Modelling the longitudinal properties of financial ratios

Abstract

Previous studies provide conflicting evidence on the time series properties of company financial ratios, claiming either that the components of ratios exhibit nonstationarity which is not eliminated by the ratio transformation, or that a unit root in the components may be rejected which implies strong persistence in their ratio. In this article, a generalized model is derived that incorporates stochastic and deterministic trends and allows also for restricted and unrestricted proportionate growth in the ratio numerator and denominator. When the individual firm series are included in a panel analysis for large N and small T, this study is unable to reject convincingly a joint hypothesis of nonstationarity. However, the ratio variables are shown to be cointegrated, which can lead to stationarity in the ratio itself. Furthermore, evidence of cotrending provides support for a parsimonious model, where the financial ratio varies lognormally around its expected value.

Acknowledgements

We acknowledge the financial contribution of the Accounting Foundation at the University of Sydney. We are grateful to Aziz Jafaar for research assistance, and to John Goddard for insightful feedback on an earlier version, and to participants at the 50 Years of Econometrics conference, Erasmus University, Rotterdam (June 2006) and the Infiniti conference on International Finance, Trinity College, Dublin (June 2007), for their helpful comments on earlier versions of this article.

Notes

¹ Whittington and Tippett (1999), Ioannides et al. (2003) and Peel et al. (2004) make use of the same sample, comprising ratio time series for 110 firms that remained listed on the London Stock Exchange from 1948 to 1990.

² Accounting variables are reported periodically as financial statement line items, and are widely available as annual time series for large numbers of firms. The cross-sectional dimension tends to be large and the time dimension tends to be relatively small in terms of the count of repeated observations. The initial observation for a given firm may relate to the first year of its activities or, if censored on the left, to the first fiscal year covered by the database; each series will continue until the demise of the firm, or until the censor date on the right (i.e. the last fiscal year covered by the database), or through to the present time. Financial ratios constructed from such variables are also widely available in commercial databases.

³ It is not the aim of this article to derive the distributional form of return on equity, a ratio that has nonconvergent moments (i.e. if book equity reaches its lower bound of zero, the ratio is infinity, and if earnings are also at break-even, the ratio is undefined). Other related work on the statistical properties of ratios includes Horrigan (1965), Deakin (1976), Lev and Sunder (1979), Whittington (1980), Barnes (1982), Frecka and Hopwood (1983), McLeay (1986), Fieldsend et al. (1987), Tippett (1990), Rhys and Tippett (1993), Trigueiros (1994, 1995), Kallunki et al. (1996), McLeay (1997) and Ashton et al. (2004). For a recent application that includes two pure financial ratios (leverage and asset turnover), see McGowan (2007).

⁴ Note that a model that allows us to relax the assumption δ1 = δ2 = 0 may be derived with recursive substitution. The derivation of this more general model is given in the Appendix.

⁵ Whittington and Tippett (1999) provide an excellent overview of cointegration and unit roots. Their evidence is based on Dickey–Fuller tests at the firm level, and suggests that cointegration between numerator and denominator does not always remove the effects of nonstationarity in the ratio components, even when drift in the ratio is accounted for with an additional trend term. In the context of panel data, research in applied economics demonstrates how the connection between firm level variables may be modelled with panel cointegration analysis – see de Jong (2007) for example, on Granger causality between capital investment and research expenditure.

⁶ The stability of the distributions of the three pure ratios is in contrast to Return on Equity, where the best fit varies between normal, Student t, logistic, beta and log-logistic, as is also shown in Table 1. However, this overfitting is attributable to small numbers of extreme values – if the extreme negative values that arise when Shareholders’ Equity approaches its lower bound of zero are removed, we find a consistently best fit by the logistic distribution.

⁷ Extreme value distributions are commonly applied in survival analysis – see George and Devidas (1992).

⁸ By construction, the mean residual is zero, and the estimate of a is not significantly different to zero (the 0.05 and 0.95 confidence limits are below and above zero, respectively, for each of the five variables of interest).

⁹ In the context of GARCH modelling, Sjölander (2008) examines the robustness of unit root tests to stationary distortions when the number of observations is high, and demonstrates that the most commonly applied unit root tests exhibit considerable sample size bias.

¹⁰ For fixed and random effects models, where the time dimension is small and the cross-sectional dimension is large, maximum likelihood estimators have also been obtained and their finite sample properties documented (Binder et al., 2005).

¹¹ The critical values of the limit distribution of the test-statistic are tabulated in Pesaran (2007) for N = 10, ···, 200 and T = 10, …, 200. When the model has an intercept but no trend, the critical values for the largest N = 200 and the shortest T = 10 are as follows:

¹² As the main interest is not in the contemporaneous correlation between ratio components but in any systematic effects in the error term, firm level estimates β_j are not required, and a panel data analysis may be undertaken to allow for between-firm variation as fixed effects. In this case, the fixed effects estimates of b are: Sales to Assets 0.3569 (SE = 0.0140); Liabilities to Equity 0.4118 (SE = 0.0129); and Costs to Liabilities 0.3436 (SE = 0.0143), and b = 1 is rejected for each of the three pairings.

¹³ Although a predicted value may be fitted to year 1 for the null and deterministic trend models, there is no initial prediction in the case of the full and stochastic trend models, which are autoregressive. Given that the firm series are short, and the degrees of freedom are a function of series length, we compare all model fits by excluding year 1 from the mean squared error of the deterministic trend models and the null. The number of observations in 2–8 years is 35 per firm when all five variables are considered jointly in the seemingly unrelated regression that restricts the proportionate growth model. The mean squared error is calculated as the sum of squared errors divided by the adjusted degrees of freedom (i.e. 35 less the number of parameters in the model), and standard F-tests are used here to compare models and to evaluate model restrictions, given the small samples by firm.

¹⁴ In this article, we do not attempt to incorporate level shifts, nor do we attempt to model the deterministic trend as nonlinear rather than nonstationary. Further research is required on these issues in order to develop robust methods in the context of small T and large N panels. One avenue of research would be to further explore the partial adjustment process documented in Ioannides et al. (2003). Other recent findings in financial economics that may inform the short panel analysis of financial ratios include those of Maki (2006), McMillan (2007) and Serrasqueiro et al. (2007). MacMillan (2007) tests financial ratio series for structural breaks with regard to linear and nonlinear processes, and Maki (2006) includes a unit root test that is applicable to a wide range of linear and nonlinear processes for time series having a mean shift at an unknown point, whilst Serrasqueiro et al. (2007) evaluate nonlinearities in firm growth and profitability using different panel estimators.