Trend analysis of India’s agricultural exports using linear and nonlinear growth models

Introduction

Agricultural exports are an important part of India's trade. They support rural income, value addition, employment, and foreign exchange earnings. Over the years, India has expanded its export basket from traditional items like rice and spices to fruits, vegetables, livestock products, and processed foods. This shows a clear improvement in the structure of agricultural trade (Basu, 2024; Priyanka et al., 2024) [1,2] . Studies also report that India's agricultural exports have increased in value and volume, although the growth pattern often shows periods of rise and fall (Globalization and Indian Agricultural Exports, 2023; Growth Trends and Instability in Agricultural Crops). Trade liberalisation after 1991 brought major changes to India's export performance. Evidence shows that these policy reforms improved India's competitiveness in global markets. Shinoj and Mathur (2008) [5] showed that India gained advantage in many agricultural commodities during the post-reform years. Commodity-specific studies such as Adhikari et al. (2016) [6] for rice and Bhopala et al. (2023) [7] for sugar also reported positive export growth. Research on horticultural crops (Keerthika et al., 2024; Growth Trends and Instability in Horticultural Crops, 2023) [8,9] highlights both new export opportunities and instability in some years. These findings show the importance of studying long-term export behaviour. To understand these long-term patterns, suitable analytical tools are needed. Earlier studies mostly used compound or exponential growth models, but recent agricultural research shows that nonlinear models are better at identifying turning points, curvature, and saturation in long-term data. Logistic and Gompertz models are useful for this because they describe S-shaped growth patterns that start slowly, increase quickly, and then stabilise (Winsor, 1932; Franses, 1994; Satoh, 2019) [10,11,12] . Other agricultural studies also show that nonlinear models fit better when growth changes over time (Srinivasa Chary et al., 2023; Growth and Instability Analysis in Indian Agriculture, 2018) [13,14] . Comparing different growth models requires proper statistical tools. Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC) help select the best model by balancing fit and simplicity (Akaike, 1974; Schwarz, 1978) [15,16] . Theil's U statistic also measures forecasting accuracy. These tools make it possible to choose the most suitable model when several alternatives are available. Although many studies discuss agricultural exports and trend modelling, most of them use only one or two growth models and focus on individual commodities. Very few studies compare a wide range of linear and nonlinear growth models for India's total agricultural exports. This is a gap in the literature because aggregate export data reflect broader structural patterns and policy effects. The present study fills this gap by applying twelve growth models-linear, logarithmic, inverse, quadratic, cubic, compound, power, S-curve, exponential, logistic, and Gompertz to India's agricultural export data. The models are compared using Adjusted R², RMSE, MAPE, AIC, BIC, and Theil's U. Short-term forecasts are also prepared, and agricultural export trends are compared with total national exports to understand their relative growth behaviour.

Materials and Methods

Data Source and Study Period

The present investigation is based on annual time-series data on India's Agricultural Exports and Total National Exports spanning a period of 36 years, from 1990-1991 to 2025-2026. The data was collected from the Ministry of Agriculture & Farmers Welfare, Government of India in the website https://www.indiastat.com. This long study horizon enables an in-depth assessment of both short-term fluctuations and longterm structural trends in agricultural export growth. All subsequent analyses, including model estimation and trend diagnostics, were conducted using the compiled time-series dataset for the specified period.

Growth Model Specifications

To analyze the underlying trend behavior, twelve alternative linear and nonlinear deterministic growth models were fitted to each export series. These functional forms have been commonly utilized in agricultural and economic trend studies due to their capacity to capture linearity, curvature, acceleration, saturation, and various other structural characteristics. The models, along with their mathematical specifications, are detailed below: To examine the temporal behaviour of India's agricultural export earnings, two widely used growth models such as Linear Growth Model and Compound Growth Model were employed.

(i) Linear Growth Model

The linear model assumes a constant absolute change in export earnings over time. It is specified as:

where Y t = Agricultural export earnings in year t, t= Time (years), a,b = Parameters estimated through Ordinary Least Squares (OLS). The Linear Growth Rate (LGR) is computed using:

where is the mean of the export series. This indicates the average annual percentage increase in exports.

(ii) Compound Growth Model

The compound model assumes a constant percentage change over time, following an exponential growth pattern:

Parameters are estimated using OLS on the log-transformed values.

The Compound Growth Rate (CGR) is computed by:

This growth rate captures accelerated or decelerated trends in export performance over the study period.

(iii) Statistical Significance of Growth Rates

The significance of LGR and CGR was verified using the Student's t-test, following:

where r = estimated growth rate (LGR or CGR), SE(r) = standard error of the estimated growth rate, N = number of observations. A growth rate is considered statistically significant if the calculated t exceeds the critical t value at the 5% probability level. Each of these models establishes a unique behavioral pattern over time. Linear and logarithmic models reflect proportional and gradually increasing trends, whereas quadratic and cubic functions facilitate shifts in curvature throughout the time period. Compound, growth, and exponential models represent multiplicative or constant-proportion increases. The logistic and Gompertz models are especially effective for identifying Sshaped trends, where initial growth accelerates before eventually stabilizing.

