The time-varying parameter vector autoregressive model (TVP-SV-VAR) used in this paper evolved from the structural vector autoregressive model (SVAR). The main difference with the SVAR model is that in the modeling assumptions, there is no assumption of homoskedasticity in the TVP-SV-VAR model, which is more in line with reality, and it assumes that the coefficient matrices and the covariance matrices change in real time, which is better able to capture the relationships and characteristics of the variables in the context of different market sentiments, avoiding undue smoothing, and is conducive to the delineation of the influence relationships between the variables and the This is conducive to characterizing the influence of variables and their nonlinear features. These features also make the model suitable for measuring the time-varying relationship between climate policy uncertainty and commodity prices.

Since the TVP-SV-VAR model is based on the SVAR model, the structural VAR model is constructed first:

where s is the number of lags; y t is an observable variable of order k × 1 . A , F 1 , ⋯ , F s are parameter matrices of k × k ; u t is used to measure structural shocks and u t ~ N ( 0 , Σ t ) .

Assuming that structural shocks obey recursive identification, matrix A is assumed to be a lower triangular matrix of order k × k , which can be obtained by Cholesky decomposition and serves to better identify the economic structure of the model parameters in the form of equation (2):

Transform equation (2) by shifting the terms and multiplying both sides of the equation by A − 1 to obtain equation (3):

where B i = A − 1 F i , i = 1 , ⋯ , s ; ε t ~ N ( 0 , I k ) ; Σ is the standard error matrix and is a diagonal matrix, which is structured as in equation (4):

In equation (4), σ is the standard deviation of the structural shock; the model transforms to equation (5) by stacking the row vector elements of the lag coefficient matrix B i to form a column vector of order k 2 s × 1 , β , and setting X t = I k ⊗ ( y ′ t − 1 , ⋯ , y ′ t − k ) , ⊗ to be the Kronecker product:

where s is the model lag order and tis the time identity. All parameters in equation (5) are time-varying, and if we assume that β t , and Σ t in the model are time-varying variables, we can get TVP-SV-VAR model as equation (6):

In equation (6), β t is the vector of time-varying coefficients, A t is the time-varying parameter matrix, and Σ t is the covariance matrix of the stochastic perturbation term ε t . a t = ( a 21 , a 31 , a 41 , ⋯ , a k , k − 1 ) ′ denotes the stacked vector of lower triangular elements of A t , where σ j t 2 = exp ( h j t ) , the stochastic volatility matrix h t = ( h 1 t , h 2 t , ⋯ , h k t ) ′ , j = 1 , 2 , ⋯ , k ; t = s + 1 , s + 2 , ⋯ , n , assumes that the parameters in the model obey mutually independent stochastic wandering processes:

where t = s + 1 , s + 2 , ⋯ , n , β s + 1 ~ N ( μ β 0 , Σ β 0 ) , a s + 1 ~ N ( μ a 0 , Σ a 0 ) and h s + 1 ~ N ( μ h 0 , Σ h 0 ) . The model has the following assumptions, assuming that the parameters all obey first-order random walks and that the random shocks to the time-varying parameters are uncorrelated, and assuming that Σ β , Σ a , and Σ h are diagonal matrices in order to simplify the estimation of the model. The traditional estimation technique based on the likelihood function is no longer applicable due to the stochastic volatility of the parameters in the model.

This study uses a Markov chain Monte Carlo (MCMC) method based on Bayesian inference to estimate the parameters. The MCMC simulation process is first constructed with samples originating from a multidimensional posterior distribution of the parameters. The sampling method then involves joint sampling using the residual parameter β = { β t } t = s + 1 n , a = { a t } t = s + 1 n , h = { h t } t = s + 1 n , followed by sampling of β and a using an analog filter. Finally, a state-space simulation is constructed that samples h using multiple shifts. An important advantage of this approach is that it effectively reduces the number of parameters that need to be estimated by modeling the parameters as random walk processes rather than autoregressive processes. This change helps to simplify the model and potentially improve the estimated.

