The double entry structural constraint on the econometric estimation of accounting variables

ABSTRACT

This paper develops a structural system for estimating accounting variables, within which the deterministic relationships inherent in financial statement articulation are clearly defined in the econometric model. The key proposition of the paper lies in the treatment of the financial statements as a matrix of codetermined information constrained by double entry, where the expected value of each of the individual items that comprise the financial statements will be mirrored elsewhere in the system with a different sign. Given that the change in net operating assets shares the same variation as the change in net financial claims, it is shown, by formally identifying the articulation, that empirical application will yield increased precision and improved efficiency by comparison to the more traditional methods that fail to specify the structural double entry property.

KEYWORDS:

• Double entry constraint
• financial econometrics
• financial statements
• identification
• structural systems

JEL CLASSIFICATIONS:

• M41
• C39
• C51

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The reconciled position at t is an audit requirement prior to the publication of the financial statements that can only be temporarily violated during equity restructures or combinations. Also, the accrual flow may include comprehensive income items.

2. Such looping processes have also been described as a source of endogeneity by Butterworth (1972) and Arya et al. (2000, 2004).

3. The particular set of variables described in Figure 1 has been identified as that for which there is sufficient disclosure by Compustat to allow for comparable coverage with a complete set of observations for each listed non-financial firm. Using the source data, a summary financial reconciliation statement has been constructed for each firm in each financial period. The Appendix provides a numerical example for Nike Inc using Compustat data. We have also produced the same articulated structure for the Datastream database which is available upon request.

4. This is a highly simplified representation for a firm that holds no inventory, where current assets are made up entirely of accounts receivable from customers, and the current liabilities are accounts payable to suppliers. The double entry structure may easily be adapted – for instance, the second column may be rewritten so that payments to suppliers (Cr) and cost of goods sold (Dr) articulate with the change in accounts payable net of inventories.

5. As in Fairfield, Whisenant, and Yohn (2003), fiscal obligations are treated here as a component of $\mathrm{\Delta }NFC$, together with the claims from equity holders and debt holders, and, given the complexities surrounding deferred taxation, we group all tax items together for the purpose of this paper. It is also assumed that any cash holdings and interest received have been offset against debt and interest charges, again in order to simplify presentation in this paper. The double entry structure could easily be adapted to treat these separately.

6. The terminology and structural forms adopted in this paper are inspired by Kmenta (1997).

7. It is not necessary that all dependent variables of the system are identified only by diagonal elements of the matrix $\mathit{\alpha }$ but it does simplify exposition considerably. Then the elements of the vector of the endogenous variables must be ordered in a way to reflect this representation.

8. For simplicity, we assume the spherical form of the structural variance-covariance (VCV) matrix, ${\mathit{\sigma }}_{m\times m}\otimes {\mathbf{\text{I}}}_N=E\left[{\mathit{\alpha }}_{m\times m}^{-1}{\mathbf{\text{u}}}_{m\times 1}{\mathbf{\text{u}}}_{m\times 1}^{\prime }\right({\mathit{\alpha }}_{m\times m}^{-1}{)}^{\prime }]$, where the identity matrix ${\mathbf{\text{I}}}_N$ is of order $N=n\times T$. For more precise empirical weight functions to the VCV see, amongst others, Greene (2011). The dimension of ${\mathit{\sigma }}_{m\times m}$ is reduced by those equations m that constitute identities given the absence of error term, as in the case of accounting identities.

9. When the structural model is exactly identified then the two estimators yield identical results, but when it is overidentified (i.e. there are more instruments than covariates) then it is worth contrasting the two using a Hausman specification test

10. The asymptotic bias of Equation (8(8) ${\mathbf{\text{y}}}_{m\times 1}={\text{β}}_{m\times l}{\mathbf{\text{y}}}_{l\times 1}+{\mathit{\gamma }}_{m\times k}{\mathbf{\text{x}}}_{k\times 1}+{\mathbf{\text{u}}}_{m\times 1}$(8) ) is at most of order 1/N (Zellner 1962), but unbiasedness in all finite states is ensured by deflating $\mathit{\sigma }$ with N + 1−l−k (Mehta and Swamy 1976).

11. Formally, $\mathbf{\text{J}}$ is an inner vector with dimension $j\times 1$, and the assigned coefficients ${\mathit{\gamma }}_{\mathit{j}}$ represent inner vectors with dimension $1\times j$, where $j=1,2,\dots ,J-1$. It follows that the industry effects are interpreted relative to the baseline effect ${\gamma }_0$ that takes on the role of the excluded industry fixed effect. Similarly for $\mathbf{\text{S}}$.

12. We recognise the potential panel data effects of the firm-year dataset leading to a likely downwards bias due to the dynamic system, but at this stage we prefer to avoid any further complication of the structural model given the need for dynamic system GMM specification.

13. Standard financial analysis relies on the use of common-size statements. For example, dividing all components on the balance sheet by total assets creates a common-size balance sheet, so that the resulting percentages eliminate the size-effect and enable comparative study. However, deflation by total assets does not create proportional measurements for a reconciliation statement since sales and expenses are often much larger. For this sample, 62.86% of the reconciliations show sales greater than total assets. Therefore, a common-size reconciliation requires a deflation by the total volume of transactions that flow within a year, i.e. the addition of the absolute value of all fourteen variables that define the structural equation. Christodoulou and McLeay (2009) demonstrate the notable benefits from considering such standardised bounded variations especially in dealing with problems related to extreme values. For completeness, we confirm that our results are robust to deflation by total assets.