IJBF PATTERNS OF DEBT USE IN SMALL BUSINESSES: A NON-PARAMETRIC ANALYSIS

This paper uses non-parametric techniques to examine patterns of debt use by small rms and how such patterns differ across rm categories. The methodological goal is to use the richness of the rm level data and allow convincing presentations with minimum of assumptions. The procedures used provide easily comprehendible graphical descriptions of the data. The procedures augment what can be discerned from descriptive statistics by accounting for differential weights and allowing for clustering that is a native feature of cross-sectional data. We also investigate how rms could bene t if credit availability improves. Though a model-based analysis would be required to provide a detailed analysis, our analysis suggests that greater credit availability will bene t all rms. Firms with low levels of equity will be better off as their credit constraints will be less binding, while rms with high levels of equity will bene t from acquiring more debt.


Introduction
Using data from the National Survey of Small Business Finances (NSSBF) (1993), this paper offers some insightful ſ ndings into the debt holdings of small businesses in the United States. The primary purpose of the paper is to examine patterns of use of debt by small ſ rms and observe how those patterns differ across ſ rm categories. Subsequently, ſ rms' debt pattern for varying levels of equity was examined. Firms' propensity to hold debt and their likelihood of being credit constrained was also studied. 3 Apart from observing debt patterns, we analyze the incidence of credit availability for small business.
Our methodological goal is to use the unusually rich and comprehensive ſ rm level data collected by the NSSBF on small business ſ nances and apply nonparametric tools to analyze it. 4 The tools impose minimum of assumptions on the underlying distribution of the data and let the data speak for itself. In time-series data analyses, with thousands of observations, one has a luxury of being able to examine and make inferences, for example by plotting the series as a function of time. With the same number of observations, in a cross sectional setting it is less obvious what the appropriate graphical tools should be. Histograms for example, are useful for single variables, but the relationship between variables is harder to describe and discern. A two-dimensional scatter diagram on the other hand is too cluttered to be informative. Instead heavy reliance is placed on cross-tabulations and linear regressions as means of summarizing the data. While scatter diagrams do not transmit information clearly, cross-tabulations and regressions tend to over summarize, rarely doing justice to the amount of information available. This paper, highlights simple non-parametric techniques for density estimation and regression to summarize data and describe relationships. These methods provide easily comprehendible graphical descriptions of the data that are informative about the problem at hand. These techniques call for little more than presentation of the data relying less on economic or econometric assumptions. The results indicated that while the non-parametric techniques used may be limited in scope, they provide important insights by revealing natural clustering in the data. This paper augments often used cross-sectional data analysis techniques with non-parametric methodology to understand the ſ rms' behavior toward using debt, debt-equity relationship and variations in debt usage. The paper examines ſ rms' probabilities of holding debt, being credit-constrained, applying for loans and incidence of loan approval. Descriptive statistics such as mean and variance capture the central tendency and dispersion of random variables, respectively. However, descriptive statistics do not incorporate the formula that accounts for differential weights and allows for clustering that may be a natural feature of cross-sectional data. Techniques like kernel density estimations provide simple mechanisms to incorporate such features of random variables in easily comprehendible visual representations. 3 We deſ ne debt (or credit) as the combined amount of total loans including trade credit, mortgages, notes, bonds, and capital leases. 4 We focus on data visualization to this end as several studies have used different models to estimates demand for debt (Cole, 1998), incidence of loan (Cole, 2010) and have found statistically signiſ cant difference among different legal forms of small businesses.
: Non-parametric methods were used to examine the relationship between debt use and levels of equity. Regressions are used to describe linear relationships between variables with some distributional assumptions. By contrast, kernel regressions are more akin to a cross-tabulation and devoid of causal signiſ cance. It is a descriptive device that is at best preliminary to a model-based analysis. Such description provides easily comprehendible 'maps' of the effects of differences across ſ rms and some distributional characteristics on ſ rms' levels of debt without imposing the linearity and distributional assumptions. While kernel regression captures the relationship between variables, it provides no impression of the variability in debt use at each level of equity. Contour maps allow one to gauge the variability in debt use at each level of equity, and provide the ƀ exibility to include observations on ſ rms that have either no debt or equity or both. Exclusion of such ſ rms renders the sub-sample non-random. 5 Therefore, any model-based analyses of small ſ rms' debt that do not incorporate ſ rms without debt either provides inconsistent results or their conclusions relate only to the sub-sample not to all small ſ rms.
