A comparison of neural network and multiple regression analysis
in modeling capital structure
Hsiao-Tien Pao
*Department of Management Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 03, Taiwan, ROC
Abstract
Empirical studies of the variation in debt ratios across firms have used statistical models singularly to analyze the important deter-minants of capital structure. Researchers, however, rarely employ non-linear models to examine the deterdeter-minants and make little effort to identify a superior prediction model. This study adopts multiple linear regressions and artificial neural networks (ANN) models with seven explanatory variables of corporation’s feature and three external macro-economic control variables to analyze the important minants of capital structures of the high-tech and traditional industries in Taiwan, respectively. Results of this study show that the deter-minants of capital structure are different in both industries. The major different deterdeter-minants are business-risk and growth opportunities. Based on the values of RMSE, the ANN models achieve a better fit and forecast than the regression models for debt ratio, and ANNs are cable of catching sophisticated non-linear integrating effects in both industries. It seems that the relationships between debt ratio and independent variables are not linear. Managers can apply these results for their dynamic adjustment of capital structure in achieving optimality and maximizing firm’s value.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Capital structure; Multiple regression model; Artificial neural network model
1. Introduction
Regarding the qualitative aspects of capital formation within the high-tech industry of the 90s, we find that begin-ning about 1995 a mob mentality set in within the invest-ment community. Essentially, no rational reason could be quantified for the ability of the high-tech companies to attract large amounts of investment capital. That is, on the surface, there seemed to be an irrational behavior within the investment community. If we mine the informa-tion deeper, it would be quite rainforma-tional for the venture cap-italists to fund the high-tech to the extent that they did. Examining the phenomenon of the high-tech, several fac-tors come into play. Firstly, the general economy was doing well and the allure of high-tech business was
irresist-ible to stock purchasers. Secondly, the thought that much of the world business would be internet/computer orien-tated took root and became the glamorous hot issue of the day. Venture capitalist read the fervor and proceeded to fund startup companies in record numbers. As a result, the capital structure of the high-tech industry seems to be significantly different from that of the traditional industry.
Ever since Myers article (1984)on the determinants of
corporate borrowing, literature on the determinants of cap-ital structure has grown steadily. Part of this literature materialized into a series of theoretical and empirical stud-ies whose objective has been to determine the explanatory
factors of capital structure. The article ofTitman and
Wes-sels (1988)on the determinants of capital structure choice take such attributes of firms as asset structure, non-debt tax shields, growth, uniqueness, industries classification, size, earnings, volatility and profitability, but found only
uniqueness was highly significant. But Harris and Raviv
(1991)in their similar article on the subject point out that
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Expert Systems with Applications 35 (2008) 720–727
Expert Systems with Applications
the consensus among financial economists is that leverage increases with fixed costs, non-debt tax shields, investment opportunities and firm size. And leverage decreases with volatility, advertising expenditure, the probability of
bank-ruptcy, profitability and uniqueness of the product.Moh’d,
Perry, and Rimbey (1998)employ an extensive time-series and cross-sectional analysis to examine the influence of agency costs and ownership concentration on the capital structure of the firm. Results indicate that the distribution of equity ownership is important in explaining overall cap-ital structure and managers do reduce the level of debt as their own wealth is increasingly tied to the firm. Moreover,
Mayer (1990)indicated that financial decisions in
develop-ing countries are somehow different. Rajan and Zingales
(1995)study whether the capital structure in the G-7 coun-tries other than the US is related to factors similar to those appearing to influence the capital structure of US firms. They find that leverage increases with asset structure and size, but decreases with growth opportunities and profit-ability. Again firm leverage is fairly similar across the
G-7 countries. Booth, Aivazian, Demirguc-Kunt, and
Maksimovic (2001)take tax rate, business-risk, asset tangi-bility, firm size, profitatangi-bility, and market-to-book ratio as determinants of capital structure across 10 developing countries. They find that long-term debt ratios decrease with higher tax rates, size, and profitability, but increase with tangibility of assets. Again the influence of the mar-ket-to-book ratio and the business-risk variables tends to be subsumed within the country dummies. Recently, some studies have explored capital structure policies using
differ-ent models on differdiffer-ent countries (Chen, 2004; Dirk, Abe,
& Kees, 2006; Fattouh, Scaramozzino, & Harris, 2006; Francisco, 2005). Furthermore, Kisgen (2006) examines
credit rating and capital structure, andJan (2005)develops
a model to analyze the interaction of capital structure and ownership structure. Otherwise, in time-series test, Shyam-Sunder and Myers (1999) show that many of the current empirical tests lack sufficient statistical power to distin-guish between the models. As a result, recent empirical research has focused on explaining capital structure choice by using time-series cross-sectional tests and panel data.
