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«Douglas O. Cook Department of Economics, Finance, and Legal Studies Culverhouse College of Business University of Alabama Tuscaloosa, AL 35487-0224 ...»

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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 35487-0224



Robert Kieschnick

University of Texas at Dallas

2601 N. Floyd Rd, SM 31

Richardson, TX 75080



* 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.

On the nature of corporate capital structure persistence and convergence


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

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.

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:

CSi,t = f ( β X i,t ) (1) where CSi,t represents the proportion of capital accounted for by debt capital for firm i at time t, and f(.) is a nonlinear function of Xt, which represents the determinants of a corporation’s capital structure. For the sake of simplicity, suppose that Xt 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:

–  –  –

where Fi, Fi*, Fi** represent different measures of firm fixed effects (e.g., Fi 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 (CSi,0) and a firm fixed effect (Fi), then one should also observe that CSi,0 and Fi will be significant; both statistically and A quadratic function is clearly inappropriate for these data as it is not bounded on the unit interval.

Note that β1, β2 and β3 represent the sum of the different cross-product terms involving powers of xi,t since the derivatives of the function evaluated at xi,0 are constants.

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.3

2.2 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 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.

See Caves (1998) or Cabral and Mata (2003) for discussions of this literature.

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 2003. 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, 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.

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