Though a model-based analysis would be required to incorporate a hypothesis that some small ſ rms are credit constrained, and what is the effect of such constraints on debt holdings of ſ rms, a simple kernel regression with dichotomous dependent variable will allows one to discern whether ſ rms with intrinsic preference for holding debt are more likely to be credit constrained. Such regressions are akin to Probit regressions and plot the probabilities. The probability plots allow one to draw simple conclusions about small businesses' behavior toward holding incremental debt if credit availability increased due to some policy changes. 6 In essence, non-parametric analyses contain a good deal less than what one would like to know. The methodology, nevertheless, allows the data to speak for itself.
This paper contributes to the existing body of literature on credit usage and small business in a several important ways. First, it documents that a large segment of small businesses in the United States are non-borrowers, i.e., they have no debt in their balance sheets. Empirical research indicated that small businesses with no debt grow slowly, hire fewer workers and invest less in productive capital (King and Levine, 1993;Rajan and Zingales, 1998). For policy makers, it raises questions such as why some small businesses use no debt and what would be the economic implications. For academicians, it would be relevant to be cognizant of a segment of small businesses that refrains from borrowing therefore skewing their empirical estimates about small business debt 5 As an example, Leece (1999) uses cluster analysis to segment market for risky ſ nancial assets and his analysis reveals that previous study may have missed some important interactions in the data. 6 Kunz (2007) presents a rich application of choice model based on latent variable models and principal component analysis to visualize competitive market structures based in individual consumer choice data.
use. Second, our use of non-parametric methodology provides some compelling indications of how small businesses that do borrow will generally beneſ t from greater availability of credit.
There are important policy implications of understanding the patterns of debt usage by small business by different sub-categories of small business owners. 7 First, a better understanding of the debt usage by different ownership types of small businesses is critical for targeting businesses that are more vulnerable to changing credit conditions. Since the debt usage differs across ownership types, policy intervention will be more effective when targeted toward groups of small businesses that are more vulnerable to changing credit availability. Any evidence on the differences on debt usage at ſ rm level will help in drawing appropriate tax and transfer policies to aid such small business operators. Second, similar information on small businesses can also affect the outcome of a comprehensive monetary policy (Gertler and Gilchrist, 1994). For example, the "credit" or "lending" view stresses the ability of monetary policy to regulate the pool of funds available to bank-dependent borrowers. 8 Our ſ ndings can be used to design monetary policies that may have a disproportionate impact on borrowers with limited or no access to capital markets.
The plan of the paper is as follows. The general organization follows the main objective of the paper and provides some methodological comments along the way. Section 2 brings up some salient features of small businesses that characterize small businesses' debt and their equity holdings. Section 3 provides the analysis of debt and some distributional features of credit availability. Section 4 concludes.

Debt in Small Businesses
There are three parties to the choice of capital structure of a ſ rm: a) the management, b) the equity holders and c) the creditors. The capital structure of a ſ rm depends on the judgment that each of the groups has about the future cash ƀ ows. In small businesses, there could be just two parties: equity holders and banks. More precisely, managers and owners constitute the equity holders and banks are the creditors. In essence, owners sell ſ nancial claims when ſ rms choose to borrow funds from banks. But, the absence of a well-formed and complete market for ownership stakes in small businesses affects ſ rms' ſ nancial decisions and differentiates them from those ſ rms operating in mature capital markets. Small businesses are inevitably more dependent on the ability of a single individual or a small number of individuals to raise either debt or equity than in the case of large enterprises. In this section, the relationship between legal 7 See Kashyap and Stein (2000) for a survey. 8 Romer and Romer (1990); Kashyap, Stein, and Wilcox (1993); and Bernanke and Blinder (1992) provide discussions.
forms of small ſ rms and other categories of small businesses and their debt holdings are discussed and the ſ rms' debt and equity holdings and their relative positions in their ſ nancial growth cycles are analyzed.
To understand the amount of equity debt holdings, ſ rst look at the legal forms of ownership. Forms of ownership are related to agency considerations. The conditions for information asymmetry arise both due to less stringent auditing requirements and because a single owner can monopolize ſ nancial information more easily than a broad management group in a large corporation. Unlike large corporations, small businesses do not allow for separation of management and ownership. Small businesses such as proprietary and partnership ſ rms can be seen as extensions of owners, especially in matters of their ſ nancial matters.
