Forecasting Area And Production Of Wheat By Using Unobserved Components Model

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G. MOHAN NAIDU* B. RAVINDRA REDDY, V. AMARNADH AND ADDANKI JOTHI BABU

Department of Statistics and Maths, S.V. Agricultural College, Tirupati – 517 502

ABSTRACT

Forecasting a time series is generally done by using autoregressive integrated moving average (ARIMA) models. The main drawback of ARIMA technique is that the time series should be stationary. In reality, this assumption is rarely met. The Unobserved Component Model (UCM) is a promising alternative to a ARIMA in overlapping this problem as it does not make use of the stationarity assumption. In addition, it breaks down response series into components such as trends, cycles and regression effects, which could be useful especially in forecasting the production of agricultural crops. The present study is aimed at using UCM for annual wheat area and production in India. Results revealed that both the trend components, level and slope were insignificant. The linear trend model zero variance slope was found to be the best fit for the data. The forecast error for the area and production are 2012 and 2013 were 0.43% and 2.73% while 0.28% and 0.94% respectively. From the fitted model, predicted annual wheat area and production for 2016 would be 32.11 million hectares and the 95% CI is 29.38 to 34.84 million hectares where as 99.90 million tones and the CI is 91.14 to 108.67 million tones. Thus the used of UCM is recom-mended for annual data.

KEYWORDS:

Non-stochastic process, trend, UCM, Ljung and Box chi-square.

INTRODUCTION

The forecasting of seasonal time series is a challenging problem. We approach the forecasting challenge from a model-based perspective and adopt the unobserved components time series model. The key feature of this class of models is the decomposition of a time series into trend, seasonal, cyclical and irregular components. Each component is formulated as a stochastically evolving process over time. The decomposition of an observed time series unobserved stochastic processes can provide a better understanding of the dynamic characteristics of the series and the way these characteristics change over time. The trend component typically represents the longer term developments of the time series of interest and is often specified as a smooth function of time.

Box Jenkins and exponential smoothing techniques are commonly used in the analysis of time series in agriculture. Main drawbacks in these models are that, they are suitable only for the stationary series (Box et al., 1994), empirical in nature and fail to explain the underlying mechanism. It is not always possible to create a time-series stationarity by differencing or by some other means.

Hence, this approach could be limited to few data sets. Also correlogram, and Partial Auto Correlation Function (PACF) specifying models are not always informative, especially in small samples. This could lead to inappropriate models and predictions.

UCM is a promising alternative approach to overcome these problems (Harvey, 1996). It provides structural time series models and it is a flexible class of models that are useful for forecasting. It decomposes the response series into latent components such as trend, cycle and seasonal effect and linear and nonlinear regression effects. The saline feature of the UCM is latent components, which follow suitable stochastic models and it provides suitable set of patterns to capture the outstanding actions of the response series. UCM can also consist of explanatory variables. Apart from the forecast, structural modeling gives estimates of unobserved components and it is found useful in practical usage. UCM can handle intensive data irregularities too. It is very similar to dynamic models and also popular in the Bayesian time series (West and Harrison, 1999).

In view of the above, the aim of this study is to investigate the possibility of using UCM for modeling and forecasting annual area and production in Indian context.

MATERIALS AND METHODS

The secondary data on wheat area (million hectares) and production (million tonnes) for a period of 63 years from 1950-51 to 2013-14 has been collected from the book entitled Agricultural Statistics at a Glance, 2014 published by the Department of Agriculture & Co-operation, Directorate of Economics and Statistics, Ministry of Agriculture, Government of India.

Description of the model

A UCM consists of trend, cycle, seasonal and irregular components, and specified of the form (Harvey and Stock, 1993).

Yt  = μt +ϕt + ωt + εt ; t = 1, 2,…….,n (1)

Description of the model A UCM consists of trend, cycle, seasonal and irregular components, and specified of the form (Harvey and Stock, 1993).
tttttY εω ϕμ +++= ; t = 1, 2,…….,n (1)

where t μ
, t ϕ
, t ω
and t ε
denote the stochastic trend, stochastic cycle, seasonal component and overall error (irregular component), which is assumed to be a Gaussion white noise with variance 2 t σ . Since the data is annual, seasonal effect cannot be identified and thus the UCM for the data can be formulated of the form
ttttY εϕ μ ++= (2)

Estimating trend effect There are two different ways to modeling the trend component in UCM. The first method is by mean of random walk (RW) model, (3). The RW model can be formulated of the form (Harvey and Koopman, 2009).
ttt δμ μ += −1 ; d iit . .~ δ
N(0, ) 2 δσ
(3) The second method involves modeling the trend as a Local Linear Trend (LLT), which consists of both level and slope (Harvey, 2001). The trend, t μ is modeled as a stochastic component with varying level and slope and it can be formulated of the form,
tttt δβ μμ ++= −− 1 1 ; N (0, (4)
ttt τβ β += −1 ; d iit . .~ τ
N (0, ) 2 δσ
(5)

where is the slope of the local linear time trend. The disturbances and are assumed to be mutually independent. Special cases of this trend model is obtained by setting one or both of the disturbances variances, and , equal to zero. If is set equal to zero, then the trend becomes linear (fixed slope). If is set to zero, then the subsequent model generally has a smoother trend. If both the variances are set to zero, then the resulting model is the deterministic linear time trend,
t t 0 0 βμ μ += (6) Thus the reduced form of a LLM is the ARIMA (0,2,2) model. Test for normality and independence of residuals The Shapiro–Wilks (1965) statistic was used to test whether the residuals are normally distributed or not. The Ljung and Box Chi-square can be used to test the residual auto-correlations are independent or not (Alan, 1983).