In this study, a distributed lag nonlinear model (DLNM) is introduced to analyze the nonlinear and lagged effects of climate policy uncertainty on the markets for bulk agricultural products and energy. The DLNM, which was initially widely used to study the effects of air pollution and temperature changes on health outcomes, is also applicable to characterize the nonlinear and lagged effects among the variables. The main advantages of the DLNM are as follows: firstly, lagged effects, which, in many real-life situations, there may be a certain lag in the effect of exposure factors on outcome events, and DLNM can estimate the effects of exposure factors at different lagged time points, thus revealing the existence of lagged effects and their durations. The second is the nonlinear relationship; the relationship between exposure factors and outcome events may be nonlinear, and DLNM allows the introduction of nonlinear terms in the model to capture the nonlinear relationship between exposure factors and outcome events. Finally, the DLNM employs natural cubic spline smoothing of the exposure factors to capture the effects of the exposure factors over the entire time horizon. This approach eliminates potential confounders and makes the model results more plausible. In this study, the predictor variable is defined as the intensity of CPU in China. In contrast, the response variables are bulk agricultural and energy markets. The cross-basis matrix R is generated through tensor-product interactions, allowing for a comprehensive exploration of the complex, delayed, and nonlinear relationship between policy uncertainty and market spillovers.

where r 1 ( x , t ) denotes the exposure response function based on natural cubic spline (node = 3, AIC optimization), and r 2 ( x , t ) is a polynomial distribution lag function with a maximum lag of L = 12trading months.

The model parameters are estimated using penalized likelihood estimation and the mathematical expression of DLNM is:

where t denotes time; x t j is the independent variable, such as China CPU; u t k is the control variable, which has a linear effect on Y . The function s j represents the nonlinear effect of the independent variable on Y where it is defined by a smooth function, such as a spline function or a polynomial function. The goal is to transform Y into a time series with several different coefficients in order to characterize its nonlinear effect. Using the smoothing function s ( C P U t ; β ) = z t T ⋅ β , the transformation expression is as follows:

Here, Z t · denotes the n × v x th row vector of the basis matrix Z of dimension t , which is designed to allow the introduction of lagged effects into the model. It is assumed Y to be affected by x t − l , where L denotes the lag period. With this construction, we obtain the n × ( L + 1 ) -dimensional matrix Q with the mathematical expression:

where Lis the maximum lag and hence q 1 · = C P U , which is the first column of Q. Then, a new DLNM can be constructed and s ( C P U t ; η ) = q t · T C η , whereCdenotes ( l + 1 ) × v l the matrix of basis variables obtained by applying a specific basis function to the lag vectorl. For example, if C== 1, it is a moving average model. 22 It is the vector of parameters that should be estimated, and the true coefficients β ⌢ = C η ⌢ .

Compared with the classical VAR model, the DLNM has three key analytical advantages: 1) it is able to identify the threshold of the intensity of policy shocks; 2) it quantifies the cumulative lagged effects and their confidence intervals; 3) it reveals the nonlinear coupling mechanism of “intensity-lag-market response” through the three-dimensional response surface. Thus, the DLNM describes the short- and long-term nonlinear lagged effects of China’s CPU on the bulk agricultural and energy markets at different levels.

The selection of the TVP-SV-VAR and DLNM models is based on their distinctive strengths in capturing complex dynamic interactions. Compared to conventional VAR or linear regression models, the TVP-SV-VAR model allows for time-varying coefficients and stochastic volatility, offering a more flexible framework to track structural changes and evolving shock responses. The DLNM model, originally developed in epidemiological research, is particularly effective in analyzing nonlinear and delayed relationships. It accommodates interactions between policy uncertainty intensity and its temporal lag effects, which are crucial to understanding commodity market responses. Nonetheless, both models entail computational intensity and require careful calibration. The robustness of results was ensured through Bayesian inference and sensitivity analyses.

From the equally spaced impulse responses of agricultural futures in Figure 2, it can be seen that in the face of the shock of climate policy uncertainty, all four agricultural products receive different impacts of shocks throughout the sample period. Among them, corn, cotton and soybean have less fluctuation and less impact in the short-term, but wheat receives unstable and more volatile short-term shocks throughout the sample period. Corn faces the impact of climate policy uncertainty, and the medium-term and long-term shocks are more volatile at particular points in time, for example, around 2017, the medium-term impact of CPU on corn futures is negative and then positive, on the contrary, the long-term impact is positive and then negative, and both of them reach the peak of the whole sample period; before 2020, the medium-term impact has an obvious positive effect, and in 2022, the long-term impact and the medium-term impact successively In 2022, the long-term and medium-term shocks successively fluctuate significantly, but the short-term shocks are more stable and consistently positive throughout the sample period. Cotton and soybeans, in the face of shocks from climate policy uncertainty, likewise have large fluctuations in medium- and long-run shocks at a few particular points in time, and, as with corn, the medium- and long-run restarts of CPU fluctuate dramatically around 2017, which may be attributed to the establishment of the Paris Agreement, which is exactly the kind of global framework that has been developed to combat climate change, aiming at reducing greenhouse gas emissions, preventing a global temperatures from rising, and increase countries’ resilience to climate change. The impulse responses of wheat and the other three agricultural products are slightly different, firstly, the short-term shocks suffered are more volatile over the whole sample period, and the short-term shocks are still larger than the medium- and long-term shocks in 2017, when the rest of the three agricultural products were particularly affected, which may be due to the fact that our country is a large producer of wheat but not a strong producer of wheat, and the share of our wheat production globally is decreasing year by year. Based on the above findings to understand that agricultural products receive different CPU impulse responses at different lags, the DLNM model is next used to observe the trend impacts received by the four agricultural products at different lags and different levels of CPU.

