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Linear valuation without OLS: the Theil-Sen estimation approach

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Abstract

OLS-based archival accounting research encounters two well-known problems. First, outliers tend to influence results excessively. Second, heteroscedastic error terms raise the specter of inefficient estimation and the need to scale variables. This paper applies a robust estimation approach due to Theil (Nederlandse Akademie Wetenchappen Ser A 53:386–392, 1950) and Sen (J Am Stat Assoc 63(324):1379–1389, 1968) (TS henceforth). The TS method is easily understood, and it circumvents the two problems in an elegant, direct way. Because TS and OLS are roughly equally efficient under OLS-ideal conditions (Wilcox, Fundamentals of modern statistical methods: substantially improving power and accuracy, 2nd edn. Springer, New York 2010), one naturally hypothesizes that TS should be more efficient than OLS under non-ideal conditions. This research compares the relative efficiency of OLS versus TS in cross-sectional valuation settings. There are two dependent variables, market value and subsequent year earnings; basic accounting variables appear on the equations’ right-hand side. Two criteria are used to compare the estimation methods’ performance: (i) the inter-temporal stability of estimated coefficients and (ii) the goodness-of-fit as measured by the fitted values’ ability to explain actual values. TS dominates OLS on both criteria, and often materially so. Differences in inter-temporal stability of estimated coefficients are particularly apparent, partially due to OLS estimates occasionally resulting in “incorrect” signs. Conclusions remain even if winsorization and the scaling of variables modify OLS.

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Notes

  1. Note that we use the marginal median—univariate medians for each variable—but there are other ways to obtain a median in multivariate cases. One such variation of multivariate medians considers the geometric median.

  2. If one changes the number of independent variables from two to an arbitrary K independent variables, yet K/N is relatively small, then it becomes impractical to consider all possible combinations of K vectors of observations. The number can be truly large since the order of magnitude is N!/{K!(N–K)!} (which is impossibly large, probably even by the standards of computers if, say, N > 1,000 and K > 5). To deal with this issue, one can simply draw K vectors of observations at random, solve for the coefficients, and then repeat the process a very large number of times, say, 100,000 times. The adequacy of this number is easily checked since one may consider the implications of changing it to, say, 200,000 times. In this context, note that, even if N or K are small, one can still use simulation; there will be no harm done by having the number of random draws much larger than N!/{K!(N − K)!}. Thus standardized software is likely to use random drawings rather than consider all possible combinations resulting in coefficient derivations. In this paper, we rely on all possible pairs of points for estimation. (To be sure, however, using the random drawing approach does not change the results.) The SAS program used to calculate TS estimates is available from the authors. It requires some adjustments to apply in other models.

  3. We check on this asymptotic efficiency claim by conducting elementary computer simulations. Specifically, we consider the case of four independent variables (all correlated), N = 100 and various N materially greater than 100, various R-squared and, critically, error terms simulated to be normally distributed with the same variance for all observations. We find no discernible evidence that OLS estimates, on average, come closer to the true coefficients compared to the TS estimates. Both estimation methods seem, basically, to perform about the same. The finding is consistent with the literature on TS (e.g., Wilcox 2004). Also, TS estimators are unbiased under certain conditions of symmetry (Wang and Yu 2005).

  4. Our appendix provides an illustration of how TS compares to OLS in terms of outliers.

  5. In addition to the appeal of not having to decide how to scale, Wilcox (2013) finds that the TS estimator performs better than alternative methods in data where heteroscedasticity is a concern, except in cases of a very small sample. Our simulations, involving relative large N, confirm that TS performs strikingly better than OLS in settings with (material) heteroscedasticity.

  6. When we refer to valuation contexts, we include cases where accounting variables (for example, earnings or net accruals) are on the right-hand side or earnings forecasting settings. It should further be noted that variables can be on a per share basis. This aspect, however, does not eliminate the heteroscedasticity issue in an OLS context. (Berkshire Hathaway is the prototypical illustration).

