Selection of markets

            Selection of cauliflower markets will be decided on the basis of maximum arrivals of cauliflower in the market. Major five markets of the state will be selected on the basis of maximum arrivals of cauliflower for the present study.

          Nature and sources of data 

            The time series data on monthly arrivals and price of cauliflower required for the study will be collected from the registers maintained by respective APMCs. These markets maintain data on daily, monthly and yearly arrivals and prices of agricultural commodities. The data on arrivals refers to the total arrivals during the month in quintals in a market place. The data on prices refer to modal prices in a month. Modal price is considered superior to the monthly average price as it represents the major proportion of the commodity marketed during the month in a particular market.

            For the studying the temporal fluctuations in arrivals and prices of cauliflower, monthly secondary data on arrivals and prices will be collected from the selected markets for period from 2008 to 2020.

Analytical Techniques

1 .Time series analysis

Time series analysis was used to study the variations in monthly prices and arrivals for the period of 10-15 years. Time series is a complex mixture of four components namely, Trend (T), Seasonal variations (S), Cyclical fluctuations (C) and Irregular fluctuations (I). These four types of movements are frequently found either separately or in combination in a time series. The relationship among these components is assumed to be additive or multiplicative, but the multiplicative model is the most commonly used method in economic analysis, which can be represented as

OT = T x C x S x I

Where,

OT = Original observation at time't'

T = Trend component

S = Seasonal variations

 C = Cyclical element

I = Irregular fluctuations

Linear trend (T)

Over a long period of time, time series is very likely to show a tendency to increase or decrease over time. The factors responsible for such changes in time series are the growth of population, change in the taste of people, technological advances in the field etc. There are different types of trends, some of them are linear and some are nonlinear in their form. For shorter period of time, in most of the situations the linear form provides the best description of trend and for longer period of time, the non-linear form generally provides a good description of the trend. Often, it may be possible to describe such movements with a structured mathematical model. In the absence of such a definite format, approximately a polynomial or a free hand could describe the movements.

Seasonal variation (S): The variation in a year is called as seasonal variation. The main causes of seasonal variations are customs, traditions, climate etc. Such seasonal components can be analyzed through harmonic analysis.

Cyclical movements (C): Cyclical movements are fluctuations which differ from periodic movements. Cyclical movements have longer duration than a year and have periodicity of several years as in business cycles.

Irregular variations (I): Here the effects could be completely unpredictable, changing in a random manner. A given observation is affected by episodic and accidental factors. These are also known as causal series and are affected by the unknown causes. These unknown causes act in an unpredictable manner.

Analysis of long-term movements (trend)

For estimating the long run trend of arrivals and prices, the method of least squares estimate Will employed. This method of ascertaining the trend in a series of annual arrivals and prices involves estimating the co-efficient of intercept (a) and slope (b) in the linear functional form. The tentative equation adopted for this purpose will specified as follows.

Yt = a + bt

Yt = Trend values at time t

t = Period

a = Intercept

b = Slope

Annual trends of prices and arrivals for the selected markets were computed and will be compared. The goodness of fit of trend line to the data will be tested by computing the coefficient of multiple determinations which is denoted by R2.

2. Estimation of seasonal indices of monthly data

To measure the seasonal variations in prices and arrivals, seasonal indices will be calculated employing twelve months ratio to moving average method.

The seasonal indices will be calculated by adopting the following steps

1. Generation of a series of 12 months moving totals
2. Generation of  a series of 12 months moving averages: A series of 12 months moving averages will be generated by dividing 12 months moving totals by 12
3. Generation of   a series of centered 12 months moving averages. This step involves taking averages of pairs of two subsequent 12 months moving averages and entering between each pair. There are no corresponding moving averages for the first six and last six months.
4. Expressing each original value as a percentage of corresponding centered moving average value. The percentage of moving average will represent indices of seasonal and irregular components combined.
5. The next  will step involve removing the irregular component.
6.  Arrange the percentages of moving averages in the form of monthly arrays.
7. Next, the average index for each month is calculated.
8. These averages will be adjusted in such a way that their sum becomes 1200. This can be done by working out correction factor and multiplying the average for each month by this correction factor. The correction factor (K) will be worked out as follows.

