To analyze how financial frictions shape R&D cost behavior, we adopt firm age and size as financing constraint proxies, consistent with the lifecycle-scale paradigm (Hall, 2005; Hadlock and Pierce, 2010; Gorodnichenko and Schnitzer, 2013). This approach leverages two established mechanisms: First, young/small firms face higher borrowing costs and collateral constraints, amplifying cash flow sensitivity. Second, large/mature firms benefit from R&D scale economies, enabling stable innovation investment.

We stratify the sample into terciles by age and size separately, avoiding conflation across dimensions. To mitigate potential confounding effects, we include employee intensity, asset intensity, and leverage as control variables. Using OLS with firm-level clustered standard errors and industry/year fixed effects, we expect $\beta$₂ to be less negative (or positive) for young/small firms, signaling anti-stickiness through aggressive R&D cuts during downturns.

Model A:

$$\mathrm{\Delta }R{D}_{i,t}={\beta }_0+{\beta }_1\mathrm{\Delta }Sale{s}_{i,t}+{\beta }_{2}De{c}_{i,t}*\mathrm{\Delta }Sale{s}_{i,t}+$$ $${\beta }_{3}De{c}_{i,t}*\mathrm{\Delta }Sale{s}_{i,t}*Controls+{\epsilon }_{i,t}$$

Although the baseline model captures the average stickiness pattern, it does not account for firm heterogeneity. Hypotheses 1 and 2 posit that financial constraints and green innovation strategies systematically alter firms’ R&D cost decisions. We follow Kama and Weiss (2013) and Chang et al. (2022) and parameterize $\beta$₁ and $\beta$₂ as linear functions of financial constraints (FC) and other covariates:

(2)$${\beta }_2={\alpha }_0+{\alpha }_{1}F{C}_{i,t}+{\alpha }_{2}A{I}_{i,t}+{\alpha }_{3}E{I}_{i,t}+{\alpha }_{4}Sde{c}_{i,t}$$ $$+{\alpha }_{5}Le{v}_{i,t}$$

(3)$${\beta }_1={\lambda }_0+{\lambda }_{1}F{C}_{i,t}+{\lambda }_{2}A{I}_{i,t}+{\lambda }_{3}E{I}_{i,t}+{\lambda }_{4}Sde{c}_{i,t}$$ $$+{\lambda }_{5}Le{v}_{i,t}$$

Substituting into Eqn. 1, we estimate:

Model B:

$$\begin{array}{c}\hfill \mathrm{\Delta }R{D}_{i,t}={\beta }_0+\{{\alpha }_0+{\alpha }_{1}F{C}_{i,t}+{\alpha }_{2}A{I}_{i,t}+{\alpha }_{3}E{I}_{i,t}+\hfill \\ \hfill {\alpha }_{4}Sde{c}_{i,t}+{\alpha }_{5}Le{v}_{i,t}\}*De{c}_{i,t}*\mathrm{\Delta }Sale{s}_{i,t}+\{{\lambda }_0+\hfill \\ \hfill {\lambda }_{1}F{C}_{i,t}+{\lambda }_{2}A{I}_{i,t}+{\lambda }_{3}E{I}_{i,t}+{\lambda }_{4}Sde{c}_{i,t}+{\lambda }_{5}Le{v}_{i,t}\}*\hfill \\ \hfill \mathrm{\Delta }Sale{s}_{i,t}+{\epsilon }_{i,t}\hfill \end{array}$$

The interaction term FC*Dec*$\mathrm{\Delta }$Sales ($\alpha$₁) provides a direct test of H1. A positive coefficient on $\alpha$₁ would suggest that financially constrained firms exhibit less cost stickiness. There is no unified standard for measuring financial constraints. Since Fazzari et al. (1987) introduced investment–cash flow sensitivity as a proxy, various indicators have been proposed. Following the literature, we adopt two widely used measures where higher values indicate more severe financing frictions. A positive coefficient on $\alpha$₁ would suggest that financially constrained firms exhibit less cost stickiness. First, we adopt the Kaplan and Zingales (1995) index (KZ index):

KZ = –1.002CF + 0.283Q + 3.139Lev + 39.367Div – 1.315Cashholdings

where the CF is the ratio of cash flow to total assets, Q is Tobin’s Q, and cashholdings are cash holdings divided by total assets.