Proprietorship provides the simplest structure with a few legal requirements. The business and the owner are treated as one entity for legal and tax purposes. The main advantage is its simplicity and ƀ exibility, but the lack of distinction puts owners' personal assets at risk in the event of default or failure of the business. Therefore, banks judge proprietary ſ rms not only by ſ rm characteristics, but also with those of the owners. The reliance of a proprietary ſ rm on its owner makes it vulnerable and, therefore, not a highly regarded prospect for a potential lender. In many instances, a ſ rm's creditworthiness is as good as that of its owner. While proprietary ſ rms provide a better recourse in case of default or failure, opacity of business activity makes them a less attractive prospect for a lender.
Partnership ſ rms have a wider ſ nancial base than that of proprietary ſ rms. Though the sources of ſ nance available to partnership ſ rms are similar to those available to proprietary ſ rms, there is a greater pool to draw from by the virtue of multiple partners. To the extent that a partnership reduces the vulnerability and riskiness of the business by providing a broader range of managerial skills and greater ſ nancial stability, partnership ſ rms may be in a better position to raise capital than a proprietary ſ rm.
The ease with which corporations acquire capital depends on their size, stability, quality of management, nature of the industry, track record and their investment opportunity sets. The nature of limited liability to shareholders is perceived as a disadvantage by potential lenders, and in many cases lenders will require additional collateral both from shareholders as private individuals and from the company. It is hypothesized that bank-ſ rm relationships matter more in credit extension to C-and S-corporations than to proprietary and partnership ſ rms. When ſ rms default, banks have greater recourse with proprietary or partnership ſ rms than with corporations.
Looking beyond the legal form of ownership, the study extended the analysis to some of the owner characteristics, such as race and gender. Research on small business lending indicates that some owners face racial discrimination in the credit market (Cavalluzo and Cavalluzo, 1998). Impact of such actions would be seen either in low loan approval rate for certain groups of owners or such groups may self-select not to apply for loans. Dummy variables for the race of owner captures the extent of discrimination, while the selection methodology used in other studies took into account the lower incidence for loan application in general (Cole, 1998;Cavalluzzo, Cavalluzzo and Wolken, 2002;Blanchƀ ower, Levine and Zimmerman, 2003). To account for sexual discrimination, we could include a dummy variable for female operators, though the literature indicated that female owners are generally not discriminated in the credit market (Cavalluzo and Cavalluzo, 1998). While ownership structure explains the legal framework and role of some owner characteristics on debt in small business, it ignores relative positions of ſ rms in their ſ nancial growth cycle and relationship between equity and debt.
The study started with a basic presumption that assets of ſ rms are highly correlated with the stages of growth. Financial growth cycle explains ſ rms' debt holdings relative to ſ rms' ſ nancial growth cycle. Firms go through ſ ve stages of growth, namely: formation, rapid growth, growth to maturity, maturity and decline (Weston and Brigham, 1981). 9 Despite limited agreement on the terminology, ſ nancial growth cycle theory provides a reasonable framework to analyze the debt of small businesses.
Major sources of ſ nance for start-up businesses are insider ſ nance and particularly owners' personal resources. So this stage of growth does not see much bank debt as deſ ned earlier, except in some instances usage of personal credit cards. Insufſ cient resources at this stage may result in under-capitalization of ſ rms. As ſ rms grow, they gain access to venture capital on the equity side and bank ſ nance on the debt side. Continued growth beyond formation is very likely to be ſ nanced by retained earnings, trade credit and bank loans -some of the only and critical sources of ſ nance small businesses have (Sahlman, 1988;Wetzel Jr., 1994). Rapid growth outstrips ſ nancial resources leading to liquidity crises and the outcome is a greater reliance on short-term ſ nance due to lack of long-term ſ nance.
The unavailability of long-term debt and equity, which is deſ ned as ſ nancial gap, will have direct effect on the ſ nancial characteristics of small businesses. In the presence of ſ nancial gap, ſ rms are presented with very different situations. Firms have to rely more on internally generated funds. Lack of long-term debt ſ nance forces reliance on short-term sources and reduces liquidity. Similarly, lack of equity ſ nance results in higher levels of leverage. Thus, short-term debt substitutes not only long-term debt but also equity.
