India as largest milk producer is with per capita availability of 427 grams per person per day. In 2018-19, the livestock sector contributed ₹8,71,884 crore to India’s Gross Value Added (GVA), accounting for 5.1% of total GVA at current prices. During the same year, the total value of output from the livestock sector was ₹11,59,636 crore, of which milk and milk products accounted for ₹7,72,705 crore (National Accounts Statistics, 2020). In addition, per capita consumption expenditure on major groups of commodities was reported to be Rs 116.38 in rural areas and Rs 187.14 in urban areas in 2011-12 (PIB, 2023). Since the white revolution, dairy sector of the country has witnessed remarkable transformation, which encouraged the nation to emerged as the global leaders in milk production (Hans et al., 2024). Over 70 million rural households are engaged in milk production making this subsidiary source of occupation more inclusive and equitable enterprise (Adari et al., 2025). In fact, dairy sector stands as the predominant agricultural commodities contributing over 21 per cent of the nation agricultural production (Katoch et al., 2024). The relevance of this sector lies not only in its production but it is also characterized by its strong backward connection which generate rural employment opportunities in fodder production, veterinary facilities, developing dairy cooperatives, processing units for value addition and cold chain distribution channels. The dairy sector is witnessing new opportunities for commercialization, reflected by a diverse demand which is influenced by consumer preference and valuable products, including cheese, probiotics, flavoured milk and ghee (NDDB, 2023). Among different states, Uttar Pradesh is the fourth largest state in India, where 16 percent of the total population, which is around 77 per cent of households are located in rural areas (Census, 2011). Approximately 47 per cent of this population is directly dependent on agriculture as a source of their livelihood, yet the average size of holding is just 0.74 hectares. This structural challenge hinders the potential of income generation through crop cultivation, leading rural households to engage in subsidiary enterprises to improve their livelihood (Yadav et al., 2022). In the year 2012-13, the average monthly income for agricultural households in Uttar Pradesh was reported as fifth lowest across Indian states, at Rs 4,923 (Gulati and Saini, 2021). In this case, livestock plays very crucial role as it complements agricultural income. Uttar Pradesh contributes approximately 17 percent of the total milk production in India, with the cattle constituting 9.88 per cent of total livestock population. Organizations such as the National Gokul Mission, National Livestock Mission and Dairy Entrepreneurship Development Scheme have facilitated in promoting breed enhancements, scientific management techniques and access to institutional loans. For farmers of Uttar Pradesh, the dairy region not only serves as a supplement enterprise, but also as a financial buffer, providing “seed fund” to meet the daily expenses of marginal, small and medium farmers (Sahu et al., 2021; Sahu et al., 2022). With a combination of high production capacity and increasing demand in both rural and urban markets, dairy is emerging as an important carrier of economic security. Such estimates ensure more informed policy formulation, infrastructure planning and income stabilization mechanisms for rural families (Amin et al., 2014).

The information on the milk production and per capita availability of milk was collected from the secondary databases of NDDB, Agricultural Statistics at a Glance, IASRI, UPDES, etc.
 
Box-Jenkins models
 
Forecasting through modelling was enriched and boosted by the development of the Box-Jenkins model in the 1970s and more specifically by the software development of it (Box and Jenkins, 1976). The sole crux of this method is that the magnitudes of the current parameters in a series are influenced by and connected to the past values in some way. In the time series, ARMA was taken as the predictive tool for the time series data Xt, along with the other tools such as auto-regressive (AR) and amoving average (MA) (Srivastava et al., 2022).
 
AR model
 
ARIMA, which is Autoregressive Integrated Moving Average, is used as a comprehensive model for forecasting. In it, the term ‘integrated’ denotes the removal of the trends from the data series, and when there is no significant trend found, then the ARMA is deployed. The scratch in the model AR(p) is of order p.
 
AR(p) model

αi = Model parameters.
c = Constant.
µ = Random term or error.     
Constant can be removed sometimes (Devi et al., 2021).
 
MA model
 
MA(q) = model of moving average for order q.
MA(q) model:

 
Where,  
θ1, ..., θ_q = Model parameters.
μ = expectation of X_t (often = 0).
e = Random error term.
 
ARMA model
 
In ARMA (p, q), p is the autoregressive terms and q is the moving average terms.

 
Cumulatively, the model consists of autoregressive and moving average models,
 
ARIMA model
 
This model is a common model used in time series forecasting. Stability of this model is ensured through transformation or differencing. Within ARIMA, the ‘auto-regressive’ means occurring, the ‘moving average’ shows appearing in the forecasting equation and an ‘integrated’ series is a stationary series that needs to be differenced. It is considered a fine-tuned version of irregular and spontaneous models. To remove any autocorrelations in the errors, the addition of lags of the forecast errors to the prediction equation is the fine-tuning.
       