Second, we follow Zhang and Wang (2013), who developed a financial constraint index tailored to Chinese firms by adapting the Hadlock and Pierce (2010) approach. The index is derived from a logistic regression model using:

Pr (FC = 1/0 Z_i⁢t) = 1/1 + e^-Z⁢i⁢t, where Z_i⁢t = $\beta$₀ + $\beta$₁Lev + $\beta$₂MB_i⁢t + $\beta$₃Div + $\beta$₄NWC + $\beta$₅EBIT

where Lev is the leverage ratio, MB is the market-to-book ratio, Div is dividends divided by total assets, NWC is net working capital divided by total assets, and EBIT is earnings before interest and taxes divided by total assets.

Similarly, we incorporate green innovation into the model. The interaction term GI*Dec*$\mathrm{\Delta }$Sales provides a direct test of H2. We further explore whether green innovation (GI) enhances firms’ ability to maintain R&D investment during downturns, acting as a moderator of cost stickiness. We measure GI using the logarithm of the number of green patent applications plus one, following Tan and Zhu (2022). Patent applications reflect contemporaneous innovation behavior more effectively than granted patents due to shorter reporting lags (Brunnermeier and Cohen, 2003). Incorporating GI into Model B, we specify:

We specify Model C:

$$\begin{array}{c}\hfill \mathrm{\Delta }R{D}_{i,t}={\beta }_0+\{{\alpha }_0+{\alpha }_{1}G{I}_{i,t}+{\alpha }_{2}A{I}_{i,t}+{\alpha }_{3}E{I}_{i,t}+\hfill \\ \hfill {\alpha }_{4}Sde{c}_{i,t}+{\alpha }_{5}Le{v}_{i,t}\}*De{c}_{i,t}*\mathrm{\Delta }Sale{s}_{i,t}+\{{\lambda }_0+\hfill \\ \hfill {\lambda }_{1}G{I}_{i,t}+{\lambda }_{2}A{I}_{i,t}+{\lambda }_{3}E{I}_{i,t}+{\lambda }_{4}Sde{c}_{i,t}+{\lambda }_{5}Le{v}_{i,t}\}*\hfill \\ \hfill \mathrm{\Delta }Sale{s}_{i,t}+{\epsilon }_{i,t}\hfill \end{array}$$

We expect $\alpha$₁ 0, indicating that firms with stronger green innovation strategies exhibit greater R&D cost stickiness, potentially due to reputational concerns, stakeholder pressure, or preferential financing.

Finally, we examine whether green innovation moderates the adverse effect of financial constraints on R&D cost stickiness. We interact GI and FC variables in the following model: In Model D, we expect $\alpha$₁ (FC*Dec*$\mathrm{\Delta }$Sales) to be positive, suggesting that financial constraints lead firms to make anti-sticky cost decisions. A negative coefficient on the interaction term $\gamma$₀ (GI*FC*Dec*$\mathrm{\Delta }$Sales) in model D would imply that green innovation strategies help alleviate financing constraints, thereby enabling firms to maintain R&D cost stickiness.

Model D:

$$\begin{array}{c}\hfill \mathrm{\Delta }R{D}_{i,t}={\beta }_0+\{{\alpha }_0+{\alpha }_{1}F{C}_{i,t}+{\gamma }_{0}F{C}_{i,t}*G{I}_{i,t}+\hfill \\ \hfill {\alpha }_{2}A{I}_{i,t}+{\alpha }_{3}E{I}_{i,t}+{\alpha }_{4}Sde{c}_{i,t}+{\alpha }_{5}Le{v}_{i,t}\}*De{c}_{i,t}*\hfill \\ \hfill \mathrm{\Delta }Sale{s}_{i,t}+\{{\lambda }_0+{\lambda }_{1}F{C}_{i,t}+{\gamma }_{1}F{C}_{i,t}*G{I}_{i,t}+{\lambda }_{2}A{I}_{i,t}\hfill \\ \hfill +{\lambda }_{3}E{I}_{i,t}+{\lambda }_{4}Sde{c}_{i,t}+{\lambda }_{5}Le{v}_{i,t}\}*\mathrm{\Delta }Sale{s}_{i,t}+{\epsilon }_{i,t}\hfill \end{array}$$

