OPINION
Petitioners Nabil and Sawsan Hassan appeal a Board of Immigration Appeals (“Board” or “BIA”) order affirming an immigration judge’s finding that Petitioners were removable under 8 U.S.C. §§ 1227(a)(1)(A) and 1227(a)(3)(D). Petitioners also appeal the Board’s denial of their motion to remand the record so that they could apply for a waiver of admissibility. Finally, Petitioners appeal the Board’s ruling that the immigration judge’s failure to recuse herself did not amount to a due process violation. For the reasons set forth below, we AFFIRM in part and REVERSE in part the judgment of the Board of Immigration Appeals.
FACTUAL/PROCEDURAL BACKGROUND
Petitioner Nabil Hassan is a 48-year-old Muslim man who was born and raised in Jerusalem, Israel and identifies himself as a Palestinian. Petitioner Sawsan Hassan (née Wadi) is a 40-year-old Muslim woman who was also born and raised in Jerusalem and identifies herself as Palestinian. It is undisputed that Nabil and Sawsan are presently married and have four children, all of whom were born in the United States. What is primarily at issue in this *918case is when the Hassans’ marriage took place.
Nabil Hassan was admitted to the United States on March 25, 1995 at New York City on an F-24 Immigrant Visa, which is reserved for unmarried children of lawful permanent residents (“LPRs”). Hassan qualified for this visa because his mother was living in the United States and had LPR status. Sawsan Hassan entered the United States on the same day as Nabil and was admitted to the country on a Nonimmigrant Tourist Visa. On April 10, 1995, Nabil and Sawsan had a small wedding ceremony at a mosque in Michigan and signed documents to certify their marriage. On May 4, 1995, Nabil filed an I-130 petition on behalf of Sawsan, requesting that her immigration status be adjusted to LPR on the basis of their marriage. On August 11, 1995, the government granted that request.
On December 29, 1999, Nabil filed an application for naturalization. Daniel Wells, then a district adjudications officer for the Immigration and Naturalization Service,1 was assigned to investigate and adjudicate Nabil’s application. On July 27, 2000, Officer Wells conducted an in-person interview of Nabil Hassan as part of his investigation. Based on Nabil’s answers to certain questions during the interview, Officer Wells became suspicious that Nabil and Sawsan had in fact married sometime before their 1995 entry in the United States. Based on the interview and the results of an investigation of the Hassans’ marital status in Jerusalem prior to their entry into the United States, Nabil’s naturalization application was denied.
On May 23, 2002, the government served Nabil Hassan with a Notice to Appear (“NTA”), alleging that: 1) he had married Sawsan before entering the United States; 21 the marriage automatically revoked his visa under 8 C.F.R. § 205.1(a)(3)(i)(I); and 3) he was removable under 8 U.S.C. § 1227(a)(1)(A) because he was actually an inadmissible alien at the time of his entry into the country. Because Sawsan’s immigration status was based on Nabil’s status, the government also issued an NTA to Sawsan alleging that she too was removable for lacking a valid immigration visa. The government later added two other charges of deportability to Nabil’s NTA, including that he was removable under 8 U.S.C. § 1227(a)(3)(D) as an alien who falsely represented himself as a U.S. citizen for any purpose or benefit under the Immigration and Nationality Act (“INA”) or any other federal or state law. This additional charge was based on an allegation that on March 27, 2001 and May 16, 2001, Nabil falsely represented himself as a U.S. citizen on a Small Business Administration loan application form. Petitioners denied the pertinent allegations, including the claim that they had married prior to their entry in the United States.
I. Merits Hearing and Testimony
At a November 29, 2005 merits hearing, Immigration Judge Marsha K. Nettles heard the testimony of seven witnesses: 1) Officer Daniel Wells; 2) Imam Mohammed Mardini; 3) Ismail, Ayoub, and Ibraham Hassan — Nabil’s three brothers; and 4) Petitioners Nabil and Sawsan Hassan. The testimony of these witnesses is summarized below.
