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Testing for incremental information content in the presence of collinearity
Journal of Accounting and Economics 6 (1984) 219-223. North-Holland
TESTING FOR INCREMENTAL INFORMATION CONTENT IN THE PRESENCE OF COLLINEARITY A Comment
William H. BEAVER Stanford University, Stanford, CA 94305, USA
Paul A. G R I F F I N Unioersity of California, Daois, CA 95616, USA
Wayne R. LANDSMAN Unioersity of California, Los Angeles, CA 90024, USA Received June 1984, final version received July 1984
Christie et al. (1984) make two major points: (1) A two-stage regression is not the econometricaUy unique way to examine the incremental explanatory power of an additional variable (e.g., an earnings variable based on ASR 190 data). 1 (2) A two-stage regression approach does not 'overcome' or 'solve' the collinearity issue. Beaver, Griffin and Landsman (1982) (BGL) or Beaver and Landsman (1983) (BL) do not claim that the two-stage approach is the unique approach or that it overcomes the collinearity issue. Yet, because Christie et al. perceive that the papers are open to such interpretations, we welcome this opportunity to elaborate upon the motivation for our choice of the two-stage approach from a larger set of econometrically equivalent approaches. We view the issue as one of presentation of results rather than econometric 'correctness'. The interesting research design issue concerns how the researcher incorporates the institutional details of the topic at hand to choose among virtually equivalent approaches from an econometric perspective. As stated on page 16, BGL examines whether the inclusion of an earnings variable based on ASR 190 data adds to the explanatory power with respect to cross-sectional differences in security returns beyond that provided by historical cost (HC) earnings. There are a number of ways to assess the statistical significance of an additional explanatory variable. These include (1) a F-test 1Throughout we will assume a setting with two explanatory variables, a historical cost earnings variable and a second earnings variable based on ASR 190 data. 0165-4101/84/$3.00©1984, Elsevier Science Publishers B.V. (North-Holland)
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on the reduction in residual variance achieved by adding the variable (an approach cited in footnote 8 of BGL); (2) a single-stage, two-variable (i.e., a multiple regression) approach suggested by Christie et al., and (3) a two-stage approach as in BGL or BL. Since they are econometrically equivalent for purposes of testing the significance of an added explanatory variable, the choice among them must rest on grounds other than statistical. The two-stage approach constitutes a convenient and reasonable choice based on two criteria: consideration of the institutional aspects of the reporting issue and comparison with previous research. (1) It explicitly reflects the institutional aspects of the ASR 190 data (i.e., their supplemental nature). (2) It presents the coefficients on the HC earnings variable in a manner that facilitates comparison with previous studies of single-variable HC earnings regressions [e.g., Beaver, Lambert and Morse (1980)]. As discussed in BGL and BL, the ASR 190 (as well as FAS 33) data are viewed as supplemental to, not a replacement for, the historical cost data. The regulatory based arguments alleged the ASR 190 data would provide information beyond that provided by historical cost data. Moreover, the ASR 190 data were constructed by making a series of adjustments to the HC data. Hence, the primary research issue is the incremental explanatory power of the adjusted data. The two-stage approach focuses on the incremental nature of the supplemental data in a structurally explicit manner. In particular, it makes explicit that in an OLS world, the additional explanatory power of the ASR 190 earnings variable is constrained to be that portion of the original variable that is orthogonal to the HC earnings variable. While the three approaches cited above are econometrically equivalent with respect to testing the significance of adding the ASR 190 earnings variable, they are not econometrically equivalent methods of reporting other aspects of the results. 2 in particular, the single-stage approach does not provide the same information about the coefficient (and t-value) associated with the HC variable. Under a two-stage approach, the HC coefficient (by construction) cannot be affected by the addition of the (orthogonal) ASR 190 variable. However, in a single-stage approach the HC coefficient and its t-value are affected by the collinearity with the ASR 190 variable. Typically, the t-value will be eroded by the presence of a collinear variable. However, we view the erosion effect on the H C variable to be of secondary interest, at best, in the context of this particular research issue. For example, suppose the HC and the ASR 190 earnings variables are virtually perfectly coUinear. The t-values on both variables in a single-stage equation could be near-zero, in spite of a significant
2 The F-test, cited in footnote 8 of BGL, does not provide the same information as the two-stage approach. It provides no information on either the coefficient or the t-value of the HC variable. It does not indicate the sign of the t-value of the ASR 190 variable. Further, it also requires two regressions.
