The United States brought suit against Arthur R. McFerrin (“McFerrin” ) seeking to recover a tax credit that the government alleges was erroneously paid by the Internal Revenue Service (“IRS”) to McFerrin. After a six day bench trial, the district court held that the government had proved by a preponderance of the evidence that the tax credit was not properly supported and, consequently, should not have been granted. The district court ordered repayment of the erroneously paid refund plus interest. For the following reasons, we vacate the district court’s judgment and remand for further proceedings.
FACTS AND PROCEEDINGS
McFerrin is a prominent chemical engineer who co-founded KMCO, Inc. (“KMCO”), a Subchapter-S corporation, in 1975. KMCO manufactures commodity and specialty chemicals, mainly for the petrochemical industry. McFerrin owns three other Subchapter-S corporations which are related to KMCO: KMCO Port Arthur, Inc. d/b/a KMTEX (“KMTEX”), South Coast Acquisition, Inc. (“SC Acquisition”), and South Coast Deleware, Inc. (“SC Deleware”). SC Acquisition and SC Deleware are the only partners in another corporation, South Coast Terminals (“SC Terminals”). This case concerns tax returns filed by these corporations and McFerrin for tax year 1999 and, specifically, tax credits for increasing research activities under I.R.C. Â§ 41. McFerrin owns all of the corporations, and their pass-through income was part of his 1999 income tax return.
In 2000, all of the corporations and McFerrin originally filed tax returns for tax year 1999 and did not claim any credits for increasing research activities. In May 2003, KMCO contracted with alliantgroup, L.P. to conduct a study to determine if it was eligible for an increasing research tax credit. Based upon this study, in September 2003, McFerrin and all of the corporations filed amended 1999 tax returns claiming a credit for increasing research activities. McFerrin’s income tax return claimed an overall credit of $472,092.00. Less than a month later, the IRS, as a result of a clerical error, issued the refund, which, including interest, was for $601,228.40.
On October 31, 2005, the United States filed suit to recover this amount plus interest. Its complaint alleged, among other things, that the amended return “included. . . no supporting documents” and thus there were no documents provided to substantiate the claimed credits. On summary judgment, the district court found that SC Terminals’ amended return was untimely filed and therefore McFerrin had to return the portion of the refund relating to SC Terminals and its owners SC Deleware and SC Acquisition. The issue of whether the tax credits to KMCO and KMTEX were substantiated was tried for six days before the district court which ruled in the government’s favor and ordered McFerrin to repay the refund with interest.
The district court held in its conclusions of law that research was only qualified research if it expanded or refined the existing principles in the field, had a high threshold of innovation, and had broad effect. In addition, the court held that qualified research only applied if a process of experimentation involving the forming and testing of hypotheses had occurred, rather than “trial and error” testing. Using these definitions, the court determined that while some of the projects “may have involved some research,” it was “unpersuaded that those few projects involved `qualified research’ for purposes of the research tax credit.” The district court also determined that there were no records of the hours worked on any given project or of the hours worked or supplies used that involved research. The court was unwilling to credit the rough estimates given by employees years after the fact.
STANDARD OF REVIEW AND APPLICABLE LAW
When reviewing a district court decision after a bench trial, we review factual findings for clear error and conclusions of law de novo. Flint Hills Res. LP v. Jag Energy, Inc., 559 F.3d 373, 375 (5th Cir. 2009); Green v. Comm’r, 507 F.3d 857, 866 (5th Cir. 2007). “Findings of fact influenced by an erroneous view of the law are entitled to no deference.” G.M. Trading Corp. v. Comm’r, 121 F.3d 977, 980 (5th Cir. 1997).