Figure 2. CPU equal-interval impulse for agricultural markets.

The nonlinear and lagged effects of CPU on corn futures were first investigated. As shown in Figure 3, CPU has a significant nonlinear and lagged effect on corn futures prices. The negative effect on corn futures prices only occurs at higher levels of CPU, and this negative effect may strengthen as CPU continues to increase. However, when the lag is about 2 to 6 months, CPU growth has a progressively significant positive effect on corn futures only at high CPU levels. Then, when the lag exceeds 6 months, this effect will again turn to become negative, before the negative effect steepens at a lag of about 10 months, and then turns positive at more than 12 months thereafter. In terms of lagged effects, a rise in CPU will have a negative effect when the lag is about 1 month, and when the lag reaches about 12 months, a low level of CPU may have a significant negative effect on corn futures. However, as the level of CPU increases, the effect of CPU on corn futures prices gradually increases and turns positive. At higher lags, CPU is more likely to have a positive effect on corn futures prices, again at high CPU levels, again positive.

The nonlinear and lagged effects of CPU on cotton futures were then investigated. As shown in Figure 4, CPU has a significant nonlinear and lagged effect on cotton futures prices. The positive effect on cotton futures prices is only observed at higher levels of CPU, and this positive effect may strengthen as CPU continues to increase. However, when the lag is about 8 months, CPU growth has a negative effect on cotton futures prices at high CPU levels. Then, when the lag ranges from 0 to 8 months, CPU at high levels shifts from a positive to a negative effect, and

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Figure 3. Impacts of CPU on corn.

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Figure 4. Impacts of CPU on cotton.

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Figure 5. Impacts of CPU on soybean.

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Figure 6. Impacts of CPU on wheat.

As shown in Figure 5, CPU has a significant nonlinear and lagged effect on soybean futures prices. Unlike corn futures and cotton futures, the effect of CPU on soybean futures prices is positive at both low and high levels of CPU, and likewise with increasing lags. The impact effect has also been increasing at increasing levels of CPU, while it is decreasing between lags of 0 and 2 months, but is in a constant state of growth thereafter. In terms of lagged effects, when the lag period is about 1 month, the increase in CPU will have a positive impact on soybean futures, and when the lag period reaches about 12 months, the increase in CPU is also likely to have a significant positive impact on soybean futures. And the impact effect is stronger when the lag period increases. At higher lags, CPU is more likely to have a positive effect on soybean futures prices, and likewise at high CPU levels. Finally, the nonlinear and lagged effects of CPU on wheat futures are examined. As shown in Figure 6, CPU has a significant nonlinear and lagged effect on wheat futures prices. The positive effect on wheat futures prices is only observed at higher levels of CPU and is likely to strengthen as CPU continues to increase. However, when the lag period is about 0 to 2 months, the increase in CPU has a progressively significant negative effect on wheat futures at high CPU levels. A small increase then occurs, but continues to decline after 6 months and increases after 8 months. In terms of lagged effects, an increase in CPU will have a negative impact when the lag period is about 1 month, and when the lag period reaches about 12 months, the impact of CPU on the price of wheat futures gradually increases as the level of CPU increases and remains positive. At higher lags, CPU is more likely to have a positive effect on wheat futures prices, again at high CPU levels.