Apart from issues of ownership and stages of growth, other factors may affect the ſ nancial characteristics of small businesses. Neck (1977) grouped such factors as hosts, agents and environment. The host are the owner-manager, agents are various ſ nancial environment institutions, and the environment constitutes legal, tax and economic institutions and market conditions. Hosts matter, because different owners using the same amount of ſ nance could produce very different 9 This classiſ cation does not imply that ſ rms necessarily go through all growth phases. Maturity and decline of a ſ rm can take place at any point after the formation. Incidentally, some of the small businesses avoid some stages by forming and ƀ oating in the stock market at the same time. In some instances, the term growth is misleading, because growth can continue to occur during the maturity stage.
results. Agents, particularly creditors, often turn out to be a major source of external funds. The environment is so broad that it is beyond the scope of this paper. But what does come up very often is the transparency of small business activities and the extent to which small businesses are legally required to disclose their ſ nancial statements publicly.

Debt Patterns of Small Businesses
The study used the data from NSSBF (1993) to describe debt patterns of small businesses. Non-parametric methods were applied to examine the relationship between debt and equity, and how does this relationship vary across legal forms of ſ rms and owners' gender and ethnicity. The study examines a general hypothesis that some small ſ rms are credit constrained and some small ſ rms do not have any debt in their capital structure. Using a simple kernel regression with a dichotomous dependent variable, the investigation attempted to predict the effect of greater credit availability on ſ rms that are credit constrained and those that do not have any debt. A similar analysis was conducted to understand the effect on ſ rms' incidence of credit application and credit approval. This paper begins with the description of ſ rms' ſ nancial data, and this forms the basis for Tables 1 and 2.  Table 1 shows some ſ rm characteristics and their descriptive statistics for different ownership categories. There are 4,637 ſ rms in the NSSBF, and the study chose four ſ rm characteristics -debt, equity, assets and age, and mapped them for different legal forms of the ſ rms and owners' gender and ethnicity. The NSSBF has proportional representation of each of the categories and throughout this paper debt is used as a measure of leverage; either the absolute amount of debt or the debt-asset ratio. Judging by the average amount of debt, C-and S-corporations and proprietary ſ rms are less levered than partnership ſ rms. When it comes to equity, proprietary ſ rms have the highest equity followed by C-and S-corporations, and partnership ſ rms have the least amount of equity. C-corporations are some of the oldest ſ rms in the sample, while other ſ rms' average age is about 14 years. 10 Minority and female owned ſ rms are the youngest ſ rms in the sample. Table 1 ranks partnership ſ rms the largest with the highest average amount of assets, followed by C and S-corporations. Lower panels of Table 1 present similar statistics for ſ rms by female and minority owned organisations. Minority owned ſ rms average about 92% debt-equity or 44% debt-asset compared to 127% debt-equity and 49% debt-asset ratios for majorities. However, such disparity is not observed among business ownership differentiated by gender. Such inferences drawn from Table 1 point to the central tendencies and can be misleading when we compare similar statistics at greater granularity. It is expected that non-parametric methods that were used in this paper will improve upon inferences that were generally gathered from descriptive statistics.
The study also examined the incidence of ſ rms' holding debt and the probability of ſ rms' being credit constrained. Some small ſ rms chose not to have any debt in their capital mix, and it is easy to identify them in the present dataset. However, it was difſ cult to identify ſ rms that are credit constrained due to the lack of any direct measure of a credit constraint. The NSSBF of 1988-89 and 1993 provides information to identify credit constraint ſ rms. A ſ rm is deſ ned be credit constrained if the ſ rm replied afſ rmatively to one of the two following questions: a) "With the most recent loan application did a bank turn down the loan application or has the ſ rm been unable to get as much as it applied for?" and b) "During the past three years, were there times when the ſ rm needed credit, but did not apply because it thought the application would be turned down?" On average C-and S-corporations are more likely to be credit constrained compared to proprietary and partnership ſ rms, and also more likely to have debt (see Table 2). Proprietary and partnership ſ rms are less likely to be credit constrained and to hold debt. It was noted that there are similar proportions of credit-constrained ſ rms irrespective of owners' gender or race. An economy wide credit crunch may affect corporations more adversely because of their greater reliance on debt than proprietary and partnership ſ rms. The ſ nal two columns of Table 2 show the percentage of ſ rms that applied for a loan in last three years and the percentage of ſ rms that had their loans approved. The incidence of holding debt and being credit-constrained separately were analyzed from the probability of applying for a loan and the application being approved for all ſ rms in the survey. This was done because ſ rms' decision to hold debt can be seen as a separate process than their decision to apply for credit. 11 10 Berger and Udell (1995) have shown that longer relationship does impact availability of credit as it lowers the likelihood of collateral being required for lines of credit. 11 See Cole (2010) for an analysis of why some small businesses do not use credit.
