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Applying thresholds effect regression to analyzeretained earnings of American manufacturingindustryPai-Lung Chou a , Chen-Hua Yao b , Jai-Jun Lin b & Pei-Shan Li ca Department of Risk Management and Insurance , National Kaohsiung First Universityof Science and Technology , No. 2 Jhuoyue Rd., Nanzih District, Kaohsiung City , 811 ,Taiwan, R.O.C. E-mail:b Institute of Management , National Kaohsiung First University of Science andTechnology , No. 2 Jhuoyue Rd., Nanzih District, Kaohsiung City , 811 , Taiwan, R.O.C. E-mail:c Institute of Management , National Kaohsiung First University of Science andTechnology , No. 2 Jhuoyue Rd., Nanzih District, Kaohsiung City , 811 , Taiwan, R.O.C. E-mail:d Department of Budget, Accounting and Statistics , Taipei City Government , 5F/6F, No.1, Shifu Rd., Taipei City , 11008 , Taiwan, R.O.C. E-mail:Published online: 14 Jun 2013.

To cite this article: Pai-Lung Chou , Chen-Hua Yao , Jai-Jun Lin & Pei-Shan Li (2009) Applying thresholds effect regressionto analyze retained earnings of American manufacturing industry, Journal of Statistics and Management Systems, 12:3,575-586, DOI: 10.1080/09720510.2009.10701408

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Applying thresholds effect regression to analyze retained earnings ofAmerican manufacturing industry2

Pai-Lung Chou ∗

Department of Risk Management and Insurance4

National Kaohsiung First University of Science and TechnologyNo. 2 Jhuoyue Rd., Nanzih District6

Kaohsiung City 811Taiwan, R.O.C.8

Chen-Hua Yao †

Jai-Jun Lin ‡10

Institute of ManagementNational Kaohsiung First University of Science and Technology12

No. 2 Jhuoyue Rd., Nanzih DistrictKaohsiung City 81114

Taiwan, R.O.C.

Pei-Shan Li §16

Department of Budget, Accounting and StatisticsTaipei City Government18

5F/6F, No. 1, Shifu Rd.Taipei City 1100820

Taiwan, R.O.C.

Abstract22

The main issue of this paper is how the financial ratios affect the operationalperformance under the fluctuation of the debt ratio. The inflexibility of the traditional24

regression is illustrated and results are compared to flexible threshold effect regression. Thisapproach calculates robust threshold value from outcome of repeated bootstrapping. The26

empirical results demonstrate the existence of single threshold value and the comparisonbetween traditional regression and threshold effect regression.28

Keywords and phrases : Retained earnings, debt ratio, threshold effect regression.

∗E-mail: plchou@ccms.nkfust.edu.tw†E-mail: u9727905@ccms.nkfust.edu.tw‡E-mail: u9727903@ccms.nkfust.edu.tw§E-mail: peishanli@gmail.com

——————————–Journal of Statistics & Management SystemsVol. 12 (2009), No. 3, pp. 575–586c© Taru Publications

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576 P. L. CHOU ET AL.

1. Introduction

When firms gains operational profits, managers have one of two2

choices; they can either retain the earnings and reinvest them inthe business, or pay it out to shareholders as a cash dividend. When4

the CEOs decide that earnings should be retained, they have to accountfor them on the balance sheet under equity. For this reason, supervisors6

and investors have to understand how much money has been put intothe business over the years, and how wisely managers are deploying and8

investing the shareholder’s money. If the firms retain reasonable divisionof its profits back into itself and experience exceptionally stable growth,10

we could ensure that the stock holders would not be better served if theboard of directors declared a dividend.12

We are also interesting in the causalities of important financial ratiosand retained earnings. So, this paper tries to explore how the financial14

ratios affect the rate of retained earnings under the fluctuation of thedebt ratio. The samples are gathered from the annual data of the listed16

companies in the second-class industry in North America. In order tomake different threshold models objectively, this study uses debt ratio18

as the threshold variable. Rivaling assumption of single sample spaceunder traditional regression, and improving fixed threshold model that20

specifies that individual observations and divided into classes basedon the value of an observed variable intuitively. Hansen [2] developed22

econometric techniques appropriate for threshold regression with paneldata. Confidence intervals for the parameters are constructed by an24

asymptotic distribution theory. The statistical significance of the thresholdeffect is evaluated by a bootstrap method. The research applies panel data26

and threshold effect regression model from Hansen’s concept. In the end,the empirical results explore the threshold numbers and also demonstrate28

the comparison with traditional model.The remaining parts of this paper are organized as follows. The next30

section presents a brief literature review. Section 3 describes data analysisand methodology. Section 4 provides analyses of empirical results. The32

final section concludes this paper.

