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1. Introduction
2. Conceptual issues and literature review
3. The LEAP database
4. Results
5. Conclusion
6. Appendix

1   Introduction

Who creates jobs? While the popular perception is that the answer is small firms, ¹  the literature is inconclusive. Some studies find a negative relationship between firm size and net job creation, but others do not.

The question is important since job creation is often a political and social imperative. Especially in recessions, but also when unemployment is deemed too high, decision-makers focus on strategies to increase employment. Knowing where jobs are being created is valuable for policy formulation.

As well, the employment growth patterns of firms provide general information about the economy. Typically, a large number of small firms coexist with a small number of large firms (Bartelsman et al., 2003). Studying these distributions and the processes that generate them can inform models of the functioning of the economy. ² 

Some studies have found at least a weak negative relationship between firm size and job growth, which would seem to confirm the popular wisdom. However, others argue in favour of the average proportionate growth condition of Gibrat's Law—the assertion, in 1931, of French economist Robert Gibrat that employment growth rates are independent of firm size.

The question of which firms create jobs can be approached in two ways. The first is to examine how the rate of employment growth is related to firm size. The second is to determine if some groupings of firms, by virtue of their weight in the economy, are responsible for more net job creation. Small firms can experience higher growth, yet be too small a fraction of the economy to be important job creators. Large firms may grow slowly, but because of their dominance, still be where most of the jobs are to be found. If the rate of employment growth is relatively constant across size groupings, then the share of aggregate job growth attributable to each size group is proportional to its importance in the economy as an employer.

This paper focuses on the first approach. The relative importance of small and large firms in the Canadian economy is the subject of future research.

Using Statistics Canada's Longitudinal Employment Analysis Program (LEAP) database, this paper explores whether annual rates of employment growth of different-sized firms in the Canadian business sector are more or less the same. The analysis is conducted at level of the business sector as a whole over a 10-year period. Two different measures of employment are used: one that corrects for job quality and one that does not.

Section 2 reviews the literature on the importance and implications of the size-growth relationship. Section 3 describes the structure of the LEAP database and its analytical advantages and disadvantages. Section 4 presents the findings of the analysis: descriptive statistics and regression results. Section 5 contains the conclusions of the study and suggests directions for future research.

2   Conceptual issues and literature review

Birch (1981) is often credited with popularizing the importance of small firms for job creation. His claim that small firms (20 employees or less) were responsible for two-thirds of all net jobs created in the United States in the 1970s captured the imagination of small business advocates and politicians and set off a vigorous debate about the validity of his results.

Birch's findings contradicted the growth condition stated in Gibrat's "law of proportional effect": that observed employment growth rates and firm size are independent. ³  Gibrat's Law is in many ways a (disputed) regularity in search of a theory. Its main implication is that whether they are large or small, firms face similar idiosyncratic shocks, the incidence and magnitude of which are independent of firm size. Firm growth rates are a consequence of these shocks. A corollary is that a firm's capacity to exploit or deal with whatever these shocks bring is positively related to its size. If it holds, Gibrat's Law should manifest itself in the relationship between firm size and employment growth rates, and in the shape of the distribution of firm size.

Accordingly, studies of the independence of growth rates fall into two categories. The more extensive literature looks directly at the relationship between firm size and employment growth rates. The other looks at the firm-size distribution, and infers whether the growth process that generates it suggests a relationship between firm size and employment growth. Both approaches are employed in this report.

Like most empirical studies, this analysis focuses on mean growth rates across firm size classes in order to address the question regarding which firms create proportionally more jobs. But the full description of the stochastic process associated with Gibrat's Law that is used to infer long-run distributions also assumes a constant variance of the distribution of outcomes (Mata, 2008). The similarity of shocks and/or outcomes across firm sizes that is reflected in average growth rates does not necessarily apply to higher moments of their distributions. And while the question of whether growth is more volatile at the small or large end of the firm size distribution is important, it is beyond the scope of the present paper.

2.1  Size and growth

The literature on firm size and employment growth is old and extensive, but researchers disagree on what it shows. Geroski (2005) claims that fifty-plus years of firm growth research suggest randomness. ⁴  On the other hand, in their summary of a half-century of empirical work, Audretsch et al. (2004) mostly document departures from Gibrat's Law for a range of countries over different periods.

This literature contains three testable propositions (Petrunia, 2008): that mean employment growth rates are independent of firm size; that the variance of employment growth rates is independent of firm size; and that shocks are uncorrelated over time. ⁵  However, the set of firms to which these propositions of size-independent processes and size-dependent reactions should apply is not clear.

It is sometimes assumed that the propositions apply only to continuing firms, with firm births and deaths treated separately. That is, conditional on existence, growth opportunities are independent of firm size, and the probability that not-yet existing firms would take advantage of them is governed by another process. ⁶ 

Alternatively, the propositions could apply to all firms. Ideally, the growth of dying firms would be measured as the rates they would have posted had they survived. The rates for entering firms, on the other hand, could be calculated according to some "if all goes well" optimal or forecast size. Of course, measuring would-be's and would-have-been's presents obvious difficulties. Surveys have been used, but the precision of the estimates derived from these sources is unknown. And this information is not available in administrative databases such as the LEAP. Without such information, entry rates will always be unbounded, because initial size is zero, or if created using an arbitrary sizing, likely to be unnaturally large. Rates of decline associated with deaths will be censored and subject to measurement error.