A. Testimony of Officer Daniel Wells
Daniel Wells testified that he was the immigration officer in charge of investigating and adjudicating Nabil Hassan’s application for naturalization. During his initial investigation, he had noted that the Has-*919sans’ first child must have been conceived prior to their April 10, 1995 wedding in Michigan.2 According to Wells, this was a very uncommon occurrence among applicants from the Middle East. Suspicious that the Hassans had actually been married at an earlier date, Officer Wells questioned Nabil about these suspicions at Nabil’s naturalization interview on July 27, 2000. According to Wells, Nabil admitted that he and Sawsan had in fact been married before they entered the United States. Wells then typed up a statement for Nabil to sign that included this admission. Nabil refused to sign the statement, however, asserting that it was untrue. Officer Wells then typed up a second statement that did not include the admission, and which Nabil signed. Officers Wells, however, later made his own notation on this statement that Nabil’s “story changed.”
Officer Wells testified that he then asked the U.S. Embassy in Israel to conduct an investigation into whether Nabil and Sawsan had been married in Israel. According to Wells, an officer at the Embassy eventually provided the results of its investigation in a letter that it sent to him via facsimile; that letter indicated that the Israeli Ministry of the Interior had an official record of Petitioners’ marriage that predated their entry into the United States.3 Based on this confirmation, the age of the Hassans’ first child, and Nabil’s own statements in the naturalization interview, Officer Wells denied Nabil’s application and initiated removal proceedings against him and Sawsan.
B. Testimony of Imam Mohammed Mardini
Mohammed Radwan Mardini, a Detroit-area imam who worked for the Michigan Department of Corrections and the American Muslim Center, testified as an expert witness regarding Muslim marriage customs. Mardini testified that Islamic marriages involve four steps, each of which must be completed before a marriage is considered finalized. The first step is called Al Fatha, and consists of the man and woman’s families meeting and reading from the Koran. The second step is called Al Khuba, which is the engagement and includes the man giving the woman a ring. The third step is called Kateb al Ketab, which involves the parties drafting and signing a marriage contract, which would include the dowry terms and any other conditions placed on the marriage. Mardini testified that the father of one of the families is typically given a copy of the contract, and the contract is also filed with the Sharia court and the civil records department. The fourth and final step is the marriage celebration and the consummation of the marriage. This is the most important step in the marriage process and must be completed before the couple is considered married.4
C. Testimony of Nabil’s Three Brothers
Nabil Hassan’s brothers — Ismail, Ayoub, and Ibraham — all testified at the merits hearing regarding Nabil and Sawsan’s wedding ceremony in Michigan on April 10, 1995 (which occurred some ten years before the merits hearing) and regarding Nabil and Sawsan’s living arrangements in *920Michigan prior to their wedding. Although there were some inconsistencies among their statements at the hearing, the three brothers generally testified that Nabil and Sawsan had a small wedding ceremony with family members at a nearby mosque and a small celebration after the ceremony at the home of Nabil’s oldest brother. Prior to the wedding, Nabil had resided with that brother, and Sawsan lived in a different dwelling with another of Nabil’s brothers and his family. Ibraham, Ayoub, and Ismail all lived in Michigan at the time when Nabil and Sawsan first entered the United States; consequently, none of the brothers had personal knowledge as to what steps Nabil and Sawsan had taken in Jerusalem with respect to their marriage process.
D. Testimony of Nabil Hassan
Nabil also testified at the merits hearing. According to Nabil, he and his family had known Sawsan and her family for years. This connection led Nabil and Sawsan to become romantically involved some time in early 1994. At the end of 1994, Nabil’s mother had filed a visa application on his behalf that would authorize his entry into the United States. In February 1995, Nabil became engaged to Sawsan. Nabil testified that on February 24, 1995, the families created a written engagement contract, and that a Sharia agent named Nasra helped the families create the document and then file it with the appropriate authorities. Nabil claimed that he and Sawsan did not finalize their marriage in Jerusalem, however. Instead, they waited until April 10, 1995, when they had their ceremony in the mosque and celebration in Michigan.
With respect to his naturalization interview, Nabil testified that he never told Officer Wells that he had been married in Jerusalem prior to his entry into the United States. He claimed instead that Officer Wells had both confused him and threatened him, yelling that Muslims do not have children out of wedlock, that Nabil and Sawsan must have been married in Jerusalem, and that Nabil would be deported.
Regarding his SBA loan application forms, Nabil did admit that there were inaccuracies on the forms and that he had signed them. Nabil claimed, however, that he did not know that the forms were inaccurate at the time he signed them. He testified that the forms were actually filled out by his financial advisor, that he did not know if his citizenship status would affect his loan status, and that he was eligible for the loan regardless of his status. Indeed, Nabil testified that he had filed two other SBA loan applications earlier, indicating on each that he was a resident alien. Each of those loans was approved.5
E. Testimony of Sawsan Hassan
Sawsan Hassan also testified at the merits hearing and essentially corroborated Nabil’s testimony with regard to the time-line of their marriage process.