Though the achievement is rich, but there are few stud-ies that evaluate the model’s ability to predict. In addition, comparisons between linear and non-linear models for firm leverage with different industries are rare. Recently, artificial neural network (ANN) non-linear models have
been widely used for resolving forecast problems (Altun,
Bilgil, & Fidan, 2007; Hill, O’Connor, & Remus, 1996; Tseng, Yu, & Tzenf, 2002). The ANN model attempts to duplicate the processes of the human brain and nervous system using the computer. While this field originated in biology and psychology, it is rapidly advancing into other
areas including business and economics (Chiang, Urban,
& Baldridge, 1996; Enke & Thawornwong, 2005; etc.). The theoretical advantage of ANNs is that relationships need not be specified in advance since the method itself establishes relationships through a learning process. Also,
ANNs do not require any assumptions about underlying population distributions. They are especially valuable where inputs are highly correlated, missing, or the systems are non-linear. A lot of research has been done to com-pare the performances of ANN and traditional statistical
models (Kumar, 2005; Pao, 2006; Wang & Elhag, 2007;
Zhang, 2001; etc.). Most researchers find that ANN can outperform linear models under a variety of situations, but their conclusions are not consistent with one another (Zhang & Qi, 2005).
Our focus is on answering three quantitatively oriented questions and proposing a qualitative comments in opti-mizing capital structure and maxiopti-mizing firm value: (1) whether if the corporate financial leverage decisions differ significantly between high-tech and traditional industries; (2) whether if the determinants of the capital structure dif-fer significantly in both industries; (3) whether if non-linear models provide better model fitting and forecasting than linear models for capital structure. The rest of the paper
is organized as follows. Section2presents the data source,
the definition of variables, and methodologies. Section 3
presents a comparative study of ANN and linear regression models and an attempt to rationalize the observed regular-ities. The final section contains the summary and conclusions.
2. Data source and methodology
In this study, corporations are classified into two cate-gories: the high-tech and the traditional corporations. High-tech corporations include electronics, telecommuni-cations, computer hardware, software, networking, infor-mation systems, and other related corporations. The rest are traditional corporations such as clothing, textile, trad-ing, agriculture, manufacturtrad-ing, etc. Leading one hundred corporations with sound financial statements are selected to create a database in each industry. Both data sets include a total of 720 firm-year panel data of public trad-ing high-tech and traditional corporations in Taiwan from 2000 to 2005. The period from 2000 to 2004 is treated as the training period and the subsequent is the out-of-sample period for models. Each corporation contains one depen-dent variable and 10 independepen-dent variables. The Taiwan Economic Journal (TEJ) compiles all variables. Basic sta-tistics are estimated to describe each variable collected and t-tests are conducted to determine if variables of high-tech corporations are different from that of tradi-tional corporations.
As for regression models, we used total debt ratio (DEBT) as the response variables, and firm size (SIZE), growth opportunities (GRTH), profitability (ROA), tangi-bility of assets (TANG), non-debt tax shields (NDT), dividend payments (DIV), and business-risk (RISK) as explanatory variables of corporation’s feature. In each model, there are three external macro-economic control variables: capital market factor (MK), money market fac-tor (M2), and inflation level (PPI).
2.1. Multiple linear regression model
In order to test the relationship between capital struc-ture and its determinants, the following multiple regression equation is proposed for the panel data.
DEBTit ¼ a0þ a1LSIZEitþ a2GRTHitþ a3ROAit þ a4TANGitþ a5NDTitþ a6DIVit
þ a7RISKitþ a8MKitþ a9M2itþ a10PPIitþ uit;
i¼ 1; . . . ; N ; t ¼ 1; . . . ; T ; ð1Þ
where N is the number of cross sections (N = the number of corporations) and T is the length of the time series for each cross section (T = the number of months in time per-iod). The following notation is used to define the variables in the empirical model:
DEBT the total book-debt/total assets; LSIZE ln (asset size);
GRTH average sales growth rate over the previous two year;
ROA the earnings before interest and tax divided by
to-tal assets;
TANG fixed assets/total assets;
NDT ratio of depreciation, investment tax credit, and
tax loss carry forward to total assets;
DIV dividend payout ratio;
RISK variance of the return on assets;
MK rate of return of the overall stock market;
M2 annual growth rate;
PPI producers’ price index.
