an entire unit, any random sampling of the same must have the exhibit ratio. If we estimate the ratio of the universe of un-audited precincts and estimate another for the audited ones, the second cannot be statistically different from the first. Otherwise, it would not be a random and representative sample.
To implement this strategy we again made use of our model that correlates signatures, voter participation rates and new voters with the number of actual votes cast. In this case, because we are not going to use the exit polls in the analysis we can use the data from all precincts. We estimated this ratio based on the universe of 4,580 precincts and we looked at the obtained coefficients. We then estimated them separately between the audited precincts and the un-audited ones and determined if the coefficients are statistically different.
To see if the results are different and to calculate the statistical significance of the difference, it is useful to estimate the equations in the following way: The results are very clear, as is indicated in Table 10. The interaction term dum i * Signatures shows that the elasticity of the signatures to votes is 10.5 percent higher in the audited precincts than in the un-audited ones, that is, the signatures collected in the audited precincts on August 18th generate 10.5 percent more YES votes than the rest of the precincts. The statistical value of Student's t is 2.73. The probability that this is by chance is less than 1 percent (shown with the three asterisks in the table). The coefficient on new voters is also different with a level of confidence of 1 percent whereas the coefficient with regard to the abstaining voters is different with a level of confidence of 10 percent.
To illustrate what is unusual with this result we constructed 1,000 random samples of 200 precincts based on the universe of un-audited precincts. We estimated the same equation and calculated the statistical value of Student's t for the term dum i * Signatures. The result is shown in Table 11. As the table shows, a value of said statistic higher than 2.48 occurs less than 1 percent of the time. In the sample of the audit on August 18, this value is 2.73.
We conclude that the data indicate that the audited precincts are statistically different from the un-audited precincts. This implies that they do not form a random sample of the entire universe of precincts (audited and un-audited). In the audited pre- cincts, the signatures are transformed into a larger number of votes than in all of the precincts (audited and un-audited) taken together. The probability that this occurs by coincidence is less than 1 percent. This result tends to confirm the doubts expressed as regards the reliability of the audit.

