On the nature of corporate capital structure persistence and convergence*

Transcription

1 On the nature of corporate capital structure persistence and convergence* Douglas O. Cook Department of Economics, Finance, and Legal Studies Culverhouse College of Business University of Alabama Tuscaloosa, AL Robert Kieschnick University of Texas at Dallas 2601 N. Floyd Rd, SM 31 Richardson, TX * We thank David Mauer and Jeff Wooldridge for comments on prior drafts. Cook gratefully acknowledges financial support from the Ehney A. Camp, Jr. Chair of Finance and Investments.

2 1 On the nature of corporate capital structure persistence and convergence ABSTRACT Lemmon, Roberts, and Zender (2008) provide evidence suggesting that corporate capital structures are surprisingly persistent; that firm fixed effects account for a significant portion of the observed variation in corporate capital structures; and that there is a pattern of convergence of corporate capital structures over event time. We argue that Lemmon, Roberts, and Zender s evidence is consistent with corporate capital structures following a nonlinear process, dictated in part by the definition of capital structure and in part by the growth pattern of firms. We find evidence consistent with these arguments. Specifically, we provide evidence that corporate capital structures follow a nonlinear process that is also followed by their determinants and that unobserved firm heterogeneity (fixed effects) is less important and prior explanatory variables are more important than suggested by Lemmon, Roberts, and Zender s evidence. JEL Codes: D20, G32 Keywords: firm dynamics, capital structure, fractional variables

3 2 1. Introduction Lemmon, Roberts and Zender (2008) examine the capital structures of firms with CRSP and Compustat data between 1965 and 2003 and find that although there is some convergence toward the mean over time, a sample firm s capital structure shows more persistence than expected, and that a significant portion of a firm s capital structure is explained by firm fixed effects. Lemmon, Roberts and Zender (denoted LRZ, hereafter) suggest that their findings concerning the importance of firm fixed effects in explaining observed corporate capital structures raises questions about the explanatory power of variables employed in prior capital structure literature. While LRZ recognize that their results imply that prior empirical models are probably misspecified, we argue that they misinterpret both the nature of the specification error and the underlying cause of their convergence and persistence evidence. As long as debt and equity are non-negative, the proportion of capital accounted for by debt capital is likely determined by a nonlinear function since it is bounded on [0,1]. However, in specifying their regression models, LRZ ignore the fact that corporate capital structure measures are likely to be nonlinear in the relevant decision variables. With respect to their convergence and persistence evidence, LRZ, like most of the authors of prior corporate capital structure papers, ignore the fact that firms do not grow in a linear fashion, but rather, in a nonlinear fashion. Going back to Evans (1987) and Hall (1987), it is accepted in the industrial organization literature that although surviving firms grow as they age, their mean growth rates decrease systematically with age. Such growth patterns are not only consistent with the type of convergence and persistence evidence reported in LRZ, but also with Clementi and Hopenhayn s (2006) model that shows how in the face of asymmetric information firms financing choices endogenize financing constraints and so generate these types of growth patterns. To demonstrate the validity of these arguments, we organize our paper as follows. Section 2 specifies both arguments in more detail. Section 3 describes our sample and sample data. Section 4 presents our analyses of the data, and Section 5 concludes with a summary of our findings.

4 3 We find evidence for the following conclusions. First, RESET tests reject LRZ s regressions as being correctly specified and provide evidence consistent with their leverage measures being determined by a nonlinear mechanism. Second, LRZ s convergence and persistence evidence is shown to be consistent with firms following a nonlinear growth path as implied by the industrial organization literature. Third, consistent with this point, we find that the differences between the highest and lowest quartiles of leverage use are distinguished by their firm characteristics. And finally, after accounting for these nonlinearities, firm fixed effects are no longer important in explaining observed corporate capital structures and prior explanatory variables are more important than LRZ s evidence suggests. 2. The motivating arguments for our study Our study is motivated by two distinct arguments: an econometric argument and an economic argument. We shall first discuss our econometric argument, and then proceed to our economic argument. These discussions will frame some of the subsequent empirical tests. 2.1 Econometric motivation LRZ, like the majority of capital structure studies, regress the proportion of a firm s capital structure accounted for by debt financing on various explanatory variables using a linear model. Prior econometric research, however, argues that the conditional expectation of a fraction, or proportion, or percentage is a nonlinear function of the explanatory variables. Papke and Wooldridge (1996) provide theoretical arguments explaining why the conditional expectation function of a fractional or proportional variable is nonlinear. Cox (1996) and Kieschnick and McCullough (2003) provide evidence that the conditional expectation of a proportion is nonlinear. Consistent with this characterization, Cox (1996), Kieschnick and McCullough (2003), Papke and Wooldridge (1996), Paolina (2001), and Ferrari and Cribari-Neto (2004) all model the conditional expectation function of proportions as a sigmoidal function. And of more specific relevance, Cook, Kieschnick and McCullough (2007) and Fattouh, Harris and Pasquale (2005) provide evidence that the conditional expectation function for the proportion of capital accounted for by debt capital is nonlinear. To see the implications of the kind of nonlinear conditional expectation employed in this research for the results in LRZ s paper, we start with the following equation:

5 4 CS = f ( β X ) (1) it, it, where CS i,t represents the proportion of capital accounted for by debt capital for firm i at time t, and f(.) is a nonlinear function of X t, which represents the determinants of a corporation s capital structure. For the sake of simplicity, suppose that X t is a scalar. Constructing a Taylor series approximation of the firm s current capital structure and using its initial capital structure as the reference point would yield: ( x, x,0) i n 1 it i i it, = ( i,0 ) + ( i,0) + i i= 1 i! CS f x f x e (2) where f(x i,0 ) represents its initial capital structure and f i (x i,0 ) is the ith derivative of f(.) evaluated at x i,0. For the logistic function, the conditional expectation function used in Papke and Wooldridge (1996), we would need at least the third-order terms to adequately approximate its shape. 1 If we assume that fourth-order and higher terms are sufficiently small, then we can rewrite (2) as: where i,0 i 1 i, t 2 it, 3 it, it, it, ( i ) ( i ) f x f x CS CS x x f x ( x x ) ( x x ) e 2! 3! ,0 3,0 it = i,0 + ( i, t i,0 ) ( i,0 ) + i, t i,0 + i, t i,0 + i = CS + F + β x + β x + β x + e = CS + F + β x + e * 2 i,0 i 1 i, t it, = F + β x + e ** 3 i 1 i, t it, = β x + e 1 it, 4 * ** Fi, Fi, F i represent different measures of firm fixed effects (e.g., i (3) F represents the sum of all the constant terms in the expanded third-order Taylor series approximation). 2 These expressions provide a rationale for several of the findings in LRZ s paper. If one follows their methodology and fits a linear regression model to these data without terms for a firm s initial leverage or fixed effects, then one should expect (as they observe) that a substantial portion of the variation in observed capital structures will be accounted for by the residual. Expanding on this simple specification to include a firm s initial leverage (CS i,0 ) and a firm fixed effect (F i ), then one should also observe that CS i,0 and F i will be significant; both statistically and 1 A quadratic function is clearly inappropriate for these data as it is not bounded on the unit interval. 2 Note that β 1, β 2 and β 3 represent the sum of the different cross-product terms involving powers of x i,t since the derivatives of the function evaluated at x i,0 are constants.

6 5 economically. Further, if much of the explanatory power of capital structure determinants comes through their higher order terms for observations away from the mean, then one can expect to find that firm fixed effects will explain a significant amount of variation in observed corporate capital structures. Before concluding this discussion, we should note that some might argue that LRZ s linear model can be viewed as a first order Taylor series approximation to a nonlinear function. Such an argument fails to fully understand its assumptions and implications. First, Stebulaev (2007) demonstrates that this assumes that all firms are in equilibrium: which is inconsistent with empirical evidence (e.g., Leary and Roberts (2005)). Further, and more importantly, such a linear approximation induces a type of endogeneity bias in such estimations, which neither LRZ nor prior research addresses Economic motivation As noted earlier, the industrial organization literature has established a number of stylized facts concerning firm survival and growth. 4 For our purposes, the primary issue is that surviving firms do grow with age, but the mean of this growth rate decreases systematically with age. In biology, such a growth pattern is often modeled using a logistic difference or differential equation which is consistent with the functional form used to model fractional data in biometrics (e.g., Cox (1996)) or econometrics (e.g., Papke and Wooldridge (1996)). Such a distinctive growth pattern implies that we should expect to observe convergence and persistence in firm characteristics, such as capital structure. As firms mature and become less opaque; their access to capital becomes less constrained. In this regard, Cabral and Mata (2001) argue that financial constraints play an important role in explaining the evolution of firm size distribution: younger firms face tighter constraints. For some firms, they will have limited access to debt financing, especially if they have characteristics that favor the use of equity financing, and so rely more heavily on equity financing (firms on the lower curve in Figure 1). For other firms, they will have limited access to equity financing, especially if they have characteristics that favor debt financing, and so rely more on debt financing (firms on the upper curve in Figure 2). Thus, one might expect that after 3 This bias is easily seen by recognizing the higher order terms show up in the error term and are correlated with the included linear terms. 4 See Caves (1998) or Cabral and Mata (2003) for discussions of this literature.