W.H. Beaoer et al., Reply
221
F-value for the entire regression. The erosion of the t-value on the HC variable induced by the addition of a second highly coUinear variable (which was constructed by making a series of adjustments to HC) is not of major interest here. In this context, the collinearity between the HC and the ASR 190 variable is of secondary interest, relative to the primary issue of incremental explanatory power. However, we will comment more about collinearity issues later. With respect to comparison with previous studies, the two-stage approach reports the coefficient (but not the t-value) on the H C variable that would have been reported in a single variable equation. The presentation of the information facilitates comparison with prior research in which a HC earnings variable is used as a single variable to explain cross-sectional differences in security returns. The research dealing with the association between security returns and historical cost earnings has a well-established heritage [e.g., since Ball and Brown (1968)]. Given the role played by H C earnings in several years of previous research, the coefficient of HC earnings in a single-variable model is of interest. The two-stage approach provides that information, while the single-stage, two-variable approach does not. Of course, some readers may be interested as well in the incremental explanatory power of HC earnings, beyond that provided by the ASR 190 earnings variable. Such information can be provided by a second set of two-stage regression, where the HC variable is constructed to be orthogonal to the A S R 190 variable. 3 If this second set had not been reported, the single-stage, two-variable approach would provide information not provided by the BGL study. However, we report such results (labeled as 'turning the tables' on historical cost). Hence, the issue is not one of disclosure of results, but rather the sequence and the relative emphasis placed on the results. Christie et al. offer two reasons for 'preferring' the single-stage, two-variable approach: (1) The two-stage approach is needlessly cumbersome and more prone to errors. (2) Unwary readers 'without a knowledge of the relations between the single-stage and two-stage procedures' might be misled. 4 Of course, if one wishes to report the coefficients that would be obtained from a single-variable HC equation, the same number of regressions have to be computed and there is no savings in this regard. In other words, to report the same information as reported in BGL, the same number of regressions would have to be conducted. The single-stage approach is not sufficient. However, more to the point, it is difficult to visualize the computational issue as being important, given current computer technology and software such as SAS. In an SAS programming format, there is one additional model instruction. The risk 3The issue could also be addressed by conductingan F-test on the reduction in residual variance achieved by adding the HC variable.Two regressions(one including only the ASR 190 variable, the other including both variables)is implied by such a test. 4As we read Christie et al., the first and third reasons provided are essentiallythe same.
222
W.H. Beaoer et ai., Reply
of programming error from this source is close to trivial. In fact, given the relationships that must be preserved between the single-variable regressions and the various regressions in the two-stage approach, there is an additional vehicle for checking for an inadvertent programming error that would not be available in the single-stage approach. In this sense, the risk of programming error could effectively be reduced. However, as we mentioned before, we have difficulty viewing this as a material issue. Alleged effects on uninformed readers are always a difficult issue to assess. For example, i~ could be argued that the two-stage approach, which explicitly focuses on the orthogonal component of the ASR 190 variable, enhances a reader's understanding by making explicit an aspect of incremental explanatory power that is implicit under the single-stage approach. We now turn to the second major point - the collinearity issue. Fortunately, we have much less to say about this issue, because the collinearity issue is subordinate to that of incremental explanatory power of the ASR 190 data. Christie et al. (page 7) suggest that by the use of the term to deal with the collinearity issue (BGL, page 16) we were offering the two-stage approach as a solution to the collinearity issue. We simply mean to deal with as opposed to to ignore. The presence of collinearity merely serves to motivate why we make an effort to explicitly orthogonalize the variables. If the variables are uncorrelated, there is no need to explicitly make the variables orthogonal (or to employ an econometrically equivalent procedure that implicitly looks only at the orthogonal component in computing the t-values). We find the risk of misinterpretation to be minimal. 5 In retrospect, we did not describe in detail the equivalence of several econometric procedures because it is of secondary interest to us an is well documented elsewhere. Christie et al. assume that the single-stage regression [eq. (5) of Christie et al.] is the 'true' model (page 210 and following). It is not clear what theory would justify such an assumption. From an informational perspective, the explanatory power provided by a HC earnings variable and an ASR 190 variable is undistinguishable from that provided by a HC earnings 5 On the one hand, the very fact that the results labeled ' turning the tables on historical cost' are also presented makes it explicit that the two-stage approach is not being offered as a means of 'overcoming' the collinearity between the HC and the ASR 190 variables. On the other hand, it could be argued that Christie et al. have misinterpreted the BL citation (page 66 of BL) appearing on page 1 of Christie et al. A reading of the original text will indicate that a comparison is being made between results based on a regression with several FAS 33 variables versus a series of regression with only one FAS 33 variable. The specific collinearity issue addressed here is that the t-values of each FAS 33 variable from the first regression may face erosion because of coUinearity with the other FAS 33 variables (the results reported in table 19 of BL), while the t-value from a model which contains only one FAS 33 variable cannot be eroded (the results reported in tables 16 and 18 of BL) by other coUinear FAS 33 variables which are not included. This statement is econometrically correct. Moreover, it in no way implies that a two-stage approach is being offered to overcome the collinearity between the explanatory variables, which is the manner in which Christie et al. offer the 'cropped' citation. The reader is referred to pages 64 through 66.