The estimation procedure involves two steps. In step one, each variable is normalized by subtracting its mean value and divided by its standard deviation to have zero mean value and unity variance for all variables. As a result, we will not have an intercept in our results and we can determine the relative importance of each independent var-iable in explaining variations of the dependent varvar-iable based on its estimated coefficient. Variance inflation factor (VIF) is estimated for each independent variable to identify causes of multicollinearity. Pending on the results of step one, model one is re-estimated in step two by deleting vari-ables with insignificant coefficient or significant VIF value one at a time (stepwise) (VIFj> 20 implies that the jth inde-pendent variable is highly correlated with other indepen-dent variables of the model).
2.2. Artificial neural network model
The back-propagation (BP) neural network consists of an input layer, an output layer and one or more intervening layers, also referred to as hidden layers. The hidden layers can capture the non-linear relationship between variables. Each layer consists of multiple neurons that are connected to neurons in adjacent layers. Since these networks contain
many interacting non-linear neurons in multiple layers, the networks can capture relatively complex phenomena.
A neural network can be trained by the historical data of a firm-year data set in order to capture the characteristics of this data set. A process of minimizing the forecast errors will iteratively adjust the model parameters (connection weights and node biased). For each training process, an input vector, we randomly selected from the training set, was submitted to the input layer of the network being trained. The output of each processing unit was propagated
forward through each layer of the network (Liu, Kuo, &
Sastri, 1995).
As shown inFig. 1, the ANN model consists of an input
layer with ten input nodes, one hidden layer consisting of h nodes, and an output layer with a single output note. The input to the ANN includes 10 variables used in the regres-sion model. During training, a set of n pairs of input
vec-tors and corresponding output,ðX ð1Þ; yð1ÞÞ; ðX ð2Þ; yð2ÞÞ;
. . . ;ðX ðnÞ; yðnÞÞ is presented to the network, one pair at
a time. A weighted sum of the inputs,
NETt¼
XN
i¼1
wtixiþ bt ð2Þ
is calculated at tth hidden node; wtiis the weight on
con-nection from the ith to the tth node; xi is an input data
from input node; N is the total number of input nodes
(N = 10); and bt denotes a bias on the tth hidden node.
Each hidden node then uses a sigmoid transfer function to generate an output,
Zt¼ ½1 þ expðNETtÞ1 ¼ f ðNETtÞ; ð3Þ
between 0 and 1. We then sent the outputs from each of the
hidden nodes, along with the bias b0on the output node, to
the output node and again calculated a weighted sum,
NET¼X
h
t¼1
vtZtþ b0; ð4Þ
where h is the total number of hidden nodes; and vtis the
weight from the tth hidden node to the output node. The weighted sum becomes the input to the sigmoid transfer function of the output node. We then scaled the resulting output, y b0 Output Layer v1 vh b1 … …...… bh 1 2 …… h Hidden Layer w11 wh10 ……… Input Layer x1 x2 ……… . x10
b
Y ¼ f ðNETÞ ¼ ½1 þ expðNETÞ1; ð5Þ
to provide the predicted output value. At this point, the second phase of the BP algorithm, adjustment of the con-nection weights, begins. The parameters of the neural net-work can be determined by minimizing the following objective function of SSE in the training process:
SSE¼X n j¼1 ðyj bYjÞ 2 ; ð6Þ
where bYjis the output of the network for jth observation.
Assume the relationship of Y and X is monotone, then
calculate the sensitivity Si of the outputs to each of the
ith inputs as a partial derivative of the output with respect
to the input (Hwang, Choi, Oh, & Marks, 1991).
Si¼ o bY oXi ¼X h t¼1 o bY oNET oNET oZt oZt oNETt oNETt oXi ¼X h t¼1 ½f0ðNETÞvtf0ðNETtÞwti: ð7Þ
Assume f0(NET) and f0(NET)
tare constants and we ignore
them. Then the relative sensitivity is bSi¼P
h
t¼1vtwti. The independent variable with higher relative positive (nega-tive) sensitivity has the higher positive (nega(nega-tive) impact on the dependent variable.