Averages such as those in Table 1 and 2 conceal as much as they reveal. Let us look at examples from each table. Table 1 ranks partnership ſ rms as the largest with the highest mean value of assets over C-and S-corporation ſ rms. More large ſ rms in C-and S-corporation categories than in partnership category were found when examined their asset values at different percentile levels. It could be true for any descriptive statistics that we stated in Table 1, and it is hoped that non-parametric methods will improve upon them. Similarly, in Table 2 the broad inter-categorical patterns of distribution tell us which group of ſ rms will be more impacted by a poor lending scenario due to their greater debt exposure. It may be concluded that corporations are more likely to be credit constrained looking at the statistic in Table 2, when compared to partnership and proprietary ſ rms. However, any direct effects of credit crunch will be difſ cult to uncover unless the methodology allows us to examine ſ rms at different levels of debt.  Figure 1 shows estimates of the distribution of total debt for four legal forms of ſ rms at different levels of debt, unlike the statistics of Table 1. Institutional features indicate that lenders have better recourse against funds lent to proprietary and partnership ſ rms than to S-or C-corporations. Corporations are characterized by limited owner liability unlike proprietary and partnership ſ rms.
It stands out that corporations have generally been in business for longer and are at later stages in their ſ nancial growth cycle than proprietary and partnership ſ rms. We expect to see different distributional attributes of debt across the ſ rms of these four legal categories. Density functions allow us to examine distribution attributes of debt at different levels. For estimation purposes, the logarithmic transformation for most variables was chosen because the distribution of debt like most of other variables is skewed, and log transformation yields a distribution that is more symmetric and closer to the normal distribution. 12

Figure 1. Distribution of Debt Across Legal Forms of Firms
The most obvious feature of Figure 1 is the relative positions of debt holdings of four types of the ſ rms. Partnership ſ rms and S-and C-corporations have higher level of average debt and some of them have very large amount of debt as evident by fat right tail of the distribution. Proprietary ſ rms tend to hold less debt. The kernel density plot shows their debt holdings and suggests how it is different from other three categories of legal forms of ſ rms. The mean level of debt of S-corporations is about 75% of that of the partnership ſ rms and 40% C-corporations. The density plots indicate that S-and C-corporations tend to have much similar distributions of debt holdings. Partnership ſ rms have greater number of outliers evident by the fat right tail and a signiſ cant mass of ſ rms around its median value. 12 We provide a brief note in Appendix 1 on bandwidth selection in kernel estimation. Univariate plots of Figure 1 reveal little about the relationship between two or more variables. While organizational structure theory of small ſ rms predicts that partnership ſ rms may have greater amount of debt among four legal categories of ſ rms because they provide a greater recourse against loan default and failure than corporations and proprietary ſ rms. The ſ rm growth cycle theory sheds light on role of relationship on borrowing as such relationships forms very gradually over extended period of time. 13 We want to gauge how debt levels differ with equity for different categories of ſ rms using kernel regressions.
