Adding Non Linear And Asymmetric Dependence To The Market Timing Theory


Fernando Francis Moreira
Krannert Graduate School of Management
Purdue University
West Lafayette, IN 47906

December 2007

Keywords: Capital Structure, Market Timing Theory, Copulas, Asymmetry.
JEL Classification: G32.


“Discovering consists of looking at the same thing as everyone else does and thinking something different.” (Albert Szent-Gyorgyz, 1937 Nobel Prize in Physiology and Medicine)

Many current theories in finance are based on the assumption of normality and in terms of dependence between variables the correlation coefficient is the most usual measure.
In their seminal work, Embrechts at al. (2002) show that linear correlation (Pearson Correlation, usually represented by rho) does not capture the dependence between variables that are not normally distributed or, more generally, spherically or elliptically distributed . The authors add that empirical research in finance has shown that the distributions of the real world are seldom in this class.
In many situations the concept of correlation has been used as synonym of dependence but correlation is only one particular measure of stochastic dependence among many.
Some recent researches have been using the concept of copulas to deal with variables not normally distributed. However, an extremely reduced number of such works is related to corporate finance.
Trying to fill this gap in the literature and to eliminate the inadequate assumptions of normality and the consequent assumption of linear dependence, we will search for the best family of copulas able to capture the dependence structure between leverage and four variables, namely: market-to-book ratio, asset tangibility, profitability, and firm size. This done, we will be able to identify potential non-linearity and asymmetry in such relationship. In other words, we will check whether the proportion of debt changes at different rates depending on the direction and level of the other four variables. For instance, it could be the case that the leverage decreases X% when equity market values increases (i.e. market-to-book ratio goes up) but increases (X-alfa)%, for alfa>0, when equity market values decreases. This hypothetical example is equivalent to say that the dependence between leverage and market-to-book ratio is stronger when the latter is at an increasing movement.

“I can’t understand why people are frightened of new ideas.
I’m frightened of the old ones.”
(John Cage)

II.1. Questions
II.1.A. Main questions

- Is the impact of market-to-book ratio, asset tangibility, profitability, and firm size on leverage conditional on the magnitude changes in those variables?
- Is such impact asymmetric? That is, does the leverage oscillate in different rates if those variables are increasing or decreasing?

II.1.B. Secondary questions
- In which scenario does the leverage change at the highest rate? In which scenario does it change at the lowest rate?
- Is such relationship constant or it varies through time?

II.2. Objectives
- To check whether the relationship between leverage and Market-to-book ratio, asset tangibility, profitability, and firm size is non-linear and asymmetric (i.e. to check whether leverage oscillations are conditional on the level and direction of changes in the other four variables);
- To test conditional copulas as a method to integrate leverage changes with market-to-book ratio, asset tangibility, profitability, and firm size (by comparing to the methods that use correlation coefficient, specially by comparing our results to the results reached by Baker and Wurgler, 2002).

II.3. Motivation/Importance of the study
The Market Timing theory on Capital Structure assumes a linear relationship between changes in leverage and changes in market-to-book ratio, asset tangibility, profitability, and firm size. This means that relationship among those variables is considered constant regardless of the magnitude and the direction of the changes. For example, in their table II (p.10), Baker and Wurgler (2002) found a linear relationship between changes in book leverage and market-to-book ratio, fixed assets (tangibility), profitability, firm size, and lagged leverage (being that the latter will not be included in this study). For the first year analyzed, the relation is:
Change in Book Leverage = -3.70*MB + 0.04*TANG – 0.10*PROFT + 3.83*SIZE
MB = market-to-book ratio;
TANG = tangibility;
PROFT = profitability;
SIZE = log sales (proxy for size).