Intuition
In this section, we would like to illustrate both our theory of fraud, as well as how we test for it.
Assume that in Florida half the precincts are Republican and half Democratic. How do we know this? Well, first we have the results of the previous presidential election in each precinct, which should be a good predictor of today's preferences, and we also know how many Republicans and Democrats are registered in each precinct. Obviously, these measures are not perfect, and they are possibly biased, but they should be related. Also assume that on election day there are exit polls. Assume that these polls are extremely noisy and biased.
Assume that fraud is going to be committed-in favor of the Republicans (just an example). How can we have a perfect fraud? In the absence of an audit, electronic fraud is simple-when the machines con- nect to the central computer, the central computer sends a program that makes the machine to report 10 percent fewer Democratic votes, and 10 percent more Republican votes. This does not change the total number of voters but changes the proportion. This is undetectable, statistically speaking. The exit poll and the party registration data will show that there is a change in votes in favor of the Republicans. The exit polls will give a different answer, but in the end, because the exit polls are so noisy, the blame will be given to the imperfection in the collection of the polls rather than use them as evidence of fraud.
The only deterrent to fraud in this case is to have an audit, and the question is how we can achieve a kind of fraud that survives the audit. Assume that the machines leave a paper trail of each vote and some of the machines will be audited.
Here is a fraud strategy that would be undetectable using standard statistical methods. Assume that the election is really evenly divided between Republican and Democratic precincts but that half of the precincts are tampered with. Let us say that 10 percent of the votes will be shifted. The tampered result of the election will now show that 3 4 of the cen-ters will have voted Republican and only 1 4 would have gone for the Democrats. Now, to pass the audit, the machines that will be checked cannot belong to the set of tampered machines. This is why it is important to be able to control the choice of the machines to be audited. Therefore, to make the fraud pass the audit the authority draws random numbers that have 3 4 weights in the Republican precincts and 1 4 in the Democratic ones. Assume this is done in the morning of the election, so the authority knows in advance which precincts to leave unaffected by fraud. In the end, the audit is passed, 3 4 are Republican and 1 4 are Democrat. This simple procedure-which only requires observing the results of previous elections, or in the Venezuelan case, to observe the number of signatures and compare them to the universe of voterswill hide fraud from an audit if the precincts are not chosen in a truly random fashion. Here, the exit polls would give a different result, but again, most of the discrepancy would be blamed on the exit polls.
Two properties worth emphasizing are satisfied by these data. First, the mean of the audited sample and that of the whole sample would be similar. Second, the correlation between votes and the prior information is the same in the two samples. This, at a first glance, could look as if this is evidence of no fraud, but that is incorrect. Remember that the correlation between two variables is unaffected if one of the variables is multiplied by a positive number. Hence, the correlation between the signatures and the votes in the Venezuelan case is exactly the same as the correlation between the signatures and 90 percent of the votes. So, a fraud of, say, 10 percent would not affect the correlation between signatures and fraud. It would, however, affect the coefficient, which is what we check for.
Therefore, how can we detect fraud? In the audited sample, the information that existed beforethe estimates of voter preference-is a better predictor of the actual votes than in the non-audited sample. For example, in the audited sample, if the precinct was Democratic in the past, it has a high likelihood of being Democratic today. But in the non-audited sample, the problem is that this relationship is weaker. In other words, we can detect if the conditional behavior between the two samples is different, and therefore, argue that something strange is happening in the data. This is the second test we run.
The first test is one in which we compare the predicted error of the votes using the two different measures of the preferences of voting. For example, the reasons why the exit poll is an imperfect measure of voter preference are different from the reasons that make the previous election a bad measure. For instance, one is affected by the turnout rate, while the other one is not; one took place several months before the other one; one is collected by the electoral committee and the other is collected by the private sector-which could possibly have a vested interest in a particular outcome, etc. The important aspect of our test is that the reasons why one of the measures is imperfect are different from the reasons why the other one also is imperfect. On the other hand, if there is fraud, then both measures have a common reason why they fail. This is our first test.

CARTER CENTER CRITIQUE
The Carter Center issued a report entitled "Report on an Analysis of the Representativeness of the Second Audit Sample, and the Correlation between Petition Signers and the Yes Vote in the Aug. 15, 2004 Presidential Recall Referendum in Venezuela" which is a response to our paper "In search of the black swan: Analysis of the Statistical Evidence of Electoral Fraud in Venezuela." In preparing their response, the Carter Center never contacted us to ask questions about our methodology or asked to see our data in order to reproduce our results, although we offered to do so.
One important aspect of our paper is that we studied whether the sample used by the Carter Center for the purpose of the audit was a random sample of the whole universe of automated voting precincts.
We also present what we believe to be evidence of fraud, but the Carter Center report does not deal with this aspect of our report. They are mainly concerned with the randomness of their subsample.
How does the Carter Center answer our claim? They make three propositions: (1) They check whether the means of the votes in the two samples are similar.
(2) They check whether the correlation between signatures and votes is similar in the audited and in the non-audited precincts.
(3) They test the random number generator program used by the Electoral Council and find that it does generate a random draw of all the precincts. They also correctly point out that the numbers are not truly random in the sense that the same initial seed number generates the same sequence of numbers.

