EDITH BROWN CLEMENT, Circuit Judge:

Appellant Alstom Transportation, Inc. ("Alstom") appeals from the final judgment entered by the district court, in which the district court partially vacated an arbitration panel's final award in Alstom's favor. For the reasons explained below, we VACATE the district court's order. We REMAND for further proceedings consistent with this opinion, with instructions to reinstate the Panel's final award.

FACTS AND PROCEEDINGS

Appellee BNSF Railway Co. ("BNSF") hired Alstom to implement and manage a new maintenance program for BNSF's locomotives. As relevant here, BNSF and Alstom's relationship was governed by a Maintenance Agreement ("Agreement"), which contained an arbitration agreement. Alstom's performance did not satisfy BNSF in the early years of the new program. As part of Alstom and BNSF's negotiations to resolve the growing conflict, the parties amended the Agreement to give BNSF the right to terminate the parties' contractual relationship "at any time, without cause." Later, BNSF informed Alstom that it planned to remove a significant percentage of its locomotives from its active fleet. These plans triggered a provision in the Agreement, which required BNSF to confer with Alstom to make a reasonable economic adjustment in Alstom's favor.1 But, before BNSF and Alstom met to discuss an adjustment, BNSF terminated the contract. After BNSF sought declaratory relief in the district court, Alstom asked the district court to compel arbitration. The district court ordered BNSF and Alstom to arbitrate their claims under the Federal Arbitration Act ("FAA"), 9 U.S.C. § 1, et seq.

The Agreement specified that the arbitration panel ("Panel") should resolve any dispute according to the Agreement's express terms. The Agreement provided that it should "be governed by and construed in accordance with the laws of ... Illinois." If nothing in the Agreement answered the question presented to the Panel, the Agreement directed the Panel to refer to Illinois law to discern the controlling rule. The Agreement also provided that the Panel could award ordinary and direct damages, but not consequential or incidental damages, such as lost profits.

The Panel found that BNSF exercised its termination rights to avoid its contractual duty to confer with Alstom regarding a reasonable economic adjustment. Thus the Panel held that, under Illinois law, BNSF breached the covenant of good faith and fair dealing when it terminated the Agreement. The Panel also held that BNSF breached the Agreement by failing to confer on a reasonable economic adjustment. The Panel then considered what remedy to give Alstom. The Panel refused to award certain remedies, such as lost profits, because the Agreement prohibited them. But the Panel interpreted the Agreement to allow out-of-pocket damages. Accordingly, the Panel awarded Alstom its out-of-pocket costs, minus penalties and liquidated damages.

Alstom moved to confirm the award in the district court. BNSF moved to vacate the award pursuant to 9 U.S.C. § 10(a)(4). The district court sided with BNSF, vacating the Panel's holding that BNSF violated the Illinois covenant of good faith and fair dealing. In the alternative, the district court vacated the Panel's remedy in part, holding that the Panel had given Alstom damages that were forbidden under the Agreement.

STANDARD OF REVIEW

We review a district court's order vacating an arbitration award de novo. See Citigroup Global Mkts., Inc. v. Bacon, 562 F.3d 349, 350 (5th Cir.2009). Our "review of the underlying award is exceedingly deferential." Rain CII Carbon, LLC v. ConocoPhillips Co., 674 F.3d 469, 472 (5th Cir.2012) (internal quotation marks omitted). "[T]he sole question for us is whether the arbitrator (even arguably) interpreted the parties' contract, not whether he got its meaning right or wrong." Oxford Health Plans LLC v. Sutter, U.S. ___, 133 S.Ct. 2064, 2068, 186 L.Ed.2d 113 (2013).

DISCUSSION

Alstom argues that the district court erred by vacating the Panel's decision and award and refusing to confirm the award in Alstom's favor. For the reasons explained below, we agree with Alstom.

I.

A.

"Arbitration is a matter of contract." Brook v. Peak Int'l, Ltd., 294 F.3d 668, 672 (5th Cir.2002). Thus, "the power and authority of arbitrators in an arbitration proceeding is dependent on the provisions under which the arbitrators were appointed." Id. (alteration and internal quotation marks omitted). This principle is enshrined in 9 U.S.C. § 10(a)(4), which provides a remedy in the federal courts "where the arbitrators exceeded their powers."

When an arbitration goes an opponent's way on the basis of questionable contract interpretation, parties often seek refuge in § 10(a)(4). But the Supreme Court has made clear that district courts' review of arbitrators' awards under § 10(a)(4) is limited to the "sole question... [of] whether the arbitrator (even arguably) interpreted the parties' contract." Oxford Health, 133 S.Ct. at 2068. This is an objective test. Cf. Stolt-Nielsen S.A. v. AnimalFeeds Int'l Corp., 559 U.S. 662, 676, 130 S.Ct. 1758, 176 L.Ed.2d 605 (2010) (holding that, despite arbitrators' "few references to [the parties'] intent," evidence showed that they could not have been guided by parties' intent). Thus a party challenging an arbitration award need not adduce hard-to-obtain evidence concerning the arbitrators' true intent. At the same time, in evaluating the available evidence, the district courts "must resolve all doubts in favor of arbitration." Brook, 294 F.3d at 672. Accordingly, the party challenging an arbitrators' award under § 10(a)(4) must carry a "heavy burden." Oxford Health, 133 S.Ct. at 2068.