Estimation Procedure

To capture the long-term behaviour, direction, and structural pattern of India's agricultural export earnings, a comprehensive trend analysis was conducted by fitting twelve linear and nonlinear growth models. This approach enables comparison across models that differ in their assumptions about curvature, saturation, acceleration, and proportional change. Such multimodel trend evaluation is widely recommended in agricultural economic research and has been applied in earlier studies on production and trade growth. Each functional form was estimated using either Ordinary Least Squares (OLS) for linearizable models or nonlinear least squares (NLS) for intrinsically nonlinear forms. The independent variable in all models was time (t), while the dependent variable was either Agricultural Exports or Total National Exports. The complete list of functional forms fitted in the present study is given below. The estimation process utilized the Python programming language along with the Statsmodels and SciPy libraries, ensuring robust numerical optimization for the nonlinear models. For each export series, fitted values were generated and used to calculate residuals, trend curves, and forecast values. No differencing or transformation was applied to the dependent variable, except where required by the functional specification.

Model Evaluation Criteria

The adequacy of each fitted trend model was assessed using a set of statistical criteria widely employed in time-series growth studies. Since no single measure is sufficient to judge goodness of fit, multiple indicators were computed to ensure a robust comparison of model performance for both Agricultural Exports and Total National Exports. Where RSS = residual sum of squares.

Lower AIC and BIC values indicate a better-fitting and more efficient model. This multi-criteria evaluation ensures that the selected function provides both strong explanatory performance and high predictive accuracy, consistent with best methodological practices in agricultural trend analysis.

Forecasting Methodology

After identifying the logistic function as the optimal model for both agricultural and national exports, the study proceeded to make projections for the four-year period from 2027 to 2030. The estimated parameters from the best-fitting logistic model were integrated into the trend equation to generate export forecasts.

Results and Discussion

Model Performance for Agricultural Exports

Twelve linear and nonlinear growth models were fitted to the time-series data of Agricultural Exports and Total National Exports of India for the period 1990-91 to 2025-26. The bestfitted model for each series was identified based on the relatively highest Adj.R², lowest MAPE, RMSE, and Theil's U-Statistic. The fitted trend lines and model summaries are described in Table 2.

A. Trends in Agricultural Exports

Among the models, the Logistic model recorded the highest Adj.R² (0.9097) along with the lowest MAPE (26.69), lowest RMSE (38,834.56) and lowest Theil's U-Statistic (0.0999). These diagnostics indicate a superior fit compared to all other functional forms, and the model was statistically significant at the 1% level. Hence, the Logistic model was identified as the best-fitting model for Agricultural Exports.

The fitted logistic equation is … (13) where t represents the time in years.

From Fig. 1, agricultural export earnings were observed to increase steadily throughout the study period, exhibiting an Sshaped growth pattern with signs of gradual stabilization in recent years. Theil's U (0.0999). While both the cubic and Gompertz models exhibited satisfactory performance, the logistic model consistently surpassed them across all error and information criteria.

The superior performance of the logistic model indicates that India's agricultural exports follow an S-shaped growth trajectory, characterized by initial slow growth, followed by acceleration, and ultimately stabilization as they approach a carrying capacity. This pattern aligns with the theoretical expectations for mature export sectors nearing market saturation.

Model Performance for Total National Exports B. Trends in Total National Exports

The Total National Export data was analyzed using the same set of growth models, and the results are presented in Table 3.

The Logistic model again outperformed all other models with the highest Adj.R² (0.9173), lowest MAPE (14.68), lowest RMSE (3,25,148.8) and lowest Theil's U (0.1012). These values indicate that the logistic form provides the most accurate representation of the long-term movement in national export earnings.

Accordingly, the following logistic model was selected

Fig. 2 shows that total exports follow a smooth, accelerating trend followed by a period of stabilization, consistent with the characteristics of a logistic curve.

The logistic model once again proved to be the best-fitting specification for total national exports, exhibiting the lowest RMSE (325,148.80), the lowest MAPE (14.68), and the highest Adjusted R² (0.9096). These results suggest that national exports also adhere to an S-shaped nonlinear trajectory, aligning with the findings observed in agricultural exports.

Trend Behaviour Analysis

The fitted curves indicated that agricultural exports have experienced distinct phases of growth throughout the study period. The initial phase showcased moderate expansion, which was subsequently followed by a period of accelerated growth and eventual stabilization, resulting in a logistic-shaped trajectory akin to other agricultural and economic time series. In contrast, total national exports displayed a more pronounced upward trend with greater intensity, yet still reflected a broadly similar growth pattern.

The S-shaped trajectory captured by the logistic model indicates that agricultural exports has three distinct phases 1. Initial Phase: Slow growth during early years due to limited market access and infrastructure constraints. 2. Acceleration Phase: Rapid expansion following trade liberalisation and policy reforms.

Maturation Phase:

Gradual deceleration as the sector approaches its current carrying capacity.

