A Descriptive Method of Firm Size Transition Dynamics Using Markov Chain

A Descriptive Method of Firm Size Transition Dynamics Using Markov Chain

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\subparagraph*{Key words:}

Firm size, job creation, Markov chain, dynamics description.

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Introduction

Firms are business organizations which carrying the vast majority of social employment. When defining the size of firms as their number of employees (Birch, 1979), the net job creation can be calculated by the firm size variations. Early studies mostly focus on quantifying net job creation of firms via cross-sectional data sets: Birch (1979, 1981, 1987), Davis & Haltiwanger (1994), Neumark et al. (2011) all look into the job creation abilities of different firm types including small, medium and large firms.

In practice, the process of firm job creation is continuous. The changes of firm job creation in various time periods can be regarded as the dynamics of firm size, making scholars examine this issue via time series approaches. One of the classic method used is the panel vector autoregression (PVAR, Love & Zicchino, 2006). By this model, the impact among firm characteristics can be clearly estimated. Davis & Haltiwanger (1999, 2001), Koellinger & Roy (2012) all deeply examine the firm employment or entrepreneurship determinants via this method.

The PVAR method holds the advantages of expressing how the dynamic changes of individual firms are influenced by its own. However, this approach cannot clearly quantify the net job creation of firms in different size categories as Birch's studies do. Moreover, data stationary and data time length is strictly required when using autoregressive models. Due to the existence of firm entry & exit, existing firm-level data sets can hardly meet the requirement. This inspires us to further propose a Markov-chain-based approach to describe the dynamics of firm size.

Markov chain, which can describe the random transfer processes is widely used in many aspects. Unlike autoregression models, any two-period transfer can be described by establishing transition matrix. Using the properties of Markov chain, multiple first-order transfer matrices can also describe the dynamics over a continuous time period in an intuitive way. In the topic of firms, there are currently not many relevant studies. Uyar (1972) forecasts the replacement demand of employees using Markov chain. Horowitz & Horowitz (1968) use a first-order Markov process to describe market change. Joining Markov first-order transfer matrices with entropy, the industrial agglomeration of the brewing industry is quantified. Kopecky & Suen (2010) propose a method that can use Markov chain to approximate autoregression models.

In this paper, a method that describes firm size transition dynamics is proposed based on Markov chain. Joint with a rich continuous panel data exacted from the Chinese Industrial Enterprises Database 1998-2013, the first order transfer matrices are firstly calculated, and the dynamic size transfer path of Chinese manufacturing firms between 1998 to 2013 are revealed. Furthermore, using the properties of Markov Chain, not only the job creation amount but also the general firm growth trend of each can be presented by calculating the probability of upward or downward transfer in the matrices. Besides, following the idea of Horowitz & Horowitz (1968), entropy of firm size are defined and measured. Through these results, the job creation probability of firms in any time period or any size category can be directly described, which may intuitively help policy maker recognize firms with which size categories may create more jobs, and in turn formulate effective employment-boosting polices for the target firms to alleviate the employment pressure of the society in an effective way.

For the rest of this paper, section 2 illustrates the methodology of the proposed method, in which the definition of firm size transition and the Markov first-order transition matrix is given. Based on the transition matrices, the transfer path, trend and entropy are defined. Section 3 shows the results of using the proposed method over a firm-level data set, which includes data set descriptions, calculations details, results analyses and robustness check. Section 4 is the conclusion and discussion of this paper.

Methodology

In this section, the methodology of the transition analysis will be presented. First, the size of the firm is divided into multiple states, which is judged by the number of employees of firms from small to large. Next, existing panel data will be rectangularized including firm entry and exit, and the Markov state transfer matrices between different consecutive years will be calculated. By analysing the obtained Markov matrices, a general trend of inter-transition in terms of firm size may be concluded, while the average job creation among all categories will be calculated based on transfer matrices.

Definition of firm size transition

The basic idea of using Markov chain is to illustrate how the size of firms, which belongs to one of the $N$ categories, are relocated after $d$ years. The entire process of the transfer constitutes multiple probability transition matrices which can be considered as a Markov chain, while all these matrices can be calculated from the sample data set.

Based on this idea, it is necessary to find that among all firms, how many of them has transferred from size category $i$ to $j$ after $d$ years. To this end, let $f_{ji}(D,D+d)$ be the possibility of the size of a firm transfers from category $i$ to $j$, from base year $D$ to year $D+d$. Let the categories in terms of firm size at year $d$ be $S_{d},\:S_{d}\in\left\{ 0,1,2,\text{\dots},N\right\} $. Then there is:

From equation (1), it is obvious to find that the firm size transition process can be described as a Markov chain. Also, for notation simplicity, in this paper, assuming base year $D=0$, which means the base year the firm size transfer begins at year $0$. Hence, there is $f_{ji}(D,D+d)=f_{ji}(0,d)$.