Performance is measured by looking at the degree to which the ANN output matches the actual value for the corresponding input values. In this study, the number of hidden nodes for the neural network was varied from one to twelve. Note that the resulting neural network models performed relatively better with six to nine hidden nodes. However, the predictive accuracy of the model improved with the in-sample data set and declined with the out-of-sample data set when more than nine hidden nodes are used. Hence, eight hidden nodes are used in the resulting ANN. In general, the need for more hidden nodes indicates big interaction of the inputs, and an enlarged ability for the neural networks to outperform other statistical models. Such a large number of hidden nodes provide assurance of the robustness of the ANN out-of-sample.
While ANNs have some limitations, several researchers have demonstrated that ANNs are excellent at developing overall models. Neural network accuracy in predicting out-comes has been documented under a wide variety of appli-cations. This study attempts to examine the usefulness of ANNs as analyses and predictions of capital structure and to compare these ANNs with multiple linear regression results.
3. Empirical results
Table 1presents descriptive statistics of all variables and t-tests for variable difference between high-tech and tradi-tional corporations. The results indicate that: (1) the total debt ration, firm size, and tangibility of the high-tech
cor-porations are insignificantly different from that of tradi-tional corporations; (2) the growth opportunities (higher), profitability (higher), non-debt tax shield (higher), dividend policy (lower), and business-risk (higher) of the high-tech corporations are significantly different from that of the tra-ditional corporations. Therefore, it can inferred that although the capital structure measured by debt ratio of the high-tech corporations is insignificantly different from that of the traditional corporations, the determinants of the capital structure of the high-tech corporations can be
significantly different from that of the traditional
corporations.
3.1. Regression results
Table 2 presents the results of standardized multiple regression models. The results indicate that: (1) all three external macro-economic variables are insignificantly asso-ciated with the capital structure for both industries; (2) the estimated VIF coefficients of all three macro-economic variables are high, i.e. VIF > 20, which would create multi-collinearity to end up with inefficient estimates; and (3) the estimated root MSE are relatively high for both industries as all variables have been normalized. To improve the esti-mates, insignificant variables with high VIF were deleted one at a time (stepwise) and the results are presented in
col-umns 2 and 4 ofTable 3. Compare to the results ofTable 3
virtually have the same implications with no statistical improvement.
3.2. ANN results
Since the results from the linear regression models are unsatisfactory, the neural network sensitivity model is employed to further analyze the possible non-linear rela-tionship. Data during the first five years (2000–2004) served as training data, while those of the remaining last year (2005) as testing data. So, training data and testing data have 600 and 120 observations in the high-tech and tradi-tional corporations, respectively. We adopted a back-prop-agation network with a {10-8-1} framework and used Eq.
(7)to compute the sensitivity of each independent variable
to capital structure.Table 3lists the results.
From the results ofTable 3, we conclude that: (1) ANN
models have lowest RMSE values for in-sample and out-of-sample forecasting. These indicate that the non-linear ANN models generate a better fit and forecast of the panel data set than the regression model, and ANNs are cable of catching sophisticated non-linear integrating effects in both industries. It seems that the relationships between debt ratio and determinant variables are not linear. (2) Clearly on each independent variable, the sign of relative sensitivity in ANN models resembles the sign of coefficient in regres-sion models. (3) The determinants of capital structure of the high tech industry are different from that of the tradi-tional industry. The most important determinants (relative sensitivity greater than 1) for capital structure in high-tech
industry are, by priority, non-debt tax shields, firm size, dividend payments, business-risk; and profitability; in tra-ditional industry are, by priority, firm size, profitability, growth opportunity, non-debt tax shields, and dividend payments. Otherwise, three macro-economic factors are insignificant on debt ratios in both industries. Based on the results of ANN models, each determinant of capital structure in both industries is discussed below.
Many previous studies (Booth et al., 2001; Harris &
Raviv, 1991) argued that the capital structure might be affected by firm size positively as larger firms are more able to borrow money to realize the benefits of financial lever-age. The results of this study are consistent with this pre-sumption. Both high-tech and traditional corporations with larger size had higher debt ratio.