In an action to recover an improperly paid refund, “the United States, as plaintiff . . ., bears the ultimate burden of proof to show not only that some amount has been erroneously refunded but also how much that amount is.” Soltermann v. United States, 272 F.2d 387, 387 (9th Cir. 1959); see also United States v. Commercial Nat’l Bank of Peoria, 874 F.2d 1165, 1169 (7th Cir. 1989) (“In an action to recover an erroneous refund . . ., the government bears the burden of proof.”). Tax credits are a matter of legislative grace, are only allowed as clearly provided for by statute, and are narrowly construed. See Stinson Estate v. United States, 214 F.3d 846, 848 (7th Cir. 2000); see also Helvering v. Nw. Steel Rolling Mills, Inc., 311 U.S. 46, 49 (1940); New Colonial Ice Co. v. Helvering, 292 U.S. 435, 440 (1934). Taxpayers are required to retain records necessary to substantiate a claimed credit. See I.R.C. Â§ 6001; Treas. Reg. Â§ 1.6001-1(a), (e). If the taxpayer can establish that qualified expenses occurred, however, then the court should estimate the allowable tax credit. See Cohan v. Comm’r, 39 F.2d 540, 544 (2d Cir. 1930) (“[T]he Board should make as close an approximation as it can, bearing heavily if it chooses upon the taxpayer whose inexactitude is of his own making. But to allow nothing at all appears to us inconsistent with saying that something was spent.”); see also Mendes v. Comm’r, 121 T.C. 308, 316 (2003) (refusing to estimate costs as taxpayer had failed to substantiate any qualified deduction); Fudim v. Comm’r, 67 T.C.M. (CCH) 3011, *12-*13 (1994) (accepting that qualified research occurred, and then estimating the time spent on that research based on “testimony and other evidence in the record”).
A. Failure to Sufficiently Plead
McFerrin first argues that the substantiation claims tried before the district court should have been dismissed because the government failed to plead with the particularity required by Federal Rule of Civil Procedure 9. We review de novo a district court’s denial of a motion to dismiss. Coop. Benefit Adm’rs, Inc. v. Ogden, 367 F.3d 323, 328 (5th Cir. 2004). We accept as true all well-pleaded facts, “viewing them in the light most favorable to the plaintiff.” In re Katrina Canal Breaches Litig.,495 F.3d 191, 205 (5th Cir. 2007) (internal quotation marks omitted). Rule 9 states that “[i]n alleging fraud or mistake, a party must state with particularity the circumstances constituting fraud or mistake.” FED. R. CIV. P. 9(b). Under the more general liberal notice standards of Federal Rule of Civil Procedure 8(a)(2), all that is required is “a short and plain statement of the claim showing that the pleader is entitled to relief.”
This case was tried based on a failure to substantiate tax claims with sufficient documentation. McFerrin argues that because the government claimed that he “misrepresented facts,” it was required to meet the more rigorous Rule 9 standards. But no fraud or mistake was alleged. Rather, the government only alleged a failure to document, which appeared several times on the face of the complaint. There was therefore no need for the government to plead a lack of substantiation with sufficient particularity to meet the requirements of Rule 9, as there was no allegation of fraud or mistake, only a lack of documentation. As such, the district court did not err in allowing the case to go to trial.
B. “Qualified Research” under I.R.C. Â§ 41
I.R.C. Â§ 41 allows a 20 percent tax credit on “qualified research expenses” over a base amount. “Qualified research” has four separate and independent requirements: (1) the expenses must be of the type deductible under I.R.C. Â§ 174; (2) the research must be undertaken “for the purpose of discovering information. . . which is technological in nature;” (3) the application of that information must be “intended to be useful in the development of a new or improved business component of the taxpayer;” and (4) substantially all of the research activities must “constitute elements of a process of experimentation.” I.R.C. Â§ 41(d)(1). McFerrin argues that the district court misinterpreted “discovering information” and “process of experimentation” in determining that no qualified research occurred.
In determining that “discovering information” meant going beyond the current state of knowledge in the field, the district court relied on cases from other circuits as well as the tax court. See Eustace v. Comm’r, 312 F.3d 905, 907 (7th Cir. 2002) (agreeing with the tax court that research must produce an “innovation in underlying principle”); United Stationers, Inc. v. United States, 163 F.3d 440, 444 (7th Cir. 1998) (“[Q]ualifying research must go beyond the current state of knowledge in that fieldâ€”expand or refine its principles.”); Norwest Corp. & Subsidiaries v. Comm’r, 110 T.C. 454, 493 (1998) (“The fact that the information is new to the taxpayer, but not new to others, is not sufficient for such information to come within the meaning of discovery for purposes of this test.”); but see Tax & Accounting Software Corp. v. United States, 301 F.3d 1254, 1262 (10th Cir. 2002) (“[T]he taxpayer must show that he discovered new information and that information must be separate from the product that is actually developed.”). The same is true of the district court’s conclusion that a “process of experimentation” requires forming and testing hypotheses. See Eustace, 312 F.3d at 907 (“[E]xperiment . . . [means] forming and testing hypotheses rather than the lay (or even engineering) sense of trial and error.”); Norwest Corp., 110 T.C. at 496 (“[T]he process of experimentation test is aimed at eliminating uncertainty about the technical ability to develop the productâ€”as opposed to uncertainty as to whether the product can be developed within certain business or economic constraints, even though the taxpayer knew that it was technically possible to develop it.”).