7 6 going public, firms will face fewer financing constraints as they mature and so their use of debt will converge to their sustainable debt carrying capacity. As a result, they will demonstrate the convergence pattern that LRZ observe for their sub-sample of firms that survive for 20 years. Somewhat consistent with this basic argument, Clementi and Hopenhayn (2006) recently develop a model of firm debt use that generates a firm growth path consistent with the stylized facts from industrial organization. In their model, an entrepreneur borrows funds to finance a project s initial investment as well as current and future investments in working capital under conditions where the lender cannot monitor the outcomes of these investments. Within the context of their model, financing constraints arise endogenously and generate a growth path that matches the above stylized facts. The key point of this discussion is that the convergence pattern observed by LRZ for their sub-sample of firms that survive for 20 years is consistent with the growth pattern of firms observed in the industrial organization literature and so we should expect a similar pattern in the determinants of corporate capital structures as well. 3. Data To test some of the implications of the above discussions, we begin by constructing a sample that identifies all public corporations on Compustat with non-zero sales from January 1965 through December After dropping all firms in financial service industries, our remaining sample is 16,246 firms. For each of these, we follow LRZ and compute the following variables identified by Compustat data number: Total Debt = Short Term Debt (34) + Long Term Debt (9), Book Leverage = Total Debt/Total Assets (6), Market Leverage = Total Debt/(Market Equity + Total Debt) Ln(sales) = Logarithm of Net Sales (12), Market-to-Book = [Market Equity + Total Debt + Preferred Stock Liquidating Value (10) Deferred Taxes and Investment Tax Credits (35)]/Total Assets (6), Profitability = Operating Income before Depreciation (13)/Total Assets (6), Tangibility = Net Property, Plant & Equipment (8)/Total Assets (6), Industry Median Leverage = median of a firm s industry leverage (book or market as required) for the Fama and French 38 industry delineations,

8 7 Although we generally construct our sample and variables similarly to LRZ, we deviate in the following ways. 5 We initially calculate Market Equity as the product of its stock price (Compustat 199) and number of shares (Compustat 54), but since there are a large number of missing values we proceed to calculate Market Equity using CRSP stock price and shares outstanding data. 6 We choose to treat a firm s book leverage as missing when the firm s book value of equity is negative since Trimbath (2001) shows that using negative book equity can distort results. Finally, rather than trimming observations that are in the 1% tail of a variable s sample distribution we follow the more common practice of winsorizing these observations. This practice increases the number of sample firms in our analyses. Table 1 contains summary statistics for the variables employed in our study. 4. Analysis 4.1 Tests of the Form of the Conditional Expectation Function We begin by first testing whether the conditional expectation function for LRZ s capital structure variables is consistent with a highly nonlinear form as suggested in the statistical literature (e.g., Cox (1996) or Papke and Wooldridge (1996)). We address the reasonableness of this characterization in several ways. First, we use a RESET test to determine whether there is evidence that LRZ s linear regression models are mis-specified and whether that mis-specification is consistent with the underlying model being nonlinear. In columns 2 and 4 of Table 2, we report our sample estimates for the linear specifications underlying columns 3 and 6 of LRZ s Table II. Although there are some differences in coefficient estimates, the signs and significance of our results and theirs is similar. Based on these regressions, we report the results of including the second and third order power of the predicted values of the dependent variables from the linear regressions in columns 3 and 5 of Table 2. The coefficients on both powers of the predicted variables are significant in both leverage equations. However, to test for their combined inclusion, we report an F test that compares a model that includes these terms to a model that excludes them. The F statistic of for the book leverage regression and for the market leverage regression implies that the coefficients are significantly different from zero at the 1% marginal significance 5 Although we believe that our reasons for deviating from Lemmon, Roberts, and Zenders procedures are appropriate, following their procedures exactly do not alter our conclusions. 6 There are 2064 missing values in Compustat that correspond to non-missing values in CRSP.