This dependence means, for example, that keeping the other variables constant, each unit increased in MB will cause a decrease of 3.70 units in Book Leverage, regardless of the level of MB and the direction of its changes (i.e. irrespective of the fact that MB is “extremely” low or high or if it is increasing or decreasing).
On the other hand, this research has the purpose of checking whether the oscillation in leverage is conditional on the level and the direction of the changes in the other variables. Taking again market-to-book ratio (MB) as an example, we want to check if the units decreased in Book Leverage are smaller or bigger when MB variation presents higher values (and vice-versa), contrary to what Baker and Wurgler (2002) advocate. This can be reach by using the (unconditional) copula theory.
Moreover, we want to check whether changes in Book Leverage are asymmetric (that is, if Book Leverage changes at different rates depending on the fact that the other variables are increasing or decreasing). To do so, we will use conditional copulas.
We tested the variables considered in this study and concluded that they are not normally distributed (the results are presented in the Appendix). Thus, the coefficient correlation is not adequate to measure dependence among such variables and the use of copula theory is an altenative since it allows us to check dependence among variables without assuming normality.
The expected contributions are:
- improvement of the comprehension of the relationship/dependence between the change in leverage and market-to-book ratio, asset tangibility, profitability, and firm size. This will help us to improve our understanding on the interaction of such variables.
- to show that managers’ decisions in terms of capital structure are not linear and conditional on the magnitude and direction of change of the factors mentioned in the prior item. This is an original contribution since, in general, the theories on capital structure do not consider this possible “non-monotonic” aspect of management’s decisions.

II.4. Originality
To best of our knowlegde this is the first research (paper) using (conditional) copula approach to analyze the dependence between leverage changes and market-to-book ratio, asset tangibility, profitability, and firm size. Actually, it is probably the first research that adopts conditional copulas to explain some specific fact related to Capital Structure .
It is interesting to point out that we will complement the findings of Baker and Wurgler (2002) by testing not only the potential non-linear dependence among the variables but also its asymmetry at the extremes (tails of distributions).
It is expected that any result we get will bring new knowledge to the study of capital structure decisions. For example, if leverage changes faster either when market-to-book ratio is decreasing or when market-to-book ratio is increasing, it will represent something never shown before.

II.5. Hypotheses
- Since book leverage, market-to-book ratio, asset tangibility, profitability, and firm size are not normally distributed, the oscillation of book leverage happens at different rates depending on the magnitude of the change in the other variables (it is expected that higher dependence occurs at the extreme variations, i.e., at the tails of the variables distributions);
- Changes in leverage also depend on whether the other variables are increasing or decreasing. That is, leverage changes are more or less dependent on the other variables according to the direction of their movements (so, the concept of conditional copula is more adequate than the simple copula one).

III.1. Theories on Capital Structure

Before Modigliani and Miller (1958) there was no generally accepted theory on capital structure. In that paper, the authors introduced the capital structure irrelevance proposition which, in sum, advocates that the leverage of a firm has no effect on its market value. It is assumed that capital markets are efficient and that firms have a particular set of expected cash flows. When each firm decides the proportion of debt and equity to be used, it is simply dividing their cash flows among investors. A key point in this theory is that investors and firms are assumed to have equal access to financial markets, so the leverage of each investment can be adjusted according to the interest of its owner. This means that investors can create any leverage even though it was not offered by firms or investors can get rid of any leverage taken on by firms but not wanted. Therefore, it does not matter the leverage decided by firms, investors can always trade in the market and adjust their position.
However the irrelevance proposition only can be applied to a world without frictions. Thus many theories try to explain capital structure in the presence of frictions, such as taxes, bankruptcy costs, transactions costs, agency conflicts, adverse selection, and time varying market opportunities.

III.1.A. Trade-off Theory
Myers (1977, 1984) add two frictions (agency costs of financial distress and the tax-deductibility of debt) to the analysis and state that the balance between them result in an optimal capital structure. Acording to this Trade-off Model, firms set a target debt-to-value ratio and then move towards the target. A key implication of this idea is that leverage exhibits target adjustment so that deviations from the target are gradually eliminated.
Frank and Goyal (2005) point out that some aspects of this theory merit additional considerations. First, the target is not directly observable and different assumptions on the structure lead to different results.
Second, the theory assumes a simplistic tax code. If different features are included, different conclusions about the target may be reached.
Third, the nature of the bankruptcy costs are not clear. For example, it is not defined if they increase with the size of the bankruptcy and if they are one-time cost or permanent cost.
Finally, there is evidence against considering tax shields as the only motivation to firms use debt is not appropriate since it is very difficult to match the observed leverage ratios in particular decades with the corporate taxes in the same period. Moreover, debt financing was common long before the introduction of corporate income taxes.