The ‘ARIMA (p, d, q)’ model is the ordered model for a non-seasonal ARIMA model.
Where,
p = Number of autoregressive terms in the prediction equation.
d = Number of non-seasonal changes.
q = Number of lagged forecast errors.
       
To examine a convenient ARIMA model, the required order(s) of differencing are calculated first to make the data stationary and to eliminate major seasonal patterns. This may be joined with a variance-stabilising transformation, such as logging or deflating. If only one differencing is enough, then the process can resume here, because with the assumption of a constant differenced series in the random trend or walk model. A random walk model assumes a constant first difference of the series, while a seasonal random walk model assumes a constant seasonal difference. In the same way, a seasonal  random trend model assumes a persistent first difference of the seasonal difference, which is mostly zero. However, autocorrelated errors could be yielded, even with a well-fitted moving average or random trend model, indicating the need for additional components in the prediction model for additional components (Mishra et al., 2020).
 
Formulation of Model
 
The whole considered time period (1950-2018) was divided, the model formulation period (1950-2009), the model Validation period (2010-2019) and the forecasting period up to 2030.
 
Model adequacy check
 
Based on maximum R2, minimum RMSE, MAPE and AIC, Box-Jenkins model was chosen. The model that meets the above criteria to a significant extent was chosen (Niranjan and Chouhan, 2016).
 
R-squared
 
R-squared is a measure of the model’s contribution to the overall variation in the data. When the data series is stationary, the metric is useful. The current model outperformed the baseline, which is confirmed by the positive numbers (Paul et al., 2013).

 
RMSE
 
The square root of the mean square error metric reveals how far a dependent series deviates from its predicted model in the same units as the dependent series.

 
MAPE
 
The mean absolute percentage error metric is used to look at how far a dependent series deviates from the level predicted by the model. It is unaffected by the units used; hence, it can be used to compare series with different units.

 
MAE
 
MAE is the mean absolute error, which measures the deviation level of the series from the level
anticipated by the model. The original series units were used to account for MAE.

 
AIC
 
AIC is Akaike’s information criterion, a predicted prediction error and statistical model quality for time series data. AIC provides a portend for model selection (Akaike, 1973).
 
Mathematically, AIC = - 2 log (maximum likelihood) + 2k
Where,
K = p + q + 1 (with intercept),
Otherwise, k = p + q (Kumar et al., 2019).
       
Diagnostic checks on the model fitted were seen by Autocorrelation function (ACF) and Partial Autocorrelation function (PACF) plots of the residuals. The forecasting model was used only if the residuals were white noise (Mishra et al., 2020).

Time-series statistics reflect a proper aging trend in India’s milk production, which increased from 146.3 million tonnes in 2014-15 to 239.3 million tonnes in 2023-24, which is an annual growth rate of about 5.7% (Table 1). This expansion is strongly correlated with parallel policy interventions, especially the participation of small farmers, progress in breed improvement and investment in dairy infrastructure. The ARIMA model demonstrates this trend and forecast a continuous growth trajectory that closely aligns with the historical patterns.


       
Additionally, the availability of per capita milk has also risen from 339 to 471grams/day during the same period from 2014-15 to 2023-24, surpassing both Indian population growth and global average (Table 1). This increase demonstrates successful adaptation at the regional level, improved animal productivity, growth of cooperatives in milk procurement and continuous formalization of supply chain have collectively contributed to wider market penetration particularly in the underserved service sector of India (Shamsuddoha et al., 2023). Data highlights the significant disparities among the states namely Uttar Pradesh, Rajasthan and Gujarat altogether contribute approximately one-third of total milk production while the eastern and north-eastern states lag behind. Such disparity identifies the opportunity to rectify the ongoing milk supply imbalance and implement the strategic interventions. Empirical evidence from the studies also supports that dairy sector in India is progressing towards enhanced productivity and adaptability influenced by favourable socio-economic factors, policy initiative and adoption of modern technology. Also, challenges related to fodder, veterinary services, and cold-chain infrastructure, continue to hinder equitable development. In order to ensure continuous growth trajectory in both milk production as well as per capita availability, it is essential for the supply chain to receive strategic focus and ongoing assistance for small-scale farmers (Basunathe et al., 2010).
 
Modelling and forecasting
 
Box-Jenkins methodology was adopted for modelling and forecasting. Data for the period 2001 to 2009 and 2010 to 2019 were used for model building and model validation, respectively. From the (0, 1, 0) model to the (1, 1, 2) models, it was found that ARIMA is suitable for modelling and forecasting the production and per capita availability behaviour of milk. None of the series was stationary in both, i.e., time series data of milk production and per capita availability. Thus, first differencing with the original data was done to make all the series stationary a constant mean (µ) and variance (σ). With the differenced series, σ, the study estimated ARIMA equations for all the parameters  using the data from 2001 to 2019 and made forecasts up to 2030, using the Gretl software. All the ARIMA models were tried among the competitive models and the best models were selected based on the minimum values of RMSE, MAE, MAPE, AIC and the maximum value of R2. However, it was also put under the diagnostic check of the residuals, with the help of ACF and PACF graphs. Fig 1 to 6 clearly showed that milk production and per capita availability in Andhra Pradesh, Uttar Pradesh, Rajasthan, Madhya Pradesh, Punjab and India showed an increasing trend in the future.