9 8 level. Thus, the RESET test implies that LRZ s regression models are mis-specified and that the underlying regression models for both their book value and market value leverage regressions are nonlinear. Since the above tests are based on LRZ s 5 factor models, it could be suggested that the RESET test simply reflects the exclusion of relevant variables. Performing the same RESET tests on their 8 factor models, we derive an F statistic of for the book leverage regression and for the market value leverage regression, both of which are significant at the 1% level. So, excluding additional variables does not explain our evidence that linear regression models of corporate leverage measures are mis-specified. Given these results, we examine the data in a different way. Specifically, we estimate: Leverage = α + Leverage + β X + β X + β X + ε (6) 2 3 it 0 1 it 1 2 it 1 3 it 1 it where Leverage 0 represents a firm s initial leverage and X i,t-1 represents a vector of lagged (to the current year) explanatory variables. 7 We do not scale our explanatory variables by their standard deviation because it would distort our test and would not be appropriate for our purposes. If our RESET evidence is determined by the nonlinearity of the leverage equations, then the vectors β 2 and β 3 should be significant. For comparison purposes, we report again in Table 3 the results from estimating the linear specifications underlying columns 3 and 6 of LRZ s Table II, and then report the results of estimating the above polynomial regression model. We first perform a Chi-Square test comparing a restricted model that excludes the second and third order terms to a model that includes these higher order terms. The Chi-Square statistic of (371.36) suggests that the coefficients on the higher order terms in the book (market) leverage regressions are significantly different from zero at the 1% marginal significance level. These results provide strong evidence that these leverage regressions follow the kind of nonlinear form that we conjecture to apply to these data. 7 We again note that a quadratic expression is not consistent with the nature of the nonlinearity that we conjecture to apply to these leverage equations.

10 9 4.3 Does unobserved heterogeneity across firms continue to be important? Given our evidence that linear regression models for LRZ s leverage measures are misspecified and that the underlying model is nonlinear, we now address the issue of whether unobserved heterogeneity across firms continues to be as important. We use Papke- Wooldridge s (1996) quasi-likelihood model to fit the data, thereby, avoiding arguments about the specific distribution underlying the data generating process which allows us to focus instead on the specification of the first moment. The conditional expectation function of this regression model is: exp( index) E( leverage X ) = (7) 1 + exp( index) where index Xβ and X is a vector of explanatory variables. This conditional expectation function is consistent with the specification in Cox (1996), Kieschnick and McCullough (2003), Papke and Wooldridge (1996), Paolina (2001), and Ferrari and Cribari-Neto (2004). Following Papke and Wooldridge (2008) we use the general estimating equations (GEE) method for panel data to estimate the β parameters. We use specifications of the regressors that are the same as those of columns 3 and 6 of Table II of LRZ s paper and report the result of these estimations for both book and market value definitions of leverage in Table 4. We will discuss two of the implications of these results now, and defer other implications for later discussion. First, both of these regression models explain more of the variation in observed corporate capital structures than the similar linear regression models reported in LRZ. Second, we find the coefficient on a firm s initial leverage measure to be significant for both leverage measures. These results are especially interesting because the solution to the type of logistic differential equation, which describes the type of growth path stylized in industrial organization, involves the initial value of the process. 8 Consequently, the regressions reported in Table 4 represent a reasonable characterization of capital structure decisions if they follow the type of nonlinear growth path typical of firms that survive. While these regressions explain a larger proportion of the variation in observed capital structures than the fixed effects models reported in LRZ s paper, one could argue that they impose the unreasonable assumption that there is no unobserved heterogeneity across firms or 8 We should acknowledge, however, that such a result can be expected for solutions to a number of differential equations with given initial values.

11 10 time. This assumption is important because LRZ argue that unobserved heterogeneity across firms is an important explanator of the observed variation of corporate capital structures at the expense of prior determinants. Following the approach discussed in Wooldridge (2005) and Papke and Wooldridge (2008), we analyze the robustness of LRZ s claim once one controls for the nonlinear nature of the conditional expectation function for their leverage measures. Specifically, we control for unobserved heterogeneity by averaging each of the explanatory variables for each firm over its history and then including these means as control variables in our prior regression model. This procedure avoids the incidental parameters problem affecting nonlinear fixed effects models. Further, we control for unobserved heterogeneity in time effects by including dummy variables for each year in our sample period. Thus, in Table 5, we estimate two sets of three regression models; one set corresponding to our book leverage measure and the other set for our market leverage measure. The first regression model in each set ignores unobserved heterogeneity across firms and time. The second regression model in each set accounts for unobserved heterogeneity across firms but not across time, and the third regression model in each set accounts for unobserved heterogeneity across firms and time. All three regression models follow the regression format employed in Table 4 (i.e., a cumulative logistic function to model the first moment of the conditional expectation function, etc.). The results reported in Table 5 are quite interesting. We observe considerable stability in the coefficient estimates across regression models for both the book or market leverage measures. We also observe substantial stability in the correlation between actual and predicted leverage measures across regression models. This evidence suggests that unobserved heterogeneity across firms and time is not as important as LRZ suggest once the nonlinear nature of the evolution of corporate capital structures is recognized, which is consistent with our earlier Taylor series interpretation of their results. This last interpretation is reinforced by the fact that the regressions without adjustments for unobserved heterogeneity either across firms or time explain more of the variation of observed capital structures than LRZ report for their linear models with such adjustments. While we have followed prior literature (e.g., Cox (1996), Kieschnick and McCullough (2003), Papke and Wooldridge (1996), Paolina (2001), and Ferrari and Cribari-Neto (2004)) and