Growth Rate Analysis C. Forecasting of Agricultural and National Exports

Based on the linear model, agricultural exports recorded a linear growth rate of 8.36 per cent, while the compound annual growth rate (CAGR) derived from the compound model was 10.89 per cent. These rates are consistent with the strong upward movement observed in the fitted curves.

The logistic trajectory suggests that agricultural exports may be approaching a steady growth phase, indicating the need for policy interventions to stimulate further acceleration through diversification, value addition, and market expansion.

Four-Year Forecasting of Exports

Using the best-fitted Logistic models, a four-year ahead projection for both agricultural exports and total national exports was generated for the period 2027 to 2030. The projected values are shown in Table 4 and Table 5. The forecasts indicate a continued increase in export earnings for both categories, though at a diminishing rate, reflecting the bounded nature of logistic growth.

D. Growth Rates in Agricultural and National Exports

The Linear Growth Rate (LGR) and Compound Annual Growth Rate (CAGR) were calculated. The linear and compound growth rates of agricultural exports and total national exports were computed and are presented in Table 6. Agricultural exports recorded a linear growth rate of 8.363 percent and compound annual growth rate of 10.890 percent, while total national exports exhibited corresponding growth rates of 8.589 percent and 10.754 percent, respectively. The relatively slower growth trajectory of agricultural exports compared to total national exports underscores the need for targeted policy measures to enhance the competitiveness of India's agricultural export sector.

Conclusion

The present study analysed long-term agricultural export trends using twelve alternative growth models and identified the logistic function as the best descriptor of export behaviour. The logistic form captures the nonlinear and saturating nature of export growth more effectively than linear or polynomial functions, as evidenced by superior performance across all evaluation criteria.

The key findings of the study are summarised as follows 1. Among the twelve growth models evaluated, the logistic model provided the best fit for both agricultural exports (Adjusted R² = 0.9097) and total national exports (Adjusted R² = 0.9096). 2. Agricultural exports exhibited an S-shaped growth trajectory characterised by initial slow growth, subsequent acceleration, and eventual stabilisation. The study acknowledges certain limitations including the deterministic nature of the models employed and the assumption of stable policy and market conditions over the forecast horizon. Future research may incorporate stochastic models and structural break tests to enhance predictive accuracy.

In conclusion, nonlinear growth models, particularly the logistic form, prove to be effective analytical tools for trend analysis and export forecasting, offering critical inputs for policymakers, exporters, and development planners.

Acknowledgement

AcknowledgementsNo external funding has been received for this study.

References

1. Basu D. Analysis of agricultural exports from Indian states: Impact of Covid-19 and the significance of infrastructure facilities. Vidyasagar University Journal of Economics. 2024;29(1):225-38.
2. Priyanka T, Patil M, Padmanabha BV. Export of fruits, vegetables and their products from India: A study on growth rate and instability index. Journal of Farm Sciences. 2024;37(1):119-26.
3. Mary A. Globalization and Indian agricultural exports: A critical evaluation. International Journal for Multidisciplinary Research. 2023;5(3):112-22.
4. Yadav R, Singh A. Growth trends and instability in export of agricultural crops from India: An economic analysis. Splint International Journal of Professionals. 2025;12(1):1827.
5. Shinoj P, Mathur VC. Comparative advantage of India in agricultural exports vis-à-vis Asia: A post-reforms analysis. Agricultural Economics Research Review. 2008;21(1):6073.
6. Adhikari A, Sekhon MK, Kaur M. Export of rice from India: Performance and determinants. Agricultural Economics Research Review. 2016;29(1):135-50.
7. Bhopala UD, Singh D, Reddy P. Growth and instability analysis of sugar exports from India. Asian Journal of Agricultural Extension, Economics & Sociology. 2023;41(7):45-54.
8. Keerthika P, Rajesh V, Kumari S. Trend and instability in the export of processed fruit products from India. Journal of Applied Horticulture. 2024;26(1):55-62.
9. Hazarika A, Bordoloi R. Growth trends, instability and export potential of horticultural crops in Assam: A comparative analysis with India. Current Agriculture Research Journal. 2023;13(2):235-46.
10. Winsor CP. The Gompertz curve as a growth curve. Proc Natl Acad Sci USA. 1932;18(1):1-8.
11. Franses PH. A method to select between Gompertz and logistic trend curves. Technol Forecast Soc Change. 1994;46(1):45-49.
12. Satoh D. Model selection among growth curve models that have the same intrinsic growth rate. J Appl Stat. 2019;46(10):1783-801.
13. Srinivasa Chary D, Supriya K, Anjaiah T, Meena A, Ramesh M. Trends in fertilizer nutrients (N, P2O5 and K2O) consumption in India. Biol Forum -An Int J. 2023;15(6):690-96.
14. Anjum S, Madhulika S. Growth and instability analysis in Indian agriculture. Int J Multidiscip Res Dev. 2018;5(11):331-37.
15. Akaike H. A new look at the statistical model identification. IEEE Trans Autom Control. 1974;19(6):716-23.
16. Schwarz G. Estimating the dimension of a model. Ann Stat. 1978;6(2):461-64.