III.1.B. Pecking Order Theory
The Pecking Order Theory (Myers, 1984 and Myers and Majluf, 1984) points out frictions due to asymmetric information between managers and outside investors that make firms prefer to use resources in the following order: internal funds, debt, and equity. The main idea is that the owner-manager of the firm knows the true value of the firm’s assets and growth opportunities. Outside investors can only guess these values. When the managers sell equity, outside investors must wonder why managers are doing so. It is plausible to infer that managers want to sell equity because it is overvalued.
In this theory, there is not a well-defined optimal debt ratio. The benefit of tax shields and the risk of financial distress are also considered but in a second-order. Changes in debt ratios are caused by the need for external funds, not by the attempt to reach an optimal capital structure.
Actually, debt is not formally included in the analysis of Myers and Majluf (1984) If debt was available and risk-free, it could be considered as internal funds. As debt is risk, when available, Myers (1984) argues intuitively that it should be somewhere between retained earnings (internal funds) and equity.
The idea of agency costs as developed by Jensen and Meckling (1976) can also be used as an argument in favor of the pecking order model. Frank and Goyal (2005) show that when a firm is owned and run by only one entrepreneur, she can get a higher payoff than in the case when part of the investment is financed by outside equity. In this latter situation, the entrepreneur is underinvesting and she bears the full cost of any perks not consumed and she must share the benefits with outside investors.
Frank and Goyal (2005) emphasize some problems of the pecking order theory.
Although it is expected that equity would be the last alternative to financing, firms issue too much equity. Also, following its ideas we can assume that it is the financing deficit that drives debt issues. However, empirical studies have shown that other factors seem to be more importantly related to the leverage of firms.
Thus, from items III.1.A and B, we see that the standard versions of the Trade-off Theory and the Pecking Order Theory need to be improved to account for the empirical findings.

III.1.C. Market Timing Theory
Baker and Wurgler (2002) propose the Market Timing Theory suggesting that managers try to time the equity markets and, therefore, the current capital structure is related to historical market values.
The authors associate market-to-book ratio, asset tangibility, profitability, and firm size with the amount of capital raised by firms (i.e, their financial deficit) . Their results suggest that market timing has large, persistent effects on capital structure and that firms tend to reduce leverage ratios by raising substantial capital when the equity market is perceived to be more favorable, that is, when market-to-book ratio are higher.
The authors use traditional capital structure regressions being that leverage is the dependent variable and the market-to-book ratio is the independent one. This latter variable is a weighted average of firm’s past market-to-book ratios . As a result they find that leverage is strongly negatively related to past market-to-book ratio. Such impact of market values goes back to until 10 years in the past, i.e. current capital structure depends strongly upon variation in the market-to-book 10 years ago.
In contrast to the efficient capital market of Modigliani and Miller (1958), Baker and Wurgler (2002) consider the inefficiency and state that market timing benefits ongoing shareholders at the expense of entering ones. Therefore managers have incentives to time the market if they care more about ongoing shareholders.
The results of the Market Timing Theory cannot be explained by the Trade-off Theory which considers market-to-book as an indicator of investment opportunities, risk, and agency cost. The Trade-off Theory states that temporary fluctuations in the market-to-book ratio or any other variable ought to have temporary effects. Notwithstanding, as said before, Baker and Wurgler (2002) found evidence that the market-to-book ratio has very persistent effects.
Pertaining to the Pecking Order Theory, adverse selection is pointed out as the reason that leads managers to avoid issuing equity. There is also a dynamic version of this theory that argues that firms with upcoming investment opportunities may reduce leverage to avoid using equity in the future. Nonetheless, it is not reasonable to imagine a version of the Pecking Order Theory that explains the observed strong relationship between leverage and the long-past market values. This theory implies that periods of high investment will push leverage higher toward a debt capacity, not lower as the Market Timing Model suggests.
Baker and Wurgler (2002) recognize that there are two versions of equity market timing that could help to explain their results. One is the dynamic version of Myers and Majluf (1984) that considers rational managers and investors. The extent of adverse selection varies across firms and/or across time and is inversely related to the market-to-book ratio. The second version is that managers think investors are irrational and raise equity when the cost of equity is unusually low. This version explains the Market Timing Theory if variation in the market-to-book ratio is a proxy for managers’ perceptions of misvaluation. However to explain the persistent effect of past valuations, both versions need that adjustment costs, maybe related to adverse selection, reduce the desirability of undoing market timing.