Debt-asset ratios for different levels of equity for each category of ſ rms are plotted in Figure 2. For each value of equity, Figure 2 shows a weighted average value of debt-asset ratios nearby. The weights are the same weights used to construct the density, but are scaled by the estimate of the density at the point. The plots of estimated regressions look similar. But, the advantage of the technique used is that the underlying data determines the shape of the function. No assumptions were made about how the data should look likefor example lie along a straight line, or along a low-order polynomial. Nonparametric regression, therefore, reveals the true relationship between debt-asset ratio and level of equity. The downward sloping curves in Figure 2 indicate that ſ rms with less proportion of debt will have greater equity, and regression plots afſ rm that the role of debt in the capital mix declines with increasing equity. 14 13 A similar model is employed by Koutmos and Booth (1995). 14 The data are available from the authors upon request. .7 .8 .9 1 It was noted that at low equity levels the debt-asset ratio is high for all types of ſ rms, and then tends to decline in a two-step fashion. Over the whole distribution, the curve is steeper at low levels of equity then ƀ attens out for ſ rms with larger amounts of equity. Debt-asset ratios could be as high as 75%, before it declines precipitously. It can be concluded that ſ rms that borrow tend to have at the minimum about one-third to one-half debt as a share of total assets. The impact of credit crunch would be very different for ſ rms if we incorporate two obvious aspects that are not being captured in Figure 2: some ſ rms choose not to have any debt or equity in their capital structure, and some of them self-select not to apply for loans fearing denial. If all ſ rms have equal access to bank credit and there is no precipitous difference in the interest rates charged to ſ rms, then in case of a general credit crunch ſ rms with high levels of debt will be more adversely affected. Apart from obvious omission of the ſ rms without debt and/or equity, the curves in Figure 2 can also be misleading for giving no impression of the variability in debt holdings at each level of equity. On average ſ rms with low levels of equity have as much as 60 to 80% debt in their capital structure as evidenced by the regression lines in Figure 2.
Contour maps bring to surface the variability in debt at each level of equity and vice-versa while incorporating observations with no equity and/or no debt (see Figures 3a & b). The graphs present estimates of the joint density of the logarithm of debt (on x-axis) and the logarithm of equity (on y-axis). While the x and y-axis are the same as in the kernel regression graphs, and the height of the graph represents the fraction of ſ rms at the levels of the log value of equity and log value of debt represented by the co-ordinates along the base. 15 Figure 3b provides the joint density surface map also known as the net-map that presents a visual impression of the surface of the joint density. The joint density map gives a better picture of relative height and the concentration of mass. The visual superiority of the contour maps is the following. First, the density surface provides information about the tails of the distribution where there may be very few observations. Second, one also observe that whether ſ rms have just debt or equity or both. The density does not fall to zero, and the presence of an open 'hole' or 'cave' indicates either ſ rms with some equity and no debt or vice-versa. The joint density graphs also bring to surface any clustering of ſ rms. It was noted that for the most part Figures 3a and b are similar, and that no new caveat emerges. Our next step is to incorporate a general hypothesis prevalent in the literature that credit constraints do exist in small businesses; as small businesses have varying levels of debt therefore they face varying degrees of credit constraints. 16 The NSSBF dataset provides an additional caveat and that is small businesses demonstrate different willingness to hold debt and some carry no debt at all. Though a model-based analysis is required to provide estimates of overall impact of credit constraints on the level of debt held by ſ rms, the data in 15 More details on contour maps can be found in Härdle (1987) and Silverman (1986). 16 See Udell (2006), Berger, Frame andMiller (2005), Petersen and Rajan (1994).
the survey allows us to visualize probability of holding debt and being credit constrained for different levels of equity. We are able to shed light on whether ſ rms with intrinsic preference for holding debt are more likely to be credit constrained. Figure (4) shows estimates of proportion of small ſ rms that chose to have debt in their capital mix and ſ rms that are credit constrained. This is a non-parametric regression using a dependent variable which takes one for ſ rms with debt and zero otherwise, and one for ſ rms that are credit constrained and zero otherwise. The regression is akin to a Probit regression, and regression lines plot the probability of holding debt and ſ rms being credit constrained, respectively. The general inference from Figure 4 is that among small ſ rms the likelihood of holding debt is greater than the likelihood of being credit constrained. As expected at low levels of equity, small businesses are more likely to have debt. The probability of holding debt reduces more rapidly as level their equity share increases. Over 60% of ſ rms are credit constrained with no or limited equity, and in excess of 40% of them continue to be creditconstrained at high levels of equity. The probability of a ſ rm being credit constrained decreases with greater levels of equity and so does the probability of holding debt; the upper line is downward sloping at higher levels of equityasset ratios. The positioning of curves also suggests that for any equity-assets ratio the probability of holding debt is higher than the probability of being credit constrained. Therefore ſ rms that are more likely to have debt are more likely to .7 .8 .9 1 be credit-constrained. Figures 3a and 3b where it was included both borrowers and non-borrowers conſ rms an upward sloping relationship between equity and debt. Figure 4 shows that ſ rms with equity-asset ratio greater than 0.90 have probability of being credit constrained at 40% or greater. Therefore, most small businesses will beneſ t from better credit availability. First, for ſ rms with lower equity-assets ratios, better conditions will lower their probability of being credit constrained. Second, ſ rms with higher equity-assets ratios can presumably lower the weighted average cost of capital by acquiring more debt for equity. 17 It is emphasized that better credit conditions may encourage more request for debt. The analysis of incidence of loan application and approval reveals that better credit environment may rarely improve the probability of loan approval, however, may increase the incidence of loan application. The process of obtaining a loan is sequential in nature. The ſ rst step in the process of applying for one is critical. Most factors that inƀ uence the incidence of loan application also affect the probability of loan approval, and NSSBF (1993) cites very similar reasons ſ rms gave for not applying for loans and banks gave for declining loan applications. .7 .8 .9 1 17 The distribution at the tails could be misleading. At the tails the estimated density becomes smaller, and since their estimate enters into the denominator of the conditional mean, the regression function is less precisely estimated.