I¥.H. Beaoer et al., Reply
223
variable and an orthogonal ASR 190 variable. As a result, the motivation for calling eq. (5) the 'true' model is not obvious to us. It is not clear that eq. (5) would flow from an income measurement perspective either. The single-stage and the two-stage approaches are viewed as econometrically equivalent for the particular context of primary interest in our study. In other contexts, the two-stage approach may be neither a convenient nor a reasonable way to present the results of interest. We make no general claim for the two-stage approach in different contexts. As stated earlier, the major issues are the supplemental nature of the ASR 190 data and the incremental explanatory power of such data. The two-stage approach is appropriate for addressing the issue in the context we faced. References Ball, R. and P. Brown, 1968, An empirical evaluation of accounting income numbers, Journal of Accounting Research, Autumn, 159-178. Beaver, W., P. Griffin and W. Landsman, 1982, The incremental information content of replacement cost earnings, Journal of Accounting and Economics, July, 15-39. Beaver, W., R. Lambert and D. Morse, 1980, The information content of security prices, Journal of Accounting and Economics, June 3-28. Beaver, W. and W. Landsman, 1983, Incremental information content of Statement 33 disclosures (Financial Accounting Standards Board, Stamford, CT). Christie, A., M. Kennelly, J. King and T. Schaefer, 1984, Testing for incremental content in the presence of collinearity, Journal of Accounting and Economics, this issue.
TESTING FOR INCREMENTAL INFORMATION CONTENT IN THE PRESENCE OF COLLINEARITY A Comment
William H. BEAVER Stanford University, Stanford, CA 94305, USA
Paul A. G R I F F I N Unioersity of California, Daois, CA 95616, USA
Wayne R. LANDSMAN Unioersity of California, Los Angeles, CA 90024, USA Received June 1984, final version received July 1984
Christie et al. (1984) make two major points: (1) A two-stage regression is not the econometricaUy unique way to examine the incremental explanatory power of an additional variable (e.g., an earnings variable based on ASR 190 data). 1 (2) A two-stage regression approach does not 'overcome' or 'solve' the collinearity issue. Beaver, Griffin and Landsman (1982) (BGL) or Beaver and Landsman (1983) (BL) do not claim that the two-stage approach is the unique approach or that it overcomes the collinearity issue. Yet, because Christie et al. perceive that the papers are open to such interpretations, we welcome this opportunity to elaborate upon the motivation for our choice of the two-stage approach from a larger set of econometrically equivalent approaches. We view the issue as one of presentation of results rather than econometric 'correctness'. The interesting research design issue concerns how the researcher incorporates the institutional details of the topic at hand to choose among virtually equivalent approaches from an econometric perspective. As stated on page 16, BGL examines whether the inclusion of an earnings variable based on ASR 190 data adds to the explanatory power with respect to cross-sectional differences in security returns beyond that provided by historical cost (HC) earnings. There are a number of ways to assess the statistical significance of an additional explanatory variable. These include (1) a F-test 1Throughout we will assume a setting with two explanatory variables, a historical cost earnings variable and a second earnings variable based on ASR 190 data. 0165-4101/84/$3.00©1984, Elsevier Science Publishers B.V. (North-Holland)
220
B".H. Beaver et aL, Reply
on the reduction in residual variance achieved by adding the variable (an approach cited in footnote 8 of BGL); (2) a single-stage, two-variable (i.e., a multiple regression) approach suggested by Christie et al., and (3) a two-stage approach as in BGL or BL. Since they are econometrically equivalent for purposes of testing the significance of an added explanatory variable, the choice among them must rest on grounds other than statistical. The two-stage approach constitutes a convenient and reasonable choice based on two criteria: consideration of the institutional aspects of the reporting issue and comparison with previous research. (1) It explicitly reflects the institutional aspects of the ASR 190 data (i.e., their supplemental nature). (2) It presents the coefficients on the HC earnings variable in a manner that facilitates comparison with previous studies of single-variable HC earnings regressions [e.g., Beaver, Lambert and Morse (1980)]. As discussed in BGL and BL, the ASR 190 (as well as FAS 33) data are viewed as supplemental to, not a replacement for, the historical cost data. The regulatory based arguments alleged the ASR 190 data would provide information beyond that provided by historical cost data. Moreover, the ASR 190 data were constructed by making a series of adjustments to the HC data. Hence, the primary research issue is the incremental explanatory power of the adjusted data. The two-stage approach focuses on the incremental nature of the supplemental data in a structurally explicit manner. In particular, it makes explicit that in an OLS world, the additional explanatory power of the ASR 190 earnings variable is constrained to be that portion of the original variable that is orthogonal to the HC earnings variable. While the three approaches cited above are econometrically equivalent with respect to testing the significance of adding the ASR 190 earnings variable, they are not econometrically equivalent methods of reporting other aspects of the results. 