2. Literature review34

As we know, threshold regression model was one kind of non-linearmodel. Its method was purely constructs the auto-regressive model by36

time series and defined threshold value by smallest AIC (Akaike Infor-

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RETAINED EARNINGS OF MANUFACTURING INDUSTRY 577

mation Criterion) (Tong [6]; Hansen [2]). Cause procedure of the abovethreshold auto-regressive model was quite trouble in searching threshold2

variable and the threshold value, therefore Tsay [8], and Tsay [9] proposedone simple and broad test and sets up standard threshold auto-regressive4

process. Cao and Tsay [5] applied this method to predict when stockvolatilities were nonlinear time-series model.6

Because AIC was designed in the linear model, but the criterions ofnon-linear model by Tong [6] using AIC as the assessment base was not8

rational. Therefore he applied Lagrange multiplier F test to calculate thesuitable lag periods. Luukkonen et al. [7] also supported the same result10

while applying logit model. The suggestion by Shen and Hakes [4] wasthat theorical economic variable could be the suitable threshold value12

instead of traditional lag period variable. But advanced literatures pointout those structural break points maybe more than one single point.14

Terasvirta [10] proposed smoothing transition model instead ofstructural break points by using the concept of auto-regressive form.16

Recently, new development on relative studies follows the main directionabove, such as the variable fuzzy time-series model. More and more18

literatures applied panel data form to carry on empirical studies as theconvenience of data collection recently. In order to evaluate measurement20

efficiency of operational performances, it is certainly important todistinguish between traditional model and thresholds model. Hansen [2]22

and Hansen [3] established relative objective criterion of threshold effectregression model in view of all the literatures development about the24

subjective or fixed categories of thresholds model.Recently, there is a set of new techniques that can be used in26

various combinations to deliver operational performances of global manu-facturing industry. Chiu et al. [12] focus on international concepts on28

operational efficiencies of high-tech manufacturing industry, and Lin andPan [11] construct combined framework to measure the impact of business30

performance.Therefore, the previous experiences would enhance our brief to adopt32

the advanced techniques. In this paper, we apply thresholds regressionmodel to substitute traditional regression model, and separate the estimate34

formulas by the different levels of firms’ debt ratios.

3. Data analysis and methodology36

This article is collected panel data from 1507 firms of Americanmanufacturing industry for the period 1999 to 2004 as empirical data. We38

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578 P. L. CHOU ET AL.

employ total 9042 balanced-panel sample to construct models in this studyand categorize the firms by standard industrial classification (SIC). The main2

data source is from “North American Compustat dataset”. Descriptivestatistics and definitions of all the variables are listed in Table 1.4

Table 1Definitions of the variables and descriptive statistics

Variables Definitions Explanations

dependent variable

Yit retained earnings The percentage of net earnings not paid out asdividends, but retained by the company to bereinvested in its core business or to pay debt

threshold variable

Tit debt ratio total liabilities/total asset

explained variable

X1t total asset turnover sales revenue/average total assets

X2t current ratio currents assets/current liabilities

X3t revenue growth rate (operating revenue-pre operating revenue)/pre operating revenue

X4t cash flow ratio net cash flow-operating/current liabilities

X5t inventory turnover cost of goods sold/average inventories

S1t operating net incomeratio

operating income/net sales

S2t total assets growthrate

(total assets-pre total assets)/pre total assets

6

Source: Excerpts from this research

The structural equation is adjusted by traditional fixed effect model8

and based on Hansen’s panel threshold effect model.