  7. See Leone et al. (2014) for a discussion of alternative robust estimators.

  8. However, especially in environmental sciences, there is an extensive list of research that applies TS to estimate trends in pollution, climate change, and rainfall, where the data do not satisfy the assumptions for classical parametric methods (e.g., Hirsch et al. 1982).

  9. We find similar results, in terms of stability of coefficients and goodness-of-fit comparison between TS vs. OLS, for a larger set of S&P 1,500 firms and also separately for S&P MidCap 400 and S&P SmallCap 600 firms.

  10. We use income before extraordinary items on Compustat, but the results do not change when we use net income.

  11. If one wants to refine the a + b = 0 hypotheses, one can perhaps argue that the liability variable should have more weight insofar unsuccessful firms tend to be more leveraged.

  12. To test the significance of TS estimates, we use the bootstrap percentile method following Wilcox (2010). From N observations, we draw with replacement a bootstrap sample of size N and compute TS estimates. This process is repeated 600 times, which has been shown to be sufficient in most situations. Lastly, we count the number of cases where estimates from the bootstrap samples are smaller (larger) than zero for positive (negative) TS coefficients and divide by 300 for a two-tailed p value.

    A more straightforward approach to derive p-values may focus on the extent to which a specific independent variable contributes to the explanation of the dependent variable. This mode of thinking is entirely consistent with classical analysis. Specifically, one can compare the accuracy of two sets of variables, one with the specific variable and one without. Two models can now be estimated. One can then assess whether the model with the variable is significantly more accurate via standard binomial tests by counting the number of times one model’s projected value is closer to the actual value than the other’s (i.e., the relative accuracy score used throughout this paper).

  13. As noted previously, experiment D provides a more complete analysis by putting assets, liabilities, and net income on the right-hand side.

  14. The relative accuracy score is computed as follows. Let I(x) equal one if x > 0 and zero otherwise. Consider two models, M and N, with fitted values yMi and yNi. Let yi denote the actual value given i = 1, …, n data points. Calculate \(1/{\text{n }}\varSigma {\mathbf{I}}\{ \left| {{\text{ y}}_{\text{Mi}} {-}{\text{y}}_{\text{i}} \left| { \, {-} \, } \right|{\text{y}}_{\text{Ni}} {-}{\text{y}}_{\text{i}} } \right|\}\)

    Given an i-independent distribution, under the null the expected value is 50 percent and the distribution is binomial. Note that the metric works regardless of the methodology that projects the dependent variable.

  15. In the first equation one can replace Pricet-1 with Pricet. Compared to OLS, results as to stability of estimates and goodness-of-fit do not change: TS dominates without any ambiguities. As an aside, we hypothesize that this kind of model will be extremely hard to beat if the objective is to forecast next period earnings as accurately as possible.

  16. The stability of estimates is also appreciated, at least implicitly, in many studies that average the cross-sectional regression coefficients across years to reach a final estimate (e.g., Fama–MacBeth regressions).

  17. The only exception is Gerakos and Gramacy (2013), who compare the accuracy of earnings forecasting models based on OLS with and without winsorization but find mixed results for scaled and unscaled earnings.

  18. LAD is a special case of a quantile regression that estimates the conditional median. Koenker and Bassett (1978) propose quantile regressions that estimate the conditional quantile as opposed to OLS regressions that estimate the conditional mean. The estimation requires minimizing the sum of weighted absolute residuals where the weight is asymmetrically given to residuals with positive and negative values. Quantile regressions can be useful to observe differences of an effect across the distribution. It is also used to test the robustness of results at the tail end of the distribution and to reduce weight of outliers via median regression (LAD).

  19. For example, all accruals estimation models (e.g., Jones 1991; Dechow et al. 1995; Dechow and Dichev 2002) deflate variables by total assets.