III.2. Copula
The concept of copula was formally published in 1959 and was first applied in Finance in 1999. Nelsen (2006) states that the study of copulas is a recent phenomenon in statistics. Moreover Rockinger and Jondeau (2001) remember that copulas have not yet found their way into empirical finance. According to Fermanian and Wegkamp (2004), “the research on relevant specifications for copulas and on their time dependence is still in its infancy” (p.1).
Copulas are functions that link a joint distribution to its marginal distributions (cumulative distribution functions, cdfs) of its variables.

F (X,Y) = C ( F(X), F(Y) )

F(X,Y) is the joint distribution of X and Y;
C is the copula (we need to calculate); and
F(X) and F(Y) are the marginal distributions of X and Y respectively (we know them).
So, by calculating the copula C we are able to find the joint distribution F(X,Y) since we already know the marginal distributions F(X) and F(Y). The name “copula” was chosen to emphasize the manner in which a copula “couples” a joint distribution function to its univariate margins.
A detailed view on copulas is given in Joe (1997) and Nelsen (2006).
The following can be seen as an informal definition of Copula : Let X1, …, Xn be continuous random variables with distribution function H(x1,…, xn) and marginal distributions FX1,…, FXn, correspondingly. For every (x1,…, xn) in [- infinite, + infinite]n consider the point in [0,1]n to [0,1] is an n-dimensional copula, or shortly n-copula. (Kolev et al., 2006).

Rockinger and Jondeau (2001) present an “intuitive” view and state that a two-dimensional copula is a function C: [0,1]2 mapped to [0,1] having three properties:
1. C(u,v) is increasing in u and v;
2. C(0,v) = C(u,0) = 0, C(1,v)= v, C(u,1)=u;
3. Any u1, u2, v1, v2 in [0,1] such that u1 < u2 and v1 < v2 we have C(u2,v2) – C(u2, v1) – C(u1, v2) + C(u1, v1) >= 0.

Property 1 means that when one marginal distribution is constant, the joint probability wil increase if the other marginal distribution increases.
Property 2 reveals expected conditions for joint distributions: if one margin has zero probability the joint occurrence also has zero probability to occur. Consequently, if on the contrary one margin is certain to occur, then the probability of a joint distribution is determined by the remaining margin probability.
Property 3 states that if both u and v increase then their joint probability also increases. This property is therefore a multivariate extension of the condition that a cdf is increasing.
Moreover, if we set u = F(x) and v = G(y), then C(F(x),G(y)) yields a description of the joint distribution of x and y. Based on this intuitive definition, further properties may be obtained.

The definition above is related to two-dimensional copulas but it also works for m-dimensional copulas, i.e. C:[0,1]m mapped to [0,1].
A central idea in copula approach is the Sklar’s Theorem which is the foundation of many applications of that theory to statistics. Such Theorem elucidates the role that copulas play in the relationship between multivariate distribution functions and their univariate margins.
Sklar’s Theorem: Let H be a joint distribution function whith margins F and G. Then there exists a copula C such that for all x,y in R,
H(x,y) = C(F(x),G(y)).
If F and G are continuos, the C is unique. Conversely, if C is a copula and F and G are distribution functions, then the function H is a joint distribution function with margins F and G.