Properties of Markov chain

According to Neumark, et al. (2011), firms are divided into 13 size categories according to their employee numbers, in which category 0 represents the non-existing firms at current year. Similar to You & Fan (2020), let $N=12$. Then, the transfer matrix of the firm size can be considered as a $[13\times13]$ matrix. Let $F(d-1,d)$ be a first order 13-state Markov transfer matrix. Based on equation (2), there is:

where elements in every column satisfies:

Note that $F(d-1,d)$ tend to be a diagonally dominant matrix if firm entry and exit (state 0) are excluded. This is because, for most companies, size change between two consecutive years is relatively small. That is to say, for any $i,j=0,1,2,\ldots,N,j\neq i$, roughly satisfying $f_{ii}(d-1,d)\geqslant f_{ji}(d-1,d)$.

Also, according to the Law of total probability, there is:

Let $p_{j}(d)=Pr\left(S_{d}=j\right)$, then, there is:

Transition path

The Gibrat Law indicates that the evolution of the firm size is only related to the size of its previous year. Following this idea, a transition path can be described by a Markov chain. Let $P(d)=\left[

\right]$$^{'}$. For any $d$ there is:

Equation (6) can also be transformed as follows:

In a special case when the Markov chain is homogeneous, define $F$ which satisfies $F=F(d-1,d)=F(d-2,d-1)=\cdots=F(0,1)$, there is:

It should be noted that the assumption given here is strong. In practice, the transfer matrices obtained from sample data can hardly be homogeneous.

Transition trend

By analysing the transfer matrices, not only the probability of small firms evolving into large firms, but also the transition trend can be discovered. For example, when prosperity comes, firms may be willing to increase recruitment, which will make the distribution of the transfer matrix elements move to the bottom left corner; conversely, when recession comes, firms may reduce recruitment or even layoffs, leading to the distribution of the transfer matrix elements moving to the upper right corner.

In order to quantify this trend, let

Next, simply examine $Q(d)=\frac{L(d)}{R(d)}$, the transition trend at year $d$ can be described.

Transition entropy

In the study of Horowitz & Horowitz (1968), the concept of entropy in information theory, which may reveal the uncertainty, are used to quantify industrial agglomeration. Following this idea, it is proper to introduce a concept of transition entropy, which reveals the randomness of firm size evolution into out topic. With this, the likelihood of firm size change in a certain category can be quantified and compared.

According to the definition proposed by Shannon (1948), the entropy of the $i-th$ column element of the transfer matrix, for each $f_{ji}(d-1,d)>0$, can be expressed as:

where $I_{i}(d)$ is the entropy of the $i-th$ firm category. Note that the greater the entropy of the $i-th$ firm category, the greater the randomness of its size change.

Evaluations over the data set

Data

The data set used in this paper is extracted from the Chinese Industrial Enterprises Database 1998-2013. Variable SIZE is defined by the average number of employees of firms in the year. The descriptive statistics of the data set used in this paper is shown in Table 1.

The data set is selected unbalanced from 1998-2013 due to firm entry and exit. The sample size in this section is 3,747,157, with its minimum value 1, maximum value 760,884 and standard deviation 1453.6. For a purpose of calculation, the unbalanced data set will be further rectangularized to a balanced data set by filling 0 into missing values, which are considered as non-existing firms.

Calculation details

First, referencing the firm size classification criterion used in our previous study (You & Fan, 2020, Table 1), firms are divided into size categories of 0-13 with employee number boundaries of 0, 20, 50, 100, 250, 500, 1,000, 2,500, 5,000, 10,000, 25,000 and 50,000.

Second, according to Section 2.2, transfer matrices $F(d-1,d)$ are built based on the probabilities $f_{ji}(d-1,d),\:i,j\in\left\{ 0,1,2,\text{\dots},12\right\} $. To calculate each $f_{ji}(d-1,d)$, according to the Borel's law of large numbers, when sample size is sufficiently large, there is:

where $Num_{i}(d-1)$ is the number of the firms whose size is in category $i$ at year $d-1$, while $Num_{ji}(d)$ is the number of firms whose size category changes from category $i$ at year $d-1$ to category $j$ at year $d$.

With this, a Markov chain that reflects dynamic transitions between firm size categories can be established.

Results

As the sample data is collected between 1998 and 2013, there are 15 first-order transfer matrices obtained, which are defined in equation (2) while calculated by equation (12). Table 2 is the first-order transfer matrix from 1998 to 1999.

As it is imagined before calculating, if firm entry and exit are excluded, these matrices tend to be diagonally dominant matrices, with most of non-zero values on the diagonal of the matrices. \footnote{See Appendix for all 15 matrices from 1998 to 2013.}

Next, based on the equations in Section 2.3, transition trend and transition entropy are calculated.\footnote{See Appendix for full results.}

Figure 1 is the obtained transition trend based on equation (9) and (10). The value of $Q(d)$ indicated a general transition trend at year $d$. It can be observed that $Q(d)$ in 2008, 2009 and 2013 are relatively low. This indicates that there is more job destruction happened in these two years. In contrast, $Q(d)$ values in 2004, 2010 and 2012 are relatively high, which means job creation is far more prevalent than job destruction in these years. The massive job destruction in 2008 and 2009 might be attributed to the subprime crisis, while the massive job creation 2004 and 2012 might be attributed to national policy changes. Note that the job destruction trend in 2010 shall be attributed to the criteria change of the database, which no longer include firms with sales less than 20 million CNY instead of 5 million.