Yit = µi +α′Xit + β′Sit + eit . (1)10

An alternative intuitive way of writing (1) is:

Yit =

{µi +α′1Xit + β′Sit + eit Tit < θ ,

µi +α′2Xit + β′Sit + eit Tit ≥ θ .(2)12

In this research, the threshold variable is debt ratio that is expressedby Tit , where Tit < θ indicates that debt ratio of the industry is lower14

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RETAINED EARNINGS OF MANUFACTURING INDUSTRY 579

than others. Contrariwise, Tit ≥ θ indicates that debt ratio of the industryis higher than others. Xit is redefined the financial variables matrix, and2

coefficients would change respectively from different groups of thresholdeffects; Sit represents matrix that is constructed from variables of non-4

mixed distribution, and coefficients would be the same in different groupsof threshold effects.6

We divide financial variables into two types, one is Xit that comprisestotal asset turnover, current ratio, revenue growth rate, cash flow ratio8

and inventory turnover, and the other is Sit that comprises operating netincome ratio and total assets growth rate.10

The traditional method to eliminate the individual effect µi is toremove individual-specific means. In the first instance, we measure devia-12

tion from group mean for every variable. Afterward the slope coefficientα and β can be estimated by least square method. However, it is insigni-14

ficant that compare difference between high and low debt ratio in diversegroups. Therefore, we use cross item of dummy variable to improve (3):16

Y∗it = α′1X∗it I(Tit < θ) +α′2X∗

it I(tit ≥ θ) + β′S∗it + e∗it , (3)

where I(g) is the indicator function. Thus we could estimate coefficient18

and parameter by least square method in (3), and the sum of squarederrors and the threshold estimation is20

E1(θ) = e∗(θ)′ e∗(θ), θ = arg minγ

E1(θ) (4)

Once θ is obtained, the residual vector is e∗ − e∗(θ) and residual22

variance is

σ2 =E1(θ)

N(d.f.). (5)24

Nevertheless, the threshold θ is an unknown parameter, and d.f.means degree of freedom. The computation of threshold θ involves the26

minimization problem (5). Since E1(θ) depends on θ only through I(Tit <

θ) , E1(θ) is a step function with at most sample steps, with the steps28

occurring at distinct values of Tit . Thus the minimization problem canbe reduced to searching over values of θ equaling the distinct values Tit30

of in the sample.

Importantly, we should determine whether the threshold effect is32

statistically significant or not. The hypothesis of no threshold effect can be

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580 P. L. CHOU ET AL.

represented by the linear constraint, which is null hypothesis H0 : β1 =β2 . If we do not accept H0 and accept alternative hypothesis β1 6= β22

which means the threshold variable to Yit will appear different thresholdeffects in different interval. The likelihood ratio test of H0 is based on4

F1 =E0 − E1(θ)

σ2 . (6)

The asymptotic distribution of F1 is non-standard, and strictly6

dominates the χ2k distribution. Unfortunately, it appears to depend in

general upon moments of the sample and thus critical values cannot8

be tabulated. Therefore, Hansen [1] suggests that use bootstrap methodto precede iterative sampling. Using the bootstrap sample, calculate the10

bootstrap value of the likelihood ratio statistic F1 . This is the bootstrapestimate of the asymptotic p -value for F1 under H0 .12

However, Hansen [2] indicates that the best way to form confidenceintervals for θ is to form the “no-rejection region” using the likelihood14

ratio statistic for tests on θ . Furthermore, we also test whether θ ofestimation is actual threshold value or not. To test the hypothesis H0 :16

θ = θ0 , where θ0 is actual threshold value and the likelihood ratio test is

LR1(θ) =S1(θ)− S1(θ)

σ2 . (7)18

The result could be tested by threshold estimates (θ) from plots ofthe concentrated likelihood ratio function (LR) . The confidence interval20

for θ is the “no-rejection region” of confidence level (1−α) is the set ofvalues of θ such that LR(θ) ≤ c(α) .22

In addition to single threshold of the above, there may be doublethreshold in the fact. The double threshold model takes the form:24

Yit = µi +α′1Xit I(Tit < θ1) +α′2Xit I(θ1 ≤ Tit < θ2)

+α′3Xit I(Tit ≥ θ2) + β′Sit + eit , (8)26

where the thresholds are ordered so that θ1 < θ2 .