F. Documentary Evidence
In addition to the testimony of the aforementioned witnesses, the immigration judge also received several documents into evidence without objection. One was an affidavit from an imam in Jerusalem outlining the Muslim marriage process. The affidavit essentially corrobo*921rated the testimony of Imam Mardini and it also affirmed that a Muslim marriage is not complete until the wedding ceremony, celebration, and consummation all take place. The affidavit also states that the time lapse between the completion of the marriage contract and the occurrence of the wedding ceremony and consummation will vary from couple to couple, and could potentially take years to complete.
Also admitted into the record was an affidavit from Abdul Jabbar Nasra, a Sharia court agent in Jerusalem. In the affidavit, Nasra stated that he “concluded” the marriage contract between Nabil and Sawsan Hassan on February 24, 1995. He indicated that the identification number for that contract was 3769. He also stated that he did not know whether or when the couple performed the final steps of their marriage process.
The IJ also admitted into evidence two letters from the United States Embassy in Israel. The first letter, dated March 4, 2002, is addressed to Officer Wells and is signed by a “Fraud Prevention Investigator” in the Fraud Prevention Unit of the Embassy. The relevant portions of the letter are as follows:
We have received your inquiry from February 27th, 2002, requesting that an investigation on subject [Nabil Hassan]’s previous marriage be conducted.
I contacted the Ministry of Interior in order to verify [sic] previous marriage. According to their database, Subject was married to Sawsan Wadi (born on the 4th of June 1969) on the 24th of Febraury 1995 in Jerusalem. Subject and his wife left Israel on the 25th of March 1995.
Please note that the information received from the Ministry of Interior should remain confidential. Subject should not know how we obtained the above information.
A second Embassy letter admitted into evidence is dated November 29, 2005. This letter is labeled as a “Statement” and is signed by the same Fraud Prevention Investigator as the first letter, as well as the “Fraud Prevention Manager and Immigrant Visa Chief’ of the Embassy’s Fraud Prevention Unit. The letter states the following:
The Israeli Ministry of the Interior (MOI) informed the Fraud Prevention Unit that Nabil Hassan holder of the identity number [number provided] and his wife, Sawsan Hassan (maiden name “Wadi”) holder if [sic] the identity number [number provided] got married in Jerusalem on February 24, 1995 (marriage certificate number 3769). They used to live in Rasud Amad, in East Jerusalem.
Our contact at the MOI has in hand a copy of Nabil Hassan’s marriage certificate. Due to the Israeli privacy act they are unable to provide us with any documents or written statement concerning the case. Therefore, we approached the Sharia court in an attempt to obtain a copy of the marriage certificate. I was told that the procedure of obtaining the information could take a week. As soon as we receive the copy we will fax it.
The government attorney assigned to the case did not receive this second letter until after the record had been closed by the immigration judge. However, the government successfully petitioned the IJ on January 22, 2007 to reopen the record so that this letter could be admitted. The IJ granted that request in a January 22, 2007 hearing held for the purposes of resolving the motion to reopen the record. The government has not stated, at the January 22 hearing or at any other point, whether it has ever received the copy of the marriage “certificate” from the Sharia court or the American Embassy. No such copy *922was admitted into evidence by either party-
II. Decision of the Immigration Judge
On April 3, 2007, the immigration judge issued an oral decision finding Petitioners removable as charged. The IJ first found that the Hassans were removable under 8 U.S.C. § 1227(a)(1)(A) because they had married prior to their arrival in the United States. To support her finding, the IJ first relied heavily on the Embassy letters, which, according to the IJ, showed that Petitioners were in fact married in Jerusalem. The IJ noted that the documents were hearsay evidence, but that the general rules of evidence do not apply in the immigration setting and that the documents were sufficiently reliable. The judge also cited Officer Wells’ conclusion that the Hassans were married prior to their entry into the United States, as well as the fact that Petitioners failed to produce the alleged marriage contract and did not adequately explain its absence. Finally, the judge relied on her conclusion that Petitioners and Nabil’s brothers were all not credible. The judge noted various discrepancies she saw in their testimony, as well as Nabil’s alleged admission of prior marriage at the naturalization interview.