12 11 focused on the cumulative logistic function to characterize the nonlinear conditional expectation function for LRZ s leverage measures, there are alternative approaches. For example, Papke and Wooldridge (2008) point out that one can also use the cumulative normal function for this purpose, which they do. To compare results across both representations (analogous to comparisons between logit and probit models), we re-estimate the regression models shown in Table 5, but this time using the cumulative normal function, and report the results in Table 6. These results bear the same conclusions that one derives from Table 5 with respect to the reduction in importance of unobserved heterogeneity either across firms or time once nonlinearity is recognized. Thus, our conclusions do not rely on one functional expression for the nonlinear shape of the conditional expectation function. One might still be skeptical concerning our results because we, like LRZ, include firms that use no debt (both short and long term) in our analysis. Cook, Kieschnick and McCullough (2007) document that the reasons firms initially issue debt differ from the determinants of how much debt they will use. Consequently, including both types of firms in a sample can produce biased estimates if one imposes the condition that both follow the same process. Thus, we repeat our analysis for the sub-set of sample firms that use debt. 9 We report these results in Table 7, and note that they are virtually identical to those in Table 5. One reason for the similarity of results is that the methods we have employed recognize that the boundary observations are from a different regime. Consequently, our inclusion of boundary observations does not explain our evidence that unobserved heterogeneity across firms or time is not as important as LRZ suggest. 4.2 Firm Growth and the Pattern of Leverage Convergence and Persistence Thus far, we have shown that LRZ s leverage measures are best explained as nonlinear functions of their explanatory variables, and that unobserved heterogeneity occupies a smaller role in explaining observed capital structures than they suggest. This last piece of evidence implies that the explanatory variables we have used account for a larger portion of the variation of corporate capital structures than credited. We now turn to the pattern of convergence and persistence that they observe in corporate capital structures. Earlier we argued that this pattern arose because of the pattern of convergence 9 This procedure is the same as that used by Orcutt, Greenberger, Korbel and Rivlin (1961) in their study of U.S. mortgage debt.

13 12 and persistence followed by firms as their age. Consequently, we should expect a similar pattern to describe the evolution of the determinants of corporate capital structures. To test this, we do the following: (1) drop all firms from the sample that do not have at least 21 years of contiguous data, (2) use these firms to create a sample in which each firm/year observation is ordered from their first year as a public firm to their 21 st year as a public firm (i.e. ordered in event time), (3) sort firms into four groups (quartiles) according to their initial book or market leverage, (4) use the estimated coefficients in Table 4 to compute an index (see equation 7) for each firm of the four initial leverage groups, and (5) follow the average index of the highest and lowest initial leverage groups over time. In essence, each index captures the underlying economic determinants of a firm s market or book leverage for that year using time constant population weights derived from the cumulative logistic function specification of the conditional mean function. Now, if our interpretation of LRZ s convergence and persistence evidence is correct, then we should observe that the indices for the lowest and highest initial leverage groups should differ significantly to begin with and then move toward each other, or converge, over time. In Figure 2, we display the average indices for the lowest and highest initial book leverage groups over 20 years after their initial public year. 10 The shapes of these indices are consistent with those displayed in LRZ s Figure 1. Specifically, they start out being very different and then begin to converge toward one another, while still remaining distinct even after 20 years. What this tells us is that the determinants of corporate capital structures are evolving as we suggested above: for either the lowest or highest quartile of initial firm leverage, they start out very differently in term of their measured characteristics, become more similar over time, and yet remain different. Further, as they become more mature, their indices change very little from year to year. So, the determinants of corporate capital structures also show convergence and persistence, which is consistent with the nonlinear growth pattern of surviving firms. Earlier, we argued that firms that started in the bottom and top capital structure groups, were different to begin with and that these differences also continued to persist. To address this issue, we compare the economic characteristics (leverage explanatory variables) of those firms in the highest quartile and lowest quartile of initial leverage each year over their event time horizon 10 The patterns for the market leverage indices are similar and are, therefore, not reported.