Another approach is to assume that Gibrat's Law applies only to firms—entering, continuing or exiting—that have reached minimum efficient scale (MES). Employment growth rates are hypothesized to be independent of firm size, except for small, young firms; if they were not born at MES, they will either grow rapidly to it, or fail. ⁷  Under this interpretation, selection bias may cause the data to show a relationship between firm size and employment growth even if employment growth is independent of greater-than-MES size, provided the probability of failure is not (Audretsch et al., 2004). If small firms with negative or slow growth rates are more likely to fall below MES and thus exit, the sample of small firms will be biased towards surviving high growth firms while the data will record the full range of growth rates for larger firms.

The literature that has addressed the "who creates jobs" question covers a range of countries over many time periods. In addition to the United States, work has been done for Canada (Baldwin and Picot, 1994; Baldwin, 1998), the United Kingdom, Italy, Portugal, Germany, Japan, and Greece. Some studies contain data that go back to the 19^th century, while others exploit contemporary sources. These studies generally examine only the mean employment growth/firm size relationship. Many of them either reject independence in favour of a negative relationship between firm size and employment growth, or find the evidence to be mixed across detailed industry classifications. Some studies include results for variances, showing that they are not independent of firm size, but rarely delve deeper.

Until recently, research has been constrained by limitations of the data that were available. Early studies rely on databases of publicly traded companies, in which small firms are under-represented. Most use only firm-level data, although some cover plants (Baldwin and Picot, 1994; Baldwin, 1998). Plant studies allow for examination of the growth characteristics of the basic building blocks of firms; firm studies have problems separating out the characteristics of growth at this most basic level from growth that is due to mergers and acquisitions.

In addition, most studies deal exclusively with the manufacturing sector (Baldwin and Haltiwanger, 1998). Audretsch et al. (2004) argue that findings contradicting Gibrat may apply to manufacturing, where entrants are often smaller than MES. Many manufacturing industries entail large sunk costs, require substantial capital investments or benefit from economies of scale. These industries are more favourable to the survival of large firms, and are particularly perilous for slower-growing firms that are below the MES boundary.

By contrast, industries that are easier to enter, for example, in the service sector, are more likely to treat large and small laggards equally. ⁸  Audretsch et al. (2004) find that many of the sub-industries of the Dutch service sector exhibit fewer deviations from the law of proportionate growth, as predicted. ⁹  Petrunia (2008), on the other hand, finds that in Canada, retail and manufacturing firms both fail to comply with all the requirements of Gibrat's Law—in particular, the constancy of variance across size classes. Neumark et al. (2008) find a similar symmetry in growth dynamics between the manufacturing and service sectors.

The results of individual studies may be affected by their coverage of industry and time. Theory and evidence on expanding industries suggest that small firms grow initially, but fail as the industry matures, leaving mostly larger firms. Declining industries, on the other hand, will likely display different dynamics. For instance, large firms may shrink as the industry shrivels, and be the first to fail when they become small. Thus, the relationship between firm size and employment growth that prevails on average may differ from that for individual industries at different periods in their evolution as they face changing macroeconomic conditions.

As well, there is little evidence suggesting that the properties and effects of technology and other shocks confronting firms are the same across industries and time. In some industries, small firms may exploit changes more easily than do less nimble large firms. Also, a relationship between firm size and growth in a particular industry during a specific period may not persist.

This paper ignores the differences across sectors and poses the question at the level of the business sector as a whole over a relatively long time period, recognizing that macroeconomic cyclical events may well impact differentially on firms of different sizes and that industry-specific technological shocks may also drive growth rates temporarily away from Gibrat's general tendency. It asks whether at a higher level of generality—across industries and over time—the rates of growth of different sized firms in the Canadian business sector are more or less the same for the time period examined.

2.2  The HJM critique

A 1993 study by Davis, Haltiwanger and Schuh and a 2010 working paper by Haltiwanger, Jarmin and Miranda (henceforth referred to as HJM) question the negative relationship between firm size and employment growth found in much research. For three reasons, they contend that the results may be misleading. First, the datasets used are problematic. The data usually fail to distinguish between employment growth resulting from organic internal job creation and job growth attributable to changes in industry structure. Second, the data exhibit regression-to-the-mean effects that should be taken into account. And third, they argue that size effects may be confused with the effects of firm age.

HJM use the U.S. Census Bureau's Longitudinal Business Database to address these problems. The authors maintain that when using better data, and when controlling for regression-to-the-mean and age, the negative relationship between firm size and employment growth disappears.

2.2.1  Data issues: Firms versus establishments

Davis et al. (1993) and HJM observe that most datasets pertain either to firms or to establishments, and therefore, pertain to different components of the firm and therefore may present different pictures of the process of growth. According to HJM, neither is sufficient on its own to address the size/growth issue.

For some purposes, firms are often more interesting than establishments when it comes to growth since they are the economic decision-making unit. HJM use the example of the retail industry, where growth occurs through the opening of new establishments, rather than the expansion of existing ones. Establishment-level studies, if used alone, show job growth in these industries comes mostly from births, and job destruction from deaths; firm studies provide an extra dimension because overall firm growth involves both birth and death, which must be jointly evaluated to capture the relationship between employment growth and firm size.

However, the firm-level data that are often used make some studies problematic. Employment changes at the establishment level are well defined—all jobs that are added or subtracted are "organic" job creation or job destruction. By contrast, job changes at the firm level may include not only "organic" changes, but also false births, deaths, expansions and contractions associated with mergers, acquisitions and divestures. These actions, many of which may amount to little more than a shuffling of existing jobs, may create the appearance of job growth when none has actually occurred. In some databases, a firm selling a plant to another firm may appear as the contraction of one and expansion of the other, and a firm spinning-off one of its parts may appear as the simultaneous contraction of a large firm, and the birth of a smaller one.