The immigration judge next found that Nabil Hassan was removable under 8 U.S.C. § 1227(a)(3)(D) based on her determination that Nabil falsely represented himself as a U.S. citizen on his SBA loan application. Based on her prior finding that Nabil’s testimony was not credible, the judge rejected Nabil’s claim that he did not know about the mistakes when he signed the application. The IJ also found that, even if the mistakes were innocent, section 1227(a)(3)(D) does not require a showing of willfulness or knowledge.
Finally, the IJ denied Petitioners’ request for voluntary departure because Petitioners did not have valid travel documents. Petitioners traveled to the United States from Jerusalem under Israeli travel documents but apparently do not have Israeli passports.
III. Decision by the Board of Immigration Appeals as to Petitioners’ Appeals
On September 11, 2007, Petitioners filed a timely appeal before the Board of Immigration Appeals. In addition to appealing the rulings of the IJ, Petitioners also sought remand on the grounds that: 1) the IJ should have recused herself due to her former role as a government immigration attorney; and 2) Petitioners were eligible for waivers of inadmissibility under 8 U.S.C. §§ 1227(a)(1)(H) and 1182(k). On February 17, 2009, the Board dismissed Petitioners’ appeal, affirmed the findings of the IJ, and denied Petitioners’ motion to remand. The Board first rejected Petitioners’ claim that the IJ should have recused herself, finding instead that the record contained no evidence that the IJ demonstrated any bias or improper conduct. The Board then affirmed the IJ’s finding that Petitioners were married pri- or to their entry into the United States. Reviewing the record de novo, the Board relied first and foremost on the fact that the “record contains Israeli documents indicating that the respondents were married, not engaged, on February 24, 1995 .6 The Board then noted that Petitioners failed to obtain a copy of the Israeli marriage certificate or “contradict the official record of the February 24, 1995 marriage.” The Board also relied on the fact *923that the testimony provided by Petitioners and their relatives was not credible.
The Board also affirmed the IJ’s determination that Nabil was removable on the ground that he misrepresented himself as a U.S. citizen on his SBA loan application forms. Without specifically addressing the issue of whether Nabil made this representation knowingly or willfully, the Board concluded that he “provided the information and signed the form.” The Board also noted that the relevant immigration statute “does not contain the requirement that the misrepresentation be material.”
The Board also denied Petitioners’ motion to remand in order to apply for a waiver of inadmissibility. The Board stated that in order to be eligible for such a waiver, an alien must be “otherwise admissible.” Because the Board had ruled that Petitioners had been married before their entry into the United States, which disqualified Nabil from being an unmarried child of a lawfully permanent resident, the Board held that the Hassans were not eligible for the waiver they sought. Finally, the Board explained that the IJ’s failure to inform Petitioners of the availability of a waiver of inadmissibility was irrelevant because Petitioners were not in fact eligible for any such relief.
This appeal followed.
ANALYSIS
I. IJ’s Failure to Recuse
Petitioners’ first argument on appeal is essentially a due process challenge to the proceedings conducted by the IJ. In particular, Petitioners argue that the immigration judge improperly refused to recuse herself from the proceedings. Because the Board in this case issued an independent opinion as to this claim, we review the decision of the Board as the final agency determination. See, e.g., Morgan v. Keisler, 507 F.3d 1053, 1057 (6th Cir.2007) (internal citations omitted). Legal conclusions, such as whether or not a petitioner has proven his or her due process claim, are reviewed de novo. See Mikhailevitch v. INS, 146 F.3d 384, 391 (6th Cir.1998) (internal citations omitted).
Although Petitioners are entitled to due process in their removal proceeding, see Castellano-Chacon v. INS, 341 F.3d 533, 553 (6th Cir.2003), they have failed to show that the IJ’s failure to recuse herself was a due process violation. “Reviewing an alleged due process violation is a two-step inquiry: first, whether there was a defect in the removal proceedings; and second, whether the alien[s] w[ere] prejudiced because of it.” Vasha v. Gonzales, 410 F.3d 863, 872 (6th Cir.2005). At different points in their brief, Petitioners point to three different alleged defects in the process they received: 1) the immigration judge’s failure to recuse herself due to her previous job as a DHS trial attorney and later as Chief Counsel of the Detroit DHS office; 2) the IJ’s failure to recuse herself due to her previous “close working relationship” with Officer Wells; and 3) the IJ’s active questioning of the witnesses during the merits hearing.
