Cross-Section Unit-Root Tests of The Returns on Equity and Invested Capital: State-Dependent Mixtures of I(1) and I(0) Dynamics in the ROE and ROIC

Cross-section finance studies virtually never test variables for unit roots, though econometric validity depends on testing. This paper develops a cross-section unit-root test and uses it on annual data for the return on equity and the return on invested capital, ROE and ROIC, key variables in the Edwards-BellOhlson (EBO) accounting and Free Cash Flow (FCF) finance valuation models. Conventional wisdom is that ROE ∼ I(0) ∼ ROIC; Siddique and Sweeney (2000) present panel evidence that ROE ∼ I(1) ∼ ROIC. For cross sections, this paper tests the null that both returns are I(1) against the alternative of I(0). Because a firm is unlikely to survive with rates persistently negative, however, another alternative considered is rates are I(1) or I(0) as the lagged rates are positive or negative. The cross-section test regresses changes in the rate on lagged levels: It easily accommodates state-dependent mixtures of dynamics as in the second alternative, and allows various types of crossand serial-correlation in the data. The test statistic is the slope’s t-value against zero. For normal errors and N firms, the t-value follows the t-distribution with k = N-2 degrees of freedom, and is approximately N(0, 1) for k ≥ 60; for errors subject to the central limit theorem, the t-value is asymptotically N(0, 1). In contrast, panel tests cannot be used on a single cross section; further, existing panel tests cannot handle state-dependent mixtures of I(1) and I(0) dynamics. For annual data, ROE ∼ I(1) ∼ ROIC for positive rates, 75-80% of the sample. But ROE ∼ I(0) ∼ ROIC for negative rates, about 20% of the sample; each year, many of the negative-rate firms have positive rates the following year. Corrections for heteroskedasticity and error cross correlations do not qualitatively affect results. * For helpful discussions, thanks are owed, in particular, to Keith Ord, and to Jim Bodurtha, Bardia Kamrad, Lee Pinkowitz, Scott Whisenant, Othmar Winckler and Teri Yohn, and participants in the Finance and Accounting Seminar at The McDonough School of Business. Georgetown University and The McDonough School of Business provided summer research support, and the Capital Markets Research Center provided summer research assistance. Part of this paper was written at Göteborgs Univeristet. . Eberhart: (O) 202-6874584; fax 202-6874031; [email protected] Siddique: (O) 202-6873771; fax 202-6874031; [email protected] Sweeney: (O) 202-687-3742; fax 202-687-7639, 4031; [email protected] Cross-Section Unit-Root Tests of The Returns on Equity and Invested Capital: State-Dependent Mixtures of I(1) and I(0) Dynamics in the ROE and ROIC Abstract: Cross-section finance studies virtually never test variables for unit roots, though econometric validity depends on testing. This paper develops a cross-section unit-root test and uses it on annual data for the return on equity and the return on invested capital, ROE and ROIC, key variables in the Edwards-BellOhlson (EBO) accounting and Free Cash Flow (FCF) finance valuation models. Conventional wisdom is that ROE ∼ I(0) ∼ ROIC; Siddique and Sweeney (2000) present panel evidence that ROE ∼ I(1) ∼ ROIC. For cross sections, this paper tests the null that both returns are I(1) against the alternative of I(0). Because a firm is unlikely to survive with rates persistently negative, however, another alternative considered is rates are I(1) or I(0) as the lagged rates are positive or negative. The cross-section test regresses changes in the rate on lagged levels: It easily accommodates state-dependent mixtures of dynamics as in the second alternative, and allows various types of crossand serial-correlation in the data. The test statistic is the slope’s t-value against zero. For normal errors and N firms, the t-value follows the t-distribution with k = N-2 degrees of freedom, and is approximately N(0, 1) for k ≥ 60; for errors subject to the central limit theorem, the t-value is asymptotically N(0, 1). In contrast, panel tests cannot be used on a single cross section; further, existing panel tests cannot handle state-dependent mixtures of I(1) and I(0) dynamics. For annual data, ROE ∼ I(1) ∼ ROIC for positive rates, 75-80% of the sample. But ROE ∼ I(0) ∼ ROIC for negative rates, about 20% of the sample; each year, many of the negative-rate firms have positive rates the following year. Corrections for heteroskedasticity and error cross correlations do not qualitatively affect results. Cross-section finance studies virtually never test variables for unit roots, though econometric validity depends on testing. This paper develops a cross-section unit-root test and uses it on annual data for the return on equity and the return on invested capital, ROE and ROIC, key variables in the Edwards-BellOhlson (EBO) accounting and Free Cash Flow (FCF) finance valuation models. Conventional wisdom is that ROE ∼ I(0) ∼ ROIC; Siddique and Sweeney (2000) present panel evidence that ROE ∼ I(1) ∼ ROIC. For cross sections, this paper tests the null that both returns are I(1) against the alternative of I(0). Because a firm is unlikely to survive with rates persistently negative, however, another alternative considered is rates are I(1) or I(0) as the lagged rates are positive or negative. The cross-section test regresses changes in the rate on lagged levels: It easily accommodates state-dependent mixtures of dynamics as in the second alternative, and allows various types of crossand serial-correlation in the data. The test statistic is the slope’s t-value against zero. For normal errors and N firms, the t-value follows the t-distribution with k = N-2 degrees of freedom, and is approximately N(0, 1) for k ≥ 60; for errors subject to the central limit theorem, the t-value is asymptotically N(0, 1). In contrast, panel tests cannot be used on a single cross section; further, existing panel tests cannot handle state-dependent mixtures of I(1) and I(0) dynamics. For annual data, ROE ∼ I(1) ∼ ROIC for positive rates, 75-80% of the sample. But ROE ∼ I(0) ∼ ROIC for negative rates, about 20% of the sample; each year, many of the negative-rate firms have positive rates the following year. Corrections for heteroskedasticity and error cross correlations do not qualitatively affect results.