This merger-and-acquisition activity may not affect plants' operations directly and may have little impact on the number of jobs actually created. It is useful to distinguish this activity from organic growth to understand whether there are differences between the two. Establishment-level data can be used to separate plant-level changes from ownership changes, and separate merger-and-acquisition activity from "organic" growth in firm-level data.

Firm- and establishment-level data may also reach different conclusions about the age of a business entity. Firms can seem to be born from venerable establishments with a stroke of pen when plant mergers occur. As well, if a firm has changed its name or address, some databases may classify it as "new," although its products, workforce and capital remain the same. Based on its plants, it would make sense to classify this firm as a mature incumbent instead of a birth, but without knowing the age of the establishments, researchers must rely on the date of firm's first appearance in the dataset. ¹⁰ 

2.2.2  Regression-to-the-mean

HJM also argue that inferences about firm size and jobs are clouded by regression-to-the-mean: the tendency for firms experiencing transitory shocks to return to their long-run averages, thereby creating a correlation between size and growth. For example, small firms may become temporarily large in response to a greater-than-normal one-off shock and then fall back to their typical size; large firms that lay off workers or cut shifts will appear small in one year, but grow the next. Depending on the method of grouping firms in size classes, such shocks can make large firms look like destroyers, and small firms, creators, of jobs. In many cases, a more accurate interpretation is that small firms create and destroy a large proportion of all jobs, but over the long-term, are responsible for few net new jobs.

Birch's base-year methodology for determining firm size (classifying firms based on their size in period t-1 and measuring their growth between t-1 and t relative to their size in t-1) is particularly susceptible to these effects. An alternative is to use the end year (t) to classify firms, but this produces the opposite bias: the small firms in the example given above will appear particularly liable to destroy jobs, and the large firms, to create them.

Davis et al. (1993) and HJM use an approach that has become standard in the literature—the "current year"—referred to here as the "average-year" method. Firms are classified according to their average size over t-1 and t, which is also used as the growth rate denominator. ¹¹  This method reduces, though it does not eliminate, regression-to-the-mean effects.

A problem it creates, however, is a tendency to misclassify permanent shocks. A small firm that permanently expands between two periods would be classified correctly by the base-year methodology as a rapidly growing small firm, but incorrectly as a rapidly growing large firm when the average-year methodology is used. Depending on how the size classes are constructed, the average-year method may get it right if the resulting average size is still small, but wrong, if it is large.

The literature is unclear about whether researchers should control for regression-to-the mean. In the strictest interpretation of Gibrat's Law, shocks are uncorrelated over time. Studies testing the properties of shocks in the context of Gibrat usually find autocorrelation, with most finding negative autocorrelation (Petrunia, 2008 and some of the industries in Audretsch et al., 2004). Negative autocorrelation is consistent with regression-to-the-mean effects, which these studies consider to violate the independence of employment growth and firm size. ¹² 

However, influences on firm growth are commonly separated into temporary and permanent shocks. It is sometimes implicitly argued that correction for the former is required, but that the latter are what is of interest in growth studies, and therefore, should not be excluded from the data. Of course, the validity of this argument depends on short-duration movements really being transitory, an issue that few studies have investigated.

2.2.3  Firm age effect

Neumark et al. (2008) use both establishment and firm data, and treat carefully the regression-to-the-mean issue. They still find a negative relationship between growth rates and firm size in the U.S. private sector, though the importance of small firms in their study turns out to be an order of magnitude less than Birch's claims.

HJM emphasize a third issue that is generally neglected in earlier studies: the role of firm age. Firms are usually born small, and new firms either grow rapidly to become larger, or else they disappear. The data record a large number of net job growth by small start-ups or young firms; but observations on the growth of slower growing small firms are often cut short by their death and thus the sample selected is either not fully representative (if just continuers are considered) or the observed growth rates are censored downward since observations of decline below zero cannot be measured. The result is the appearance of a negative relationship between size and growth, when growth is actually more a function of characteristics that are summarized more robustly by age. Unfortunately, most data products contain little information on firm age, making a richer characterization of firm growth difficult. ¹³ 

2.3  Firm size distributions

Another body of literature examines the size distribution of firms. Different growth processes generate different distributions of firm size. Gabaix (1999; 2009) shows that growth rates that are independent of size generate a power law distribution, specifically, a Zipf distribution.

Indeed, most studies find that the distribution of firms by size is skewed to the right and that its upper tail follows such a "power law." The tail distribution of firms is often said to follow "Zipf's Law" or to have a "Pareto" distribution. Zipf's Law states that the probability of a firm of size s being larger than a given size S is Description: The probability that s is greater or equal to S is equal to a times S raised to the power xi. where a is a scaling parameter and Description: xi is the Zipf exponent. If Zipf's Law holds, then Description: xi equals -1.

In general, the Zipf distribution for firm sizes seems to be a good fit for firms larger than a minimum size, with the possible exception of the very largest. Axtell (2001) finds that the universe of U.S. firms (including zero employers, that is, self-employed) had an exponent of -1.06 in 1997, and thus, is well approximated by a Zipf distribution. He states that "the Zipf distribution is an unambiguous target that any empirically accurate theory of the [U.S.] firm must hit," and suggests that the Zipf distribution may describe firm sizes in other countries as well. Fujiwara et. al. (2003) report similar results for European firm employment, although their data consist only of larger firms.