None of the allegations on which Petitioners rely amount to a procedural defect, however. Petitioners’ first allegation is based on the IJ’s allegedly inherent bias due to her previous roles at Homeland Security. The government counters that Petitioners have not pointed to any evidence in the record that suggests that the IJ even had a previous position at DHS. While we agree that Petitioners have pointed to no evidence of this assertion, in other proceedings we have taken judicial notice of the fact that Judge Nettles did hold such positions before becoming an IJ. See Kawuwung v. Holder, 356 Fed.Appx. 854, at 856 (6th Cir.2009) (noting that Judge Nettles had once been Chief Counsel at DHS). Nevertheless, her former *924role at DHS does not, in itself, amount to a procedural defect. Abdulahad v. Holder, 581 F.3d 290, 296 (6th Cir.2009); see also Petrov v. Gonzales, 464 F.3d 800, 803 (7th Cir.2006) (“The Chief Counsel of a large [government immigration] office is unlikely to play any role in routine decisions____”). An immigration judge does have an obligation to recuse him- or herself when he or she has “served in governmental employment and in such capacity participated as counsel, adviser or material witness concerning the proceeding or expressed an opinion concerning the merits of the particular case in controversy.” 28 U.S.C. § 455(b)(3). But there is nothing in the record to suggest that the immigration judge here, while at DHS, ever participated in Petitioners’ case in any of the capacities listed in the relevant statute. Petitioners have not cited to any evidence in the record that suggests otherwise.
Petitioners’ second complaint is that the IJ had a “close working relationship” with Officer Wells during her tenure at DHS. But this is a bald allegation with no support in the record. In fact, Petitioners fail to cite any evidence that indicates that Judge Nettles even knew Officer Wells while she was at DHS.
Petitioners’ final allegation is related to the IJ’s questioning techniques used at the merits hearing. Petitioners assert that Judge Nettles “actively questioned” Officer Wells, rehabilitated his credibility through her questions, and impermissibly helped DHS meet its burden of proof. Immigration judges, however, have “broad discretion in conducting” removal proceedings. Castellano-Chacon, 341 F.3d at 553. Furthermore, IJs are statutorily authorized to “interrogate, examine, and cross-examine the alien and any witness.” 8 U.S.C. § 1229a(b)(1); see Abdulahad, 581 F.3d at 296. Judge Nettles’ behavior did not amount to an abuse of this broad discretion or of Petitioners’ due process rights. Although the record indicates that the judge took a somewhat active role in the examination of several, if not all, of the witnesses, the judge was well within her rights to do so. Accordingly, we affirm the Board’s decision rejecting Petitioners’ claim that the IJ should have recused herself.
II. Removability Findings
Petitioners also appeal the Board’s affirmance of the IJ’s findings that Petitioners were removable. The IJ found, and the Board affirmed, that the Hassans were removable for two alternative reasons, either of which is sufficient on its own to justify their removal. First, the IJ found that Petitioners were removable under 8 U.S.C. § 1227(a)(1)(A) because they were married prior to Nabil’s entry into the United States. Second, the IJ found the Hassans removable under 8 U.S.C. § 1227(a)(3)(D) because Nabil made a false claim of United States citizenship.
A Standard ofRevieio
In a removal proceeding for a previously admitted alien, the government “has the burden of establishing by clear and convincing evidence that” Petitioners are removable. 8 U.S.C. § 1229a(c)(3)(A); see, e.g., King v. Holder, 570 F.3d 785, 787 (6th Cir.2009). Where, as here, the BIA reviewed the IJ’s decision de novo and issued its own separate opinion, we review the BIA’s opinion as the final agency determination. Morgan, 507 F.3d at 1057. We review an immigration court’s removability determination under the “substantial evidence” standard. Stolaj v. Holder, 577 F.3d 651, 657 (6th Cir.2009). Under this deferential standard of review, the Board’s decision must be affirmed if it is “ ‘supported by reasonable, substantial, and probative evidence on the record considered as a whole.’ ” Mullai v. Ashcroft, 385 F.3d 635, 638 (6th Cir.2004) (quoting Koliada v. *925INS, 259 F.3d 482, 486 (6th Cir.2001)). “We are not entitled to reverse ‘simply because [we are] convinced that [we] would have decided the case differently.’ ” Id. (quoting Adhiyappa v. INS, 58 F.3d 261, 265 (6th Cir.1995)). Instead, the agency’s findings of fact are “conclusive” unless “any reasonable adjudicator would be compelled to conclude to the contrary.” 8 U.S.C. § 1252(b)(4)(B); see also Koulibaly v. Mukasey, 541 F.3d 613, 619 (6th Cir. 2008). In other words, “ ‘in order to reverse the BIA’s factual determinations, the reviewing court must find that the evidence not only supports a contrary conclusion, but indeed compels it.’ ” Mullai, 385 F.3d at 638 (quoting Klawitter v. INS, 970 F.2d 149, 152 (6th Cir.1992)).