Variable Definitions:

$\mathrm{\Delta }$Sales_i,t = logarithmic change in sales revenue from year t-1 to year t;

$\mathrm{\Delta }$RD_i,t = logarithmic change in R&D costs from year t-1 to year t;

Dec_i,t = a dummy variable that equals to 1 when sales revenue decreases from year t-1 to t, and 0 otherwise;

EI_i,t = the number of employees*100,000/sales revenue;

AI_i,t = logarithm of the ratio of total assets to sales revenue;

Sdec_i,t = a dummy variable that equals to 1 when sales have decreased in two consecutive years and zero otherwise;

Lev_i,t = the ratio of total debts to total assets;

GI_i,t = logarithm of (green patent applications + 1);

FC_i,t = financial constraint index measured by the KZ and Zhan-Wang indices;

Age_i,t = the number of years since establishment;

Size_i,t = logarithm of total assets.

To test the robustness of the regression results, we sequentially control for different fixed effects. In Column (1) of Table 8, we include firm and year fixed effects, while excluding industry fixed effects. The results show that the coefficient of $\alpha$₁ (Dec*$\mathrm{\Delta }$Sales*FC) remains positive, suggesting that under financial constraints, firms tend to reduce their R&D expenditures when facing a decline in sales revenue, indicating anti-cost stickiness. Further, in Column (2), where we control simultaneously for firm, industry, and year fixed effects, the sign and significance of $\alpha$₁ remain consistent. This provides additional support for our core hypothesis: the more severe the financial constraint, the more likely firms are to cut R&D spending during economic downturns.

To relax the assumption of a linear relationship between financial constraints and cost stickiness, we group the financial constraint indicator into quartiles (Q1–Q4) and assign ordinal values from 0 to 3, then include them in the main regression model. As shown in Column (3) of Table 8, the coefficient of $\alpha$₀ (Dec*$\mathrm{\Delta }$Sales) is negative, indicating that R&D expenses exhibit a certain degree of stickiness across the full sample. Meanwhile, although the coefficient of $\alpha$₁ (Dec*$\mathrm{\Delta }$Sales*FC) is not statistically significant, its positive sign suggests a weakening of cost stickiness—and potentially a shift toward anti-stickiness—as financial constraints intensify. This result further supports the hypothesis that financial constraints are negatively associated with cost stickiness.

Given the potential bidirectional causality between financial constraints and R&D cost stickiness (e.g., reductions in R&D investment may also deteriorate financing conditions), we adopt two methods to mitigate endogeneity bias.

First, we use the lagged value of the financial constraint indicator as a substitute for the contemporaneous value in the regression. As shown in column (3) of Table 8, using the lagged KZ index as an example, the coefficient of $\alpha$₁ is 0.057 and statistically significant at the 1% level, indicating that firms facing stronger financial constraints are more likely to cut R&D expenditures in response to revenue declines, demonstrating anti-stickiness behavior.

Second, we further use the one-period lagged financial constraint variable as an instrument and conduct a two-stage least squares (2SLS) regression. The first-stage regression results indicate that the instrument is strong, with an adjusted R-squared of 0.7969 and a first-stage F-statistic of 46.88 (p 0.01), exceeding the conventional threshold of 10. The partial R-squared is 0.1518. The endogeneity test yields an F-statistic of 5.49 (p = 0.0192), rejecting the null hypothesis of exogeneity and supporting the use of instrumental variable estimation. The second-stage results, reported in Column (4) of Table 8, show a positive and significant coefficient on $\alpha$₁ (coefficient = 0.216, t = 3.030), indicating that greater financial constraints significantly strengthen the anti-stickiness effect. This further supports the causal interpretation of the relationship between financial constraints and cost stickiness.

A growing number of papers argue that intangible investment is created not only by R&D expenses but also by SG&A expenses (Enache and Srivastava, 2018). We extend the discussion of R&D costs to SG&A costs in an additional analysis. We find that $\alpha$₁ in Table 8 is significant and positive, which means that as financing constraints increase, firms’ SG&A costs no longer show stickiness but anti-stickiness. That is, the negative relationship between financing constraints and cost stickiness remains valid for SG&A cost behavior.