3   The LEAP database

This analysis uses Statistics Canada's Longitudinal Employment Analysis Program (LEAP) database. ¹⁴  The LEAP, an administrative database, includes all firms that have some payroll, and therefore, issue at least one "statement of remuneration paid"—a T4-slip. It covers incorporated and unincorporated businesses, but excludes self-employed individuals or partnerships where the participants do not draw salaries. The LEAP data cover the 1983-to-2008 period. Using the information gathered by Statistics Canada's Business Register, the administrative data in LEAP are structured at the level of the "statistical enterprise," which is the level associated with a complete set of financial statements. ¹⁵  In the current analysis, this statistical unit is referred to as "the firm."

Although the LEAP database contains information only at the level of the firm, it has several advantages. Because the LEAP pertains to the entire economy, using it avoids the manufacturing bias of many other studies. And whereas many studies rely on data with limited coverage at the smaller end of the firm size scale or impute data for small firms, the LEAP has data on almost all firms. ¹⁶ 

The chief advantage of the LEAP database is that a major form of corporate restructuring, which creates problems in employment growth studies, can be removed. In a longitudinal panel, mergers, acquisitions, spinoffs and divestitures can result in the spurious "creation" and "destruction" of firm identifiers. For example, if a large firm acquires a small firm, the small firm seems to have died, and the consolidated firm might appear in the data with its own identifier, thereby looking like a new firm. In reality, the acquisition resulted in neither a firm birth nor a death. The employees of the large firm still work for the same large firm, and most employees of the small firm, for the same establishments.

These false births and deaths are removed from LEAP through "labour-tracking." Clusters of employees of appearing and disappearing firms are compared with the clusters of employees of other firms in the previous year (for appearing firms) or the following year (for disappearing firms). If substantial portions of the employees of appearing and disappearing firms are found previously or subsequently in another firm, a connection is made between those two firms, and the structure of the firm in year t is applied to the data of both firms in year t-1 and in all preceding years. Thus, employment growth between any two years will be attributed to firms of different size classes based on the structure of the firm in the final year (year t). ¹⁷ 

The LEAP database is produced in vintages. When an additional year of data becomes available, a new longitudinal file is created, but the previous longitudinal files are also retained. Microdata files are available for each vintage starting in 1999 up to the most recent, 2008. LEAP 2008 tracks firm employment from 1983 to 2008; 2007 tracks employment from 1983 to 2007, and so on back to 1999, which track employment from 1983 to 1999. Each vintage holds the structure of the firm constant, based on information for the most recent year available. For example, 2008 aggregates all earlier annual data based on the firm's structure in 2008; that is, firms that merged in 2008 are consolidated in all previous years in the 2008 vintage.

This practice makes it possible to measure only "organic" employment growth, and ignore employment changes caused by merger and acquisition activity. For example, between period t-1 and t, firm B acquires firm A's 100 workers (growth due to mergers and acquisitions) and also adds 25 employees to the combined entity (organic growth) (Appendix 1). Had the LEAP file not been altered as described above, it would record a sharp jump in firm B's employment in year t (the 100 employees from firm A, plus 25 new positions), and a 100-employee decrease in firm A's employment due to its (false) death. ¹⁸ 

Instead, the change in employment recorded between t-1 and t removes the acquisition transaction and reflects only the employment growth that occurred "within" the now-larger firm B, which rises from 700 employees (B's original 600 plus 100 from firm A) in year t-1 to 725 employees in year t. This larger firm existed in both year t-1 and year t. Classifying growth over the previous year to the consolidation recognizes only that this growth is attributed to the newly created entity, since it was responsible for operations over this period. Thus, year-on-year, LEAP abstracts almost completely from plants changing hands, recording only "organic" growth in the overall entity defined in terms of its structure in year t.

The fact that each vintage pushes the market structure of its latest year back in time creates an extra challenge. The practice risks misclassifying the sources of growth by firm size for every year except the last two. The structure of a firm in t-2 is clearly different than the longitudinal structure, and it is not the case that the firm in year t was in any sense responsible for the consolidated operations in year t-2.

In the t vintage of the data in the previous example, the records for firms A and B are merged into one unique record for all years preceding year t, adding up the employment of both firms. This synthetic record is then linked with firm B's record in year t. Firm A disappears completely from the t vintage, and its employees are treated as though they were always part of firm B—from their first appearance to the end of the vintage.

To deal with the potential for misclassification in earlier years, this analysis uses only the last two years of each LEAP vintage from 1999 to 2008. The result is a series of ten cross-sections, pooled to estimate the distribution of firm sizes, firm ages, and employment growth rates. This reduces the comparisons to one-year periods, with the firm structure held constant for the second year of each cross-section. This approach should yield accurate inference, provided that the distribution is stationary and the sample period is not characterized by large abnormal shocks. As will be explained later, the longitudinal data prior to 1998 back to 1983 are used only to calculate firm age.

3.1  Measures of employment

No measure of employment fits all purposes, and every measure has limitations. Estimates of the number of workers tend to be either a count employment at a fixed date each year or a calculation of the average level of employment during the year using infra-annual estimates. ¹⁹  The Longitudinal Business Database used by HJM measures employment directly once a year as the number of workers at each establishment in the week that includes March 12. While this measure is consistent, late March start-ups are identified as firm births the following year although they are almost a year old. Also, employees with more than one job can be double-counted. And finally, seasonal patterns that vary across firms and industries are missed.

The LEAP, by contrast, contains annual payroll information (total payroll and total number of workers who received a T4 slip over the calendar year). Based on these data, employment levels are inferred through two measures: Average Labour Unit (ALU) and Individual Labour Unit (ILU).