RESULTS AND DISCUSSIONS

UCM was employed to fit the trends in area and production of wheat annual data. The findings are discussed in sequence as under. Identification of trend: The time series plot of annual wheat area and production showed the positive trend. Linear regression model was employed to the area and production data set of the wheat crop to test whether linear trends exist in the time – series data set or not. The existence of linear trend factor was tested through the linear regression ++= tY 10 ββ e where e is the residual which is independently normally distributed with mean zero and variance ó2 ; Y is the area and production; t is the linear trend factor, â0 and â1 are the intercept and slope respectively. The hypothesis for testing of linear trend is H0 : β1 = 0 (Non-existence of linear trend factor) H1 : β1 ≠ 0 (Existence of linear trend factor) For the area and production of wheat crop the estimated linear trend model is shown in Table-1 and all the estimated parameters values were found to be highly

Table 2, reveals that for the area, the disturbance variance of irregular and slope components were found to be non-significant whereas the level component was significant. In the case of production irregular and the level components error variances were found to be significant and the slope component was non-significant.

The slope component was non-significant in area and production of wheat crop and the estimates suggested that the slope can be treated as constant, i.e., has zero variance. Since the slope component was non-significant, it might be useful to check if it could be dropped from the model. This could be checked by examining the significance analysis of the components.

The significant analysis of component helps to decide if slope can be dropped from the model after testing the following hypothesis,

H0 : Given component is non-significant

H1 : Given component is significant

The goodness of fit of the analysis of components is shown in the Table 3.

The results observed from the Table 3, the slope component was significant; it could not be dropped from the model and could be made deterministic by holding the value of its error variance fixed at zero. Also the level component was significant and cannot be dropped from the model, thus the model is a stochastic model. The contribution of irregular component is insignificant, but since it is a stochastic component, it cannot be dropped from the model.

Now the slope variance is fixed and free parameters are estimated and given in the Table 4.

After fixing the slope, the MSE, RMSE, MAPE and MPE values were calculated and presented in the Table

5. The highest adjusted R2 was obtained for production (99%) followed by area (98 %). The adjusted R2 for the estimated model has shown the closeness of the estimates to the actual value.

Brintha et al., (2014) employed UCM model for forecasting coconut production in Sri Lank during the period from 1950–51 to 2012-13. Their results revealed that both the trend components, level and slope, have non-stochastic process. Further, it revealed that the level was significant and slope was insignificant.

Residual analysis: The results presented in the Table 6 revealed that, the residuals due to UCM model were normally distributed for area and production, since the Kolmogorov-Smirnov as well as Shapiro – Wilks test statistics values were found to be non-significant at 5% level of significance.

The Ljung – Box test was applied to the residu-als of the fitted model. Results are shown in Table 7. The results reveal that all p-values at different lags exceeded 0.05 which indicated that the acceptance of model accu-racy at 95 % level.

Forecasting: The forecasted values of area and production by using UCM model are given Table 8, respectively along with 95% Confidence Limits. The first four forecast values are just to compare with the observed series to get an idea of the quality of the forecast. The residuals were found to be very small. The forecast’s values were very close to the actual data. This implies that the model specification is adequate.

From the Table 8 it is observed that forecast using UCM shows an increasing trend for area and production of Wheat. The validity of these forecasts can be checked when the actual data is available for the lead years. The trends in area and production are depicted in the Figures 4 and 5 respectively.

CONCLUSION

UCM with slope variance seems to fit the annual wheat area and production data well. Forecasted error percentage for the year 2012-13 and 2013-14 for wheat area 2.73 while 0.94 respectively. Obtained model predicted the annual wheat area of 30.34 million hectares and the 95 % CI is from 28.66 to 32.03 where as production is 95.01 million tones with CI is from 88.80 to 101.21. Adjusted R2 for the estimated model was reveals the closeness of the estimates to the actual (observed) value. The results revealed that an increasing trend has been observed in area and production of wheat crop in India. UCM models can effectively be utilized for the time series modeling of agricultural crop area and production, especially that are of non-stationary.

REFERENCES

  1. Alan, P. 1983. Forecasting with univariate Box-Jenkins Models- Concepts and Cases. John Wiley & Sons, New York
  2. Box, G.E.P., Jenkins, G.M and Reinsel, G.C. 1994. Time series analysis: Forecasting and Control. Third edition. Prentice Hall, New Jersey.
  3. Brintha, N.K.K., Samita, S., Abeynayake, N.R., Idirisinghe, I.M.S.K and Kumarathunga, A.M.D.P. 2014. Use of Unobserved Components Model for Forecasting Non-Stationary Time Series: A case of Annual National Coconut Production in Sri Lanka. Tropical Agricultural Research. 25(4): 523 – 431.
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