B. Prior Marriage Finding
Nabil was granted an F-24 Immigrant Visa as an unmarried child of a lawful permanent resident. Under 8 C.F.R. § 205.1(a)(3)(i)(I), Nabil (and Sawsan, derivatively) would be deportable (via 8 U.S.C. § 1227(a)(1)(A)) if Nabil and Sawsan were actually married when Nabil entered the United States. Accordingly, the government had the burden to show by clear and convincing evidence that the marriage between Petitioners occurred before their entry into the country. The validity of a marriage is determined by the law of the place of celebration. Matter of Luna, 18 I. & N. Dec. 385, 386 (BIA 1983). Pursuant to Israeli law, the Sharia courts (and Sharia law) control personal status matters of Muslims residing in Jerusalem. Matter of Darwish, 14 I. & N. Dec. 307, 308 (BIA 1973).7 The IJ found, and the BIA affirmed, that the government met its burden of showing that the Hassans were previously married under this law. We hold, however, that the evidence on the record, when viewed as a whole, compels the contrary conclusion that the government did not offer clear and convincing evidence that Petitioners had completed the steps required for a Muslim marriage under Sharia law before entering the United States.
The crucial evidence offered by the government (and relied upon by the Board in its decision) to prove the prior marriage was the two Embassy letters, which state that the Israeli Ministry of the Interior had on file a document certifying that Petitioners were married in Jerusalem prior to their entry into the United States. For the purposes of our review here, we will assume that these documents were admissible in the proceedings despite the fact that they were entirely hearsay statements.8 See Dallo v. INS, 765 F.2d 581, 586 (6th Cir.1985) (noting that the Federal Rules of Evidence do not apply in immigration proceedings); but see Alexandrov v. Gonzales, 442 F.3d 395, 404-05 (6th Cir.2006) (noting that hearsay statements still pose problems in the immigration context when those statements are highly unreliable). However, even if these documents were perfectly reliable, they simply *926do not show that all four steps required of a Muslim marriage had occurred in February 1995, as the Board asserts. The Embassy letters indicate that the Israeli Ministry of the Interior has on file a “marriage certificate” of Petitioners that is dated February 24, 1995 and that has an identification number of 3769. However, the letters indicate nothing more. Most importantly, they entirely fail to specify whether the “certificate” attests to the completion of the entire marriage process, or whether it simply attests to the completion only of the engagement contract, the third step of the four marriage requirements. The record contains an uncontested affidavit from Abdul Jabbar Nasra, a Sharia court agent in Jerusalem. In that affidavit, Nasra states that he concluded the marriage contract on February 24, 1995, and that the contract was given the identification number 3769 — the same number as the “certificate” to which the Embassy letters refer. Nasra also states that he did not know whether or when Nabil and Sawsan completed their marriage process. This affidavit strongly suggests that the “certificate” referenced in the Embassy letters is nothing more than the filed copy of the marriage contract to which the Sharia agent refers in his affidavit, as opposed to some certificate of a completed marriage.9 As noted above, uncontested record evidence indicates that the completion of the marriage contract is only one step in the marriage process, and its filing does not complete the steps necessary for a Muslim marriage. Because the Embassy letters, when viewed against the other evidence in the record, show only that the marriage contract was filed, the Embassy letters on their own do not amount to substantial evidence supporting the Board’s conclusion that the government met its burden of proving by clear and convincing evidence that Petitioners’ marriage was finalized in Jerusalem.