Rosenberg and Schuermann (2005) clarify that the dependence relationship between the variables is determined by the copula, while the scale and the shape of their distributions (that is, mean, standard deviation, sknewness, and kurtosis) are completely determined by the the marginals
Schmidt (2006) adds that all marginal distributions are transformed into uniform ones so that all variables get the same type of distribution. Hence the intuition behind copula idea is that the equally marginals (after transformation) are used as the reference case and the copulas express the dependence structure according to this reference.
However such transformations are done by the sake of simplicity. Frees and Valdez (1997) state that the marginal distributions can be of any type. Moreover unlike many simulation applications, the random variables in the copula function are not assumed independent. Moreover
Nelsen (2006) points out many advantages of using copulas: their flexibility in modeling dependence (various copulas represent different dependence structures between variables and they allow us to separately model the marginal behavior and the dependence structure), they represent a more informative measure of dependence between linear correlation (copulas tell us not only the degree of the dependence but also the structure of the dependence and can directly model the tail dependence), and copula functions are invariant to transformations of the underlying variables while the correlation is not (i.e. the same copula function can be used, e.g., for both the prices of assets and their logarithm). Moreover copulas do not require normality of the variables studied, which is useful when dealing with dependence between asset returns (specially with high frequence data).
Fermanian and Scaillet (2005) stress that one attractive property of copulas is their invariance under strictly increasing transformations of the margins, being that this is not true for the Pearson’s correlation coefficient. The use of copulas allows us to solve a difficult problem, namely to find a whole multivariate distribution, by performing two easier tasks. The first task is to model each univariate marginal distribution either parametrically or non-parametrically. The second task consisits of specifying a copula, which summarizes all the dependencies between margins. The authors add: “However this second task is still in its infancy for most of multivariate financial series, partly because of the presence of temporal dependencies (serial autocorrelation, time varying heteroskedasticity,…) in returns of stocks indices, credit spreads, or interest rates of various maturities” (p.3).
This research will deal with one of this problems by using conditional copulas that try to capture time-varying dependencies.
Charpentier at al. (2006) warn that copulas empiral estimation is “a harder and trickier task. Many traps and technical difficulties are present, and these are, most of the time, ignored or underestimated by practitioners” (p.1). The association of marginal distributions to an estimated multivariate distribution produces unexpected and unusual effects with respect to the usual statistical procedures, such as non-standard limiting behavior and noisy estimations.
Jouanin et al. (date not specified) emphasize that Sklar’s Theorem, in spite of indicating how to impose one form of dependence between random variables with given margins, it does not give any hint on how to choose a specific family of copulas. This is pointed out as a “severe shortcoming of the copula approach” (p. 1). Also, it is mentioned that usually the choice of copulas in financial institutions is either arbitrary or justified by convenience and tractability (Student Copulas, for instance, are often easy to simulate by using Monte Carlo method).
There are “families” of copulas in the same way that there are types of distributions. One challenge is to find out which family of copula better fits our data. Some authors defend that it is necessary to create a “Financial” family of copulas. Some examples of families of copulas are: Normal, t-Student, Clayton, and Archimedean.
A fundamental aspect in the empirical use of copulas is the fit of the data. McNeil et al. (2006) present examples of fitting data to copulas by using different three different methods: Method-of-Moments based on Ranking Correlation, estimation of copula parameters by Maximum Likelihood (ML) from a “pseudo-sample”, and estimation of copula parameters by Maximum Likelihood Estimation (MLE). As an example, the authors analyse five years (1996-2000) of daily log-return for some stocks. They first estimate the marginal distributions empirically by using the ML method and then plot the data representing the copula.
Kole et al. (2006) propose some goodness-of-fit tests that may be applied to any copula.
Instead of showing the various steps involved in modeling copulas, Genest and Favre (2006) present graphical tools and numerical techniques for selecting appropriate copulas, for estimating its parameters and checking its goodness-of-fit.

III.3. Conditional Copula
Schmidt (2006) warns that the traditional concept of copulas refers to a static concept of dependence whereas many applications in Finance are related to time series events which demands a dynamic concept of dependence is needed.
Patton (2002) introduced the conditional copula by adding a conditioning variable to the distributions studied. For example, if we are analyzing the pair (X,Y) and want to express the dependence of their joint distribution on another variable (W), we must consider each marginal conditional distribution on W (i.e. X|W and Y|W) and then estimate the conditional copula related to (X,Y)|W.
The main properties of unconditional copulas and the Sklar’s Theorem also apply for the conditional case. However, Patton (2006) points out that there is one limitation when extending the Sklar’s Theorem to conditional distributions: the conditional variable (W) must be the same for both marginal distributions and the copula. Failure to do so results in joint distributions that present properties incompatible with the expected conditions of such distributions.
In the previous sections, it was said that the use of copulas allows us to distinguish the dependence among variables of the tails from the “middle” of their joint distribution being that the relationship is the same at both tails. Conditional copulas go beyond and allow us to discriminate the behavior of the variables at the tails of the joint distributions. For instance, it could be the case that two variables have stronger dependence at the right tail than at the left one which means to say that they are more correlated when they present higher values.
Fermanian, Wegkamp (2004) extend the concept of conditional copulas to cover a larger scope of situations. They introduce the concept of pseudo-copula.
A d-dimensional pseudo-copula is function C:[0,1]d mapped to [0,1] such that:
(a) For every u in [0,1]d, C(u) = 0 when at least one coordinate of u is zero,
(b) C(1,…,1) = 1,
(c) For every u and v in [0,1]d such that u <= v, the C-volume of [u,v].
Thus a pseudo-copula satisfies all the properties of a copula except that C(u) is not necessarily uk when all coordinates of u except uk are one.
From this concept, the authors developed “conditional pseudo-copulas” that are less restricted in terms of conditional laws when compared to “conditional copulas”.