Myers (1977)identified growth opportunities had signif-icant and negative impact on capital structure based on the argument that firms with higher investment in intangible assets are to use less debt to reduce the agency costs asso-ciated with risky debt. In contrary, this study found that growth opportunities had insignificant impact on capital structure for the high-tech corporations and positive and significant impact on capital structure for the traditional
corporations. In combining with the results of Table 1, it
seemed that most high-tech corporations are characterized by high growth opportunities (homogeneity) and therefore we could not separate and elicit the impact of high growth opportunities on capital structure statistically. Traditional corporations with higher growth opportunities had higher demand for capital to sustain their growth opportunities and borrowed more than their peers with lower growth opportunities.
Myers (1984)suggested managers have a pecking-order in which retained earnings represented the first choice, fol-lowed by debt financing, and then equity to meet their financial needs. If this is true, it would imply a negative relationship between profitability and the capital structure. The results of this study are consistent with previous stud-ies and confirmed that both the high-tech and traditional corporations’ profitability had negative impact on capital structure.
Since higher collateral value would enable firms to bor-row more, previous studies suggested that firms’ collateral value had a positive relationship with their capital struc-ture. The results of this study indicated that the relation-ship between firms’ collateral value and capital structure
was positive for both the high-tech and traditional corpora-tions. As non-debt tax shield could lower the benefit of financial leverage, previous studies suggested a negative relationship between the non-debt tax shield and the capi-tal structure. The results of this studies confirmed that both the high-tech and traditional corporations had a negative and significant impact on capital structure. As higher cash dividend payments reflected lower capital demand, previ-ous studies suggested that the relationship between cash dividend and capital structure should be negative. The results of this study confirmed that both the high-tech and traditional corporations had a negative relationship between cash dividend and capital structure.
In general, business-risk is a variable that includes finan-cial distress costs. It has been supposed that firms having greater business-risk tend to have low debt ratios, as show byBathala, Moon, and Rao (1994), Homaifar, Zietz, and Benkato (1994) and Prowse (1990). But results of this study indicate that there is a positive and significant relationship between business risk and capital structure for the high-tech corporations, but insignificant relationship for the tra-ditional corporations. In combining with the results of
Table 1, it seemed that most traditional corporations are characterized by relatively low business-risk (homogeneity) and therefore we could not separate and elicit the impact of business-risk on capital structure statistically. The busi-ness-risk is positively related to debt ratio for high-tech corporations. This is because of the attribute of high-tech industry. Generally, in high-tech industry, more specula-tion is associated with high risk and high investment opportunity. Firms with higher investment opportunity have higher demand for capital to sustain their investment. Therefore, business-risk is positively related to debt ratio. 4. Conclusion and further work
This paper uses standardized linear regression and non-linear ANN models with panel data to explain firm charac-teristics that determine capital structure in Taiwan. Results partly answers the questions posed in the introduction. It offers some hope, but also some skepticism. First, on each independent variable, the sign of relative sensitivity in ANN models resembles the sign of coefficient in regression models. And ANN models have lowest RMSE values for in-sample and out-of-sample forecasting. These indicate that the non-linear ANN models generate a better fit and
Table 1
The average of each variable in high-tech and traditional corporations
DEBT LSIZE GRTH ROA TANG NDT DIV RISK MK M2 PPI
HT corp. 0.45 6.71 0.26 0.10 0.31 0.10 0.28 4.68 0.19 9.01 94.27 TR corp. 0.49 6.93 0.08 0.08 0.35 0.07 0.59 2.51 t-test 1.12 1.49 5.01* 3.00* 1.45 2.83* 3.98* 3.59* HT: high-tech corporation. TR: traditional corporation.
t-test for H0: l1= l2(high-tech corporation = traditional corporation). * Significant at 5% level.
forecast of the panel data set than the regression model, and ANNs are cable of catching sophisticated non-linear integrating effects in both industries. Secondly, the empir-ical evidences obtained from the ANN model corroborate the following expected relationships in both industries: (1) a direct relationship between firm size and debt ratio; (2) an inverse relationship between profitability and debt; (3) an inverse relationship between non-debt tax shields and debt; and (4) an inverse relationship between dividend payments and debt. The positive coefficients on SIZE indicate that debt ratios of larger firms are less limited by the costs of financial distress, because they have more
diversification than smaller firms (Smith & Watts, 1992).
The negative coefficients on ROA indicate that the more profitable the firm, the lower the debt ratio. This finding is consistent with the Pecking-Order Hypothesis. It also supports the existence of significant information asymme-tries. This result suggests that external financing is costly and therefore avoided by firms. However, a more direct explanation is that profitable firms have less demand for
external financing, as discussed by Donaldson (1963)
and Higgins (1997). This explanation would support the argument that there are agency costs of managerial discre-tion in high-tech industry. The negative coefficients on NDT indicate that tax deductions for depreciation and investment tax credits are substitutes for the tax benefits of debt financing. Firms with large non-debt tax shields relative to their expected cash flow include less debt in their capital structures.