In determining whether the arbitrator exceeded her authority, district courts should consult the arbitrator's award itself. The award will often suggest on its face that the arbitrator was arguably interpreting the contract. Several pieces of relevant evidence can be gleaned from the award's text, including but not limited to: (1) whether the arbitrator identifies her task as interpreting the contract; (2) whether she cites and analyzes the text of the contract; and (3) whether her conclusions are framed in terms of the contract's meaning. See id. at 2069.

B.

The question before us is whether the arbitrators "(even arguably) interpreted" the Agreement in reaching their award. Id. at 2068. The parties' arguments implicate three provisions in the Agreement. Sections 13.2, 15.6, and 18.1, respectively, give BNSF the right to terminate the Agreement "without cause"; authorize the Panel to consider Illinois law when the Agreement does not contain an answer to a question in dispute; and provide that the Agreement should be generally governed by and construed according to Illinois law. The Panel's discussion of the duty of good faith and fair dealing is framed as an interpretation of the meaning of the "without cause" provision. The Panel appears to have concluded that the "without cause" language in the Agreement was ambiguous. Even if they had not, they could have turned to Illinois law in the first instance to aid their interpretation of the "without cause" language under § 18.1. In either event, the arbitrators were "arguably interpreting" the agreement when they construed the term "without cause" by reference to Illinois law. See Oxford Health, 133 S.Ct. at 2068. BNSF fails to show that the Panel could not have been interpreting the Agreement when it concluded that Illinois law imposes a limitation on the right to terminate "without cause" based on the covenant of good faith and fair dealing.

BNSF argues that the Panel should have interpreted the termination provision as giving it a right to terminate the Agreement for any reason whatsoever. BNSF also contends that the Panel "completely botched" the application of Illinois law when it applied the covenant of good faith and fair dealing. But error in interpreting a contract is not grounds for setting aside an arbitrator's award.

Hill v. Norfolk & W. Ry. Co., 814 F.2d 1192, 1194-95 (7th Cir.1987). As such, BNSF's arguments are misplaced. The sole question is whether the arbitrators even arguably interpreted the Agreement in reaching their award; it is not whether their interpretations of the Agreement or the governing law were correct. See Oxford Health, 133 S.Ct. at 2068. That we might interpret the Agreement differently than the Panel is entirely beside the point because it is not our interpretation that the parties bargained for. See id. ("Because the parties bargained for the arbitrator's construction of their agreement, an arbitral decision even arguably construing or applying the contract must stand, regardless of a court's view of its (de)merits." (citation and internal quotation marks omitted)).

Accordingly, we reverse the district court.

C.

The Panel's award also suggests that the Panel was interpreting the Agreement when it calculated Alstom's remedy. The Panel held that BNSF breached the Agreement in two ways, and that BNSF should pay damages to Alstom according to the terms of the Agreement's remedial provision. The Panel made clear, either expressly or by implication,2 that it interpreted the Agreement to allow out-of-pocket costs as direct damages, and that Alstom's remedy was for its out-of-pocket damages.

Most of BNSF's arguments about the damages the Panel awarded to Alstom boil down to a complaint that the Panel misinterpreted the Agreement. BNSF argues that the Panel exceeded its powers by creating a remedy based on the "reasonable economic adjustment" clause. We recognize that the Panel phrases its award in those terms. But the Panel also tailored the "economic adjustment" remedy to fit the Agreement's remedial provision. And BNSF fails to adduce any evidence that the Panel referred to sources besides the Agreement or relevant Illinois case law in determining the amount of Alstom's award.

BNSF contends that the Panel awarded Alstom lost profits, which were prohibited under the Agreement. As evidence, BNSF cites the Panel's statement that BNSF's bad faith termination "denied [Alstom] the opportunity to earn additional revenues through the remainder of the contract period." But the Panel clearly mentions Alstom's revenue expectations only as a component of what Alstom had a reasonable right to expect according to the covenant of good faith and fair dealing.3 The Panel's award also conveys that the Panel believed the Agreement allowed out-of-pocket damages, that Alstom "reasonably proved that they sustained legitimate out-of-pocket costs" exceeding the amount of the award, and that the award was for out-of-pocket damages pursuant to the Agreement's remedial provision.4 Given what appears on the face of the Panel's award, BNSF must show that the Agreement cannot even arguably be interpreted to allow out-of-pocket damages, or that the award itself cannot even arguably be construed as an out-of-pocket remedy. BNSF cites this court's opinion in Amigo Broadcasting, LP v. Spanish Broadcasting System, Inc., 521 F.3d 472 (5th Cir.2008), to show that out-of-pocket damages are the same as lost profits damages. But, even if that case supported that proposition, it was based on Texas rather than Illinois law. See id. at 479. BNSF fails to cite any Illinois case law — which is controlling when the Agreement does not define a given term — that holds that out-of-pocket damages are the same as lost profits. Thus BNSF fails to carry its burden to show that the Panel did not even arguably interpret the Agreement when it fashioned Alstom's remedy.

