EXPLORING THE RELEVANCE OF TWO-PART MODELS IN INNOVATION RESEARCH: TOWARDS A BETTER UNDERSTANDING OF INNOVATION SALES

Introduction

Innovation is a widely studied outcome in business and management research in the past decades (Damanpour et al., 2012). Commonly, innovation is operationalised as what we under a collective term choose to call “innovation sales”, described as, e.g., the proportion of sales attributable to new and improved products (Love and Roper, 2001), the fraction of the firm’s turnover relating to products new to the world market (Laursen and Salter, 2006), or the share of sales derived from new services (Leiponen, 2006). The proportion of innovation sales to total sales indicates the importance of innovative products in a firm’s portfolio (Love et al., 2014). One argument frequently used in the innovation literature is that it reflects not only firms’ ability to introduce innovative products to the market but also reflects their short-term commercial success (e.g., Berchicci, 2013; Hashi and Stojčić, 2013; Roper et al., 2013). The share of turnover from innovation sales tends to show the actual level of innovativeness because it assesses the end result of the innovation process and includes the entire innovation process (Mihalache et al., 2012). Thus, despite the diverse foci of the innovation literature, empirical research is often relying on standardised and comparable measures of innovation as a dependent variable. Using a standardised, commonly accepted dependent variable allows for cumulative knowledge building as authors can easily relate new research to the existing one and compare and contrast research findings.

While innovation research for many years have relied on proportional indicators of innovation’s contributions to an aggregate total outcome, researchers have used remarkedly varying approaches when it comes to statistically estimating innovation sales. This is important as using different statistical approaches to model the same outcome reduces the comparability of research results and may reflect different underlying theories about the determinants of innovation. One particular issue is whether innovation sales should be considered as a singular or two-part process (Ramalho and da Silva (2009), see also Mullahy (1986)). The former approach treats innovation as a bounded continuous variable (Hecker, 2016), whereas the latter considers it a divided process where firms initially either introduce a new product or not, and conditional on a new product introduction attempt to derive sales from the new product (Weterings and Boschma, 2009). That is, the process generating participation in innovation is separated from the process generating the magnitude, i.e., the commercial success. In this paper, we set out to outline how innovation is currently studied in research using the fractional innovation-to-sales ratio as the outcome and explore when and how innovation should be studied as a two-part process and how this approach can leverage new opportunities for theorising innovation as a multi-step process.

To investigate how innovation scholars analyse the share of sales due to innovative products or services, we conduct a systematic methodological review (Aguinis et al., 2023) of 20 years of research on innovation using innovation sales as the dependent variable. We isolate our search to studies where firm size is among the covariates in order to create comparability and follow how the relationship between the same variables changes across different choices of estimations. For our purposes, firm size is relevant as the firm size-innovation sales relationship can potentially be described using two-part process logic. Larger firms have greater access to financial and human resources (see detailed descriptions of this relationship in Vaona and Pianta (2008) and Audretsch et al. (2014). These factors might be related to explaining the decision to innovate or not, and, conditioned on that, the degree of innovation. While easier access to resources such as capital may help the firm overcome the threshold of introducing new products, it might not affect the short-term commercial success of the newly introduced product in terms of innovation sales. Firm size also serves as an example of a variable for which competing hypotheses might be posited. For instance, large companies may have higher marketing budgets leading to higher sales revenue from new products. Therefore, firm size serves as a useful illustrative example of how a two-part framework allows for directly testing competing hypotheses about the influence of a well-known variable on innovation sales. Further, firm size is a variable that is often found in studies as an independent or control variable. We identify not less than 141 articles using innovation sales as the dependent variable. Among these studies, a total of 27 employ a two-part model, however, only a subset of these use a justification related to understanding how different factors are related to different parts of the innovation process (see, for instance, Weterings and Boschma, 2009). Our comprehensive screening of the literature reveals considerable heterogeneity in the estimation techniques used and disagreement about the fundamental question of the choice between a singular and a two-part process that reflects not only a statistical practice but an underlying theoretical idea about how firms’ innovation should be conceptualised.

The state of this research reflects a limited recognition of the significant potential of using two-part models in innovation research. In this paper, we develop a two-part modelling framework suitable for innovation sales. Contrary to other research fields, it is central to understanding innovation sales as a two-part process that zero innovation sales can arise in two different ways: (1) A firm did not introduce a new product, or (2) a firm introduced a new product, but this product failed to generate any sales. We argue that a two-part model for innovation sales needs to be able to make this distinction. In this paper, we propose a framework that is different from the ones usually used in econometrics. The framework differentiates between zeroes originating from the innovation introduction process and zeroes from the sales process and we demonstrate its usefulness using simulations and an empirical example. The empirical example serves to illustrate central points about two-part processes and as an easy-to-use guide for future research. Our objective is not to propose new methods of measuring complex innovation processes, but rather to illuminate how two-part statistical approaches can help further leverage the insights obtained from innovation indicators that are already well known to innovation researchers and widely used in the field.

We aim to make several important contributions to the innovation literature and the methodological literature on two-part models. For research on innovation, we show the relevance of considering a two-part model when modelling innovation sales relative to total sales. While some theoretical frameworks, such as those proposed by Crepon et al. (1998) and Ramalho et al. (2011), offer insights, there is a concurrent need for empirical models adept at accurately handling this distinct type of innovation output. We develop a suitable two-part framework that can handle the kind of data arising when analysing innovation sales and thereby hope to contribute to improving the statistical practices used to estimate innovation to improve the trustworthiness of results. We further speculate that increased use of two-part models enables researchers to generate new theoretical insights about when firms engage in innovation efforts and how successful they are.

This paper is structured as follows. First, we introduce the intuition behind two-part models for studying innovation sales. Next, we present our review of the existing research and identify this research practice. Based on this, we build our framework for understanding innovation sales as a two-part process and illustrate it using an empirical example.

Understanding Innovation Sales as a Two-Part Process

In this section, we address in a non-technical fashion why two-part processes are often be relevant to describe innovation.

How to think about two-part processes for innovation sales

When we measure innovation sales, the innovation sales data are a mixture of two processes where there are different causes for the same observation, namely, observations at zero. First, zeroes can happen structurally. Zero innovation sales mean that nothing happened. And nothing can happen either because the probability of making an innovation sale is low, or because the process that generates innovation sales never got started. Zero-observations occurring due to the firm not developing any new products are called structural zeroes (Tang et al., 2018). Alternatively, zeroes may occur naturally when the probability of selling the new product is low. For instance, a firm has developed a new product, but three years later this product has not been commercialised and has yet to make any sales. These firms developed and possibly even introduced a new product to the market, but their innovation sales just happened to be zero. These zeroes are also known as random zeroes (Tang et al., 2018). To summarise, zeroes can arise from two separate data-generating processes: (1) either a firm does not introduce new products or (2) a firm innovates, but the innovation failed to generate any sales.

Data for the share of sales derived from new products that have been collected without excluding non-innovators, i.e., firms who did not introduce any innovative products, often contain a large number of observations at zero. For example, D’Angelo and Baroncelli (2020) found that 60$\%$ of the firms in their sample have no sales of innovative products. Similarly, Berchicci (2013) reports that one-third of the observations have a value equal to zero for the dependent variable innovation sales. If a relatively large proportion of the firms in a sample report zero innovation sales, it seems implausible that such a large share of firms would have no success at all with selling their new products. Instead, innovation data with a large amount of zeroes should make us suspect that there are different causes for the same observation. It seems more logical that the overwhelming majority of firms with zero innovation sales have not developed any new products. Structural zeros may account for most of the zero observations, but the amount of random zeros is not insignificant. For example, Weterings and Boschma (2009) found that out of the 219 firms in their sample, 42 did not introduce any new products (structural zeros) and 12 firms did not yet gain any turnover from the sales of the new products that they developed (random zeros).

The logic of qualitatively different zeroes is illustrated in Fig. 1, where y is the amount of innovation sales revenue, t is the total sales, and s is the fraction of sales derived from the innovative products, i.e., innovation sales revenue divided by total revenue. Each firm introduces a new innovation with probability $p_1$ and does not innovate with probability $1-p_1$. When a firm does not introduce a new product it always produces zero innovation sales, $s=0$ (a structural zero). Firms that innovate may produce either $s=0$ (a random zero) or $1\ge s>0$. Note that this structure implies that we have two distinct processes, where the second process is conditional on the outcome of the first. Generating innovation sales revenue, whether zero or a positive amount, is naturally conditional on the firm bringing a new product to the market in the first place. As the name signifies, a two-part process has two parts. The first process of introducing an innovation is commonly referred to as the participation decision, whereas the second process resulting in the amount of innovation sales is called the magnitude decision (Wooldridge, 2010b, Chap. 17.6).

Interpreting the zeroes as arising from two different processes is not a challenge encountered in adjacent fields. In finance, a common example is leverage, i.e., the question of whether a firm should issue debt or not and if yes how much debt it should issue (Cook et al., 2008). Once a firm has decided to issue debt, leverage is never zero, but strictly positive (Ramalho and da Silva, 2009). In international business, another example is globalisation, i.e., the share of business done abroad (Wulff and Villadsen, 2020). If a firm is global then as per definition has some degree of globalisation (e.g., Ganotakis and Love, 2012), which causes zeroes to be generated solely by the first process of deciding whether to go global or not. For these more traditional two-part processes, it is only necessary to treat strictly positive outcomes in the magnitude part. However, it is completely possible for a firm that has introduced a new product to experience zero sales and it is important to distinguish between the often notable amount of observations at zero.

Several studies of innovation sales use large-scale survey data from the European Community Innovation Survey (CIS) or surveys using similar measurements.¹ CIS is based on the definitions and measurement concepts for innovation data as laid down in the Oslo Manual (OECD and Eurostat, 2018) and constitutes the official innovation statistics for the EU. It is designed to measure innovation inputs, innovation outputs, and innovation-relevant characteristics of firms and their market environment. The CIS questionnaire is set up in a way that gives rise to a selection problem. Specific filters ensure that non-innovators do not have to answer detailed questions about innovation activities such as their share of sales of innovative products. Thus, in this case, the magnitude part can only be estimated for firms that report at least one innovation. This also means that by construction, only the two-part process fits the understanding of innovation as detailed in the CIS data collection. If this selection is not properly taken into account, the estimation of the innovation intensity equation will be biased.

If the data at hand contain a large share of zeros (or other boundary values) practitioners should consider the explanation, resonate whether separate processes are in place, and consider using two-part frameworks thereby allowing the two processes to be explained differently (Ramalho and da Silva, 2013). Having two separate processes enables researchers to investigate whether and how variables associated with engaging in the early phases of innovation, e.g., new product development are also associated with the amount of innovation sales, i.e., commercial success. While the process of innovation is not independent from the commercial success of new products or services, the same explanatory variable may have a different influence on the introduction of an innovative product and the share of sales derived from innovations, both in effect size, magnitude, and functional form. Current approaches are not well suited to understanding the complexities and interactions between different aspects of firm innovations.

Hypothesising about a two-part process for innovation sales

Innovation researchers have argued that a two-part modelling approach allows us to obtain a fuller picture of innovation success (Leiponen and Helfat, 2010). The first part provides information about whether the firm engages with innovation or not. If a firm has introduced any new or improved products it reflects a baseline engagement with product innovation. The second part reflects not only the firm’s ability to introduce new products to the market but also the new product’s short-term commercial success (Roper and Hewitt-Dundas, 2015). Although both parts reflect some innovation output, the existence of a novel product is hardly a good predictor of the economic performance of an innovation. A product innovation represents only a premarket result of innovation activity (Barlet et al., 2000). In other words, firms that generate innovative products are not guaranteed to have their products welcomed by the marketplace. Because only the second part contains information about market acceptance and actual performance, it is crucial to untangle the effects from the two parts, when we hypothesise about an explanatory variable having an effect on innovation.

As discussed in the previous section, innovation sales data are often characterised by a large number of observations at zero. In such cases, researchers have to decide whether one- or two-part models should be used. Clearly, this decision depends crucially on the assumptions and interpretations of the source of the observed zeros. In cases with structural zeros only, if the zeros may be interpreted as a result of a utility-maximising decision, a one-part model is appropriate. For example, Wagner (2001) argued that firms choose the profit-maximising volume of exports, which might be zero or a positive quantity, and therefore uses a one-part model to explain the exports-to-sales ratio. If the zeros and the positive values may be best described by different mechanisms it is more reasonable to model separately the participation and the amount decisions using two-part models (Ramalho et al., 2011). In other words, this idea that the separation of a participation and magnitude part would reflect two different choices also means that they could be affected in different ways or by different explanatory variables.

Mairesse and Mohnen (2002) explained the innovation production function as a simplified representation of the innovation processes which links innovation outputs, such as innovation sales, to innovation inputs. The authors base their work on CIS data and specify the innovation production function with a two-equation model, where the first equation, accounting for the propensity to innovate, includes explanatory variables such as industry dummies and firm size, and the second equation, accounting for the share of innovative sales, includes additional variables such as variables related to R&D efforts and the competitive environment in which the firm operates. Choosing the relevant output, input, and control variables for the specification and estimation of such a function requires a solid understanding of actual innovation processes at work within firms (Robin and Schubert, 2013). As exemplified by Mairesse and Mohnen (2002), the inputs for the two distinct processes may differ. The construction of the CIS questionnaire sets some practical constraints for the inputs, as for the first equation, the independent variables need to be available for all firms, whereas, for the second equation, independent variables only need to be available for firms that introduced a new product (Kampik and Dachs, 2011). More interestingly, a theory may lay the ground for the authors to argue a two-part setup where the explanatory variables play different roles in each process. For instance, Roper et al. (2013) tested whether externalities of openness have different effects on the propensity and intensity of innovation. Similarly, Weterings and Boschma (2009) examined which characteristics of relationships with customers may affect the likelihood that a firm brings a new product or service to the market and which characteristics affect the innovative performance. Thinking in terms of fixed cost barriers to innovation may also be a way to generate hypotheses that motivate a two-part model. A two-part model arises naturally when fixed costs affect the decision to enter a particular state (Wooldridge, 2010b, Chap. 17.6). For a firm, the decision to innovate a new product may depend on several considerations, including the availability of sufficient capital to invest. The way that capital availability affects the decision to introduce a new product may be quite different from how it affects the relative success of the innovation once it starts selling.

In the empirical application at the end of our paper, we provide an example where we suspect a different impact of firm size on the introduction of an innovation and the short-term commercial success measured by the share of innovation sales.

Current Practices for Modelling Innovation Sales

In order to understand how the existing research has modelled innovation sales as an outcome, we conducted a systematic literature review (Aguinis et al., 2020). Our objective with the review is two-fold. First, we want to create an overview of how frequently a two-part model is employed and how the choice of a two-part model is justified in the literature. While innovation processes can legitimately be theorised as one-part or two-part processes, we aim to explore the potential two-part logic used when scholars hypothesise about an explanatory variable having an effect on innovation sales. Second, we want to create an overview of the statistical modelling strategies used to model innovation sales as a two-part process and pinpoint potential pitfalls before we develop a two-part framework suitable for innovation sales in the following section.

To accomplish these objectives, it is useful to not only focus on the dependent variable. Focusing on an independent variable as well enables us to make more precise comparisons between the existing studies and follow how the relationship between the same variables changes across different estimation techniques. Therefore, we focus specifically on studies including organisation size as an explanatory variable. As we explain in detail later, the firm size-innovation sales relationship can potentially be described using two-part process logic: Larger firms have greater access to financial and human resources. These factors might be related to explaining the decision to innovate or not, but not necessarily to the amount of innovative sales. Further, the size of the firm is commonly used as a covariate in innovation studies. Among internal organisational factors determining an innovation process, firm size measured as the number of employees is one of the most prominent and has been presented as a key determinant of innovation since Schumpeter (1983). We note that firm size measured as the number of employees is such a widely used variable that we effectively include most research on innovation sales. In our initial comprehensive survey of top journals, we include around 82% of articles focused on innovation sales, therefore, only excluding a minority because they do not include firm size as a covariate.

To establish an overview of the literature using innovation sales as the dependent variable, we scoped a broad review aiming to include as many studies as possible (Aguinis et al., 2020). We focus on the recent research published between 2000 and 2019. We followed a two-step sequence for identifying relevant studies. Initially, we did a thorough screening of the four top journals Research Policy, Journal of Product Innovation Management, Strategic Management Journal, and Organization Science. Based on this, we identified relevant search terms and conducted a broader search in the Web of Science database. In total, we identified 141 studies fulfilling our search criteria of studying innovation sales and including size among the covariates. A detailed description of the search for relevant research articles is available in Table A.1, the specific search terms can be seen in Table A.2, and an overview of all studies can be found in Table A.3. By reviewing the papers, we identified two main areas for improvement: (1) Theoretical justification and hypothesising of a two-part process for innovation sales, and (2) statistical modelling of a two-part model suitable for innovation sales.

Theoretical reasoning and hypothesising

In the 27 two-part papers, we identify five main justification practices used by practitioners to justify their use of a two-part model. In Table 1, we summarise how practitioners justify their use of a two-part model using one or more types of justification. Only relatively few papers relied on the main arguments discussed in the second section for using a two-part setup to model innovation sales: If the data at hand contain a large share of boundary values or if theory dictates substantial and/or directional effect differences, we should consider two-part models. No studies explicitly link recognition of a significant proportion of zero-observations in the specific dataset to the choice of a two-part model, although several studies recognise a large share of observations at zero. About 29.6% of the studies, on the other hand, resonate with the possibility of explanatory variables having different effects on the propensity to innovation and fractional innovation sales outcome and relate this to their separate estimation method. This justification practice has no overlap with the practice used by almost two-thirds of the papers describing the two outcome measures as complementary. Whereas the justification practice resonating effect differences is an acknowledgement of the innovation sales process as two different data-generating processes, the latter is merely an argument of using two outcome variables intended to reflect different aspects of innovation outcome.

About 22.2% of the papers referred to selectivity bias as an argument for separate estimations, but only half of these studies also considered potential effect differences. About 33.3% of the studies partly justified their choice of using two innovation outcomes due to the availability of measures in the data source, e.g., CIS or IIP. Further, all of these papers also relied on the justification practice of describing how the two measures complement each other. About 7.4% of the papers in our review used one of the outcome measures in a robustness test.

Overall, scholars do not seem to be aware of the stream of the two-part literature and none of the 27 studies refer to their setup as a “two-part model”.

Although more than one-fourth of the papers reflect potential effect differences as seen in Table 1, our review revealed that this is not necessarily apparent in the hypothesis statements. Among those studies formulating hypotheses, only three studies explicitly mention both of the dependent variables, the propensity to innovate and the share of innovative sales, in their hypotheses. Only in the study by Spithoven et al. (2013) do the hypotheses hold a different association between the explanatory variable in question and the two different outcomes.

Regression strategies and treatment of zero-observations

So far, we have not addressed the statistical modelling strategies for the two-part model for innovation sales. Despite studying the same dependent variable, we observe that the researchers employ a range of different estimation techniques in their studies.

Researchers have to make other decisions than whether to treat innovation sales as a one or two-part process. An important choice relates to modelling the specific outcome, innovation sales. As described, it is fractional in nature and our review shows that authors use a variety of statistical techniques. We found that the Tobit model is the dominating choice for modelling the magnitude part of a two-part model. About 21 studies relied on the Tobit model while five used linear regression. Some studies also combine methods in different two-stage IV-methods such as Heckman models. No studies used a binomial regression model and only one study used a fractional regression model to model the magnitude part, although these methods are designed for fractional outcomes (Papke and Wooldridge, 2008; Villadsen and Wulff, 2021a). Like it is the case in international business (Wulff and Villadsen, 2020), innovation scholars are likely unaware that the Tobit makes distributional assumptions about the conditional density. This means that if the errors are heteroskedastic or non-normal, the Tobit estimator is inconsistent (Newey, 1987). As we explain later, innovation sales data is by nature heteroskedastic. Innovation scholars are therefore currently placing the bulk of their empirical two-part evidence on a potentially inconsistent estimator. A linear regression strategy is less problematic, but still not optimal. A linear regression might provide a good approximation of the average partial effect, but there is no guarantee that we obtain good estimates of the partial effect for a wide range of covariate values (Wooldridge, 2010b, Chap. 15.2). If researchers are interested in valid estimates of partial effects across the range of the variable of interest, they should instead rely on fractional or binomial regression.²

Next, we turn to the processing of random and structural zeros in this research. We intended to summarise the treatment of zero-observations in the two-part papers. However, as it turned out, in many studies it is unclear how the random and structural zeros are treated. For example, some studies describe how the regression for the magnitude part includes “innovators only”. Still, they estimate two separate models for the participation and the magnitude part with the same sample size making it difficult to discern how they treat zero-observations. Also, whereas some studies only consider firms with a positive amount of innovation output as innovators, others include firms that did not yet gain any turnover from the sales of the new products that they developed as innovators. Accordingly, there seems to be no consensus in the literature about whether firms producing a random zero are considered innovators or not. We identified three different versions of two-part models for innovation sales: (1) No zeros were included in the magnitude part, (2) random zeros were included in the magnitude part, and (3) both types of zeros are included in the magnitude part. To exclude firms that introduced new products, but had no sales attributable to innovations as in the first scenario is potentially problematic. These studies may be wrongly classifying innovators as non-innovators thereby potentially distorting the two-part model results. This corresponds to practitioners following the more traditional two-part setup of looking at strictly positive outcomes in the magnitude part. The second scenario is ideal and follows the ideas presented in the second section. The third scenario is equivalent to a one-part regression, but the authors still acknowledge the participation state by including the binary regression.

What can we learn from the studies using two-part models?

As discussed, two-part models separate the decision to participate in innovation from the process generating the magnitude of innovation sales. One-part models that do not make this separation implicitly assume that both processes are generated by the same mechanisms. One way to assess whether a two-part model is relevant is by comparing the coefficients for the regression predicting innovation participation with the coefficient predicting the magnitude of innovation. One-part models make the implicit assumption that the signs of the coefficients for the explanatory variables’ effect are the same for the probability of being an innovator and the extent of innovation sales. If these coefficients are in a different direction, it would indicate a serious threat to the assumptions behind one-part models. In the 27 studies uncovered in our literature review using two-part models, we identified 35 separate tests where we could identify the coefficient of firm size for both the first-part and second-part regression.

Only in two studies are the coefficients for innovation participation and magnitude both positive and significant. They are both negative in an additional one study. This suggests that only around 9% of the tests do not violate the assumptions behind a one-part model. In 22 tests (63%) the coefficient either changes from positive to negative or is insignificant in one regression and significant in the other. These results suggest that the mechanisms generating participation in innovation may be different from the mechanisms explaining the magnitude, and indicate that researchers should often consider the relationship between firm size and innovation sales a two-part process. While we do not have this information available for the majority of studies using one-part models, we have no reason to believe the picture would be radically different. We found none of the one-part studies describing that they have tested the premises for the assumption to treat innovation sales as a singular process. We summarise these results in Table 2.

In sum, our review reveals an unexploited potential of two-part models in management and business research. Scholars seem to lack arguments as to why a two-part model is even necessary or theoretically relevant. The literature review also indicates confusion about how to properly model a two-part innovation sales process.

Statistical Modelling of a Two-Part Model for Innovation Sales

In this section, we start building a two-part framework suitable for innovation sales.

The basic innovation sales model

Before we move to describe the two-part model of innovation sales, it is important to consider the basic statistical properties of the innovation sales variable. The model we build for innovation sales should be able to tell us about how a variable of interest is related to innovation sales. Partial effects are a very useful tool for this. A partial effect describes how the fraction or amount of innovation sales changes when some variables of interest change while others are held fixed. For each measurement, we can think of there being some proportion, p, of the total amount of sales, t, that is attributable to recent innovations. The higher this proportion, p, the more of a firm’s sales are attributable to innovation. This makes modelling p of great interest: If we can figure out which factors are related to p, firms will know which factors to manipulate to increase their innovation performance.

The model we build will make it possible to obtain valid estimates of the partial effects. By valid, we mean that (i) the estimator of partial effects is consistent (i.e., that the estimated partial effect converges to the true value as the sample size increases) and (ii) that inferences based on p-values and confidence intervals are correct (i.e., that the 95% confidence interval contains the true value 95% of the time). The model will also respect the bounded nature of the outcome allowing the partial effects to level off towards the bounds. In this way, we can obtain valid partial effects not just at the average, but for values across the range of the variable of interest.

Statistically, we seek to connect observed amounts of innovation sales, say in dollars, to variables associated with different average innovation sales. A good place to start is to think of this as a binomial probability model.³ Usually, we use the binomial distribution to model the number of “successes” out of a fixed number of possibilities. Here, we can think of it as drawing dollars (revenue) from a big pile of sales. Each dollar can either be attributable to recent innovations (success) or not. The model can be stated formally like this

where $y_i$ is the innovation sales revenue for the ith firm, $t_i$ is the total revenue, $p_i$ is the probability of any particular dollar of revenue being attributable to recent innovations, and $s_i$ is the proportion of innovation sales. By using a binomial likelihood we make a series of pre-observation assumptions about how the values are constrained. First, each data point has a natural limit, i.e., $t_i$ is the maximum. In words, the amount of innovation sales can never exceed the amount of total sales. Second, both innovation sales revenue and total sales must be non-negative. A firm can have zero dollars of innovation revenue, but never negative. The constraints of the binomial likelihood bound the proportion of innovation sales as well but to the unit interval. This seems equally reasonable: A firm can never have more than all of its sales arise from the recent innovations or less than none.

In real data, $y_i$, $t_i$, and $s_i$ vary depending on firm and industry characteristics. For instance, larger firms will tend to have higher sales, but usually also a higher proportion of innovation sales (Camisón-Zornoza et al., 2004). As our interest lies in the latter, the last line in Eq. (1) sets up a regression model that allows the proportion to vary across relevant characteristics. Specifically, F maps the probability parameter, p, onto the linear model space of $X\beta$ where $X$ is a matrix containing the relevant characteristics and $\beta$ is a vector of regression weights. F is any link function that compresses the linear model space to make it fit within the probability space. Interestingly, we do not even need to know $t_i$ in order to learn about the relationship between $X$ and $y_i$.⁴

The two-part model

We now extend our basic model into a two-part model. For such cases, where the observations at one or both boundaries of a fractional dependent variable occur with a too large frequency than seems to be consistent with a simple model, or if theory dictates potential effect differences, a better approach may be the employment of a two-part model with a participation and magnitude part. Here, the discrete component is modelled as a binary model and the continuous component as a fractional regression model (Ramalho et al., 2011). The magnitude decision is what we have covered in the basic model. We combine the two parts in one two-part setup in the following way:

The zero-inflated binomial distribution (ZIBinomial) uses the parameters $t_i$ (total sales), $p_{1i}$ (the probability of participation, i.e., introducing a new product), and $p_{2i}$ (the magnitude, i.e., the proportion of innovation sales). Because each part has its own likelihood, two-part models are also known as mixture models. Our two-part model uses two simple probability distributions to model a mixture of causes. $y_i$ is generated as $\mathrm{\text{Binomial}}(1,p_{1i})\times \mathrm{\text{Binomial}}(t_i,p_{2i})$, hence when the first part is zero, then $y_i=0$, and when the first part is one, we get $t_i\ge y_i\ge 0$. Note that this is different from the most common two-part setup described in the econometrics literature (Ramalho et al., 2011). Two-part models have been proposed as a way to model leverage and globalisation, but in such applications, the model does not differentiate between different types of zeroes. As described, once a firm has decided to issue debt, leverage is never zero, but strictly positive (Ramalho and da Silva, 2009), and if a firm is global, then as per definition has some degree of globalisation (e.g., Ganotakis and Love, 2012). This causes zeroes to be generated solely by the first process making the outcome for the second part strictly positive. In this more traditional setup, a binary variable is used to determine whether y is zero or strictly positive and the second part only considers the magnitude of $y_i$ when $y_i>0$. However, this categorisation does not fit well with the innovation sales process.

With a two-part model, we can connect a linear model and link function to each process. The parameters differ because any one predictor may be associated differently with each part of the model. Even the link functions and the variables that are present in each model do not need to be the same. In the two-part specification above we allow the variables that predict innovation participation and magnitude to be different. Further, if an explanatory variable, x is contained in both $Z_i$ and $X_i$, the two-part model allows for x to be distinctively related to the two processes, both in terms of sign and magnitude. As discussed previously, this is very advantageous for modelling innovation sales. As we show later in the empirical example, we might have variables that are associated with one or both processes inside the zero-inflated binomial mixture.

Two-part estimation

The proposed two-part model uses one part to predict participation and one part to predict the magnitude, but where the magnitude part is allowed to be zero. Participation is simply a binary choice model of the probability of introducing a product or innovation:

where $y_i^{*}$ is an indicator variable that determines if there was no innovation, $y_i=0$, or an innovation with none or some success, $1\ge y_i\ge 0$ (or $t_i\ge y_i\ge 0$ in the binomial case). Since this is a standard binary choice model, we can estimate it by maximum likelihood (ML) using the whole sample.

The second part of the model governs the magnitude of innovation success. If $t_i$ and $y_i$ are known, we assume the following conditional mean specification based on our binomial model:

which makes sure that the expected amount of innovation sales, $\mathrm{\text{E}}(y_i|X_i,t_i)$, is between zero and total sales, $t_i$. If only the fraction, $s_i$, is known, the conditional mean specification becomes where the fitted values will be between zero and one, and $s_i$ is allowed to take on values not only between zero and one but also at the endpoints. Note that the conditional mean specifications differ only in terms of $t_i$, which is used to scale the mean on the fractional scale to the innovation sales scale. These specifications would be exactly the same for the basic innovation sales model we covered earlier, except that we now conditional on innovation participation being equal to 1. The estimation of Eq. (4) or (5) is therefore on a subsample containing only firms that have introduced an innovation.

To consistently estimate partial effects, we use quasi-maximum likelihood estimation (QMLE). QMLE fits very well with our desire to invoke as few restrictions as possible and still obtain valid estimates. As long as we pay careful attention to specify the conditional mean correctly, QMLE is consistent even if our distributional assumptions are completely wrong (Gourieroux et al., 1984; Papke and Wooldridge, 1996). Estimation of Eq. (4) by Binomial QMLE is also known as (aggregated) binomial regression, while estimation of Eq. (5) by Bernouilli QMLE is called fractional regression (Wooldridge, 2010b, Chap. 18). Once $\beta$ has been estimated, average partial effects (APEs) can be estimated by plugging in the estimated parameters and computing the partial effect for each firm in the sample. Finally, we take the average thereby obtaining a consistent estimator for the APE.

When each part has been estimated, other quantities may also be computed. It is possible to compute the total expectation based on both processes similar to the traditional two-part model as described in Ramalho et al. (2011). It can be shown that the total expectation is simply $F_2(X_i{\beta }_2)\times F_1(Z_i{\beta }_i)$.

A major advantage of the proposed model is its simplicity. Assuming that the two parts are independent given the covariates, the estimation and interpretation of the two can be done independently. Because these models are standard in modern statistical software, the two-part model is straightforward to implement in practice.

Our model framework’s superiority over the more common two-part setup not taking random zeros into account is demonstrated in Fig. 2. Again, we simulate 20,000 repetitions, this time only from a beta-binomial process with $N=400$. We vary the number of random zeros and compare the bias and 95% CI coverage for our model to the traditional specification. The participation part of the traditional model suffers the largest bias (D1) increasing steadily as the number of random zeroes increases. This makes sense because the traditional specification does not distinguish between random and structural zeroes. Increasing the share of zeroes that are misclassified as structural, increases the APE bias. The magnitude part shows a slight bias, but it is mostly affected in terms of CI coverage. Importantly, the proposed two-part framework is unbiased with 95% coverage and is completely unaffected by the amount of random zeroes. This demonstrates the clear advantage of implementing a two-part model that can tell the difference between when a firm did not introduce a new product and when it did, but could not attribute any sales to the product.

Empirical Application: The Effect of Firm Size

In this section, we present an empirical application of the use of the proposed two-part framework for modelling innovation sales. The purpose is to exemplify how a two-part model of innovation sales can be justified and modelled and provide a practical guide for future research. We also show how the results of the two-part model can be interpreted and yield insights not available in a one-part model.

The firm size-innovation sales relationship

The two-part model we propose is still just a statistical model. If you wish to draw valid causal inferences, it is recommended to base the choice of variables on theory using causal graphs (Hünermund et al., 2022; Hünermund and Louw, 2023; Mändli and Rönkkö, 2023). Earlier, we argued that the firm size-innovation sales relation might best be described in terms of a two-part process. This process is depicted as a directed acyclic graph (DAG) in Fig. 3.

Larger firms have access to greater financial and human resources. Such resources may be regarded as barriers to innovation. Without the necessary financial muscle and know-how, introducing new products becomes difficult for the firm. Thus, larger firms have a greater ability to achieve at least one innovation (Leiponen and Helfat, 2010). This is an example of the fixed-cost barrier that we discussed earlier. Also, recall from our previous discussion that innovation magnitude differs by also indicating short-term commercial success. While easier access to resources such as capital may help the firm overcome the threshold of introducing new products, it might not affect the short-term commercial success of the newly introduced product.

With respect to the direct effect of firm size on innovation magnitude, one may pose two competing hypotheses. On the one hand, the resource-based view of the firm posits that a resource needs to have certain characteristics in order to provide a competitive advantage (Dierickx and Cool, 1989). Since firm size is more a proxy of resource quantity than quality, we suggest that firm size is likely to matter little for short-term commercial success as indicated by innovation magnitude. On the other hand, larger firms may have higher marketing budgets resulting in a stronger brand presence and hence higher sales revenues. In sum, we may pose two competing hypotheses about the effect of firm size on innovations sales:

$H_1$:	Firm size has a positive effect on innovation participation, but no effect on innovation magnitude.
$H_2$:	Firm size has a positive effect on innovation participation and a positive effect on innovation magnitude.

Note how these competing hypotheses both have two conditions that must hold to gain support: a positive effect on innovation participation and a positive ($H_2$) or no ($H_1$) effect on innovation magnitude. Having two components makes our predictions more precise thereby putting them at a higher risk of falsification (Aguinis and Edwards, 2014).

We limit our ambition of the empirical application to controlling for two important possible confounding variables driving innovation in general, namely, industry and country dummies. We cannot rule out that some remaining unobserved factors may drive innovation outcomes or firm size. The goal of this empirical example is not to establish any evidence of causality. We see our empirical application as an example where we suggest a potential two-part process relationship to demonstrate the possibilities and usefulness of our proposed two-part framework.

Because $H_1$ implies that firm size is not a direct cause of the magnitude of innovation sales, we set up and perform an equivalence test. In essence, an equivalence test helps determine whether an observed effect is too small to matter and therefore practically equivalent to zero (Lakens, 2017). Equivalence testing is a procedure that enables evaluating informative null results (Harms and Lakens, 2018). First, we need to quantify the smallest effect size of interest (SESOI, Lakens, 2017). We choose the lower-end confidence bound of 0.0467 around the meta-analytic effect size documented by Camisón-Zornoza et al. (2004). For simplicity, we round the SESOI to 0.05. We then use the two one-sided tests (TOST) procedure to test two composite null hypotheses (Schuirmann, 1987): $\mathrm{\text{H}}{0}_1:\mathrm{\text{APE}}\le -0.05$ and $\mathrm{\text{H}}{0}_2:\mathrm{\text{APE}}\ge 0.05$. Rejecting both these one-sided tests enables us to conclude that the APE falls within the equivalence bounds and is close enough to zero to practically equivalent (Seaman and Serlin, 1998).

Data and variables

The empirical application relies on data from 2009 from the Management, Organisation, and Innovation Survey, a joint initiative by the European Bank for Reconstruction and Development, and the World Bank Group (EBRD-World Bank, 2010). Descriptive statistics are available in Table 3. After deletion of missing data,⁵ we have 1458 firm-level observations from 12 countries and 11 industries.

Respondents are asked whether their firm has introduced any new products and services in the last three years. This question acts an indicator variable of participation. In this survey, 948 (65%) introduced new products or services and are therefore asked about the proportion of sales over the past three years attributable to new products and services. This is effectively the magnitude part of innovation sales. In Table 3, we can see that the minimum innovation sales for the innovators is zero. Eight firms introduced new products but had these new products account for zero proportion of their total sales. As explained earlier, it is expected that some firms with a low probability of making innovation sales will experience no short-term commercial success. Thirty three firms had all their newly introduced innovation account for all of their total sales. The median innovation sales among the innovators is 0.2 indicating that newly introduced products and services account for a minority of the total sales.

Firm size is measured as the number of employees. From Table 3, it is clear that firms that introduced new products are generally larger than non-innovators. The median number of employees for non-innovators is 111 while it is 165 for the innovators. This is an early indication that firm size seems to matter for the propensity to introduce new products or services. The descriptive statistics on the two possible confounding variables country and industry are also available from Table 3.

Data analysis

Our data analysis follows the same basic steps outlined in Villadsen and Wulff (2021a) for fractional one-part models. After having specified the model, we perform specification tests of the conditional mean. Finally, we analyse the results relying on APEs and graphical plots.

Step 1: Model formulation

Based on our DAG (Fig. 3), we specify the following conditional mean specification for the fractional part:

and the following for the binary part: where both models contain country and industry dummies in $X_i$ and $Z_i$, and both models use the probit link function, $\mathrm{\Omega }$. In essence, the model for the fractional part is a fractional probit model and the model for the binary part is a standard probit model. As explained we estimate the former using QMLE and the latter using ML. This secures that valid estimates of APEs across the range of firm size are obtained using the fewest possible constraints.

Our DAG is agnostic about the directional and functional form relationship between firm size and innovation sales. We expect that firm size is positively and logarithmically related to introducing new products. While larger firms are more likely to introduce a new product, the effect of adding additional size once a firm has reached a certain size is likely to wear off. Eventually, the positive effect of being larger wears off due to inertia (Mihalache et al., 2012). In other words, we expect the effect to increase at a decreasing rate. To allow diminishing returns to size, we log-transform firm size (Villadsen and Wulff, 2021b).

Step 2: Specification tests

Because the consistency of our estimates depends on the conditional mean specifications being correct, we test the specifications. We use Ramsey’s (1969) RESET test using two powers (Ramalho and Ramalho, 2012) and the generalised goodness of functional form (GGOFF) test (Ramalho et al., 2014). The test results are available in Table 4. Neither the RESET (Participation: ${\chi }^2=3.63$, $p=0.057$; Magnitude: ${\chi }^2=0.60$, $p=0.441$) nor the GGOFF (Participation: ${\chi }^2=3.72$, $p=0.156$; Magnitude: ${\chi }^2=0.81$, $p=0.667$) tests reject the null hypothesis that the models are correctly specified at an alpha of 5%.

Step 3: Results

Coefficients from binary (Mood, 2010) and fractional (Ramalho and Ramalho, 2010) models are sensitive to unobserved heterogeneity. Consequently, we should not compare coefficients across models or assess their magnitude (Hoetker, 2007). Instead, we avoid direct interpretation of coefficients by relying on summary measures and graphical illustrations of APEs (Wulff, 2015). Table 4 includes the main results. We observe a clear difference in the APEs when comparing the participation and magnitude parts. Firm size is positively associated with the probability of introducing new products or services. Increasing the size of log firm size by one unit is associated with around a 0.07 increase in the probability of new introduction. We can reject the null of no effect ($z=5.08$, $p=0.000$) and cannot reject that the effect is larger than the SESOI of 0.05 ($z=1.31$, $p=0.9051$). We thus conclude that the effect of log firm size on participation is substantial. For the firms that introduced a new product, the APE is around 0.01, meaning that a one-unit increase in log firm size is associated with around a 0.01 increase in the proportion of innovation sales. We cannot reject the null of no effect ($z=1.55$, $p=0.121$) and can reject that the effect is larger than the SESOI of 0.05 ($z=4.13$, $p=0.0000$). We thus conclude that the effect of log firm size on the magnitude part is practically equivalent to zero. This lends support to $H_1$, but not $H_2$.

Finally, we can compute the total change in innovation sales participation and magnitude. The estimated APE is about 0.03 and while we can reject the null of no effect ($z=3.94$, $p=0.000$), we also reject that the APE is large enough to be interesting ($z=3.06$, $p=0.0011$). Thus, we can conclude that the total change is different from zero, but not large enough to be of interest. Graphical illustrations of APEs across the range of log firm size are available in Fig. 4. Figure 4(a) shows that the APEs of log firm size are non-constant across the range of log firm size. As expected, the APEs decrease as firms get larger. This is consistent with the idea of diminishing returns to firm size The gained access to resources matters most when firms are small and matters less when firms are already large. Figure 4(b) shows that the APEs are statistically equivalent to zero across the entire range of log firm size. This enforces the conclusion that the effect of log firm size on the magnitude part of innovation sales is equivalent to zero.

While log firm size has a substantial effect on introducing new products, log firm size has no practical significance on the magnitude of innovation sales for firms that made new product introductions. In other words, firm size might matter quite a bit for getting a new product launched, but it might not matter for the short-term commercial success of the product. Our analysis also demonstrates the importance of separating the two processes to entangle product introduction effects from actual performance effects. In this case, we reach three different conclusions about the impact of firm size depending on how we compute the APE. This information cannot be obtained by the use of a one-part model.

Examples of traditional models

In this section, we briefly examine what would have happened had we used traditional approaches on the example dataset (Table 5). First, we examine what happens if we drop the random zeroes and estimate the fractional probit model on the magnitude part. The estimates are very close to our proper two-part specification from before. This is not surprising, as this datasets contain only 8 (0.8%) random zeroes limiting the amount of damage an improper handling of random zero can afflict. However, as shown in our simulation earlier, in datasets with a higher percentage of random zeroes this approach can result in a severe loss of statistical power.

Second, we examine what would happen if we did not discriminate between random and structural zeroes in the magnitude part. In this model, the estimated APE of 0.0270 is twice as large compared to the proper two-part specification. Furthermore, the estimate is now significant ($p=0.0000$) suggesting a nonzero effect. This leaves us with a different conclusion than earlier and highlights the benefits of our two-part specification.

Discussion

In a systematic literature review of two decades of management and innovation research, we focused distinctively on the relation between firm size and innovation sales to understand if this widely studied relationship should be understood as a two-part process and to illustrate the value of this estimation technique. We observe a lack of justification for the choice between singular and two-part models, as well as significant heterogeneity in the estimation techniques. To address this gap, we develop a two-part modelling framework specifically designed for innovation sales. The framework presented in this paper is a simple way for researchers to model two-part innovation sales processes while correctly distinguishing structural from random zeroes. It builds on rigorously tested ideas and is easy to implement using modern statistical software. By applying simulations and an empirical example, we demonstrate the usefulness and practicality of our framework, offering guidance for future research endeavours. Our contributions extend to both the innovation literature and the methodological literature on two-part models. We emphasise the relevance of considering a two-part modelling approach when analysing innovation sales relative to total sales, aiming to enhance statistical practices and improve the trustworthiness of research outcomes. To our knowledge, no prior studies have examined the use of two-part models for fractional outcomes with qualitatively different zeros such as innovation sales. This study represents a novel exploration of two-part models for fractional outcomes such as innovation sales, where zero-observations can occur due to two distinct data-generating processes: (1) a firm does not introduce an innovation, or (2) a firm innovates, but the resulting innovation fails to generate any sales.

Practical implications

While our core focus of this paper is to illustrate the underutilised research potential in two-part models for innovation research, our analysis also holds important practical implications. The two-part modelling framework has several implications for advancing our understanding of innovation for both researchers and managers. Relying on the logic of two-part models, managers are provided with valuable insights into the factors influencing the decision to engage in innovation efforts and the subsequent success of those efforts. This potentially provides a more comprehensive understanding of their firm’s innovation outcomes. By applying this framework, managers can identify key factors that influence both the initiation and success of innovation efforts. This can be particularly valuable for strategic planning, as it allows firms to allocate resources more effectively between innovation activities and commercialisation strategies. A focus on how different factors may contribute to the success of internal innovation activities and the external market performance of new products or services allows for more fine-grained decision-processes and a better understanding of firms’ strengths and weaknesses concerning fundamental business processes. By paying attention to the qualitatively different zeroes and distinguishing between the presence of innovation and its commercial success, managers can assess the effectiveness of their innovation strategies more accurately. This distinction helps firms tailor their strategies, whether by fostering a more innovation-conducive environment or by addressing market-entry barriers for new products and services. We suggest that increased utilisation of two-part models opens up avenues for generating new theoretical insights into firms’ engagement in innovation efforts and their levels of success.

Methodological implications for research

The model framework presented in this paper is a simple way for researchers to model two-part innovation sales processes while correctly distinguishing structural from random zeroes. It builds on rigorously tested ideas and is easy to implement using modern statistical software. In our review, we identified a total of 27 papers that relied on a two-part process. However, many did not follow the practices we recommend in this paper. In the remainder of the discussion, we focus on the methodological implications of the analysis and consider some possible extensions and alternative approaches.

Our model framework assumes that we have data that can distinguish between structural and random zeroes. However, this difference might not be directly observable based on the data we have. In such cases, it would be possible to rephrase the magnitude part by including into y the structural zeroes leaving us with $y^m$. Next, we include an indicator variable, $r_i$, on the right-hand side to tease out the effect of the structural zeroes: $\mathrm{\text{E}}(y_i^m|X_i)=F_2(X_i{\beta }_2+\alpha r_i)$. Following Tang et al. (2018), we could treat $r_i$ as a latent variable to be estimated by a different model. Estimation could possibly be done by quasi-limited information likelihood (Wooldridge, 2014), or maybe even the less computationally demanding control function approach (Wooldridge, 2015). Further, we would need the specification of a model for $r_i$, which is likely to be challenging. In sum, the proposed model could potentially be expanded to account for cases where the observed data does not contain information on the difference between structural and random zeroes.

Another possible extension is to assume that the two processes are dependent. Two-part dependency happens when a separate, unobserved process governs the types of firms that innovate. This problem is analogous to the well-known sample selection problem often encountered in management (Clougherty et al., 2016). If unobserved factors affecting the participation process are correlated with factors that influence the magnitude process, the framework proposed here will fail. Schwiebert and Wagner (2015) discussed a generalised approach to accounting for selection effects in fractional models. If a valid instrument is available, researchers may rely on the implementation in Wulff (2019) to cover the magnitude part thereby accounting for process dependence.

The model framework presented in this paper is built around traditional econometric principles. A popular alternative approach for mixture modelling is to expand the focus from the conditional mean to the full distribution of the outcome (Kneib et al., 2023). A distributional approach for innovation sales would assume a distribution consistent with $p_2$, typically a beta distribution, and estimate the model parameters using maximum likelihood (Kieschnick and McCullough, 2003). When combined with a binary model for the participation part, the model is known as the zero-inflated beta (ZIB) model (Cook et al., 2008). However, a ZIB model is likely not a good approach for modelling two-part process innovation sales. First, the beta distribution is not defined for values at the boundaries, i.e., 0 and 1, and as in our empirical example, real data often includes both these values in the magnitude part. Second, even if our data contains no boundary values, the ZIB is inconsistent if $y_i$ follows a different distribution than the beta distribution (Wooldridge, 2010a, Chap. 18.6). A feature of the approach proposed in this paper is that it is based on QMLE and thus consistent no matter the true distribution of $y_i$. If innovation researchers wish to apply the ZIB, they should have an actual interest in the conditional distribution. Otherwise, they might be giving up the feature of consistency provided by QMLE.

In summary, our study advances the understanding of innovation by suggesting a framework for analysing innovation sales as a two-part process. This approach not only deepens insights into firms’ innovation decisions and outcomes but also emphasises the value of distinguishing between the intent to innovate and its commercial success. Relying on this understanding, researchers and managers can gain a clearer picture of the complexities of innovation. We hope an increasing use of two-part frameworks can help inspire future innovation management research and ultimately support more effective innovation strategies in practice.

ORCID

Camille Pedersen  https://orcid.org/0009-0002-4722-0142

JESPER N. Wulff  https://orcid.org/0000-0002-7976-0939

Anders R. Villadsen  https://orcid.org/0000-0001-9935-6151

Notes

¹ For example, data from Irish Innovation Panel (IIP), Swiss Innovation Panel (SIP), UK Innovation Surveys (UKIS), and the Spanish Technological Innovation Panel (PITEC) survey use measures corresponding to the European Community Innovation Survey.

² For a detailed discussion of alternative estimators for fractional outcomes, see Wulff and Villadsen (2020) and Villadsen and Wulff (2021a).

³ The binomial assumption that we build our model on is almost always too restrictive for practical applications (Gelman et al., 2020, Chap. 3). In fact, the binomial assumption fails if unobserved firm effects play an important role in the firm’s amount of innovation sales. This phenomenon is also known as unobserved heterogeneity and we suspect that it is unlikely that researchers can convincingly rule it out when modelling innovation sales. We can add this unobserved heterogeneity to the model by applying a beta-binomial density assuming that each firm has its own unique, unobserved proportion of innovation sales As indicated by its name, the beta-binomial is a mixture of a binomial and a beta distribution. We can think of the beta-binomial as a hierarchical model where we start with a binomial distribution and then let p follow a beta distribution. Because of the extra distributional layer, the beta-binomial distribution has a larger variance for a fixed p. In Fig. B.1, we visualise the characteristics of innovation sales as a bounded variable with natural limits and compare the binomial density to the more realistic beta-binomial density for the basic innovation sales model. In Fig. B.2, we demonstrate the more realistic case with unobserved heterogeneity in the two-part model by visualising the zero-inflated beta-binomial (ZIBeta-Binomial). In Fig. B.3, we show the QMLE is consistent even in the face of unobserved heterogeneity.

⁴ As long as $t_i$ is exogenous, we can focus exclusively on the fractional variable, $s_i$ (Wooldridge, 2010b, Chap. 18). While we would prefer to know both $y_i$ and $t_i$, knowing the proportion of innovation sales, $s_i$, is enough. This is very convenient since some data sources only contain data on $s_i$, not on $y_i$ and $t_i$.

⁵ Data are missing for around 7.8% of respondents with respect to the dependent variable, while around 10.4% are missing information on the size of the firm. Due to the low level of missingness, these observations are removed using listwise deletion.