Neither of the Board’s two other justifications help to support its finding that the government satisfied its burden of proof. After discussing the Embassy letters, the Board proceeded to the absence of the marriage contract in the record. According to the Board, the contract could have contradicted the government’s claim that the marriage process was completed on February 24, 1995. The Board stated that Petitioners had access to the marriage contract, and that their failure to submit the contract was never explained. In other words, the Board seemed to negatively infer the absence of the marriage contract against Petitioners. But such an approach improperly shifts the burden of proof to Petitioners. If anything, the absence of the marriage contract is an inference against the government, the party that had the burden to prove that Petitioners were married (and not just contracted to be married) prior to their entry into the United States. In its opinion affirming the IJ’s findings, the Board suggests that the government could not obtain the marriage contract due to privacy concerns. But this statement mischaracterizes the evidence on record. In fact, the 2005 Embassy letter indicates that the government expected to obtain a copy of the marriage “certificate” from the Sharia court, which also apparently had a copy of it. In that letter, the Embassy explicitly states that it would fax the “certificate” to the government lawyer assigned to Petitioners’ case as soon as the Embassy received it. The government, however, neither submitted the “certificate” nor explained its absence from the record. Thus, if a negative inference is appropriate at all, it should fall on *927the government. Either way, the absence of the marriage contract does not help the government meet its affirmative burden of proving the completion of the marriage.
The Board also relies on the IJ’s finding that much of the testimony provided by Petitioners and their other witnesses lacked credibility. Although the Board notes several of the minor inconsistencies made by the various witnesses, these inconsistencies are simply not relevant to the question at hand. The lack of credibility of the testimony offered by Petitioners, even if assumed, does not help the government meet its burden of proof. At best, it would hurt Petitioners’ ability to rebut the government’s case. But the testimony and the credibility of Petitioners’ witnesses play no part in the analysis of whether the government has met its affirmative burden of providing clear and convincing evidence that Petitioners were married prior to their entry into the United States. The Board’s reliance on the IJ’s credibility finding improperly shifted the burden of proof again to Petitioners.
Since the Board cited no other evidence in support of its finding, the only evidence that was offered by the government as affirmative support for its charge of removability was the Embassy letters. As noted above, however, those letters did not amount to clear and convincing evidence that Petitioners had performed all the steps required for a marriage under Sharia law before they left for the United States. But even if we were to accept arguendo that a reasonable jurist could conclude that the Embassy letters alone satisfied the government’s burden here, we could not rely on them to affirm the Board’s ruling. The Sixth Circuit has previously explained that hearsay documents — although generally admissible in immigration proceedings if reliable and timely provided to opposing counsel — may not form the sole basis of a removal finding. See Alexandrov, 442 F.3d at 406-07; see also Kasa v. Gonzales, 128 Fed.Appx. 435, 440-41 (6th Cir.2005). As this Court noted in Alexandrov, this rule is particularly applicable when the hearsay document in question is a product of the State Department, because the government should not simply be allowed to “use the prestige of the State Department letterhead to make its case and give credibility to the letter’s contents.” Alexandrov, 442 F.3d at 405 (quoting Ezeagwuna v. Ashcroft, 325 F.3d 396, 407 (3d Cir.2003)) (internal quotation marks omitted).
Based on the framework set out in Alexandrov, the Embassy letters in the instant case are problematic. Admittedly, Petitioners were provided with the Embassy letters in a timely fashion, and Petitioners were at least arguably given the opportunity to rebut the letters. The reliability of the letters, however, is troubling. What is particularly problematic is that the letters lack any real degree of detail. See id. (noting that lack of detail within challenged letter cuts against reliability). After reading the letters, we do not know who provided the Embassy with the information.10 We do not know how the files were kept or the procedure for releasing them to the Embassy. We do not know what the Ministry of Interior contact point stated to the Embassy officials about the “certificate”; we have no idea how the “certificate” was characterized or whether the official even explained what the “certificate” actually said. We do not know whether the official who talked to the Embassy was familiar with Muslim marriage customs, or knew the difference between a marriage contract and a completed mar*928riage.11 “There is not much that we do know aside from the apparent conclusions of the mysterious investigation,” id. at 407, namely, that a document is on file and that it is an official record of Petitioners’ marriage. Finally, as noted above, the Board’s finding was based entirely on these letters. See id. Pursuant to Alexandrov, they cannot form the sole basis of the government’s offer of proof.