The ALU divides each firm's annual payroll by the average annual earnings of a representative worker in the same industry, province and firm size class ²⁰  because earnings have been found to vary along these dimensions (Statistics Canada, 2011; Baldwin, 1998; Drolet and Morissette, 1998). The average annual earnings for the various industry, province and firm size class combinations are determined from Statistics Canada's monthly Survey of Employment, Payrolls and Hours. For a detailed description of the ALU calculation, see Lafrance and Leung (2010).

The ALU can vary along two dimensions—number of workers, and average remuneration, which differs across firms because of longevity of employment over a year, hours worked and wages levels. Firms with a small ALU may, indeed, have fewer employees, and firms with a larger ALU may have more. However, firms may appear small because they pay wages below the industry average, or large because they pay more. The ALU, therefore, takes several dimensions of job 'quality' into account. ²¹ 

As a result, compared with simple job count measures, the ALU may enlarge differences between small and large firms. Whether this is undesirable depends on the objective—to study raw job creation or quality-corrected job creation. Because large firms are not numerous and have many employees, they dictate the average annual earnings in their provincial industry and size class. Average annual earnings for small firms are defined over a much larger population of firms, which implies more variation.

The ALU calculation considers only four firm size classes: 200 employees or more, 50 to 199, 20 to 49, and fewer than 20. Although the smallest category groups all firms with fewer than 20 employees, in many industries, firms with fewer than 5 employees offer lower earnings than do those with 5 to 19 employees (Statistics Canada, 2011). Many small firms, therefore, have an ALU value below 1. This could also occur if firms are active for only part of the year, which is more likely for start-ups created at the end of a calendar year. ²² 

To check the robustness of the results, the Individual Labour Unit (ILU) is also used. The ILU is closer to a head count—every individual who receives at least one T4 slip in a given year is assigned 1.0 ILU. If individuals worked for different firms during the year, their 1.0 ILU is split proportionately across firms according to the share of their total annual payroll earned in each. For example, if someone earned $10,000 in firm A, and $40,000 in firm B, then firm A is assigned 0.2 ILU, and firm B, 0.8 ILU.

The ILU ignores differences in the quality of jobs. Because it counts the number of T4 slips issued, regardless of the number of hours actually worked, it can treat a four-hour part-time worker in the same way as a forty-hour full-time staffer. Firms that hire many part-time or casual workers will appear to have a large number of jobs, even if the total number of hours worked is small.

On average, a firm's ILU value in the LEAP file is 3.0 times greater than its ALU employment (Appendix, Table 9). This average is heavily influenced by the employment patterns of tiny firms (less than one ALU), for which the average ILU value is 5.7 times greater than its ALU value. By contrast, the ILU for firms with 20 or more employees is, on average, only 1.1 times greater than their ALU. The wider disparity in the measures at the lower end of the size scale is likely a function of the lower wages earned by the self-employed, start-up workers, and employees of part-year firms.

Considering how the two measures are obtained, it is not surprising that more ILUs than ALUs are enumerated annually in the business sector (Appendix, Chart 13). For example, in 2008, ILUs numbered 13.5 million, compared with 11.5 million ALUs. The magnitude of gross job creation and gross job destruction is also larger based on ILUs than ALUs (Appendix, Chart 14). However, the levels of net job growth are similar. The dynamics of net job growth tend to follow the same trend using the ALU and ILU—when one rises, so does the other. Some evidence suggests that the measures respond differently over the business cycle, with increases in ILUs leading ALUs. This could be the result of firms responding to increased demand by hiring (lower-wage) part-time employees, and then making them permanent if demand persists.

3.2  Defining age

Because the LEAP employment data go back to 1983, it is possible to obtain a measure of firm age. HJM define firm age in the following way: ". . . when a new firm ID arises for whatever reason, firm age is based upon the age of the oldest establishment that the firm owns in the first year the new firm ID is observed. The firm is then allowed to age naturally . . . regardless of mergers or acquisitions and as long as firm ownership and control do not change" (p.13).

The definition of firm age in this analysis is similar. For every firm ID in the study, the first year it is observed in the 1999-to-2008 period is identified, and the corresponding LEAP vintage is used (for example, the 2000 vintage if the firm first appeared in 2000). In this vintage, the earliest year in which the firm had employees becomes its birth year. ²³ , ²⁴  Thereafter, the firm is aged naturally. ²⁵ 

An age of zero corresponds to a birth. A birth is identified when all the constituents of the new firm have no employment history. This definition fits well with that of HJM and the definition of enterprise birth proposed by the OECD and Eurostat: "A birth amounts to the creation of a combination of production factors with the restriction that no other enterprises are involved in the event. Births do not include entries into the population due to mergers, break-ups, split-off or restructuring of a set of enterprises" (Eurostat/OECD, 2007).

In the 1999 vintage of LEAP, the birth year of firms can be identified back to 1984 (15 years old in 1999), but the exact age of older firms is not known. For consistency, a separate age category, "16 or older," is created for all subsequent vintages. The age variable used by HJM is also right-censored at 15, with a group for firms aged 16 or more years.

3.3  Empirical framework: Growth

For the main measures of firm size and employment growth, the "current" or "average" year method of HJM is adopted here. The average size of firm i is defined as Description: X sub script i,t equals E sub script i,t plus E sub script i,t-1, all divided by 2. The size of a firm in period t is defined as its average employment in periods t-1 and t., and employment growth is defined as Description: g sub script i,t equals E sub script i,t minus E sub script i,t-1, all divided by X sub script i,t. The employment growth rate of a firm in period t is defined as the change in its employment between periods t-1 and t divided by its size in period t., where g_i,t is firm i's growth rate from period t-1 to period t, and E is employment. Note that the results on rates are not presented as percentage changes in the text and tables: a value of 1 corresponds to 100%, a value of 0.05 to 5%.

