In the scope of the Stiglitz and Weiss model, the existing literature has documented the use of a priori criteria to classify firms in terms of the likelihood of being financially constrained or not. In particular, size and age are significant factors regarding the variations in the investment opportunity set (Fazzari et al., 1988) which is highly correlated to proxies capturing informational opacity (e.g., Oliner & Rudebusch, 1992; Carpenter & Rondi, 2000; Audretsch & Elston, 2002; Freel et al., 2012; Xiang et al., 2015; Ferrando et al., 2017; Rostamkalaei et al., 2020; Kallandranis, 2020 etc.).

Indeed, the mainstream of the credit rationing literature supports the hypothesis that smaller firms tend to be disadvantaged relative to the larger ones, in terms of access to capital (e.g., Carpenter & Rondi, 2000; Audretsch & Elston, 2002; Drakos & Kallandranis, 2005; Garcia-Teruel & Martinez-Solano, 2007; Psillaki & Daskalakis, 2009; Hashi & Toci, 2010; Drakos & Giannakopoulos, 2011; Farinha & Félix, 2015; Kallandranis et al., 2023 etc.). Thus, SMEs are more likely to have less access to external finance and to be more constrained in their operations, which is intensified when firms adopt innovation. Indeed, large firms can capitalize on their innovation and create their product or service in high quantities versus the low capacity of smaller firms. This also helps large firms spread their products’ fixed costs more efficiently and make their innovation investment a more appealing offer to banks, contrary to SMEs.

However, the literature has conflicting results regarding the role of size on innovation activities (Khosravi et al., 2019) . Even though previous research has shown that large and small firms conduct different forms of innovation (Salavou & Avlonitis, 2008) , there is also evidence that points to size having a positive effect on innovation since they are resourceful (Mol & Birkinshaw, 2009) . Moreover, the opposite results have also been spotted (Vaccaro et al., 2012) , possibly because small firms adapt more easily to their environment than larger firms (van de Vrande et al., 2009) .

In addition, firm age is expected to be related to the degree of informational asymmetries as its long past record would indicate quality and therefore reduce asymmetric information. The effect of age has been investigated in several empirical studies (e.g., Oliner & Rudebusch, 1992; Schaller, 1993; Beck et al., 2006; Serrasqueiro & Nunes 2011; Xiang et al., 2015; Anastasiou et al., 2022; Kallandranis et al., 2023 etc.), the majority of those reporting that older firms report less financing obstacles, while the younger ones face higher premiums or even their loan application is declined (e.g., Serrasqueiro & Nunes, 2011; Xiang et al., 2015; Bongini et al., 2021 etc.).

As for its effects on innovation, the results are again conflicting. Though there are studies that hint at the age not having a significant effect on innovation (Laforet, 2013; Öztürk & Ozen, 2021) , other studies have found either adverse effects (Huergo & Jaumandreu, 2004; Rosenbusch et al., 2011) or positive ones (Winters & Stam, 2007) . This controversy arises from particular characteristics of older and younger firms. Larger firms have more experience, knowledge and established relationships that help in promoting innovation while also being bureaucratic by nature (Bierly III & Daly, 2007) and less willing to adapt (Öztürk & Ozen, 2021) , which serve as hampering factors. Young firms, on the other hand, are more flexible and willing to produce innovations but may still be immature and might lack the necessary knowledge (Sørensen & Stuart, 2000; Bierly III & Daly, 2007) .

So far, we have highlighted that even though there is a voluminous number of studies with quite diverse features, they produce a set of predictions that seem to be robust across alternative setups within the context of innovation (e.g., Savignac, 2008; Mancusi & Vezzulli, 2010; Francis et al., 2012; Lee et al., 2015; Khan et al., 2017; Chundakkadan & Sasidharan, 2020; Santos & Cincera, 2022 ): 1) under asymmetric information and not fully collateralized loans, external funds are more expensive than internal funds, and 2) this cost differential varies inversely with borrower’s net worth and especially traits. The empirical literature can be classified into three main categories depending on the focus of the relationship of innovation and financial constraints as well as access to bank finance. In particular, we focus on 1) a set of variables measuring credit rationing and how innovation affects it (Francis et al., 2012; Mina et al., 2013; Mushtaq et al., 2022) , 2) measurements of innovation activity relative to finance related variables (Adegboye & Iweriebor, 2018; Fombang & Adjasi, 2018; Chundakkadan & Sasidharan, 2020) and 3) models of simultaneous calculation of finance and innovation variables to counter endogeneity issues (Savignac, 2008; Blanchard et al., 2013; Brancati, 2015; Santos & Cincera, 2022) . The most prominent studies across the three categories are presented in Tables 1-5, giving the reader a comprehensive classification.

Regarding the adopted methodology, a number of steps were taken in order to sort out the appropriate literature. First, we set out the basic criteria for our search, which is related with how access to finance affects all fonts of innovation activities of a firm, focusing on SMEs and bank financing. Second, we chose the ResearchGate and Google Scholar engines to find relative papers along with

academic publishing companies specializing in related scientific fields. Our search was conducted in English and all the papers used in our review were published within the last 20 years (2002-2022). The keywords we used for our research were: “innovation and financial constraints”, “innovation and credit rationing” and “innovation and access to finance”. We also run the same keywords followed by “in SMEs” and “banks”, to monitor better relevant studies. Third, we started the basic screening procedure, by reading abstracts of relevant papers and reading in detail those that were close to the topic we were researching. After assessing the complete relevance of these papers, their quality in terms of results and the journals under which they were published, we ended up with the studies presented in Sections 2.2-2.4. In the fourth and final step, we classified those papers into 5 separate categories, based on the dependent variable of their econometric models. We included only the variables that were relevant with our study and excluded other independent and dependent variables for reader’s convenience, as they are not within the concept of our review.

One common factor across categories that should be mentioned is the employment of survey data for innovation and finance variables along with databases related to firm specific characteristics, like patent databases or balance sheet data. We also need to note that when research is not focused on a specific region or a small number of countries researchers mostly choose the World Bank Enterprise Survey for their analysis (Ayyagari et al., 2011; Khan et al., 2017; Chundakkadan & Sasidharan, 2020; Mushtaq et al., 2022) , due to its worldwide data, a plethora of questions across different fields and its open accessibility. However, for individual countries or small cluster of countries analysis, there is a lot of diversity in data sources. Finally, EPO’s Worldwide Patent Statistical Database (PATSTAT) has been used extensively when researchers incorporate patent measurements in their models (Francis et al., 2012; Hall, 2014; Lööf & Nabavi, 2016) and the Community Innovation Survey (CIS) for models on stages of innovation (Canepa & Stoneman, 2003; Galia & Legros, 2004; Mohnen et al., 2008) .

The selected studies cover a time span of at least 20 years and include a wide variety of countries and samples (with the majority ranging in sample above 1000 firms), while the years researched range from 1987-2020, covering 33 years in total. Following this setup allows us to examine whether the correlation of finance and innovation variables follows a common trend across different nations and continents and under distinct economic cycles. The dependent variables used are an overwhelming majority of binary variables, thus, must be modeled by a Probit or Logit model providing evidence on the marginal effects of an independent variable on the dependent one. In some cases, the Ordinary Least Squares estimator is used, when the dependent variable is not binary. In order to test for endogeneity among financing and innovative variables, bivariate probit models are usually employed. As for countering potential endogeneity issues, the use of instruments and two-stage models become essential in the literature.

Continuing with the main analysis, Table 1 shows some representative research of empirical models that use financial measurements as dependent variable. This category of papers investigates how banks comprehend different innovation measurements like R & D, innovation outputs and innovation indexes and if they are related with their decision to undertake a loan offer. Within the literature, there is a variety of measures of finance. Here, we refer to three main approaches: 1) Loan application success, 2) Reported access to finance and 3) Loan spreads.

The loan application success method (Freel, 2007; Mina et al., 2013) tests whether a firm actually gets the demanded loan provision with the desired loan terms like spreads and partial or total amount of money. In particular, it is examined whether firms with an innovative characteristic have a lower or higher likelihood of loan application success relative to non-innovative firms. Freel (2007) concludes that R & D expenditures and novel products have a negative impact on loan application success. Similarly, Mina et al. (2013) found that R & D intensity exerts a negative effect on the probability that firms obtain finance, a result that gradually disappears when more innovation factors are included. They report that separate forms of R & D measurements and innovation outputs can yield varying results on loan application success. Softer forms of innovation seem to be perceived negatively by banks, as well as earlier stages of innovation measurements like if the firm conducts R & D or not, even though the effects of its intensity are vague. On the other hand, core innovation outputs seem to positively affect loan application success.

The second way of measurement relates to perceived access to finance. Contrary to loan application success, where a certain benchmark is set in the form of successfully acquiring a loan or not, perceived access to finance is a more general measurement. It can also include problems that are non-loan related, like issues with line of credit and bank accounts, and even firms that were discouraged from applying for a loan in fear of possible rejection. In this measurement method, we observe contradicting results as Lee et al. (2015) mentioned that innovative firms are more likely to be turned down for finance, while Mushtaq et al. (2022) reported a positive correlation between innovation outputs and their innovation index with access to finance. A such contradiction of results may arise due to the different time frames of the samples, as Lee’s et al. (2015) research was launched during the period of the economic crisis, while the dataset of Mushtaq et al. (2022) facilitates both an increased number of countries and a prolonged time frame.

The third measurement method involves loan spreads. Relative to the other two categories, loan spread is the only measurement that is non-binary. Through this method, it is tested whether firms with an innovative characteristic face higher or lower loan spreads compared to non-innovative firms. Francis et al. (2012) found that R & D productivity and patents seem to help firms alleviate loan spreads. Specifically, patents are more important for SMEs than large firms, as they can signal the quality of their innovation to banks in otherwise impossible ways, contrary to large firms that can counter such problems with extensive audits and the use of collateral.

All in all, we deduce that credit rationing and access to finance is harder for firms bearing innovative characteristics that are premature, like the conduct of R & D and R & D expenditures as well as soft innovations whose results in profitability are vague and intangible. On the other hand, banks possibly presume core innovations positively, though when the economic cycle is not favourable, evidence shows the opposite.

The second category of empirical models explores the impact of access to finance and financial obstacles on different innovation metrics. Compared to the other two categories, this is the one that most researchers have been attracted to; hence for reader’s convenience we present the quite rich output into three subcategories: 1) stage and obstacles of innovation models, 2) R & D models and 3) innovation output models.

Table 2 summarizes the most influential studies of this first subcategory. It was initiated due to one of the most famous surveys regarding innovation, as stated earlier, the Community Innovation Survey (CIS) conducted by the EU. Through this survey, researchers can distinguish among different stages where firms face problems with innovation and which obstacles they encounter. Specifically, in its earlier rounds (CIS 2), the survey distinguished 3 stages: postponed, uninitiated, and abandoned. In later surveys (CIS 3.5), prematurely stopped projects were also incorporated. Obstacles to innovation include innovation costs, lack of financing, lack of skilled personnel, lack of information on markets and technologies, lack of demand, regulatory issues, and organization rigidity. Researchers have exploited these two measures to test how obstacles to innovation (and, in our case, how obstacles to finance) affect a firm differently depending on the innovation stage (Canepa & Stoneman, 2003; Mohnen et al., 2008) . Other researchers chose to investigate how firms that perform R & D perceive those obstacles relative to those that do not (Galia & Legros, 2004) , while part of this literature focuses on characteristics of firms that decided to stop an innovation project (Paunov, 2012; Segarra-Blasco et al., 2018) .

In terms of findings, financial obstacles were found to be more crucial on uninitiated, stopped and postposed innovation projects but not for a project to be abandoned (Canepa & Stoneman, 2003; Mohnen et al., 2008) , suggesting that firms will not give up an innovation idea mainly for financial reasons. Furthermore, firms engaging in internal R & D and those that postponed their innovation projects are more prone to perceive financial costs of innovation as an important impediment (Galia & Legros, 2004) . This result hints towards the importance of revealed innovation barriers which can only be perceived after the firm’s engagement in innovation activity due to the increase in awareness of the hampering factors involving innovation (D’este et al., 2012) . Finally, regarding abandoned projects, firms seem to quit them during the concept stage and not after its initiation (Segarra-Blasco et al., 2018) , probably due to high sunk cost and investment commitment, while firms that have access to public financing are less likely to abandon a project (Paunov, 2012) .

The second subcategory consists of R & D models (see Table 3). Generally, innovation as a process is measured by 3 main ways. First, R & D is one of those measurements and it is a sign of innovation input for a firm with uncertain output, since just the conduct of R & D does not equate to a result. In order to create an innovation output, R & D costs are often required, thus making it a good proxy in innovation models as a representative measurement of a firm in its early stages of innovation. It is mostly researched under 4 forms: 1) Conduct of R & D (whether a firm carries out R & D activities), 2) R & D Expenditures, 3) R & D Intensity (mainly by using R & D expenditures relative to some other balance sheet measurement, like assets, turnover or sales) and 4) R & D productivity (measured as R & D expenditure relative to patents or as the number of employees on R & D relative to patents). R & D productivity, however is used by researchers mostly as an explanatory variable for models, rather than as a dependent one and it is not a very common measurement. As an innovation variable, patents are considered an intermediate innovation output, between R & D and innovation output, which helps alleviate information asymmetries for SMEs, as stated earlier. However, the fact that only 4% of SMEs apply for a patent (Hall et al., 2013) makes it a problematic source of measurement, resulting to researchers using alternative metrics of R & D instead.

In this literature, the conduct of R & D and R & D expenditures measurements have been found to positively affect the difficulty of access to finance and financial constraints (Adegboye & Iweriebor, 2018; Chundakkadan & Sasidharan, 2020) . Moreover, innovative activities increase with a firm’s diversification of financial instruments (Ferrando & Lekpek, 2018) . Firms that use several financing instruments are more likely to invest in R & D and software activities and invest more in the R & D to turnover ratio. Generally, R & D models are the most straightforward in terms of results. The use of R & D, without an innovation output, increases the riskiness of financing firms from the side of banks, exacerbating their financial problems and constraints. However, there are ways to alleviate this problem like using multiple sources of financing, as well as patents, especially for smaller firms, that signal quality and help in increasing R & D intensity (Hall, 2014) .

The third and final subcategory, consists of innovation output models (see Table 4). Innovation outputs are the end results of the innovation process and are associated with the least faced risk from banks, as there is a small amount of information asymmetry among them and the firms. Arguably, it is the most commonly used measure of innovation, as contrary to previous subcategories, it can yield a variety of results at times since an external financing entity can interpret a firm that produces innovation outputs both positively and negatively depending on the form of output (Mina et al., 2013; Adegboye & Iweriebor, 2018) . When researchers refer to innovation output models, they mostly include new products, processes and technology incorporated into the firm recently (Lööf & Nabavi, 2016; Ferrando & Lekpek, 2018; Chundakkadan & Sasidharan, 2020) . Most survey questions set a limit of either one year or three years to consider a product or process new. Aside from this classification, the more meticulous analysis includes product and process newness relative to the firm or the market (Khan et al., 2017) , measuring if the innovation is internal, national or global, and if it was completely new or an upgraded version of a previous product or process of the firm. In some cases, softer forms of innovations like organizational innovations and marketing innovations are included, as well as ICT (Information and Communication Technology) measurements (Clausen, 2008; Adegboye & Iweriebor, 2018; Fombang & Adjasi, 2018) .

Occasionally, innovation output measurements are also conducted in the form of indexes (Ayyagari et al., 2011; D’este et al., 2012; Fombang & Adjasi, 2018) to measure how intensive a firm is in its innovation outputs. These indexes are similar in nature to the baseline measurement, but in the form of sums, meaning that if a firm made both a product and a process innovation, then its index score is higher than a firm that created only a new product. Just like in the measurements above, these indexes can be spotted in the literature under two categories: i) those that account for only core innovations (product, process, technology) and ii) those that incorporate softer forms of innovation as well.

As for research approaches, we observe researchers either testing how access to finance and financial constraints affect innovation outputs (Ayyagari et al., 2011; Lööf & Nabavi, 2016; Fombang & Adjasi, 2018) or how firms, based on their output intensity, report potential financial obstacles they face (D’este et al., 2012; Santos & Cincera, 2022) . The first branch of this literature reported that external finance is crucial in introducing all forms of innovation outputs, especially for new and upgraded product innovations relative to other core innovations (Ayyagari et al., 2011) . The effect of external finance, especially in the form of overdrafts, overwhelmingly drives innovation across all countries (Fombang & Adjasi, 2018) , while cash flow has been reported as important for innovation outputs only for high-technology exporters (Lööf & Nabavi, 2016) . On the other hand, regarding reported financial problems relative to innovation intensity, it is found that the most intense innovators report the highest financial barriers to innovation together with non-innovators (D’este et al., 2012) .

All in all, innovation dependent variable models, as observed, can be examined in many forms. All forms of innovation correlate positively with easier access to finance and less financial constraints, as expected. However, the effect that different innovation formats have on access to finance depends on the measure employed and the innovation stage. Hence, in order for a researcher to perform a complete analysis she/he should explore all the above variables. In particular, the part of this literature that gathers the most interest is innovation output measurements and how banks’ financing decisions are affected after observing different innovation outputs from firms. Thus, researchers might want to elaborate more on this subcategory of innovation measurements relative to others. From the following tables, it can be observed that the majority of recent papers use innovation output variables for their models.

In these two categories, we analysed the existing literature regarding the bidirectional relationship between finance and innovation. This relationship also holds importance for policymaking, since knowledge of how innovation responds to economic environment changes helps identify potential future policies that will promote financial development and economic prosperity. Financial assistance may reduce the high costs of innovation and the accompanied risk that leads to suboptimal investment levels (Arrow, 1972) , especially for SMEs that face higher risk and worse funding requirements.

Table 5 depicts models of simultaneous calculation of finance and innovation variables. The difference, when compared to the previous categories, is that researchers create explanatory models for both finance and innovation. Doing such, they succeed to address the endogeneity issues in this relationship, which derives possibly from two sources: 1) the fact that innovative firms are more conscious about potential financial problems relative to non-innovative ones and 2) the more innovative projects a firm undertakes, the easier it is to face financial constraints, as more cash is committed. To test for endogeneity in such bivariate probit models, one of the two variables must be assumed to have zero explanatory power over the other on its model. Some researchers choose financial variables to have zero explanatory power over their innovation variable (Brancati, 2015; Santos & Cincera, 2022) while others exactly the opposite ( Savignac, 2008; Mancusi & Vezzulli, 2010; Khan et al., 2017) . There are also cases where both variables are used as explanatory variables on two models, however, one of the two dependent variables is loosely related to innovation or finance. For instance, Segarra-Blasco et al. (2018) used “abandon an innovation project” as a loosely related variable to innovation.

Nevertheless, in most studies, the covariates of the errors are found to be different from zero, thus pointing to endogeneity issues within the models and, in general, among innovation and finance variables. The most effective countermeasure was introduced recently by excluding from the sample the non-innovators that didn’t have financial constraints, meaning firms that did not want to engage in innovation (e.g., Savignac, 2008; Brancati, 2015; Khan et al. 2017 , etc.). On the other hand, Mancusi & Vezzulli (2010) preferred to use an IV Tobit model and fitted values of R & D spending to counter the endogeneity issues of their model.

The results of this set of studies are in line with those mentioned in the categories above, as the relationship between financial constraints and innovation, after controlling for endogeneity, is found to be negative (Savignac, 2008; Blanchard et al., 2013; Santos & Cincera, 2022) and is more evident in product innovations than process upgrades (Brancati, 2015) as well as in less novel products than more novel ones (Khan et al., 2017) . However, we decided to create a separate category for these papers since Savignac (2008) solved a prevalent problem within the literature. Up to that point in time, researchers had found a positive correlation between financial constraints and innovation, a counterintuitive result, as typically financially constrained firms have less leeway to begin innovative activities. This problem was often apparent in Community Innovation Survey (CIS), that was used extensively to measure this relationship between 2000 and 2010. It seems that the empirical literature has appreciated Savignac’s (2008) approach since then as it provides the most robust results in a well-defined sample.

This strand of papers also has important implications for policymaking. The classification of firms that Savignac (2008) first implemented helps identify more precise targets for innovation policy. Specifically, firms that seek to innovate, relative to those that do not have an incentive to innovate should be targeted differently for innovation policies. Regarding the first group, governments should focus on uplifting potential obstacles that hamper innovation efforts, while in the second group, governments should find incentives that encourage firms to seek innovation (Blanchard et al., 2013) .