14 13 using multinomial logistic regressions. We report the results of these comparisons for their first year and 20th year in Table Table 8 shows that there are significant differences in economic characteristics (i.e., determinants of corporate capital structure) between the highest and lowest initial leverage quartile firms at the beginning and after 20 years that are consistent over time. Firms that use more leverage tend to be larger, with slower expected future growth, and more tangible assets than firms that use less leverage. So, while the capital structures of firms in the top and bottom quartiles converge towards one another over time, they still are very different firms in terms of the standard determinants of corporate capital structures. 4. Conclusions Lemmon, Roberts and Zender (2008) examine the capital structures of firms with CRSP and Compustat data between 1965 and 2003 and find that although there is some convergence toward the mean over time, a sample firm s initial capital structure continues to be an important determinant of its current capital structure. Just as importantly, they find that a significant portion of a firm s capital structure is explained by firm fixed effects, and they raise doubts about the explanatory power of the variables employed in prior literature. While Lemmon, Roberts and Zender recognize that their results imply that prior empirical models are probably mis-specified, we argue that they misinterpret both the nature of the specification error and the underlying cause of their convergence and persistence evidence. In specifying their regression models, they assume that corporate capital structures are linear in the relevant decision variables. However, as long as debt and equity are non-negative, then the proportion of capital accounted for by debt capital is likely determined by a nonlinear function since it is bounded on [0,1]. While analyzing their convergence and persistence evidence, they do not consider the fact that firms grow in a nonlinear fashion, as demonstrated by the tendency of surviving firms to grow but at a rate that diminishes with age. To test our re-interpretation of their evidence and conclusions, we examine data on a sample of firms with Compustat data from 1965 through Using this sample, we find evidence for the following conclusions. First, based upon the Ramsey s RESET test, the linear 11 We do not report results for all years because they do not provide any additional insights from those derived by just observing these two years.

15 14 regression models estimated by LRZ are mis-specified. More importantly, the nature of the misspecification is not so much due to the exclusion of relevant variables as it is due to the inherent nonlinear nature of the leverage equations. Second, once one accounts for the possibility that the variation of observed corporate capital structures follows a nonlinear dynamic process, accounting for unobserved heterogeneity across firms or time does not significantly increase one s ability to explain this variation. Third, the economic determinants of corporate capital structures in firms with the highest and lowest initial leverage measures also show a pattern of convergence and persistence as they mature that is consistent with LRZ s observed pattern of convergence and persistence in corporate capital structures. Thus, we find evidence that LRZ s convergence and persistence evidence is consistent with the manner in which firms grow as they age.

16 15 References: Baltagi, B. and Q. Li, 1999, Unequally spaced panel data regressions with AR(1) disturbances, Econometric Theory 68, Bass, F., 1969, A new product growth model for consumer durables, Management Science 15, Cabral, L. and J. Mata, 2003, On the Evolution of Firm Size Distribution: Facts and Theory, American Economic Review 93, Caves, R., 1998, Industrial Organization and New Finding on the Turnover and Mobility of Firms, Journal of Economic Literature 36, Clementi, G. and H. Hopenhayn, 2006, A Theory of Financing Constraints and Firm Dynamics, Quarterly Journal of Economics, Cox, C., 1996, Nonlinear quasi-likelihood models: applications to continuous proportions, Computational Statistics & Data Analysis 21, pp Cook, D., R. Kieschnick, and B.D. McCullough, 2007, Regression analysis of proportions in finance with self selection, forthcoming in the Journal of Empirical Finance. Cook, Douglas O. and Tang, Tian, 2008, "Macroeconomic Conditions and Capital Structure Adjustment Speed," De Vries, G., T. Hillen, M. Lewis, J. Muller, and B. Schonfisch, 2006, A Course in Mathematical Biology, Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. Dixon, C., 1981, Advanced Calculus, Brisbane: John Wiley & Sons, Ltd. Evans, D., 1987, The Relationship between Firm Growth, Size and Age: Estimates for 100 Manufacturing Firms, Journal of Industrial Economics 35, Fattouh, Bassam, Harris, Laurence and Scaramozzino, Pasquale, 2005, "Non-Linearity in the Determinants of Capital Structure: Evidence from UK Firms," SSRN: Ferrari, S. and F. Cribari-Neto, 2004, Beta regression for modeling rates and proportions, Journal of Applied Statistics 31, Flannery, M. and K. Rangan, 2006, Partial adjustment toward target capital structures, Journal of Financial Economics 79, Gintis, H., 2000, Game Theory Evolving, Princeton, NJ: Princeton University Press.