In the first place, we hypothesize the single threshold θ1 is known.28

Similarly, let the second threshold sum of squared errors function E2(θ2)asymptotically converges to a limit function which has two local minima30

θ1 and θ2 .

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RETAINED EARNINGS OF MANUFACTURING INDUSTRY 581

Fixing the first stage estimate θ1 , the second stage criterion is

Eθ2(θ2) =

{E(θ1,θ2) if θ1 < θ2 ,

E(θ2, θ1) if θ2 < θ1 .(9)2

And the second stage threshold estimate is

θθ2 = arg min

θ2Eθ

2(θ2) . (10)4

Both null and alternative hypothesis are separately H0 : there onlyhas single threshold; H1 : there is the presence of a double threshold. Thus6

an approximate likelihood ratio test of one versus two thresholds can bebased on the statistic:8

F2 =E1(θ1)− Eθ

2(θθ2)

σ2 , where σ2 =E2(θ2)N(d.f.)

. (11)

However θθ2 is asymptotically, but θ1 is not. This is because the10

estimate θ1 was obtained from E1(θ) which was contaminated by thepresence of a neglected regime. The asymptotic efficiency of θθ

2 suggest12

that θ1 can be improved by a third-stage estimation. Consequently, we usethe “refinement estimator”. Fixing the second-stage estimate θθ

2 , define14

the refinement criterion:

Eθ1(θ1) =

{E(θ1, θθ

2) if θ1 < θθ2 ,

E(θ2,θ1) if θθ2 < θ1 ,

there θθ1 = arg min

θ1Eθ

1(θ1) . (12)16

Similarly, we also test whether both θθ2 of estimation and θθ

1 ofrefinement estimate are actual threshold values or not. Both of the likeli-18

hood ratio tests are respectively

LRθ2(θ) =

Eθ2(θ)− Eθ

2(θθ2)

σ2 and LRθ1(θ) =

Eθ1(θ)− Eθ

1(θθ1)

σ2 . (13)20

Our asymptotic (1−α) confidence intervals for θ2 and θ1 are the setof values of θ such that LRθ

2(θ) ≤ c(α) and LRθ1(θ) ≤ c(α) , respectively.22

In conclusion, the result is tested by threshold estimates from plots of theconcentrated likelihood ratio function.24

4. Empirical results

We estimate threshold value applying plots of the concentrated26

likelihood ratio function, defining the average debt ratio of Americanmanufacturing industry as threshold base, and discovering that the debt28

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582 P. L. CHOU ET AL.

ratio in 0.366 is the possible value of the threshold value. Carries ontesting the single threshold effect with bootstrap method, the test statistic2

is 226.2371, and the 5% critical value is 197.8657. The result reveals thatthe model exists on single threshold effect while the debt ratio in 0.3664

is the threshold value in 5% significance level. The Figure 1 showedthat the threshold parameter is 0.366 while the likelihood ratio equals6

zero. Thus the threshold value in 0.366 is the retained earnings of entiremanufacturing industry in single threshold model.8

Figure 1Confidence interval in single threshold model10

Furthermore, we search for the second threshold value by fixingthreshold values 0.366. Carries on testing the second threshold with boot-12

strap method, the 5% limit value is 173.1424. The result reveals that themodel does not exist on double threshold effect fewer than 5% significance14

level. (As Table 2, Table 3)

Table 2Tests for threshold effects16

Single threshold effect Double threshold effect

F test statistic 226.2371 160.2058p-value 000.0467 000.063310% critical value 122.8076 118.10325% critical value 197.8657 173.14241% critical value 407.4181 281.3722

Note: This research is regarded as the competence of18showing assayed with 5% significance level