2 in particular, the single-stage approach does not provide the same information about the coefficient (and t-value) associated with the HC variable. Under a two-stage approach, the HC coefficient (by construction) cannot be affected by the addition of the (orthogonal) ASR 190 variable. However, in a single-stage approach the HC coefficient and its t-value are affected by the collinearity with the ASR 190 variable. Typically, the t-value will be eroded by the presence of a collinear variable. However, we view the erosion effect on the H C variable to be of secondary interest, at best, in the context of this particular research issue. For example, suppose the HC and the ASR 190 earnings variables are virtually perfectly coUinear. The t-values on both variables in a single-stage equation could be near-zero, in spite of a significant
2 The F-test, cited in footnote 8 of BGL, does not provide the same information as the two-stage approach. It provides no information on either the coefficient or the t-value of the HC variable. It does not indicate the sign of the t-value of the ASR 190 variable. Further, it also requires two regressions.
W.H. Beaoer et al., Reply
221
F-value for the entire regression. The erosion of the t-value on the HC variable induced by the addition of a second highly coUinear variable (which was constructed by making a series of adjustments to HC) is not of major interest here. In this context, the collinearity between the HC and the ASR 190 variable is of secondary interest, relative to the primary issue of incremental explanatory power. However, we will comment more about collinearity issues later. With respect to comparison with previous studies, the two-stage approach reports the coefficient (but not the t-value) on the H C variable that would have been reported in a single variable equation. The presentation of the information facilitates comparison with prior research in which a HC earnings variable is used as a single variable to explain cross-sectional differences in security returns. The research dealing with the association between security returns and historical cost earnings has a well-established heritage [e.g., since Ball and Brown (1968)]. Given the role played by H C earnings in several years of previous research, the coefficient of HC earnings in a single-variable model is of interest. The two-stage approach provides that information, while the single-stage, two-variable approach does not. Of course, some readers may be interested as well in the incremental explanatory power of HC earnings, beyond that provided by the ASR 190 earnings variable. Such information can be provided by a second set of two-stage regression, where the HC variable is constructed to be orthogonal to the A S R 190 variable. 3 If this second set had not been reported, the single-stage, two-variable approach would provide information not provided by the BGL study. However, we report such results (labeled as 'turning the tables' on historical cost). Hence, the issue is not one of disclosure of results, but rather the sequence and the relative emphasis placed on the results. Christie et al. offer two reasons for 'preferring' the single-stage, two-variable approach: (1) The two-stage approach is needlessly cumbersome and more prone to errors. (2) Unwary readers 'without a knowledge of the relations between the single-stage and two-stage procedures' might be misled. 4 Of course, if one wishes to report the coefficients that would be obtained from a single-variable HC equation, the same number of regressions have to be computed and there is no savings in this regard. In other words, to report the same information as reported in BGL, the same number of regressions would have to be conducted. The single-stage approach is not sufficient. However, more to the point, it is difficult to visualize the computational issue as being important, given current computer technology and software such as SAS. In an SAS programming format, there is one additional model instruction. The risk 3The issue could also be addressed by conductingan F-test on the reduction in residual variance achieved by adding the HC variable.Two regressions(one including only the ASR 190 variable, the other including both variables)is implied by such a test. 4As we read Christie et al., the first and third reasons provided are essentiallythe same.