But in Eustace, the Seventh Circuit also recognized that:
In the long run, neither our view nor the tenth circuit’s has staying power. Both United Stationers and Tax & Accounting Software analyzed Â§ 41 without the benefit of the regulations that are supposed to illuminate the path to decision. Section 41’s predecessor was enacted in 1981, and Â§ 41 has been on the books in its current form since 1986, but the Internal Revenue Service has yet to promulgate the regulations that are important to this statutory design. (Section 41 refers ten times to regulations that the Secretary of the Treasury is to develop and issue.)
Eustace, 312 F.3d at 908. Those regulations were finally promulgated in December of 2003 (the “2003 Regulations”). Treas. Reg. Â§ 1.41-4. Of course, we afford Chevron deference to “an executive department’s construction of a statutory scheme it is entrusted to administer.” Chevron U.S.A., Inc. v. Natural Res. Def. Council, Inc., 467 U.S. 837, 844 (1984).
According to the 2003 Regulations, “discovering information . . . does not require the taxpayer be seeking to obtain information that exceeds, expands or refines the common knowledge of skilled professionals in the particular field of science or engineering in which the taxpayer is performing the research.” Treas. Reg. Â§ 1.41-4(a)(3)(ii). Rather, “[r]esearch is undertaken for the purpose of discovering information if it is intended to eliminate uncertainty concerning the development or improvement of a business component.” Id. Â§ 1.41-4(a)(3)(i). Thus, under the new regulations, qualified research must be intended to “eliminate uncertainty.” “Uncertainty exists if the information available to the taxpayer does not establish the capability or method for developing or improving the business component, or the appropriate design of the business component.” Id. “Process of experimentation” under the new regulations is also different, as it now involves three steps: (1) “the identification of uncertainty concerning the development or improvement of a business component,” (2) “the identification of one or more alternatives intended to eliminate that uncertainty,” and (3) “the identification and the conduct of a process of evaluating the alternatives (through, for example, modeling, simulation, or a systematic trial and error methodology).” Id. Â§ 1.41-4(a)(5)(i). The 2003 Regulations thus have different definitions for “discovering information” or “process of experimentation” than the definitions adopted by the district court.
The 2003 Regulations are, however, only effective for taxable years “ending on or after December 31, 2003.” Id. Â§ 1.41-4(e). Nor can McFerrin claim reliance on them since he filed his amended tax return in September 2003 while the new regulations were not promulgated until October 2003. But in promulgating Treasury Regulation Â§ 1.41-4, the IRS stated that “[f]or taxable years ending before December 31, 2003, the IRS will not challenge return positions that are consistent with these final regulations.” T.D. 9104, *26 (January 2, 2004). McFerrin clearly was relying on the 2001 Proposed Regulations, which had similar definitions for “discovering information” and “process of experimentation.” Prop. Treas. Reg. Â§ 1-41, 66 Fed. Reg. 66362-01 (Dec. 26, 2001). In the 2001 Proposed Regulations, the IRS specifically recognized that its earlier interpretation of “discovering information” did “not fully address Congress’ concerns regarding the importance of research activities to the U.S. economy.” Id. at 66363. The 2001 Proposed Regulations apply only to taxable years ending on or after December 26, 2001, but specifically state that:
Notwithstanding this prospective effective date, Treasury and the IRS believe that these rules prescribe the proper treatment of the expenditures they address, and the IRS generally will not challenge return positions consistent with the proposed regulations. Therefore, taxpayers may rely on these proposed regulations until the date final regulations under Â§ 1.41-4 are published in the Federal Register.
Id. at 66367 (emphasis added). This appears to be exactly what happened, as McFerrin relied on the proposed regulations in his amended returns.
The government conceded at oral argument that McFerrin could rely on the definitions from the 2003 Regulations in defending this suit. See also Union Carbide Corp. & Subsidiaries v. Comm’r, 97 T.C.M. (CCH) 1207, *77-*78 & n.42 (2009) (accepting that the previous definition of “discovering information” no longer applies after the 2003 Regulations). The district court thus erred by not reviewing the evidence under the definitions from the 2003 Regulations.
C. Harmless Error
The government argues, however, that any error was harmless and that no new trial is merited because: (1) even under the 2003 Regulations definitions, the work done by KMCO and KMTEX is not qualified research; and (2) the district court’s finding of a failure to substantiate did not depend upon whether there was qualified research.