This formulation, which is a standard definition of employment growth in the empirical literature on Gibrat's Law, has a number of desirable properties. First, it has the advantage of symmetry, in that it is invariant to the choice of a base year. ²⁶  Second, it yields similar results to log growth rates, at least for firms increasing or reducing employment by up to 50% (by log/average measures). The chief difference from log growth is that the "average-year" method is linear and bounded by -2 and 2 at the extreme ends. The "average-year" method assigns births and deaths a growth rate of 2 and -2, respectively, by construction. As a result, births and deaths constitute a distinct subset that is not directly comparable to other observations and is best analyzed separately. In addition, average employment in year t is the average of (0,E_it) for births and (E_it-1,0) for dying firms, meaning that the denominator for these firms arbitrarily corresponds to half their actual size upon entry and exit.

The third advantage of the "average-year" method is that this measure is easily aggregated. For example, defining the growth rate in aggregate employment as

(1) Description: g sub script t equals the summation over all s of X sub script s multiplied by g sub script s,t. This is also equal to the summation over all s of X sub script s multiplied by the summation over all i that belong to s of X sub script i divided by X sub script s multiplied by g subscript i.

where s = 1,..., S is some grouping of observations according to some identifiable characteristics, such as size or age. X_s and g_s are defined using the group's total employment in t-1, t. The X_s's are a measure of the group's relative size importance in the economy, and g_s is the group's growth rate.

3.4  Empirical framework: Size classification

The average-year methodology of HJM differs from Birch's base-year methodology in two respects: the denominator used in the growth rate calculation, and the method used to assign a firm to a size class. Okolie (2004) demonstrated that, when calculating net and gross employment growth, the method of classifying firms into size categories matters more than the denominator used. For this reason, only one denominator is used in this study, namely the average size in t-1 and t. To determine whether regression-to-the-mean is present in the LEAP data, the results obtained when firms are classified according to their base-year size (size in t-1) are compared with results using their average size (average of size in t-1 and t).

The average-size classification mitigates, although it does not eliminate, regression-to-the-mean problems. It takes care of short-term regression-to-the-mean of small or medium magnitude, but major transitory shocks or shocks lasting more than one period may not be completely averaged out. However, as previously discussed, it is not clear that persistent effects of shocks should be defined as transitory and eliminated. The method used here also tends to misclassify enduring shocks. For instance, a small firm that permanently adds jobs may become, on average, a medium-size firm. But its correct classification is that of a small job creator. In this case, the base-year methodology produces the right answer, and the average size does not.

3.5  Econometric framework

The present analysis is restricted to the business sector. This is done by excluding the cross-sections classified in NAICS 91 (public administration), 61 (education services) and 62 (health care and social assistance). ²⁷ 

To get a picture of the statistical relationship of employment growth rates to firm age and size, the following regression model is specified:

(2) Description: g sub script i,t equals the transposed vector gamma multiplied by vector zeta plus a series of betas multiplied by dummy variables for size classes plus a series of gammas multiplied by dummy variables for age classes plus xi sub script i,t. This is a multiple regression equation that regresses a vector of control variables, zeta, and a series of dummies variables for size and age classes on growth rate.

where s = 1,..., S are firm size classes, a = 1,..., A are firm age classes, and 1[^.] is an indicator function that equals 1 if firm i = 1,...,N belongs to the given size/age class. The vector Z contains control variables (dummies for year and the 2-digit NAICS industry). Omitted categories are year 2003 and manufacturing (NAICS 33).

Five different variations of the regression are employed, using both the ALU and ILU measure of employment:

1. Size controls only, using base size classification
2. Size controls only, using average size classification
3. Age controls only
4. Size and age controls, using base size classification
5. Size and age controls, using average size classification

In each specification, industry and year dummies are included. The coefficients on the dummy variables (indicator functions) are, in effect, conditional means of the growth rates of particular groups. Of interest is whether the Description: beta's or Description: gamma's differ across firm size and age groups, respectively.

In the regressions, the cross-sectional observations with a growth rate equal to 2 (entry), but for which age is not zero, are excluded. ²⁸  Even though they are not numerous, these observations could bias conclusions about the effect of firm age.

4   Results

Firms with less than one ALU or ILU are designated "tiny" firms. "Small" firms are defined as having fewer than 500 employees, and "large" firms, more than 500 employees (using either the ALU or the ILU measure). "Start-ups" are aged zero; "young" firms are less than 10 years old; and "mature" firms are 10 or more years old.

4.1  Descriptive statistics

4.1.1  Distribution of firms and employment across size and age classes

As of 2008, the business sector comprised more than 1 million incorporated and unincorporated firms with employees (Table 1). Almost 4 in 10 (38%) of these firms had less than one ALU, which may indicate very few workers (one self-employed or working part-year only) and/or especially low wages compared with other small firms. Indeed, 25% of these tiny firms had exactly one ILU employee, and almost two-thirds had one or less. However, firms that start up late in the year with employees who earned the bulk of their incomes with other employers could also yield this result. Close to half (48%) the firms—slightly more than half a million—had more than one but fewer than 10 ALUs. Another 11% (around 115,000) had 10 to 49 employees. Firms with 50 or more employees represented 2.7% of the total, and large firms with 500 or more employees, only 0.2%.

About 12% of firms were start-ups, and another 49% were less than 10 years old. The remaining 39% were mature firms. Small firms were disproportionately young (65%), and large firms, predominately old (85%) (Table 1).