17 16 Hall, B., 1987, The Relationship between Firm Size and Firm Growth in the U.S. Manufacturing Sector, Journal of Industrial Economics 35, Hovakimian, A., T. Opler, and S. Titman, 2001, Debt-equity choice, Journal of Financial and Quantitative Analysis 36, Imbens, G. and J. Wooldridge, 2007, What s New in Econometrics?, Lecture Note 4: Nonlinear Panel Data Models, National Bureau of Economic Research. Kieschnick, R. and B. D. McCullough, 2003, Regression Analysis of variates observed on (0,1): percentages, proportions, and fractions, Statistical Modeling 3, Leary, M. and M. Roberts, 2005, Do Firms Rebalance Their Capital Structures?, Journal of Finance, Lemmon, M. M. Roberts, and J. Zender, 2008, Back to the Beginning: Persistence and the Cross-Section of Corporate Capital Structure, forthcoming in the Journal of Finance. Li, K. and J. Prabhala, 2005, Self-Selection Models in Corporate Finance, Chapter 2 in B. Espen Eckbo (ed.) Handbook of Corporate Finance: Empirical Corporate Finance (Handbooks in Finance Series), New York: Elsevier/North-Holland. Loudermilk, M., 2007, Estimation of Fractional Dependent Variables in Dynamic Panel Data Models with An Application to Firm Dividend Policy, Journal of Business and Economic Statistics 25, Maddala, G.S., 1991, A Perspective on the Use of Limited-Dependent Variables in Accounting Research, The Accounting Review 66, pp Orcutt, G., M. Greenberger, J. Korbel, and A. Rivlin, 1961, Microanalysis of Socioeconomic Systems: A Simulation Study, New York: Harper and Row. Paolina, P., 2001, Maximum Likelihood Estimation of Models with Beta-Distributed Dependent Variables, Political Analysis 9, Papke, L., and J. Wooldridge, 1996, Econometric Methods for Fractional Response Variables with an application to 401(K) Plan Participation Rates, Journal of Applied Econometrics 11, pp Papke, L and J. Wooldridge, 2008, Panel Data Methods for Fractional Response Variables with an Application to Test Pass Rates, Journal of Econometrics 145, Stebulaev, I., 2007, Do Tests of Capital Structure Theory Mean What They Say, Journal of Finance 57, pp

18 17 Trimbath, S., 2001, Lemmings to the Sea: The Inappropriate Use of Financial Ratios in Empirical Analysis, Wooldridge, J., 2005, Simple Solutions to the initial conditions problem in dynamic, nonlinear panel data models with unobserved heterogeneity, Journal of Applied Econometrics 20,

19 18 Table 1 Sample summary statistics The master sample consists of all firms on Compustat (excluding financial firms) from Book Leverage equals total debt divided by total assets where total debt is short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Ln(sales) represents the logarithm of net sales. Market-to-Book equals the ratio of [Market Equity + Total Debt + Preferred Stock Liquidating Value Deferred Taxes and Investment Tax Credits] to Total Assets. Profitability represents the ratio of operating income before depreciation to total assets. Tangibility equals the ratio of net property, plant & equipment to total assets. Industry Median Leverage represents the median of a firm s industry s leverage (book or market) for the Fama and French 38 industry delineation. # of Obsevations represents the number of firm-years with data on a variable. Mean Median Standard Deviation Minimum Maximum # of Observations Book Leverage ,760 Market ,824 Leverage Ln(Sales) ,760 Market-to ,824 Book Profitability ,760 Tangibility ,760 Industry ,376 Median Leverage (book) Industry Median Leverage (Market) ,376

20 19 Table 2 Linear leverage regression: RESET tests The sample consists of all firms on Compustat (excluding financial firms) from The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. PBook_leverage and PMarket_leverage represent the predicted values of book leverage from the linear model for each. All regressors, except the initial leverage variables, are lagged one period relative to the current period. The standard errors are estimated using Rogers estimators adjusted for clustering at the firm level. The RESET test represents an F test of whether all the coefficients on the quadratic and cubic power of the predicted leverage variables are equal to zero. We report p-values associated with the null of no significance are reported within parentheses. Book Leverage Market Leverage Initial leverage Market-to-Book (0.54) Ln(Sales) (0.51) Profitability Tangibility (0.02) PBook_leverage PBook_leverage PMarket_leverage PMarket_leverage Year fixed effects Yes Yes Yes Yes # of observations 144, , , ,854 Adj. R RESET test