Similar Sample Means
With respect to the first point, the question that the Carter Center asks is whether the unconditional means of the two samples are similar. By unconditional we mean that they do not control for the fact that precincts are different in the four dimensions we include in our equation or in any other dimension.
To see the importance of conditioning, let us imagine that there is fraud and let us suppose that the fraud is carried out in a large number of precincts but not in all of them. The question is: is it possible to choose an audit sample of non-tampered centers that has the same mean as the universe of tampered and un-tampered precincts? The answer is obviously yes. Let us give an example using a population with a varying level of income, say from US$ 4,000 per year to several million. Assume that half of them have been taxed 20 percent of their income while the other half have not. Is it possible to construct an audit sample of non-taxed individuals whose average income is similar to that of those that have been taxed? Obviously the answer is yes. However, if one controls for the level of education, the years of work experience and the positions they hold in the companies they work in, it should be possible to find that the audited individuals actually had a higher net income than the non-audited group. That is the essence of what we do. Now, let us go back to the case in point. Precincts vary from those where the YES got more than 90 percent of the vote to those where it got less than 10 percent. This is a very large variation relative to the potential size of the fraud, say 10 or 20 percent. It is perfectly feasible to choose a sample that has the same mean as the rest of the universe.
However, the non-random nature of the sample would be revealed if we compare the means but control for the fact that each precinct is different. That is what we do and this is the randomness test that the audited sample failed.

Similar Correlation Coefficients
The second check consists of comparing the correlation between signatures and votes in the two samples, which they find to be very similar. But a simple correlation is not a test of causality or strength.
To see this, suppose that in the audited sample there is a perfect relationship in which each signa-ture becomes 2 votes, and in the non-audited sample, because of fraud, the relationship implies that each signature becomes only 1 vote. The correlation coefficient in both samples is 1. This is due to the fact that the correlation coefficient is affected by whether the two variables move up and down together, but not by whether they do so in a relationship of 1-to-1, 2-to-1 or 10-to-1. This procedure is certainly no proof of randomness or of the absence of fraud.

Test of the Sample Number Generator
The final point is that the random number generator actually generates a sample that can potentially pick all the universe of precincts and that it was tested and appeared to actually generate random numbers. However, there are many ways in which this kind of analysis is weak. The most obvious one is that the program does not really generate random numbers but a predetermined set of numbers for each seed-number that initiates the sequence. By putting a known seed-number the Electoral Council would know beforehand which precincts would come up, and could thus decide which precincts to leave unaltered. It is our understanding that in the audit conducted on August 18-20, the seed number was provided by the Electoral Council and implemented in their computer. It does not matter if, as reported by the Carter Center, after 1,000 draws, the likelihood of any precinct being chosen looks reasonably random. The point is that the first draw is completely predetermined by the seed number.

Response
The Carter Center report does not address the two main findings of our report. It completely disregards the evidence we put forth regarding the statistical evidence for the existence of fraud in the statistical record. It only addresses the issues we raise regarding the randomness of the sample used for the audit they observed on August 18-20, 2004. They show that the unconditional means between the audited sample and the rest of the universe are similar. However, this is no proof of randomness. Conditional on the characteristics of the precincts, we show them to be different and this result is not challenged or addressed by the report. The report also argues that the correlation coefficient between signatures and votes in the audited sample is similar to that in the rest of the precincts, but this is an irrelevant statistic for this discussion. Finally, the report checks the source code of the software used but leaves open wide avenues for fraudulent behavior. 6