222
W.H. Beaoer et ai., Reply
of programming error from this source is close to trivial. In fact, given the relationships that must be preserved between the single-variable regressions and the various regressions in the two-stage approach, there is an additional vehicle for checking for an inadvertent programming error that would not be available in the single-stage approach. In this sense, the risk of programming error could effectively be reduced. However, as we mentioned before, we have difficulty viewing this as a material issue. Alleged effects on uninformed readers are always a difficult issue to assess. For example, i~ could be argued that the two-stage approach, which explicitly focuses on the orthogonal component of the ASR 190 variable, enhances a reader's understanding by making explicit an aspect of incremental explanatory power that is implicit under the single-stage approach. We now turn to the second major point - the collinearity issue. Fortunately, we have much less to say about this issue, because the collinearity issue is subordinate to that of incremental explanatory power of the ASR 190 data. Christie et al. (page 7) suggest that by the use of the term to deal with the collinearity issue (BGL, page 16) we were offering the two-stage approach as a solution to the collinearity issue. We simply mean to deal with as opposed to to ignore. The presence of collinearity merely serves to motivate why we make an effort to explicitly orthogonalize the variables. If the variables are uncorrelated, there is no need to explicitly make the variables orthogonal (or to employ an econometrically equivalent procedure that implicitly looks only at the orthogonal component in computing the t-values). We find the risk of misinterpretation to be minimal. 5 In retrospect, we did not describe in detail the equivalence of several econometric procedures because it is of secondary interest to us an is well documented elsewhere. Christie et al. assume that the single-stage regression [eq. (5) of Christie et al.] is the 'true' model (page 210 and following). It is not clear what theory would justify such an assumption. From an informational perspective, the explanatory power provided by a HC earnings variable and an ASR 190 variable is undistinguishable from that provided by a HC earnings 5 On the one hand, the very fact that the results labeled ' turning the tables on historical cost' are also presented makes it explicit that the two-stage approach is not being offered as a means of 'overcoming' the collinearity between the HC and the ASR 190 variables. On the other hand, it could be argued that Christie et al. have misinterpreted the BL citation (page 66 of BL) appearing on page 1 of Christie et al. A reading of the original text will indicate that a comparison is being made between results based on a regression with several FAS 33 variables versus a series of regression with only one FAS 33 variable. The specific collinearity issue addressed here is that the t-values of each FAS 33 variable from the first regression may face erosion because of coUinearity with the other FAS 33 variables (the results reported in table 19 of BL), while the t-value from a model which contains only one FAS 33 variable cannot be eroded (the results reported in tables 16 and 18 of BL) by other coUinear FAS 33 variables which are not included. This statement is econometrically correct. Moreover, it in no way implies that a two-stage approach is being offered to overcome the collinearity between the explanatory variables, which is the manner in which Christie et al. offer the 'cropped' citation. The reader is referred to pages 64 through 66.
I¥.H. Beaoer et al., Reply
223
variable and an orthogonal ASR 190 variable. As a result, the motivation for calling eq. (5) the 'true' model is not obvious to us. It is not clear that eq. (5) would flow from an income measurement perspective either. The single-stage and the two-stage approaches are viewed as econometrically equivalent for the particular context of primary interest in our study. In other contexts, the two-stage approach may be neither a convenient nor a reasonable way to present the results of interest. We make no general claim for the two-stage approach in different contexts. As stated earlier, the major issues are the supplemental nature of the ASR 190 data and the incremental explanatory power of such data. The two-stage approach is appropriate for addressing the issue in the context we faced. References Ball, R. and P. Brown, 1968, An empirical evaluation of accounting income numbers, Journal of Accounting Research, Autumn, 159-178. Beaver, W., P. Griffin and W. Landsman, 1982, The incremental information content of replacement cost earnings, Journal of Accounting and Economics, July, 15-39. Beaver, W., R. Lambert and D. Morse, 1980, The information content of security prices, Journal of Accounting and Economics, June 3-28. Beaver, W. and W. Landsman, 1983, Incremental information content of Statement 33 disclosures (Financial Accounting Standards Board, Stamford, CT). Christie, A., M. Kennelly, J. King and T. Schaefer, 1984, Testing for incremental content in the presence of collinearity, Journal of Accounting and Economics, this issue.