On its first argument, however, the government concedes the point in its brief. The government argues that “most of taxpayers’ activities did not constitute research at all.” The district court accepted that a “few of the project[s] identified . . . may have involved some research,” but, applying the erroneous definitions of “discovering information” and “process of experimentation,” was unpersuaded that these were “qualified research.” As this finding of fact is based on an erroneous interpretation of law, it is accorded no deference. G.M. Trading Corp., 121 F.3d at 980. The government “bears the ultimate burden of proof to show not only that some amount has been erroneously refunded but also how much that amount is.” Soltermann, 272 F.2d at 387. If most of the activities were not qualified research, then the responsibility lies with the government to show what portion of the refund is attributable to those activities. On the other hand, if, using the correct definitions, some activities were “qualified research,” then McFerrin is entitled to keep at least a portion of the refund.
The government next argues that even if qualified research occurred, McFerrin failed to provide adequate documentation to substantiate the costs associated with that research. But this goes against the longstanding rule of Cohan v. Commissioner that if a qualified expense occurred, the court should estimate the allowable tax credit. 39 F.2d at 544. If McFerrin can show activities that were “qualified research,” then the court should estimate the expenses associated with those activities. The district court need not credit McFerrin’s reconstruction of expenses from years after the fact. See Eustace v. Comm’r, 81 T.C.M. (CCH) 1370, *5 (2001). But the court should look to testimony and other evidence, including the institutional knowledge of employees, in determining a fair estimate. See Fudim, 67 T.C.M. (CCH) 3011, *12-*13.
Further proceedings are warranted so that the district court may apply the correct definitions of “discovering information” and “process of experimentation” to the facts of this case, determine whether any qualified research occurred, and, if so, estimate the expenses related to that research.
D. McFerrin’s Bonus
The government argues that even if we remand for a new trial, it is entitled to the part of the refund attributable to McFerrin’s $6.4 million bonus in 1999. The government contends that this bonus was not attributable to the research work done by McFerrin, and is not a “qualified research expense.”
Under I.R.C. Â§ 41(b), “qualified research expenses” include “any wages paid or incurred to an employee for qualified services performed by such employee.” “Wages” are “all remuneration . . . for services performed by an employee for his employer,” I.R.C. Â§ 3401(a), and include “salaries, fees, bonuses,” etc., even when paid as a “percentage of profits,” Treas. Reg. Â§ 31-3401(a)-1. If a portion of the bonus was part of McFerrin’s wages for “qualified services” he performed at KMCO and was “reasonable under the circumstances,” I.R.C. Â§ 174(e), then it would be part of KMCO’s “qualified research expenses” and justify a tax credit.
The district court did not make any findings of fact on these questions, however, and only determined that the bonus was calculated based on the “profits and cash flow of KMCO in 1999” and not on the research performed that year. We decline to usurp the role of the district court as a fact-finder in this case, and leave it to the district court to make these factual determinations on remand.
In summary, in the lengthy bench trial, the court used incorrect definitions for “discovering information” and “process of experimentation.” Because there is the possibility that applying the correct definitions would result in at least some of the tax credit being found to have legitimately issued to McFerrin, we VACATE the judgment of the districtcourt, and REMAND for further proceedings consistent with this opinion.
 Arthur McFerrin’s wife, Dorothy F. McFerrin, is also a party to this case because of their joint income tax return.
 In 2005, Texas A&M University renamed its chemical engineering department the Artie McFerrin Department of Chemical Engineering.
 McFerrin further argues that the government only raised the substantiation argument after motions for summary judgment had been filed, and that the IRS revenue agent did not refer the case to IRS counsel based on a lack of substantiation. These arguments go beyond the face of the complaint and, to the extent that McFerrin is making an argument that the government may have waived its substantiation claim, he has failed to sufficiently appeal or argue that issue.
 The 2003 Regulations were promulgated following a two year period of comment on proposed regulations circulated in December of 2001 (the “2001 Proposed Regulations”). See Prop. Treas. Reg. Â§ 1-41, 66 Fed. Reg. 66362-01 (Dec. 26, 2001).
 The main difference is that the 2001 Proposed Regulations do not have the three step “processof experimentation” test described above, and instead require a “facts and circumstances” test to show a “process of experimentation.”
 As we are vacating the judgment and remanding for further proceedings, we need not consider McFerrin’s argument that the government was allowed to use an impermissible “project” based accounting methodology.