CONCLUSIONS
This report rejects certain hypotheses about fraud in the Venezuelan referendum of August 15, 2004, but not others. We did not find empirical validity for the much-discussed hypothesis of numerical caps. We were also unable to prove any hypothesis that implies differentially tampering with the (voting) machines of the same precinct. A manipulation of this kind would alter the percentile differences in such a way as to violate the expected variance at the precinct, and would have been detected by this analysis. All hypotheses of fraud must presuppose a similar tampering in all the machines of a precinct. If this had been done in a homogeneous manner in all of the country's precincts, none of the methods applied in this study-as well as other statistical methods-would have been able to identify it. What allows us to develop a test of the possible existence of fraud is precisely the heterogeneous treatment of the different precincts. To carry out this test we used two imperfect, random and independent indicators of the intent to vote. Our definition of fraud consists of the existence of a difference between the voters' intended vote and the votes registered by the CNE. Our two indicators, as imperfect as they might be, are correlated with the intended vote of the elector, but not with the tampering. If both are used independently in regressions to estimate the relationship between them and the vote count, the error term or deviation will reflect not only the imperfection of the instrument applied but also the fraud. If both deviations are correlated, it shows there is a common element of deviation in both. This element is our evidence of fraud. Furthermore, to be consistent with the hypothesis of fraud, this correlation has to be positive; that is, in those precincts where fraud was larger, both measures would project more votes than were actually registered. This is precisely what we find. Our two indicators are the number of registered voters in each precinct who signed in the Reafirmazo of November 2003, and the exit polls held by Súmate and Primero Justicia on August 15th, day of the Recall Referendum. The result holds if we control for the changes in the electoral register and the abstention rate. Furthermore, the result holds up well to changes in the functional form of the ratio (linear, logarithmic, percentile). The result is not due to spurious statistical effects (errors in the variables or the possible presence of random coefficients), as they hold up when we correct for these factors using estimators based on instrumental variables.
We note that this technique identifies fraud insofar as it is carried differentially across precincts. It allows us to test for the presence of fraud, but it does not allow us to estimate its magnitude as the average fraud will be reflected in the parameters of the relationships we estimate while the differential fraud will be reflected in the error terms. We use the error terms in the identification of fraud, not the estimated parameters. Again, any hypothesis of fraud must presuppose that the results of all the machines in the same precinct were tampered with proportionally. This requires some coordination mechanism. In theory, this coordination could be in the software or in the communication with the central computer hub. For these reasons, it is useful to point out the following precedents: • The machines had the capacity to communicate bidirectionally with the central computer server or hub and this communication took place. • The machines communicated with the hub before printing the Certificates, which opens the possibility that they were instructed to print results different from the real ones. • The entrance of witnesses from the opposition or of the international observers to the computer hub during election day was not allowed.
The voting system implanted in Venezuela generates voting ballots that are checked by the voter and placed in boxes, which are subject to audit in a random manner. A fraud scheme must take into account how to avoid detection during an audit. One possibility is to leave some precincts unaffected and to direct the audit to those precincts. The choice of which precincts to affect can be done systematically or at random. This generates two kinds of precincts: those that were tampered with and those that were not. Now, if the program that selects the boxes to be opened in an audit process can be controlled, then it will be possible to select the boxes of those precincts that were not tampered with and this sample might seem random in all aspects except as to the question of fraud.
Our analysis shows that the sample selected to carry out the audit on August 18, 2004 was not random nor representative of all the precincts. In this sample, the elasticity of the signatures compared to the votes is 10 percent higher and the possibility that this is random is significantly less than 1 percent. We repeated our analysis randomly selecting 1,000 samples from un-audited precincts and this result does not hold.
One important fact is that the CNE refused to use the random number-generating program offered by the Carter Center for the August 18th audit and instead used its own program installed in its own computer and initialed with their own seed.
In conclusion, this study rejects certain hypotheses of fraud, but indicates others that are compatible with the statistical data.
In statistics, it is impossible to confirm a hypothesis, but it is possible to reject it. As Karl Popper said when observing 1,000 white swans: this does not prove the accuracy of the thesis that all swans are white. Nevertheless, observing a black swan does allow one to reject it.
Paraphrasing Popper, our white swan represents no fraud. The results we obtain make up a black swan. The alternate hypothesis that there was fraud is consistent with our results, which is why we are unable to reject it.

ACKNOWLEDGMENTS
This study was requested by Súmate who also provided the databases we used. We appreciate the great information gathering effort carried out by this organization. We are equally indebted to a hardworking collaborator who, because of institutional reasons, must remain anonymous. We thank Andrés Velasco as well for his useful comments. The opinions expressed in this report and the errors we may have incurred are our responsibility and do not compromise either Súmate, or the universities to which we belong.