As shown in Figure 7, the response of energy markets to shocks to climate policy uncertainty exhibits strong negative shocks, particularly in the 2016-2018 period, indicating that crude oil markets responded more sharply to climate policy uncertainty during this period. The response curves in periods 1 and 6 are relatively small, suggesting that the market reacts more quickly to short-term shocks with greater volatility. The response is smoother up to period 12 (solid line), suggesting that the market gradually adapts to climate policy uncertainty over time, and the impact of the shocks diminishes. The response of the new energy market to climate policy uncertainty shocks exhibits larger negative shocks, especially in 2016 and 2018, which may be related to major adjustments in new energy policy or other related events. The response curves are smaller in periods 1 and 6, but the response of the new energy market to climate policy uncertainty gradually intensifies over time, especially in period 12, when the magnitude of the shock increases. This trend suggests that new energy markets are more sensitive to shocks to climate policy uncertainty and that their impacts gradually increase over time. The sharp reaction of crude oil markets to climate policy uncertainty may be explained by the sensitivity of global energy demand and supply. As one of the major global energy sources, crude oil’s price and production are vulnerable to the direct impact of policy changes. 2016-2018 may have experienced changes in international energy policies, climate change agreements, or adjustments in carbon emission policies, making crude oil markets more volatile. In the long term, the crude oil market has become more resilient to climate policies, and its response has gradually smoothed out with the rise of new energy sources and the gradual adaptation of traditional energy markets to the green transition. New energy markets have a relatively larger response to climate policy uncertainty, which may be related to fluctuations in policy support and uncertainty in technology development. New energy markets are often strongly influenced by factors such as government subsidies and green policies, and in particular, uncertainty about climate change-related policies may exacerbate their market volatility. In 2016 and 2018, there may have been key policy changes or major events that led to increased volatility in the new energy market. Over time, as new energy technologies mature and policies gradually stabilize, the market’s response tends to level off. Based on the above findings to understand that agri-energy prices receive different CPU impulse responses at different lags, the DLNM model is next used to observe the trend impacts received at different lags and different levels of CPU.

Figure 7. CPU equal-interval impulse for energy markets.

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Figure 8. Impacts of CPU on oil.

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Figure 9. Impacts of CPU on NE.

Figure 8 and Figure 9 present the DLNM results, analyzing the nonlinear and lagged effects of climate policy uncertainty on the two markets, crude oil and new energy, respectively. The left side of the figure shows the nonlinear and lagged effects of the crude oil market on climate policy uncertainty, presenting a three-dimensional surface plot, as well as the nonlinear effects at different lags. The right side of the figure shows a similar analysis for the new energy market, containing the nonlinear effects at different lags. The three-dimensional surface plots in Figure 8 and Figure 9 show the nonlinear impact of climate policy uncertainty on crude oil and new energy markets. The vertical axis represents the market response, and the horizontal axis represents climate policy uncertainty as well as the lag period (number of months). The surface plot shows that when climate policy uncertainty changes, the market response shows a non-linear relationship, which implies that the impact of policy uncertainty on the market is not a simple linear change. The nonlinear response of the crude oil market to climate policy uncertainty suggests that the volatility of crude oil prices or other market variables increases significantly when the CPU changes significantly. In Figure 8 and Figure 9, it can be seen that the impact of higher CPU levels (right part) on the market increases significantly. The nonlinear response of the new energy market shows a similar trend, i.e., the volatility and response of the new energy market increases under the influence of high levels of CPU. In chart (b) for each market, the nonlinear effects are shown at different lags (1 month and 12 months). Here one can see the variation in lag effects, especially the impact of different time horizons on market response. The crude oil market reacts more rapidly with shorter lags (1 month), with the effect diminishing over time. In contrast, at longer lags (12 months), the market response to climate policy uncertainty becomes more persistent and slower. This suggests that crude oil markets may adjust quickly in the short term, but in the longer term the response to policy uncertainty may take longer to materialize. New energy markets show a similar trend, with more pronounced impacts in the short term (1 month) and more sensitive responses, especially at low CPU levels. In a longer lag (12 months), the new energy market’s response gradually smoothed out, which may be related to the long-term stability of new energy policies or the market’s adaptability. The nonlinear effect of climate policy uncertainty on the market suggests that policy changes will not always affect the market in the same way. Even the same level of climate policy uncertainty may have different levels of impact in different markets and over different time periods. This may be related to the sensitivity of markets to policy changes. Energy markets, especially crude oil markets, typically show a stronger reaction to sudden policy changes. The lag period analysis shows the process of market adaptation to climate policy uncertainty shocks. In the short term, market reactions are usually more dramatic, especially in crude oil and new energy markets. However, long-term lagged effects reveal market adaptation, suggesting that markets gradually absorb the impact of policy changes after the initial shock.

Short-term shocks to crude oil markets are usually large, reflecting the fact that crude oil, as an important component of the global economy, is very sensitive to factors such as policy changes and demand volatility. Over time, however, the market gradually adjusts and adapts, resulting in smaller long-term lagged effects. The new energy market, on the other hand, has a different response pattern than crude oil. The new energy market usually relies on long-term policy support, such as green energy incentives and carbon emission policies, so while it reacts more in the short term, it is more adaptable in the long term. The lagged response of new energy markets to climate policy uncertainty is more muted. The analysis shows that there are significant differences in the responses of different markets to climate policy uncertainty, and that these responses change gradually over time. Crude oil markets are more volatile in the short term in response to policy uncertainty, while new energy markets are more adaptive, with diminishing effects in the long term.