However, while examining the probability of loan approval, the probability of applying for a loan should be taken into consideration. 18 Figure (5) suggests that if credit conditions turn favorable, the increase in loan application approval may not be as large as the increase in number of ſ rms that apply for loans.

Conclusions
The main conclusion of this paper is that there are some marked differences among small businesses in their debt holdings across different ownership types. The contour maps uncover an important detail -some small businesses do not carry any debt in their balance sheet and some have no equity. This becomes an important distributional feature to be recognized in any empirical work. The often-used practice of excluding such ſ rms may provide results that cannot be generalized. Cole (2010) noted that this sub-sample of small businesses has received limited attention in academic work. It is important to recognize that several empirical strategies can be used to incorporate these ſ rms with no debt or equity. For example, Heckman's two stage estimation technique can ameliorate the selection bias of leaving out a subsample of data. It was also observed that greater credit availability is likely to beneſ t all small businesses. However, the beneſ t would come from reducing the probability of being credit constrained for ſ rms with no or low equity, allowing ſ rms to switch to less expensive debt or by increasing the probability of application for loans. On the methodological front, the study was able to view some distributional features of debt in small businesses using a large data set. Though some of the substantive conclusions will require rigorous model based analyses, the study was able to use the data to draw conclusions about the impact of favorable credit conditions. By analyzing the sub-samples for different categories of ſ rms, we are able to shed light on how different is allocation of debt within the small business sector. It was observed that larger corporations have more debt and more equity, but are also more likely to be credit constrained.
The equations (1) -(3) are used for non-parametric regressions in the paper. In some ſ gures, the dependent variable y is either logarithm of debt, or debt-asset ratio or simply one or zero depending on whether the ſ rm does or does not have debt or is credit constrained. The graphs such as in Figure 2 of text are constructed by calculating (1) for 20 equally spaced values of the log of equity and plotting the results. To select bandwidth, we start with trial and error. The basic criterion behind this procedure is to select a bandwidth that appears to give enough smoothness without obscuring detail. We do experiment with crossvalidation, one of the procedures generally followed in the literature. However, we ſ nd that the informal methods were unlikely to be misleading, at least for the graphical purposes of this paper.
For the density estimations, such as Figure 1 in the text, we follow a similar procedure. At each point on the x-axis, we take into account 50 nearby observations. We obtain an estimate of density by taking a ratio of the count and of the sample size. Like kernel regression, it is sensible to use a weighing arrangement that gives closer ſ rms greater weights. We use the following kernel function to achieve this. (4) where K h (.) integrates to unity; a condition required to generate a proper estimate of the density. Equation (4) is used for both the univariate densities and also for the kernel regressions.
Contour maps and surface plots are calculated using similar general principles. We ſ rst construct a grid over the range of the two variables. Second, at each point on the grid, a weighted count is made of the observation within a neighborhood of the point. We use a kernel weighting function just as we did for kernel regressions and densities. The bivariate Epanechionov is given by, where d i is a two element vector of deviations of X i -X and Y i -Y each divided by the bandwidth h. An observation for estimation is included only if it is within a circular region centered at the current point and with radius h. The density estimate at a point is given by, where k(d'd) = K(d), and S is the sample variance covariance matrix of the two variables.