21 20 Table 3 Linear leverage regression: third order tests The sample consists of all firms on Compustat (excluding financial firms) from The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Ln(sales) represents the logarithm of net sales. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. All regressors, except the initial leverage variables, are lagged one period relative to the current period. The standard errors are estimated using Rogers estimators adjusted for clustering at the firm level. The cubic Chi-Square test represents a test of whether all the coefficients on the cubic terms are equal to zero. The quadratic and cubic Chi-Square test represents a test of whether all the coefficients on the quadratic and cubic terms are equal to zero. We report p-values associated with the null hypotheses (i.e., coefficient equal zero) within parentheses. Book Leverage Market Leverage Initial Leverage Market-to-Book Ln(Sales) (0.12) Profitability Tangibility Initial Leverage Market-to-Book Ln(Sales) (0.61) (0.09) Profitability (0.52) Tangibility (0.90) (0.94) Initial Leverage Market-to-Book (0.24) Ln(Sales) (0.11) (0.50) Profitability Tangibility (0.08) (0.71) Year fixed effects Yes Yes Yes Yes Adj. R Quadratic and Cubic Chi-Square test

22 21 Table 4 Cumulative logistic conditional expectation function The sample consists of all firms on Compustat (excluding financial firms) from The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. Each regression uses a cumulative logistic conditional expectation function and is estimated using the GEE approach developed in (2008). All regressors, except the initial leverage variables, are lagged one period relative to the current period. We adjust the standard errors for clustering at the firm level and report p-values associated with the null of no significance within parentheses. Book Leverage Market Leverage Constant Initial leverage Market-to-Book Ln(Sales) Profitability Tangibility # of observations 144, ,854 Chi-Sq statistic Correlation between Actual and predicted

23 22 Table 5 Cumulative logistic conditional expectation function with controls for unobserved heterogeneity across firms and time The master sample consists of all firms on Compustat (excluding financial firms) from The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. Each regression uses a cumulative logistic conditional expectation function and is estimated using the GEE approach developed in Papke and Wooldridge (2008). All regressors, except the initial leverage variables, are lagged one period relative to the current period. We adjust the standard errors for clustering at the firm level and report p-values associated with the null of no significance within parentheses. Book Leverage Market Leverage Constant Initial leverage Market-to-Book Ln(Sales) Profitability Tangibility Controls for unobserved No Yes Yes No Yes Yes heterogeneity across firms Controls for unobserved No No Yes No No Yes heterogeneity across time Chi-Sq statistic Correlation between Actual and predicted

24 23 Table 6 Cumulative normal conditional expectation function with controls for unobserved heterogeneity across firms and time The master sample consists of all firms on Compustat (excluding financial firms) from The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. Each regression uses a cumulative normal conditional expectation function and is estimated using the GEE approach developed in Papke and Wooldridge (2008). All regressors, except the initial leverage variables, are lagged one period relative to the current period. We adjust the standard errors for clustering at the firm level and report p-values associated with the null of no significance within parentheses. Book Leverage Market Leverage Constant Initial leverage Market-to-Book Ln(Sales) Profitability Tangibility Controls for unobserved No Yes Yes No Yes Yes heterogeneity across firms Controls for unobserved No No Yes No No Yes heterogeneity across time Chi-Sq statistic Correlation between Actual and predicted

25 24 Table 7 Robustness check using only firms that use debt The master sample consists of all firms on Compustat (excluding financial firms) from with non-zero debt. The dependent variables are Book Leverage and Market Leverage. Book leverage equals total debt divided by total assets. Total debt represents short term debt plus long term debt. Market Leverage equals total debt/(market equity + total debt) where market equity is the stock price at fiscal year end times the number of shares outstanding from CRSP. Initial Leverage is the first value of leverage during the interval Market-to-Book is the ratio of [market equity + total debt + preferred stock liquidating value deferred taxes and investment tax credits] to total assets. Ln(sales) represents the logarithm of net sales. Profitability equals the ratio of operating income before depreciation to total assets. Tangibility is the ratio of net property, plant & equipment to total assets. Each regression uses a cumulative logistic conditional expectation function and is estimated using the GEE approach developed in Papke and Wooldridge (2008). All regressors, except the initial leverage variables, are lagged one period relative to the current period. We estimate the standard errors using Rogers estimators and adjust them for clustering at the firm level. Book Leverage Market Leverage Constant Initial leverage Market-to-Book Ln(Sales) Profitability Tangibility Controls for unobserved No Yes Yes No Yes Yes heterogeneity across firms Controls for unobserved No No Yes No No Yes heterogeneity across time Chi-Sq statistic Correlation between Actual and predicted