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Figure 2 is the obtained empirical result based on equation (11). In the figure firms in categories 1-3 are defined as small firms, firms in categories 4-6 are defined as mediums firms and firms in categories 7-12 are defined as large firms. From different size categories it can be observed that small firms tend to have greater transfer entropy while large firms tend to have smaller transfer entropy. The entropy of small firms fluctuates around 1.2. For the entropy of medium firms, there is a rough entropy difference of 0.1 in average compared to small firms. The entropy of large firms have the lowest value in average, which fluctuates around 1. From the perspective over time it can be observed that in 2001, 2004, 2008 and 2010, the transition entropy values increase significantly, which means the likelihood of size transfer increase. This might be attributed to the instability of industrial economic environment. Note that in 2012, due to missing data, the transition entropy cannot be accurately estimated.

Robustness check of Markov transfer matrices

According to Neumark et al. (2011), the obtained results should be checked by 2-year periods instead of 1 in the consideration of robustness. To this end, the 2nd-ordered Markov transfer matrices will be calculated and compared with the previous results. According to equation (7), when $P(0)$ is given,

where $d\geq2$ and $F(d-2,d)$ is the second-ordered Markov transfer matrix from $d-2$ to $d$. This means the second-ordered Markov transfer matrices should completely equals to the product of two continuous first-order Markov transfer matrices. Table 3 is the second-order transfer matrix 1998-2000 calculated from the product of two first-order transfer matrices 1998-1999 and 1999-2000. Table 4 is the second-order transfer matrix 1998-2000 calculated by the cross-sectional data from 1998 to 2000.

Note that the later cannot be calculated by only the two cross-sectional data sets of 1998 and 2000. This is because the firms that enter at 1999 while exit at 2000 are ignored, which are considered as 0-to-0 state transfers from 1998 to 2000. To include all 0-to-0 state transfer in the second-order matrices, the three cross-sectional data sets of 1998, 1999 and 2000 must be rectangularized to a strongly balanced data set with missing values replaced by 0. With this change, all firms that exist in any of the three years are included, and it can be found that the second-order transfer matrix obtained from data set equals to the product of two first-order transfer matrix, which means the obtained first-order transfer matrices are correctly calculated.

Compare the second-order transfer matrix with the first-order transfer matrix, it can be found that there are no significant differences between most of the elements. The most obvious difference between the two kinds is that compared to the second-order matrices, the diagonal elements of the first-order matrices have larger values. This can be understood as when time interval length increases, the probability that firms remain in the same categories may gradually decrease. But in general, it can be considered that the results obtained are not very sensitive to the time interval length.

Conclusion and Discussion

In this paper, a Markov-chain-based descriptive method is proposed for presenting firm size transition dynamics. By dividing firms into multiple states according to their size categories, a Markov chain that reveals the internal transfer between size categories is established. Using the properties of Markov chain, the definition of transition path, transition trend and transition entropy are introduced based on first-order transition matrices. By calculating the transition path, the growth probabilities of firms in terms of size for a certain time period can be clearly demonstrated; by evaluating the transition trend, the tendency of firm size transition towards large firms or small and medium firms in any time period can be revealed; by calculating transition entropy of a certain firm size category, the uncertainty of firm size transition, or in other words, the likelihood of firm size change can be quantified.

Furthermore, the proposed descriptive method is used on the rich firm-level data set extracted from the Chinese Industrial Enterprises Database 1998-2013. From the transition matrices it can be observed that most firms have minor size changes, while a small number of firms may grow or deteriorate significantly. From the result of transition trend, it can be found that in 2004, 2010 and 2012 Chinese manufacturing firms tend to transfer into larger firms, which implies economic growth; in 2008, 2009 and 2013 however, firms tend to transfer into smaller firms, which implies economic recession. From the result of transition entropy, it can be observed that compared to large firms, small and medium firms in China are much more likely to change in terms of size. In general, evidence concludes that small and medium manufacturing firms in China have greater job creation potentials compared to large firms over the time period.

Nevertheless, the research remains imperfect. First, the proposed Markov-chain-based method is only applicable to describe the fact that has occurred. There is no function of prediction so far. In addition, currently there is no sufficient precedent paper fully investigate job creation issue via Markov chain. So, whether the property of Markov chains is applicable to empirical analysis and forecasting under all circumstances needs further discussion. Finally, this paper follows the classification standard proposed by Neumark et al. (2011) to classify firm size, which is extensively used but not linear. According to previous research, it can be roughly considered that this classification conforms to a log-normal function (You & Fan, 2020), yet this standard might not be the best on describing categories of small firms.

\part*{Appendix}

Obtained First-order Markov Transfer Matrices

The obtained first-order Markov transfer matrices from 1998 to 2013 are presented as follows:

Obtained Transition Trend and Entropy Results

The calculated transition trend and transition entropy of each year/category are presented as follows: