Chesbrough first introduced the concept of open innovation and contended that open innovation is a new paradigm of innovation management in which firms maximize the use of resources inside and outside the organization to innovate and earn profit. Guo et al. (2019) pointed out that in a business environment, innovation resources and experience will be gained through accelerated openness once a firm is under the pressure of a competitor. Regarded as interaction, integration and synergy between elements of innovation, open innovation realizes the free flow of crucial resources to cross the organizational boundary (Ritala et al., 2017). Although scholars have reached a general consensus on the importance of open innovation while a unified standard for related concepts is still missing. Alam et al. (2022) considered that the mechanism of the role of open innovation cannot be fully explained from the single view of firms that a study in which the management theory and the resource dependency theory should be conjoined. Based on the above research, this paper analyzes the concept of open innovation from diverse perspectives by summarizing the existing literature and concludes with the following aspects: (1) From the perspective of innovation resource, resource theory regards open innovation as an asset. Vanhaverbeke et al. (2008) deem that the acquisition of productive resources such as knowledge, customers or suppliers, and infrastructure services is the main purpose of open innovation; (2) From the perspective of knowledge, Lichtenthaler (2009) mentioned that open innovation is a process of knowledge search and enrichment, in which internal and external knowledge development, integration and utilization are systematically carried out through innovation activities; (3) From the perspective of resource dependence theory, the inability of firms to fully integrate the entire resources needed for innovation by internal integration leads to the inevitable demands of production resources provided by other firms (Chesbrough, 2003).

In addition, the influencing factors of firms’ open innovation are also highlighted in this paper. (1) Firm’ capability. Keupp and Gassmann (2009) thought that a series of innovation obstacles, such as rising R&D costs within firms and the shortening trend of the product life cycle, make certain firms more inclined to carry out open innovation than others; (2) Organizational Environment. In the process of innovation, Chesbrough and Crowther (2006) considered that external resources and knowledge contribute to innovation performance even in firms with strong R&D and innovation capabilities; (3) Senior management characteristics. According to Salter et al. (2014), the thinking ability and innovative ideas of R&D individuals have a nonlinear relationship with the openness of external knowledge sources. Meanwhile, the knowledge-based theory shows that knowledge, as the core element of innovation, is the basis of open innovation (Hilbert and López, 2011). Ye (2022) deemed that the improvement of innovation capabilities depends not only on the firm’s R&D investment but also on the diffusion or spillovers of external knowledge and R&D capital. Knowledge spillovers with no compensation or compensation less than the actual value of knowledge can acquire valuable core technologies from other firms (Cassiman and Veugelers, 2002). Through knowledge spillovers, the knowledge base and resources of other firms are used to increase firms’ own R&D investment and to achieve more innovative output with less R&D cost (Dai et al., 2022). As the unique modality of knowledge acquisition makes knowledge spillover an essential factor affecting open innovation, this paper analyzes how to optimize the use of knowledge spillover to promote firms’ open innovation from the perspective of knowledge spillover.

Since the concept of knowledge spillover is introduced in the analysis of economic problems by Arrow in 1962, knowledge spillover has become an important research direction in economics and management. However, concepts related to knowledge spillover such as knowledge diffusion, knowledge transfer and knowledge flow are confused by scholars in various fields, in fact, these concepts are quite different. First, knowledge spillover and knowledge transfer belong to the same category of knowledge flow. Fallah and Ibrahim (2004) put forward that knowledge communication may occur in every interaction between knowledge subjects. “Knowledge transfer” is considered to be the knowledge exchange that occurs consciously among people or organizations while any unconscious knowledge transmission beyond such communication belongs to “knowledge spillover.” Whereas, contradictory to previous studies that regard knowledge spillover as an unconscious knowledge dissemination process, Alexy et al. (2013) found that to augment the possibility of acquiring valuable knowledge in the future, firms can consciously select some internally developed knowledge to provide free use to external participants through carriers such as technical drawings, meeting minutes and contracts. pointing out that defining knowledge spillover in terms of “conscious and unconscious” is not accurate. Furthermore, Keller (2010) emphasized that as a method of knowledge diffusion, knowledge spillover mainly refers to the part of knowledge that diffused through externalities. The externality feature of knowledge spillover suggests that knowledge recipients can develop and create new knowledge by combining the acquired knowledge with their knowledge without compensating the knowledge creator or below the compensation of knowledge creation cost (Zhu and Xu, 2019). In a word, this paper states that knowledge spillover refers to knowledge being acquired by subjects other than the knowledge creator in the form of no compensation or compensation less than the value of knowledge created.

Moreover, Angeles Montoro-Sánchez et al. (2011) contended that distinct types of knowledge spillovers lead to differences in the quantity and quality of external resources acquired by firms, which may eventually affect the effect of firms’ innovation. According to the source of knowledge spillover, knowledge spillover can be classified as domestic knowledge spillover and foreign knowledge spillover (Chen et al., 2012). According to the direction of spillover, it is categorized as outward and inward knowledge spillover by Cassiman and Veugelers (2002). Gaur et al. (2019) asserted that knowledge flow depends on the attributes and context of knowledge being transferred. The characteristics of tacit and explicit are the most common factors to affect knowledge flows (Dhanaraj et al., 2004). Jensen (2007) classified knowledge into explicit and tacit knowledge based on the degree of modifiability. As formalized knowledge, explicit knowledge is easy to encode, retrieve and transfer (Hansen, 2002). Inversely, as an informal form of knowledge-based primarily on personal experience and skills, tacit knowledge is difficult to compile and communicate (Bibi and Ali, 2017; Bogers et al., 2018). The huge difference between explicit knowledge and tacit knowledge leads to distinct spillover effects. Consequently, for the purpose of launching an advanced exploration related to the distinct effects of various types of knowledge spillovers on firms’ open innovation, this paper divides knowledge spillover into two dimensions: tacit knowledge spillover and explicit knowledge spillover. In specific, explicit knowledge spillover refers to new technologies or products carried by language, text or graphics that are acquired by other subjects other than knowledge creators for free or at a small cost. Tacit knowledge spillover refers to all kinds of information, experience and skills based on individuals acquired by other subjects other than knowledge creators for free or at a small cost.

Knowledge spillover is considered an inevitable knowledge transfer phenomenon in open innovation (Cassiman and Veugelers, 2002). Specifically, Knowledge spillover not only provides the raw materials needed for open innovation but also strengthens the communication between the firm and external knowledge, as well as the increment of value extension on spilled knowledge (Yang et al., 2010). However, open innovation is affected differently due to distinct attributes of knowledge. Seidler de Alwis and Hartmann (2008) believed that compared with explicit knowledge, tacit knowledge capable of adapting to the rapidly updating trend of innovative technologies in the knowledge economy plays a more significant role in knowledge spillovers and innovation. Based on the statement above, this paper explores the role played by tacit knowledge spillover and explicit knowledge spillover in open innovation.

Scholars have not yet reached a consensus on the definition of trust, due to the characteristics of abstraction and complexity (Rutten et al., 2016). Based on psychology (Lau et al., 2007), individuals generate trust in their psychological consciousness, which is reflected in their positive expectations of other others’ behavior. Jones et al. (2010) suggested that goodwill and competence are two essential components of trust, in which goodwill is typically present in interpersonal relationships and competence is an evaluation of other firms’ skills. The existing literature generally regards trust as a subjective belief that plays a crucial role in facilitating and stabilizing relationships between various subjects. Accordingly, this paper defines psychological trust as a state of psychology, that is, a subject’s optimistic prediction of others’ behavior as well as a subject’s view of the fairness and security of the organizational environment. According to Bottazzi et al. (2016), it is challenging for firms to constrain opportunistic behaviors during open innovation through formal means. Firms engaged in open innovation must simultaneously rely on other means to ease the conflict between various types of knowledge. Thus, during the process of cooperation, psychological trust helps to union different subjects together, to remove informational obstacles and to decrease opportunism. In addition, psychological trust contributes to the establishment of open knowledge environment where a variety of required resources for knowledge subjects are provided (Von Krogh et al., 2012). Inter-firm cooperation based on trust is beneficial to the establishment of an organizational environment for mutual understanding and communication. Such cooperation can motivate organizational members to share their explicit and tacit knowledge (Nonaka et al., 2016; Millar et al., 2017). In this paper, we analyzed the changes in the relationship between knowledge spillover and open innovation in firms under different psychological trust levels.

Duan et al. (2022) thought that manufacturing is an important area of comprehensive national power competition and technological competition among countries in the world. According to the current industrial classification of the Chinese national economy, the high-tech manufacturing industry mainly includes firms in pharmacy, aerospace, transportation equipment manufacturing, computer and other electronic equipment manufacturing. Meanwhile, high-tech manufacturing firms are knowledge-intensive, and their production and operation status, R&D activities, new product development and sales, and patents all account for important ratios (Duan et al., 2021). Based on the above analysis, we collected the financial statements of Chinese A-share listed manufacturing firms in the CSMAR database, the sample firms were screened as follows: (1) Firms with “ST” and “*ST” marks are excluded; (2) Firms in financial industries with apparent differences in accounting standards from other industries are excluded; (3) Firms with missing information are also excluded. Since the primary data for this paper were obtained from the CSMAR database and the 2015 China General Social Survey (CGSS), where the CSMAR database is only updated to 2020 and the CGSS data were collected in 2015, we aimed to measure with the most recent sample data based on the availability and completeness of the data. As a result, the data of 180 firms from 2014 to 2020 were selected as the research sample.

We analyze the relationships between firms’ open innovation, psychological trust, and knowledge spillover in high-tech manufacturing firms. To test the proposed hypotheses H1-H4, we establish the following regression models:

To test hypotheses H3,H4, based on the model (1), (2), we add the interaction terms of psychological trust, tacit knowledge spillover, and explicit knowledge spillover square to establish model (3), (4), respectively, as follows:

We performed a descriptive statistical analysis of all variables and Table 2 presents the means, standard deviations, minimum and maximum values of the variables. The statistical results show that there is a large gap between the minimum and maximum values of firms’ open innovation. In terms of the two dimensions of knowledge spillover, the mean value of explicit knowledge spillover in firms is higher than that of tacit knowledge spillover, indicating that most firms mainly use explicit knowledge spillover to promote innovation.

The matrix of correlation coefficients between the variables is displayed in Table 3. The table demonstrates a significant positive correlation between firms’ open innovation and tacit knowledge spillover (TKS) and explicit knowledge spillover (EKS). Regression analysis must, however, be used to examine the relationship between the independent and dependent variables in further detail. The analysis of the data reveals a significant positive link between psychological trust (TR) and open innovation practices across businesses. The correlation coefficients between the variables were also typically lower than 0.5. However, certain correlation coefficients were higher than 0.5, thus a variance inflation factor test (VIF test) was carried out in this paper to avoid the problem of multicollinearity. Table 4 shows the VIF test results of the variables in which the maximum value of the VIF test result is 2.050 < 5 and the mean value of the VIF test result is 1.59 < 5, so there is no serious co-linearity problem between the variables.

Drawing on Haans et al. (2016), this paper proposes three criteria for testing the validity of the inverted U-shaped relationship: first, the primary term coefficient is positive and the quadratic term coefficient is significantly negative; second, the slope at both ends of the definition domain should be significantly positive or negative; and third, the confidence interval at the 95% level of the turning point needs to fall within the definition domain. The test results meet the above requirements, and the details are shown in Figures 2, 3 and Table 5.

Table 5 shows that Model 1 is a regression of the relationship between the primary term of explicit knowledge spillover, the quadratic term of explicit knowledge spillover, and open innovation of firms. The results show that the coefficient of the primary term of explicit knowledge spillover is: 0.005(P < 0.01),and the coefficient of the quadratic term is -0.000 (P < 0.1). When the minimum value of explicit knowledge spillover (0), the slope of the curveβ₁ + 2β₁ = EKS_max 0.005 > 0; when the maximum value of the explicit knowledge spillover (1633), the slope of the curve β₁ + 2αβ₁ = EKS_min -0.004 < 0, and the slope of the sample boundary has the opposite sign. And, the inflection point of the curve is 1254.326, which is within the sample range. Open innovation shows an upward and then downward trend with the rise in explicit knowledge spillover of the firm. The inverted U-shaped association between explicit knowledge spillover and open innovation is supported by the data, supporting hypothesis H1. Model 2 is the regression of the relationship between the primary term of tacit knowledge spillover, the quadratic term of tacit knowledge spillover and open innovation. The results show that the coefficient of the primary term of tacit knowledge spillover is: 3.648 (p < 0.05) and the coefficient of the quadratic term is -0.094 (p < 0.05). When the tacit knowledge spillover takes the minimum value (11.36924), the slope of the curve α₁ + 2α₁ = TKS_min 1.511 > 0;when the tacit knowledge spillover takes the maximum value (22.71756), the slope of the curve α₁ + 2α₁ = TKS_max -0.623 < 0, and the slope of the sample boundary has the opposite sign. The curve’s inflection point, which falls inside the sample range, is 15.87979. There is an inverted U-shaped correlation between tacit knowledge spillover and open innovation, which supports the hypothesis H2 that open innovation shows a tendency of growing and then declining as tacit knowledge spillover of firms grows.

According to Haans et al. (2016), the moderating variable can affect the inverted U-shaped relationship in two different ways: first, it can move the turning point to the left or to the right, and second, it can flatten or steepen the curve. The inflection points of the curves are as follows after including the interaction term of psychological trust in models (3) and (4) along with explicit knowledge spillover, tacit knowledge spillover, and their squared terms in the equation.

Apparently, when psychological trust shifts, so does the positioning of the inflection point. Calculate the model (5)’s number of partial guidance:

We set the first derivative with respect to X to zero. Model (6) shifts the curve’s inflection point when it is not equal to 0, 2(λ₂ + λ₄ = TR)² and 2(γ₂ + γ₄ = TR)² are always greater than 0. Therefore, when λ₁λ₄−λ₂λ₃ 0 and γ₁γ₄−γ₂γ₃ 0, the inflection point shifts to the left; λ₁λ₄−λ₂λ₃ 0 and γ₁γ₄−γ₂γ₃ 0 the inflection point shifts to the right. Hypotheses H3 and H4 demonstrate that psychological trust plays a moderating role in the inverted U-shaped relationship between explicit knowledge spillover and tacit knowledge spillover and open innovation. The higher the level of psychological trust, the stronger the positive impact of explicit and tacit knowledge spillover on open innovation. By performing the same test for model (3) and calculating λ₁λ₄−λ₂λ₃ 0 and γ₁γ₄−γ₂γ₃ 0 therefore the whole curve shifts to the left. The coefficients of the psychological trust moderating term are: -87.202 (p < 0.1), -248.227 (p < 0.01) respectively, which proves the existence of a moderating effect. Columns (3) and (4) in Table 5 show the model regression results after adding the interaction terms of explicit knowledge spillover, tacit knowledge spillover and its quadratic term with psychological trust, respectively. The interaction coefficient between psychological trust and EKS is 3.059 (p < 0.01), while the coefficient of the quadratic term is -0.009 (p < 0.01); the coefficient of the interaction term between psychological trust and TKS is 4043.868 (p < 0.01) and the quadratic term coefficient was -105.394 (P < 0.01). Therefore, the inverted U-shaped relationship between explicit knowledge spillover, tacit knowledge spillover, and open innovation remains valid after including the interaction terms. We plotted the moderating effects of psychological trust based on the results in Table 5 (Figures 2, 3).

Before taking into consideration the ease of psychological trust, Figure 2 illustrates changes in the relationship curve between explicit knowledge spillover and open innovation: (1) Before the inflection point, the curve for psychological trust with a lower level is relatively flat, and the curve for psychological trust with a higher level is steep, suggesting that firms with high psychological trust improve their open innovation performance more quickly as explicit knowledge spillover increases. (2) The inflection point’s location shifts. The apex of the high psychological trust curve is higher than the inflection point of the low psychological trust curve, demonstrating that explicit information spillover is more beneficial to improving open innovation performance at higher levels of psychological trust. (3) After the inflection point, the open innovation curve of the firms has begun to decline. Although the flatness of the various psychological trust curves is comparable, the high psychological trust curve is always higher than the low psychological trust curve, showing that open innovation performance decreases with increasing explicit knowledge spillover regardless of psychological trust; even with the same explicit knowledge spillover, the high psychological trust firms still outperform low trust firms in terms of open innovation performance. The relationship between explicit knowledge spillover and open innovation cannot be considered to be positively regulated by psychological trust because this regulatory effect is dependent on the amount of explicit knowledge spillover. In the link between the two, psychological trust has a negative impact at the level of low explicit knowledge spillover. When the level of explicit knowledge spillover is high, psychological trust positively moderates the relationship between the two.

Calculating coefficient λ₁λ₄−λ₂λ₃ 0, results in the turning point of the curve moving to the left, as shown in Figure 3. The curve shows that regardless of psychological trust, open innovation performance declines when tacit knowledge increases; even if the same tacit knowledge overflows, high psychological trust firms do better in open innovation than low psychological trust ones. H4 is therefore confirmed.

To test the reliability of the regression model and empirical results, this paper uses two ways to test the robustness of the regression results: (1) changing the regression model, i.e., using Poisson regression; (2) lagging the explanatory variable firms’ open innovation by one period, and the robustness results are shown in Tables 6, 7: Model 1 and Model 2 are regressions of the relationship between explicit knowledge spillover, tacit knowledge spillover quadratic term and open innovation, respectively. The results show that the primary coefficients of the explicit knowledge spillover quadratic term are both -0.000 and significant at the 1% and 10% levels, respectively. Also, primary coefficients of the tacit knowledge spillover quadratic term are -8.793 and -0.101 and significant at the 5% and 10% levels, respectively, and hypothesis H1 and hypothesis H2 are verified. Model 3 adds the interaction term of explicit knowledge spillover and psychological trust on the basis of model 1, and the results show that the coefficients of the interaction term of the secondary term of explicit knowledge spillover and psychological trust are -0.008 and -0.006, and both of them are significant at the 1% level, saying that the inverted U-shaped relationship between explicit knowledge spillover and firms’ open innovation is strengthened with the increase of psychological trust; model 4 adds Model 4 adds the interaction term of tacit knowledge spillover and psychological trust on the basis of model 2, and the results show that the coefficients of the interaction term of tacit knowledge spillover and psychological trust are -15.230 and -99.530, which are significant at the 5% and 1% levels, respectively, indicating that the inverted U-shaped relationship between tacit knowledge spillover and firms’ open innovation is strengthened with the increase of psychological trust. The above findings are generally consistent with the regression results and the signs of each variable are the same as the empirical results in the previous paper, proving that the conclusions are robust.

Collected from Chinese listed manufacturing firms in Shanghai and Shenzhen A-shares from 2014 to 2020 as research samples, this paper systematically expounds on the impact of knowledge spillovers on firms’ open innovation. Dividing knowledge spillover into explicit knowledge spillover and tacit knowledge spillover, psychological trust, an important variable in the field of psychology, is introduced as a moderator variable to analyze the influence mechanism of these two different types of knowledge spillovers on firms’ open innovation. The following findings were obtained:

First, this paper found the “double-edged sword” effect of knowledge spillovers on firms’ open innovation. On the one hand, knowledge spillovers can significantly improve the innovation process by lowering R&D costs and market risks; on the other hand, knowledge spillover could hinder both the recipient firm’s ability to innovate and the spillover firm’s passion for innovation. This paper proposed and verified the hypothesis that an inverted U-shaped relationship exists between explicit knowledge spillovers, tacit knowledge spillovers and open innovation. The empirical results demonstrated that the promotion effect of knowledge spillover on firms’ open innovation only functions at a limited level that excessive knowledge spillover damages the promotion effect of open innovation.

Second, although we conclude that knowledge spillover and firms’ open innovation are nonlinear related, different types of knowledge spillover have distinct mechanisms to influence open innovation. In contrast, tacit knowledge spillover takes individuals with knowledge as carriers to acquire valuable experience and technology, which has a subtle impact on open innovation. Meanwhile, we propose that knowledge spillover producers are not always in a disadvantageous position in the spillover process, and proper knowledge exchange is also beneficial to knowledge creators.

Third, the findings indicated that psychological trust is a crucial moderator in the relationship between information spillover and firms’ open innovation. Employees will face resource loss owing to excessive emotional resource consumption when psychological trust is low (Hobfoll, 2001; Halbesleben et al., 2014). On the contrary, the increment of psychological trust can not only promote the exchange of experience and knowledge among organizational members to a large extent but also improve the willingness of organizational members to share knowledge (Yuan et al., 2016). By strengthening and fostering trust amongst knowledge subjects, firms may encourage more collaborative communication and enhance the effectiveness of open innovation.

First, regarding the influence mechanism of knowledge spillovers, this paper breaks through the discussion of a single level that knowledge spillovers only positively or negatively affect firms’ open innovation. While the inverted U-shaped relationship between explicit knowledge spillover, tacit knowledge spillover and firms’ open innovation is verified, the balance of positive and negative effects of knowledge spillover in different contexts is also explored in this paper. As a further refinement of the relationship between knowledge spillovers and innovation, these findings not only help researchers understand the relationship between knowledge spillovers and firms’ open innovation more comprehensively, but also improve knowledge management theory and innovation theory. Moreover, emphasis on the dynamic role of the knowledge spillover producers in the process of knowledge spillover shows that the knowledge spillover producers, as the initiator of knowledge spillover and the bearer of knowledge spillover, has a decisive role in the scale and channel selection of knowledge spillover.

Second, by integrating knowledge spillover, firms’ open innovation and psychological trust into the same framework for research, this paper provided a new understanding vision for the development of open innovation in specific contexts. From the perspective of psychological trust, which effectively combines management science and psychology, we examined psychological changes in knowledge subjects between knowledge spillover and firms’ open innovation. Encouraging the integration of knowledge across disciplines, the above perspective further verified the logical analysis that psychological trust can increase organizational competitiveness more thoroughly while a reasonable explanation of management behavior is provided. What’s more, a rational explanation of management behavior from the perspective of psychology enriches and expands the related research on psychological trust.

First, as the main subject of innovation activities, firms need to raise their awareness of open innovation and strive to break their own development boundaries. Firms actively cooperate with universities, research institutions and even competitors to obtain the complementary resources they need. Also, a comprehensive and integrated understanding of the relationship between knowledge spillovers and open innovation is necessary for market innovation activities. At the micro level, spillovers are not entirely harmful to firms on the knowledge spillover producers. Turning passive into active, the observation of innovation activities or other knowledge reorganization behaviors that happens in recipient firms can help the knowledge spillover producers to accumulate relevant innovation experience as well as reduce the uncertainty and risk in the subsequent innovation process. Accordingly, recipient firms must be cautious in their resource selection for digestion, and absorption. The reuse of the new external knowledge should be based on their development situation to optimize the benefits of the firms’ knowledge spillover.

Second, firms should pay more attention to the trust relationship between organizations, including the mutual trust between individuals. Furthermore, in addition to paying attention to the trust construction of the entire organization in all aspects of human resource management, the micro-level such as the individual psychology of employees needs more attention. In particular, knowledge-based employees are strongly motivated to achieve their goals with a strong sense of self-worth. To address the needs of knowledge-based talent, firms need to create a competitive compensation and welfare system by focusing on the growth and development of knowledge-based employees.

Third, at the national macro level, market regulators must establish a favorable institutional environment for firms’ innovation. The state must provide knowledge guarantee and institutional support for the transition from Made in China to Create in China, relying on policies and systems such as intellectual property protection to effectively maintain the profits of innovative firms, to limit the “speculative” behavior in innovation activities, and to motivate firms’ innovation activities as well as the whole society.

First, while this research mainly analyzes the impact of employees’ psychological trust from a static perspective, recent studies indicate that psychological trust shows some fluctuation in the temporal dimension (Qin et al., 2018; Ju et al., 2019). Therefore, future research might investigate the effects of psychological trust in a dynamic temporal context. Second, focused on the positives of psychological trust, potential negatives are neglected. It is possible that psychological trust will lead to emotional tiredness or even destruction. To develop a more comprehensive and dynamic understanding of the relationship between knowledge and innovation, it is meaningful to investigate the detrimental impacts of psychological trust in terms of its function between the two in the future. Third, based on the availability of data, the data sources in this paper cannot be updated to the most recent year, and other measurement methods such as questionnaires or replacement of other databases can be used in the future to make the findings of this paper more comprehensive.