Private sector jobs in Canada totalled 11.5 million in 2008, up from 10.2 million in 1999. Firms with fewer than 100 employees accounted for roughly half of private sector employment.

Based on the ALU measure, tiny firms make up less than 2% of all employment, and large firms, 35% (Table 2). Based on the ILU measure, small firms' share of employment is slightly larger, and large firms', correspondingly lower. This, again, suggests that small firms generally have more low-wage and part-time jobs.

4.1.2  Mean growth rate by size and age

Employment growth rates by firm size classes depend on whether the base-size or the average- size method is used to classify firms (Charts 1, 2 and 3). With the base-size classification, the average growth rate is negative for all size classes except tiny firms. ²⁹  The opposite occurs with the average-size classification: except for tiny firms, all size classes have a positive growth rate. These divergent results illustrate the importance of the sizing method used. The different sign for the rate of growth indicates how much regression to the mean occurs—and that firms that are likely to change direction are more likely to have larger negative than positive growth, perhaps because decline possibilities are less likely to be bounded.

With the average-size classification, the rate of growth rises over the smallest firm size classes, is relatively constant for the middle size classes, and declines in the largest size classes. This pattern prevails whether employment is measured with the ALU or the ILU, and whether all firms or just continuing are examined.

With the base-size classification, the rate of employment growth is larger at the low end of the firm size distribution, namely, among firms with less than 1 ALU or ILU. However, with the average-size classification, the rate of growth is lower for these micro-firms. In either case, micro-firms are probably sufficiently different from the rest of the population of firms that conclusions about the superior growth of small firms should probably not rest on this category.

The specific size categories for this analysis were chosen to provide a picture that allows us to evaluate whether there are monotonic trends from small to large groups that reveal a uniform superiority of ever smaller firms. It is difficult to see a particular strong trend in average growth rates that suggests its existence using the average size classification. When the ILU measure is used for all firms, middle sized firms grow faster than both small and large firms. When the ALU measure is used, the pattern is somewhat irregular above 1 ALU, though the largest firms have slower growth rates than all but micro firms.

It should be recognized that differences across size classes are rarely investigated in this amount of detail. Indeed, very aggregative groupings are often used that arbitrarily divide the universe of firms into three groups: less than 100 employees, 100 to less than 500 employees, and 500 employees or more. Using these broad size classes, smaller firms are found to have higher growth rates than middle sized firms and the largest firms (Table 3). Aggregations such as these hide whether it is all small firms that have superior performance or only a subset, which the more detailed categories used here allow us to examine. But all of these categories (the very aggregate ones and the more detailed ones) fail to take into account differences that might be expected in growth rates across groupings that are due to compositional effects, that is, whether small firms are found more in some industries that might be experiencing more rapid growth over the period studies, or whether time periods chosen affect the relative importance of small versus large firms or whether differences in age is an important factor and affects differences found across size classes.

Chart 3, which presents the average employment growth rate by age, demonstrates the importance of a firm's early years. Many start-ups do not survive to the next year, but those that do survive grow—the average growth rate for continuing one-year-old firms is positive. ³⁰  Thereafter, average employment growth rates are, on average, negative. The growth rate is less for firms overall than for continuing firms, because exits reduce the average substantially. Accounting for firm age, therefore, provides more information about employment growth than does firm size alone.

The next section make use of multivariate analysis to consider what happens to differences in growth rates across size classes once differences in industry, time period and age characteristics are considered.

4.2  Regression results – Conditional means

The regression results are presented in Tables 4 to 7. The parameters from each of these regressions are graphed in Charts 4 to 11. To each parameter, the unconditional mean of the growth rates for the excluded size class (at least 1,000 employees) and/or age category (16 years old or more) is added. As noted above, the unconditional mean for the larger size class is negative with the base-size classification, but positive with the average-size classification.

The parameters on the dummies for each firm size class are plotted in Charts 4 to 7 for both the base-year and average-year sizing methods, including and not including age dummies. Three results stand out. The first is that, regardless of method, mean employment growth rates at the higher end of the size distribution differ very little.

The second result is that a negative relationship between firm size and employment growth prevails only for the base-year method and only for the smallest firms. Employment growth rates of small firms are greater only when regression to the mean is ignored. And even then, the relationship disappears at relatively small firm sizes (5 to 10 employees). The average-year method yields a positive relationship over the same interval; that is, small firms have no advantage. The difference between the results of the two methods suggests that regression-to-the-mean matters, at least for the smallest firms. ³¹ 

The third result is that, as in HJM, firm age matters. With the base-year method, including age reduces or eliminates the negative relationship between firm size and employment growth rates, depending on whether all firms or only continuing firms are considered. The higher growth rates of the smallest firms are attributable to the younger age of this group. With the average-year method, the positive relationship holds over a wider range of firm sizes when firm age is controlled, but disappears beyond the 10-to-20-employee range. Here, growth rates actually rise with increasing firm size, at least for the smaller size classes. In both cases, smaller firms do not have proportionately higher growth rates.

The parameters on the age variables are presented in Charts 8 to 11. The vast majority of estimates are significant at 0.01%, indicating a difference between the employment growth rates of the omitted group (age 16 or older) and younger firms. Since growth rates for births are 2 by construction, they are excluded from the chart. Two features stand out. First, controlling for firm size does not influence the nature of the relationship between firm age and employment growth, although it does influence its magnitude. In all cases, employment growth rates decline sharply between the first and second year of operation. ³²  When both continuing and exiting firms are included, a pronounced positive relationship between firm age and employment growth after the second year is evident. However, for continuing firms, there is only a small negative relationship after the second year. This indicates a higher exit rate for young firms compared with old firms. HJM call this the "up or out" dynamics of start-ups and very young firms.