For all of the above reasons, we REVERSE the judgment of the Board with respect to its ruling that the government met its burden of proving Petitioners were married prior to their entry into the United States.
C. False Statement Finding
The government also charged Petitioner Nabil Hassan as removable under 8 U.S.C. § 1227(a)(3)(D). The IJ found, and the Board affirmed, that the government met its burden of proving that charge. The relevant statutory language states the following: “Any alien who falsely represents, or has falsely represented, himself to be a citizen of the United States for any purpose or benefit under this chapter ... or any Federal or State law is deportable.” 8 U.S.C. § 1227(a)(3)(D)®. Thus, the government had the burden to show not only that Nabil misrepresented himself as a U.S. citizen, but also that he did so “for any purpose or benefit” under some law. Id. The federal courts have begun to provide examples of what qualifies as a “purpose or benefit” under section 1227(a)(3)(D). See, e.g., Theodros v. Gonzales, 490 F.3d 396, 402 (5th Cir.2007) (obtaining private-sector employment qualifies as “purpose or benefit”); Sowah v. Gonzales, 196 Fed.Appx. 576, 577 (9th Cir. 2006) (applying for U.S. passport qualifies as “purpose or benefit”); Jamieson v. Gonzales, 424 F.3d 765, 768 (8th Cir.2005) (obtaining entry into the United States qualifies as “purpose or benefit”). Petitioners claim that the government failed to establish that Nabil’s submission of the SBA loan application had any such “purpose or benefit.” We agree.
In the instant case, the government never attempted to show the “purpose or benefit” Nabil received, and the Board does not discuss the issue in its opinion. Both the government and the Board focus on the misrepresentation itself, highlighting that it must have been knowingly made by Nabil. True, its finding that Nabil had knowledge of the misrepresentation passes muster under our standard of review. But that does not change the fact that the government simply offered no evidence of the “purpose or benefit” of the misrepresentation, despite the fact that it is one of the necessary elements that must be shown to establish removability under section 1227(a)(3)(D). Presumably, the government may have assumed that the “purpose or benefit” was the obtainment of the SBA loan. However, the government made no attempt to explain how, if at all, Nabil’s immigration status would affect the loan application. In fact, the only evidence on record with regard to this issue came from Petitioners and indicates that his immigration status would have no effect on the loan.12 (AR 721, 724) (copies of Nabil’s *929previous loan applications, which were granted, and on which Nabil indicated that he was not a U.S. citizen). Admittedly, Nabil might still be removable notwithstanding the lack of effect if his “purpose” was to affect his application (even though his immigration status was irrelevant). There are two fatal problems with this reasoning, however. First, the government never made this argument or presented any testimony to prove it. Second, the only evidence on record leads to the opposite conclusion. For Nabil had actually applied for and received multiple similar SBA loans as a resident alien prior to the applications in question. And in those previous applications, he (properly) asserted that he was not a U.S. citizen. In other words, Nabil’s past conduct suggests that he did not subjectively believe that his immigration status affected his application. Thus, the record compels the conclusion that the government failed to show that Nabil misrepresented himself for “any purpose or benefit” under state or federal law. Accordingly, we must also REVERSE the Board’s finding that Nabil was removable under 8 U.S.C. § 1227(a)(3)(D).
D. Conclusion
The Board (and the IJ) failed to provide a legally sufficient basis for finding the Hassans removable. Indeed, the record before the agency at the time the removability rulings were made did not contain evidence to support those rulings. “Had the BIA properly applied the facts before it to the law, it would have held that the Petitioner[s were] not deportable. The proper course in the circumstances before us is to reverse without a remand for further consideration.” Pickering v. Gonzales, 465 F.3d 263, 271 (6th Cir.2006) (internal citation omitted) (reversing and remanding with instructions to end removal proceedings in analogous situation). Accordingly, we will do so in this case as well.
III. Waivers/Motion to Remand
Petitioners also appeal the Board’s ruling denying their motion to remand the record in order to apply for a waiver of inadmissibility. The Board’s decision and Petitioners’ appeal with respect to these claims are moot, however, in light of our previous rulings here. Therefore, we decline to address them now.
CONCLUSION
For the foregoing reasons, we AFFIRM in part and REVERSE in part the opinion of the Board of Immigration Appeals. The petition is granted, and the case is REMANDED to the Board of Immigration Appeals so that it may quash the removal order and terminate the removal proceedings against Petitioners.