Source: Excerpts from this research20

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RETAINED EARNINGS OF MANUFACTURING INDUSTRY 583

Table 3Threshold estimates and confidence interval

Threshold estimate 95% confidence interval

γ1 0.366 [0.357, 0.386]2

Source: Excerpts from this research

Conclusively, the detecting process ensures the existence of single4

threshold in manufacturing industry and the threshold value ought tobe 0.366. The debt ratio lower than 0.366 is in first sector, and higher6

should be the other sector. Firms’ percentage falls into two sectors eachyear. Evidences from the first research year, the ratio of firms in the up8

sector has a tendency toward gradual increase slightly from 37.03% to41.67% between 1999 and 2004, and the ratio of firms in the bottom sector10

has a tendency toward gradual decrease slightly from 62.97% to 58.33%.These evidences could demonstrate that firms alter the level of high debt12

ratio and gather the potential threshold value. (Table 4)

Table 4Percentage of firms in each regime by years14

Years 1999 2000 2001 2002 2003 2004

debt ratio<0.366 37.03% 39.42% 39.08% 39.48% 39.75% 41.67%debt ratio>0.366 62.97% 60.58% 60.92% 60.52% 60.25% 58.33%

Source: Excerpts from this research16

By the process of determining threshold value, we try to establish thethreshold model as this form:18

Y∗it = α′1X∗it I(Tit < 0.366) +α′2X∗

it I(Tit ≥ 0.366) + β′S∗it + e∗it , (14)

The variables are defined as follows: Xit is the variables consist20

of “total asset turnover”, “current ratio”, “revenue growth rate”, “cashflow ratio” and “inventory turnover”; Sit is the matrix is constructed by22

“Operating net Income ratio” and “total assets growth rate”; Tit is the debtratios; and Yit is the retained earnings.24

The coefficient of determination of traditional regression is lowerthan R2 of threshold effect regression (As Table 5). Thus new method26

provides relative better goodness of fit. Instead of common analysis, underthreshold effect regression, the firms might focus on the cash flow ratio28

to ensure promotion of retained earnings if their debt ratio is higher than0.366. Oppositely, while the debt ratio is lower than 0.366, the firms should30

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584 P. L. CHOU ET AL.

promote revenue growth rate and inventory turnover primarily to elevatethe retained earnings and to gain the outstanding operational benefits.2

Table 5Parameter estimation of traditional and threshold model

Retained earnings

traditional threshold model

model debt ratio < 0.366 debt ratio>0.366

total asset turnover – 0.4163 – 0.3080 – 0.5130

(0.000)∗∗∗ (0.000)∗∗∗ (0.000)∗∗∗

current ratio 0.0466 0.0439 0.0071

(0.000)∗∗∗ (0.000)∗∗∗ (0.505)

revenue growth rate 0.0029 0.0034 0.0026

(0.000)∗∗∗ (0.000)∗∗∗ (0.000)∗∗∗

cash flow ratio 0.0521 – 0.0013 0.2982

(0.000)∗∗∗ (0.931) (0.000)∗∗∗

inventory turnover 0.0024 0.0040 0.0011

(0.124) (0.076)∗ (0.530)

operating net income ratio – 0.0004 – 0.0005

(0.228) (0.228)

total assets growth rate 0.0011 0.0010

(0.000)∗∗∗ (0.000)∗∗∗

constant term 0.1446 0.1866

(0.004)∗∗∗ (0.004)∗∗∗

R2 0.0002 0.00244

Note: ∗ , ∗∗ , ∗∗∗ represents 10%, 5%, 1% significance level, respectivelySource: Excerpts from this research6

5. Conclusion and recommendation

This paper explores American manufacturing industry as the target8

observations and adopts the debt ratio to determine the threshold modelsfor ROA of these industries by collecting the dataset from the annual data10

of the listed companies in the second-class industry in North Americafrom 1999 to 2004. The threshold model which is determined by the12

debt ratio could get higher coefficient of determination. The empiricalanalysis supports double outstanding threshold values under repeated14

threshold effect test and these thresholds could separate debt ratios of

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RETAINED EARNINGS OF MANUFACTURING INDUSTRY 585

all the samples into two conditions. Applying identical threshold effectregression to confirm management policy might enhance the accuracy of2

prediction.In the one way, firms in low debt ratio should increase short-4

run liquidity target and speed the turnover of inventory and revenuegrowth rate to elevate the retained earnings and to gain the outstanding6

operational benefits. In the other way, the firms which are high debt ratioshould focus on net cash flow management and make more effort to8

evaluate future retained earnings.

References10

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Received November, 2008

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