Thirdly, the determinants of capital structure of high-tech industry are different from that of the traditional industry. The major different determinants are business-risk and growth opportunities. The coefficients on busi-ness-risk are positive/negative for high-tech/traditional corporations, and traditional corporations have substan-tially lower ratios of business-risk. This is because of the characteristic of high-tech industry. Generally, in high-tech industry, more speculation is associated with high risk and high investment opportunity. Firms with higher invest-ment opportunity have higher demand for capital to
sus-tain their investment. Therefore, business-risk is
positively related to debt ratio. In traditional industry, business-risk is an estimate of the probability of financial distress. It notes that low business-risk enhances a firm ability to issue debt. The coefficients on growth opportuni-ties are in-significant/positive for high-tech/traditional cor-porations, and traditional corporations have substantially lower growth opportunities. It seems that most high-tech corporations are characterized by high growth opportuni-ties (homogeneity) and therefore we can not separate and elicit the impact of high growth opportunities on capital structure statistically. Traditional corporations with higher growth opportunities have higher demand for capital to sustain their growth opportunities and borrowed more than their peers with lower growth opportunities.
Finally, crucial determinants affecting capital structure in high-tech industry are, by priority, non-debt tax shields,
Ta ble 2 Res ults of standar dize mult iple line ar regression mode ls DEB T LSIZ E GRTH ROA TANG NDT DIV RISK MK M2 PPI RMS E Hig h-tech 0.45 (0.14) * 0.36 (0 .15) * 0.38 (0.13) * 0.27 (0.14) 0.72 (0.17) * 0.16 (0.17) 0.25 (0 .14) * 0.72 (0.51) 1.75 (1. 02) 1.95 (1.31) 0.83 VIF 1.91 1.70 2.91 2.84 3.30 1.84 2.08 31.48 132.36 183.9 1 Tra ditiona l 0.74 (0.12) * 0.37 (0 .15) * 0.29 (0.08) * 0.19 (0.07) * 0.39 (0.11) * 0.27 (0.07) * 0.19 (0 .08) * 0.20 (0.47) 0.41 (0. 71) 0.63 (0.83) 0.56 VIF 2.90 1.71 1.75 2.85 3.68 1.39 2.37 23.89 126.81 158.3 8 * Signifi cant at 5% level.
firm size, dividend payments, business-risk; and profitabil-ity, in traditional industry are, by priorprofitabil-ity, firm size, profit-ability, growth opportunity, non-debt tax shields, and dividend payments. Otherwise, three macro-economic fac-tors are insignificant on debt ratios in both industries.
Managers can apply these results for their dynamic adjustment of capital structure in achieving optimality and maximizing firm’s value. For example, a manager may be able to enhance or reduce the benefit of financial leverage if the corporation becomes larger or profitable. Consequently, there is much that needs to be done, both in terms of empirical research as the quality of databases increases, and in developing theoretical models that pro-vide a more direct link between profitability and capital structure choice in different industries.
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Table 3
Results of improve multiple regression and sensitivity from ANN
Indep. variable High-tech Traditional
Multi-reg. ANN sensitivity Autoreg. ANN sensitivity
LSIZE 0.42 (0.15)* 2.48 0.81 (0.07)* 4.09 GROWTH 0.11 (0.12) 0.16 0.32 (0.06)* 1.98 ROA 0.30 (0.14)* 1.03 0.36 (0.08)* 2.86 TANG 0.28 (0.18) 0.78 0.21 (0.08) 0.85 NDT 0.74 (0.21)* 3.84 0.35 (0.07)* 1.67 DIV 0.27 (0.14) 2.06 0.24 (0.10)* 1.08 RISK 0.41 (0.15)* 1.32 0.17 (0.06)* 0.84 MK N/A 0.89 N/A 0.71 M2 N/A 0.50 N/A 0.40
PPI N/A 0.27 N/A 0.05
RMSE of out-of-sample 0.86 0.58
RMSE of training sample 0.065 0.061
RMSE of testing sample 0.078 0.072
N/A: independent variable is deleted stepwise.
*
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