       
The ACF and PACF plot of first difference, the value of State wise milk production in considered state and India, which suggests that the tentative value of p and q that would be suitable for milk production were p=1 and q=2 for Andhra Pradesh, p=1 and q=0 for Uttar Pradesh, p=1 and q=2 for Rajasthan, p=1 and q=1 for Madhya Pradesh, p=1 and q=4 for Punjab and p=1 and q=0 for India. Thus, ARIMA (1,1,2), (1,1,0), (1,1,2), (1,1,1) and ARIMA (1,1,0) were found to be the best ARIMA models for milk production in the considered states and India. And, as shown in Table 2, milk production has low RMSE, MAPE, MAE and AIC values in Andhra Pradesh, Uttar Pradesh and India, in Madhya Pradesh, has low MAPE and MAE, while the least RMSE, MAPE value and largest R2 value in Rajasthan and Punjab, so the best-fitting models are ARIMA (1,1,2), ARIMA (1,1,0), ARIMA (1,1,2), ARIMA (1,1,1) and ARIMA (1,1,0). In 2019-2020, the milk production in Andhra Pradesh, Uttar Pradesh, Rajasthan, Madhya Pradesh, Punjab and India was 15263 thousand tonnes 31864 thousand tonnes, 25573 thousand tonnes, 17109 thousand tonnes, 13348 thousand tonnes and 198440 thousand tonnes, respectively, compared to 15247.00 thousand tonnes, 31781.30 thousand tonnes, 24799.56 thousand tonnes, 7061.85 thousand tonnes, 13345.81 thousand tonnes and 198854 thousand tonnes, predicted. For the years 2030-2031 andhra Pradesh, Uttar Pradesh, Rajasthan, Madhya Pradesh, Punjab and India are expected to have 20115.52 thousand tonnes, 43002.19 thousand tonnes, 37575.04 thousand tonnes, 27787 thousand tonnes, 19742.11 thousand tonnes and 302025.10 thousand tonnes, respectively.


       
The ACF and PACF plot of first difference, the value of State wise per capita availability in considered states and India, which suggests that the tentative value of p and q that would be suitable for per capita availability were p=1 and q=2 for Andhra Pradesh, Uttar Pradesh and India, but in p=1 and q=1 for Rajasthan and Punjab, while p=1 and q=0 for Madhya Pradesh. Thus, ARIMA (1,1,2), ARIMA (1,1,1) and ARIMA (1,1,0) were shown to be the best ARIMA models for per capita availability in the considered states and all India. And, as shown in Table 3, per capita availability has low RMSE, MAPE, MAE and AIC values and the largest R2 in Andhra Pradesh, Madhya Pradesh and India, in Uttar Pradesh and Rajasthan have low RMSE, MAPE and MAE, while the least MAPE, MAE value in Punjab, so the best-fitting models are ARIMA (1,1,2), ARIMA (1,1,1) and ARIMA (1,1,0). In 2019-2020, the per capita availability in Andhra Pradesh, Uttar Pradesh, Rajasthan, Madhya Pradesh, Punjab and India was 799 gms per day, 387 gms per day, 904 gms per day, 568 gms per day, 1221 gms per day and 406 gms per day, respectively, compared to 834.97 gms per day, 386.55 gms per day, 882.81 gms per day, 567.43 gms per day, 1224.51 gms per day and 407.10 gms per day, predicted. For the years 2030-2031 andhra Pradesh, Uttar Pradesh, Rajasthan, Madhya Pradesh, Punjab and India are expected to have 1121.46 gms per day, 497.03 gms per day, 1246.76 gms per day, 789.28 gms per day, 1494.54 gms per day and 518.07 gms per day, respectively.

As per the study, for modelling and forecasting of Milk production; ARIMA (1,1,2), ARIMA (1,1,0), ARIMA (1,1,2), ARIMA (1,1,1) and ARIMA (1,1,0) and per capita availability of milk; ARIMA (1,1,2), ARIMA (1,1,1) and ARIMA (1,1,0); were selected as best best-fitted model for the considered state and India, respectively, with a 95 percent accuracy level in the prediction. The findings of employing the best models to anticipate milk production and per capita  availability of milk output from 2020 to 2030 reveal that production and per capita availability will increase in all considered states and India. The findings of this study prove beyond a shadow of a doubt that India will have the highest anticipated value in 2030.

The authors declare no conflicts of interest regarding the publication of this article.