While we have endeavored to construe BNSF's arguments according to the proper standard — whether the Panel even arguably interpreted the Agreement — we may be giving BNSF's arguments more solicitude than they deserve. The gist of most of its arguments is that the Panel erred, and this error proves that the Panel failed to interpret the Agreement. But "convincing a court of an arbitrator's error — even his grave error — is not enough." Oxford Health, 133 S.Ct. at 2070.

Because BNSF failed to carry its burden to show that the Panel was not even arguably interpreting the Agreement, "[t]he [Panel]'s construction holds, however good, bad, or ugly." Id. at 2071. We reverse the district court on its alternative holding that the Panel failed to even arguably interpret the Agreement when it awarded Alstom damages.

II.

BNSF argues that, even if the Panel's award would stand under the FAA, we should overturn it under the Texas Arbitration Act ("TAA"), Tex. Civ. Prac. & Rem.Code § 171.001, et seq., or the Illinois Arbitration Act ("IAA"), 710 Ill. Comp. Stat. 5/1, et seq. BNSF admits that the Agreement does not expressly reference the TAA or IAA. "FAA rules apply absent clear and unambiguous contractual language to the contrary." Action Indus., Inc. v. U.S. Fid. & Guar. Co., 358 F.3d 337, 341 (5th Cir.2004). "[T]his Court permits arbitration under non-FAA rules if a contract expressly references state arbitration law...." Id. (emphasis added).

BNSF contends that Hall Street Associates v. Mattel, Inc., 552 U.S. 576, 128 S.Ct. 1396, 170 L.Ed.2d 254 (2008), supersedes this part of Action.5 We disagree. In Hall, the Supreme Court noted that parties may obtain more searching review of arbitration decisions by stipulating in the arbitration agreement that state statutes or common law rules apply. See id. at 590, 128 S.Ct. 1396. Action is consistent with Hall. Action simply states that the FAA provides the default standard of review, and that parties must unambiguously express their agreement to non-FAA standards to obtain more searching review. See Action, 358 F.3d at 341. Because the Agreement does not refer to the TAA, IAA, or any other body of law offering a competing standard of review, we hold that the FAA's standard of review controls.

BNSF also maintains that the district court erred when it ordered the parties to arbitrate gateway questions of arbitrability. BNSF argues that it never agreed to submit these gateway questions to arbitration. The problem with this argument is, BNSF did not file a cross-appeal. "An appellee who does not take a cross-appeal may `urge in support of a decree any matter appearing before the record, although his argument may involve an attack upon the reasoning of the lower court.'" Jennings v. Stephens, ___ U.S. ___, 135 S.Ct. 793, 798, 190 L.Ed.2d 662 (2015) (quoting United States v. Am. Ry. Express Co., 265 U.S. 425, 435, 44 S.Ct. 560, 68 L.Ed. 1087 (1924)). "But an appellee who does not cross-appeal may not `attack the decree with a view either to enlarging his own rights thereunder or of lessening the rights of his adversary.'" Id. (quoting Railway Express, 265 U.S. at 435, 44 S.Ct. 560); see also Weaver v. Tex. Capital Bank N.A., 660 F.3d 900, 905 (5th Cir.2011) (per curiam) ("Although an appellee may argue any ground available to support affirmance of a judgment, he may not argue for a ruling that would expand his legal rights."). Here, the district court's final judgment provided that "the arbitration panel's award is PARTIALLY VACATED, in that the panel's decision that the termination of the parties' agreement by plaintiff BNSF Railway Company (`BNSF') breached Illinois's duty of good faith and fair dealing and its assessment of a reasonable economic adjustment against BNSF is VACATED." By now arguing that the arbitration at issue in this case should have never occurred because the parties did not agree to it, BNSF seeks full — not partial — vacatur of the arbitral award. Thus BNSF asks for an expansion of the judgment without filing a cross-appeal, which it may not do.

Accordingly, we may not consider BNSF's arguments on this ground.

III.

Alstom asks us to confirm the Panel's final award, but it fails to cite any case law showing that we have the power to do so in the first instance. The FAA provides that parties should seek confirmation of an arbitration award in the court specified in the agreement, or if no court is specified, in the federal district court in the district where the award was made. See 9 U.S.C. § 9. Alstom does not argue that this court is named in its agreement. Thus we do not appear to have the power to confirm the Panel's final award.

CONCLUSION

For the reasons explained, we VACATE the district court's order. We REMAND for further proceedings consistent with this opinion, with instructions to reinstate the Panel's final award.