K = 1200/ S

Where, K is correction factor and S is sum of average indices for 12 months, multiply K with the percentage of moving average for each month to obtain the seasonal indices.

Regression analysis

            This analysis will be carried out to ascertain the response of the prices on the arrivals. The arrivals and prices are subjected to change over period of time due to innovations, supply of more inputs and increase in population. The econometric model used for the purpose can be stated as:

y = a + bx + e

where,

y = Arrivals

a = Intercept

b = Slope or regression coefficient

x = Prices

As shown in the equation, y will be assumed to be the dependent variable while,x will be taken as an independent variable

e = Error

The Karl Pearsons correlation coefficient

The Karl Pearsons correlation coefficient will be computed to estimate the degree of relationship between markets arrivals and prices.

r =     XY-XYnX2-(∑X) ² n                  Y2- (∑Y) ²n

Where,

r = Correlation coefficient

x = Prices of cauliflower in selected markets

y = Arrivals of cauliflower in selected markets

N = Number of observations or time in number of years

3. Estimation of market integration

For examining the integration between the markets the co- integration technique will be used. Integrated markets are those markets, where prices are determined independently.

To examine the price relationship between two markets, the following basic relationship commonly used to test for the existence of markets integration

Pij = aij + aiPj + Ei

Where,

                        Pi and Pj are the price series of a specific commodity in two markets iand j

                        E = residual term

Before further analysis, the data will be  checked for the stationary of variables. The AngmentedDicky Fuller test (ADF test) will be used to determine the stationarity of the variables.

If 0 < rank (II) = r < n, and there are n × r matrix of α and β such that π = αβ, then it is said that markets are integrated.

4. Estimation of price forecasting models

ARIMA Model

     In general, an ARIMA model is characterized by the notation ARIMA (p,d,q) where,p, d and q denote orders of auto-regression, integration (differencing) and moving average respectively. In ARIMA parlance, TS is a linear function of past actual values and random shocks. For instance, given a TS process {yt}, a first order auto-regressive process is denoted by ARIMA (1,0,0) or simply AR(1) and is given by

yt  =  µ+ φ1y t-1 + ɛt

and a first order moving average process is denoted by ARIMA (0,0,1) or simply MA(1) and is given by

yt  = µ - ѳ1ɛt-1 + ɛt

Alternatively, the model ultimately derived, may be a mixture of these processes and of higher orders as well. Thus a stationary ARMA (p, q) process is defined by the equation

yt =  φ1y t-1+φ2y t-2+…+ φp y t-p - ѳ1ɛt-1-ѳ2ɛt-2 -…-ѳqɛt-q +ɛt

where ɛt‘s are independently and normally distributed with zero mean and constant variance σ2 for t = 1,2,...n. Note here that the values of p and q, in practice lie between 0 and 3.

Suppose that Yt is a time series the ARIMA (p, d, q) model is written as:

φ(????)(1−????)????????????=????(????)???????? …(1)

Where,

????????~(0,????2),???????? indicates white noise,

φ(????)=1−????1????−????2????2−∙∙∙∙∙∙∙∙∙−????????????????and

(????)=1−????1????−????2????2−∙∙∙∙∙∙∙∙∙−????????????????.

The integration parameter d is non-negative integer.

            The ARIMA methodology is carried out in three stages, viz. identification, estimation and diagnostic checking. Parameters of the tentatively selected ARIMA model at the identification stage are estimated at the estimation stage and adequacy of tentatively selected model is tested at the diagnostic checking stage. If the model is found to be inadequate, the three stages are repeated until satisfactory ARIMA model is selected for the time- series under consideration. A detail  discussion on various aspects of this approach is given in Box et al. (2007).

ARCH Model

Autoregressive conditional heteroscedastic (ARCH) model, was introduced by Engle in 1982. This model allows the conditional variance to change over time as a function of squared past errors leaving the unconditional variance constant. The presence of ARCH type effects in financial and macro-economic time series is well established fact. The combination of ARCH specification for conditional variance and the Autoregressive (AR) specification for conditional mean has many appealing features, including a better specification of the forecast error variance.