IV.1. Procedures

According to prior tests, we found that the variables are not normally distributed . However, it is worth noting that even if they were normal, it would not mean that their joint distributions would be normal as well.
The data will get from Compustat Annualy. The sample of firms will exclude financial institutions (SIC code between 6000 and 6999). Following Baker and Wurgler (2002), we will exclude firms with book value of assets below $10 million but differently from those authors we will not exclude outliers for capital structure and market-to-book ratio.
We will compare changes in leverage from period “t” to “t+1” with changes in other variables from period “t-1” to “t”.
For each year, we will calculate the average of each variable (from all companies considered in this study) and consider it as the point that will take part in the time series data.
After these steps, the data will be ready to be fitted to copulas that will express the relationship between capital structure (debt variation) and each of the other four selected variables (their oscillation) .
Next, we will fit a multidimensional copula for all five variables so that we can define the scenarios in which the capital structure has the slowest and the quickest adjustment.
As a final step, some robustness checks suggested in the literature (specially in Genest and Favre, 2006 and Kole et al., 2006) will be run and the results provided by the copula approach will be compared to results from traditional methods based on coefficient of correlation.

IV.2. Data
The period considered here ranges from 1950 to 2006 (which represents all data available on COMPUSTAT).
Based on Baker and Wurgler (2002) , we will get the annual data from Compustat and will calculate the following variables:
- Book leverage is defined as book debt to total assets; being that:
book debt = total assets (Compustat item 6) minus book equity
book equity = total assets less total liabilities (item 181) and preferred stock (item 10) plus deferred taxes (item 35) and covertible debt (item 79). If preferred stock is missing, it will be replaced with the redemption value of preferred stock (item 56).
- Market leverage is the book debt divided by the result of total assets minus book equity plus market equity; where:
market equity = common shares outstanding (item 25) times price (item 199).
- Market-to-book ratio is defined as asset minus book equity plus market equity all divided by assets.
- Asset tangibility is net plant, property, and equipment (item 8).
- Profitability is earnings before interest, taxes, and depreciation (item 13) divided by total assets and expressed in percentual terms.
- Size is defined as the log of net sales (item 12).

IV.3. Further research
Further research could be related to include other variables to check if they have a non-linear and asymmetric relationship with leverage. Also, as soon as more data are available, the tests can be extended to longer periods.
Actually, the (conditional) copula theory can be applied to several topics in Corporate Finance in which there is some possibility of non-linear and asymmetric dependence between variables.
In the theoretical field, the main contribution demanded by the Conditional Copulas Theory at this point seems to be the possibility of using more than one conditioning variable, i.e. to allow each marginal to be conditioned on one different variable. Naturally, such improvement would be beneficial to several areas of study other than Finance.


Spring/2008 = Classes and preparation for Prelim Examination
Summer/2008 = Prelim, classes, and reading about Copula Theory
Fall/2008 = Classes and Literature Review
Spring/2009 = Research (Literature review and fitting data to copulas)
Summer/2009 = Research (Literature review and fitting data to copulas)
Fall/2009 = Research (Empirical tests: fitting data to copulas and goodness-of-fit tests)
Spring/2010 = Dissertion elaboration and defense

PS: Partial results of the research may be presented at Conferences at any time before the Final Defense.


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Variables Mean Median Max Min Std Dev Skewness Kurtosis
D_Lev 0.0418784 0.0048427 0.9963223 -0.6703008 0.2205491 1.688617 9.0613423
D_MB 0.1107286 0.0387828 2.9953920 -0.9026572 0.4920875 3.730953 21.6062141
D_AT 0.0535717 0.0425059 0.4952885 -0.2506515 0.0873642 1.584452 13.3662508
D_Size 0.0571839 0.0710487 0.2063169 -0.3720816 0.0888496 -2.01392 8.8247978
D_Profb 0.0524394 0.0422595 0.3771890 -0.2957139 0.1116809 0.190129 1.8238491

D_Lev = (decimal) change in leverage
D_MB = (decimal) change in market-to-book ratio
D_AT = (decimal) change in asset tangibility
D_Size = (decimal) change in size
D_Profb = (decimal) change in profitability

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