  20. OLS has a breakdown point (i.e., the maximum proportion of outliers in the sample that the estimator can handle without yielding incorrect results) of 1/n. Guthrie et al. (2012) shows how two outliers in a sample of 865 firms can result in different inferences when relying on OLS.

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Acknowledgments

We thank Sudipta Basu, Ilia Dichev, Stan Markov, Stephen Penman, Kam-Ming Wan, and Rand Wilcox for helpful comments. Kim gratefully acknowledges financial support from the Samsung Scholarship.

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Authors and Affiliations

  1. Hong Kong Polytechnic University, Kowloon, Hong Kong

    James A. Ohlson

  2. Cheung Kong Graduate School of Business, Beijing, China

    James A. Ohlson

  3. New York University, New York, NY, USA

    Seil Kim

Authors
  1. James A. Ohlson
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Correspondence to James A. Ohlson.

Appendix: An illustration of OLS and TS

Appendix: An illustration of OLS and TS

The Theil-Sen (TS) method computes the slopes of all possible pairs of observations and takes the median value. OLS can also be expressed using the slopes of all possible pairs of observations (Heitmann and Ord 1985; Wilcox 2010):

$$b_{OLS} = \frac{{cov\left( {x,y} \right)}}{var\left( x \right)} = \frac{{\frac{1}{{2n^{2} }}\mathop \sum \nolimits_{j} \mathop \sum \nolimits_{k} \left( {x_{j} - x_{k} } \right)\left( {y_{j} - y_{k} } \right)}}{{\frac{1}{{2n^{2} }}\mathop \sum \nolimits_{j} \mathop \sum \nolimits_{k} \left( {x_{j} - x_{k} } \right)^{2} }} = \frac{{\mathop \sum \nolimits_{j < k} \left( {x_{j} - x_{k} } \right)\left( {y_{j} - y_{k} } \right)}}{{\mathop \sum \nolimits_{j < k} \left( {x_{j} - x_{k} } \right)^{2} }} = \mathop \sum \limits_{j < k} w_{j,k} \frac{{(y_{j} - y_{k} )}}{{(x_{j} - x_{k} )}}$$

where

$$w_{j,k} = \frac{{\left( {x_{j} - x_{k} } \right)^{2} }}{{\mathop \sum \nolimits_{j < k} \left( {x_{j} - x_{k} } \right)^{2} }}.$$

More specifically, the OLS slope equals the weighted average of slopes from all possible pairs where the weight for pair (j,k) is wj,k. Note that the weight is proportional to the squared difference of the two x-values and the sum of all weights equals one.

From this expression of OLS, it is not obvious why we would prefer a particular weighting based on only x-values or use any weighting at all when fitting a linear model. The OLS weighting scheme can be problematic if undue weight is given to outliers. The below example illustrates how even a single outlier can influence OLS estimates.Footnote 20

Consider a set of five data points plotted in Fig. 1: (5, 10), (2, 7), (1, 3), (7, 15), and (19, 1). With the exception of an outlier in the lower right corner, the data show an increasing trend. However, the OLS slope is −0.26. As shown in Table 8, the outlier generates negative slopes when paired with other points and has the greatest weight, driving down the estimate. Note that the four slopes based on the outlier account for 92 % (=0.19 + 0.28 + 0.31 + 0.14) of total weights while representing 40 % of all possible slopes, only because the outlier has an x-value distinct from that of other points. In contrast, the TS slope is the median of the 10 slopes and equals 1.30. The TS slope it is not influenced by the outlier and well represents the increasing trend.

Fig. 1
figure 1

Plot of OLS and TS slopes

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Table 8 Example of OLS slope calculation using all possible pairs of slopes
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Ohlson, J.A., Kim, S. Linear valuation without OLS: the Theil-Sen estimation approach. Rev Account Stud 20, 395–435 (2015). https://doi.org/10.1007/s11142-014-9300-0

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