The ARCH (q) model for series (Ɛt)is defined by specifying the conditional distribution of  Ɛt giveninformation available up to time t-1. Let ψt-1denote this information. It consists of the knowledge of all available values of the series and anything which can be computed from these values. In principle, it may include knowledge of the values of other related time series, and anything else which might be useful for forecasting and is available by time t-1.

The process Ɛtin ARCH (q), if the conditional distribution of Ɛtfor given available information ψt-1is

Where (Ɛt) is stochastic error condition on the realized values of the set of variables ψt-1 = (yt-1, xt-1, yt-2, xt- 2,+.......), ht is conditional variance and N is number ofvariables.

GARCH Model

            The ARCH(q) model for the series {εt} is defined by specifying the conditional distribution of εt given the information available up to time t-1. Let ψt-1 denote this information. ARCH (q) model for the series {εt} is given by

            ɛt/ψt-1 ~ N(0,ht)            ….(2)

ht=a0+i=1qɑiεt-12               ….(3)

            where, a0>, ai≥ 0, for all i and<1  are requiredto be satisfied to ensure non-negative and finite unconditional variance of stationary {εt} series.

            However, ARCH model has some drawbacks. Firstly, when the order of ARCH model is very large,estimation of a large number of parameters is required. Secondly, conditional variance of ARCH(q) model has the property that unconditional autocorrelation function (ACF) of squared residuals; if it exists, decays very rapidly compared to what is typically observed, unless maximum lag q is large. To overcome the weaknesses of ARCH model, Bollerslev (1986) proposed the Generalized ARCH (GARCH) model in which conditional variance is also a linear function of its own lags and has the following form :

            εt= ξtht1/2

            ht=ɑ0+i=1qɑiεt-12+j=1pbjht-j             ….(4)

which shows that the denominator of the prior fraction is positive.

EGARCH Model

            The EGARCH model will be developed to allow for asymmetric effects between positive and negative shocks on the conditional variance of future observations. Another advantage, as pointed out by Nelson and Cao (1992), is that there are no restrictions on the parameters. In the EGARCH model, the conditional variance, ht, is an asymmetric function of lagged disturbances. The model is given by

            εt= ξtht1/2

where, ξt ~ IID(0,1). A sufficient condition for the conditional variance to be positive is:a0> 0, ai≥0, i = 1,2,...,q. bj≥0, j = 1,2,..., p

The GARCH (p, q) process is weakly stationary if and only if

i=1qɑi+j=1pbj<1

                The conditional variance defined by Equation (4) has the property that the unconditional autocorrelation function of εt2; if it exists, can decay slowly. For the ARCH family, the decay rate is too rapid compared to what is typically observed in financial time-series, unless the maximum lag q is long. As Equation (4) is a more parsimonious model of the conditional variance than a high-order ARCH model, most users prefer it to the simpler ARCH alternative.

            The most popular GARCH model in applications is the GARCH (1,1) model. To express GARCH model in terms of ARMA model, we denote ηt = εt2 – ht. Then from Equation (4), we get,

            εt2=α0+i=1Max(p,q)(αi+bi)εt-i2+ƞt+j=1pbjƞt-j          ….(5)

Thus, a GARCH model can be regarded as an extension of the ARMA approach to squared series {εt2}. Using the unconditional mean of an ARMA model, we have

            Eεt2=α01-i=1Max(p,q)(αi+bi)                   ….(6)  

            lnht=α0+1+biB+…+bq-1Bq-11-αiB+…+αpBpg(εt-1)                    ….(7)

where,

            gεt=θ+γεt-γEεt,   ifεt≥0,θ+γεt-γEεt,   ifεt<0,

B is the backshift (or lag) operator such that

            Bgεt=g(εt-1)

The EGARCH model can also be represented in another way by specifying the logarithm of conditional variance as

lnht=α0+βlnht-1+αεt-1ht-1+γεt-1ht-1          ….(8)

            This implies that the leverage effect is exponential, rather than quadratic, and the forecasts of the conditional variance are guaranteed to be non-negative.

Progress of the project: Data collection work in progress