Together, the results suggest that short-run regression-to-the-mean effects matter in Canada. Little evidence supports a relationship between firm size and employment growth, except for the very smallest firms. On the contrary, the results suggest that Gibrat's Law holds for mean employment growth rates of firms at or above minimum efficient scale; that relationship sees higher rates of employment growth as firm size increases toward minimum efficient scale; and what relationship does exist is driven by births and very young firms, with little difference among older firms.

These results hold whether employment is measured by ALU or ILU. This suggests that firms modify quality-adjusted and quality-unadjusted employment in a similar way year-on-year. Idiosyncratic shocks, therefore, have a similar effect on these two concepts of employment.

4.3  Size distribution

Additional evidence supporting the applicability of Gibrat's Law to the employment growth rates of Canadian businesses is provided by the size distribution of Canadian firms. If Gibrat's Law holds, firm size should follow a Zipf distribution, with a Zipf exponent of -1. ³³  Whether the Zipf exponent Description: xi x is significantly different from -1 was tested by regressing the log of the rank of the firm (ordered from largest to smallest) on its log size (Gabaix,1999):

Description: The log of the rank is equal to the log of a plus xi multiplied by the log of the size. This is a simple regression equation that regresses the log of the size on the log of the rank.

If Zipf's Law holds, firms should fall along a straight line with a slope of -1.

Chart 12 suggests that Zipf's Law is a good approximation for firms with 5 employees or more (log of firm size of 0.7). Smaller firms, however, clearly do not fit the pattern.

Table 8 presents the Zipf regression coefficients for firms active in 2008. At the aggregate level, when all firms are included, the coefficient is significantly different from -1. Restricted to firms with 5 or more employees, the coefficient is only slightly above 1 using both ALU (-1.03) and ILU (-1.06) measures of employment. These results are close to the coefficient of -1.06 found by Axtel for the United States (Axtell, 2001). Thus, Zipf's Law is a good approximation of the upper tail of the distribution of firm size for Canada.

5   Conclusion

Which firms grow and which shrink is an increasingly important question. In countries with high unemployment, especially in the wake of the financial crisis, policy-makers benefit from knowing what kinds of firms create jobs.

An extensive literature has examined the relationship between firm size and employment growth. Many studies find a negative relationship. Others, however, argue that the evidence in support of this result relies on inadequate data, omitted variables (firm age), and misleading interpretations, and report little or no relationship between the number of employees in a firm and its propensity to create jobs.

This paper analyzed the employment growth patterns of Canadian firms using Statistics Canada's LEAP database. The results show that regression-to-the-mean effects are important, at least for the smallest firms. Age also matters, especially for the youngest firms. When the influence of transitory shocks is taken into account and when firm age is controlled, little evidence emerges to support the contention that small firms have proportionately higher employment growth rates. Gibrat's Law holds for the mean growth rate; that is, above a 20-employee threshold, little relationship is evident between firm size and employment growth. These results are qualitatively similar to those of Haltiwanger et al. (2010).

The size distribution of firms that results from the pattern of employment growth rates was also examined. Gibrat's Law should produce a distribution of firms into size categories that follows Zipf's Law—a small number of large firms and a large number of small firms. And in fact, the size distribution of Canadian firms is well approximated by a Zipf distribution, which is consistent with the independence of growth rates and firm size.

It is sometimes claimed that small firms grow faster than large firms. Others argue that this is not so. This paper finds that both positions are correct. It is true that average growth rates are larger for a broad swath of firms defined arbitrarily as smaller. But it is not true that as we move across the spectrum of size classes from the very smallest to the very largest that growth rates decline monotonically with size. Indeed, there is evidence that average growth rates increase initially. Having proportionately more smallish firms in this circumstance will not lead to proportionately more job growth.

Even this finding needs to be qualified because growth rate averages conceal considerable differences across the underlying population brought about by industry composition and year effects. Once these are considered using multivariate analysis, there is little in the way of a relationship between job growth and size of the firm, especially after age is considered. That still means that taken together small firms create proportionately more jobs but that much of this occurs in the population studied because particular industries where small firms are heavily represented were growing more quickly, or the time period studied was one with a particular concentration of years where small firms did better than large firms, or because there was a particular concentration of younger firms that were small and it is age that is closely related to growth. What was driving growth in the aggregate of small firms were industry-specific factors, time-specific circumstances or the age of the participants. The circumstances encouraging growth are to be found in these areas, not the size of a firm per se.

This study also points to a new set of research questions on the start-up process and early development. For instance, what are the characteristics of start-ups and young firms that are successful? Do they respond to opportunities differently than their failing competitors? Related to this is the topic of firms smaller than minimum efficient scale (MES). A positive relationship between firm size and growth rates was found for firms below 20 employees. This positive relationship is still found even when only continuing firms are considered. Being closer to this size not only improves the survival probability of firms below it but also is associated with higher employment growth. Further studies could be devoted to examining how growth is affected by the initial distance of entrants from this size and how it relates to MES in general.

Another topic that deserves attention is the variance of growth rates. The literature for Canada and other countries finds that the variance of growth rates is not independent of firm size. Therefore, Gibrat's Law might not hold for variances in Canada even though it holds for means. This issue could be studied with the database developed in this paper, which would also allow the exploration of the relationship between the variance of growth rates and firm age.

6   Appendix

Figure 1: Merger example

6.1  Comparison between Average Labour Unit (ALU) and Individual Labour Unit (ILU)