A fractional cointegration approach to testing the Ohlson accounting based valuation model

Abstract

We examine the long-run relationship between market value, book value, and residual income in the Ohlson (Contemp Acc Res 11(2):661–687, 1995) model. In particular, we test if market value is cointegrated with book value and residual income in light of their non-stationary behaviors. We find that cointegration applies to only 51 % of the sample firms, casting doubt that book value and residual income alone are adequate in tracking variations in market value, yet we find that market value is fractional cointegrated with book value and residual income for 89 % of the sample firms. This implies that the long-run relationship follows a slow but mean-reverting process. Our results therefore support the Ohlson model.

Appendix

Estimating the relationship between market value, book value, and residual income under fractional cointegration.

Based on Barkoulas et al. (2000), let ARFIMA (p, d, q) denote the model of an autoregressive fractionally integrated moving average process of order (p,d,q), with constant μ. Using operator notation, it can be expressed as:

$$ \Upphi (L)(1 - L)^{d} (y_{t} - \mu ) = \Uptheta (L)u_{t} ,\,\,u_{t} \sim i.i.d.\,(0,\sigma_{u}^{2} ), $$

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where L is the backward-shift operator, $\Upphi (L) = 1 - \varphi_{1} L - \cdots - \varphi_{P} L^{P} ,$ $\Uptheta (L) = 1\,+$ $\vartheta_{1} L + \cdots + \vartheta_{q} L^{q} ,$ and $(1 - L)^{d}$ is the fractional differencing operator defined by:

$$ (1 - L)^{d} = \sum\limits_{k = 0}^{\infty } {\frac{{\Upgamma (k - d)L^{k} }}{\Upgamma ( - d)\Upgamma (k + 1)}} , $$

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with $\Upgamma ( \cdot )$ denoting the gamma function. The parameter d takes a real value.

The arbitrary restriction of d to integer values gives rise to the standard autoregressive integrated moving average (ARIMA) model. If all roots of $\Upphi (L)$ and $\Uptheta (L)$ lie outside the unit circle and $\left| d \right|$ < 0.5, then the stochastic process $y_{t}$ is both stationary and invertible. Assuming that $d \in (0,0.5)$ and $d \ne 0$, Hosking (1981) shows that the correlation function, $\rho ( \cdot )$, of an ARFIMA process is proportional to $k^{2d - 1}$ as $k \to \infty$.

The autocorrelations of the ARFIMA process consequently decay hyperbolically to zero as $k \to \infty$, contrary to the geometric decay of a stationary ARMA process. For $d \in (0,0.5)$, $\sum\limits_{j = - n}^{n} {\left| {\rho (j)} \right|}$ diverges as $n \to \infty$, and the ARFIMA process is said to exhibit a long memory, or long-range positive dependence. For $d \in ( - 0.5,0)$, it is said that the process exhibits intermediate memory, or long-range negative dependence. The process is said to have short memory for d = 0, corresponding to the stationary and invertible ARMA modeling. For $d \in (0.5,1)$, the process is non-stationary (having an infinite variance), but is mean reverting since there is no long-run impact of an innovation on future values of the process.

To estimate the long-memory parameter, we use Robinson’s Gaussian semiparametric method. Robinson (1995) proposes a Gaussian semiparametric estimate, (GS hereafter), of the self-similarity parameter H, which is not defined in closed form. The spectral density of the time series, denoted by $f\left( \cdot \right)$, can be described as:

$$ f(\xi )\sim G\xi^{1 - 2H} \,{\text{as}}\,\xi \to 0^{ + } , $$

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for $G \in (0,\infty )$ and $H \in (0,1)$. The self-similarity parameter H relates to the long-memory parameter d by H = d + 1/2. The estimate for H, denoted by $\hat{H}$, is obtained through the minimization of the function:

$$ R(H) = \ln \hat{G}(H) - (2H - 1){\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 v}}\right.\kern-0pt} \!\lower0.7ex\hbox{$v$}}\sum\limits_{\lambda = 1}^{v} {\ln \xi_{\lambda } } , $$

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with respect to H, where $\hat{G}(H) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 v}}\right.\kern-0pt} \!\lower0.7ex\hbox{$v$}}\sum\limits_{\lambda = 1}^{v} {\xi_{\lambda }^{2H - 1} I(\xi_{\lambda } ),}$ $I(\xi_{\lambda } )$ is the periodogram of $y_{t}$ at frequency $\xi_{\lambda }$, and v is the number of Fourier frequencies included in estimation (bandwidth parameter). The discrete averaging is carried out over the neighborhood of zero frequency.

Asymptotic theory assumes that ν goes to infinity much slower than T. The GS estimator of $v^{1/2}$ is consistent. Its variance of the limiting distribution is free of nuisance parameters and equals 1/4ν. The GS estimator appears to be the most efficient semiparametric estimator. It remains consistent with the same limiting distribution under conditional heteroscedasticity (Robinson and Henry (1999)).