The regression analysis of non-stationary time series produces spurious results, which can be misleading (Ghafoor et al. 2009). The most appropriate method to deal with non-stationary time series for estimating long-run equilibrium relationships is cointegration, which necessitates that time series should be integrated of the same order. Augmented Dickey-Fuller (ADF) is used to verify the order of integration for each individual series. The ADF test, tests the null hypothesis of unit root for each individual time series. The rejection of the null hypothesis indicates that the series is non-stationary and vice-versa (Dickey and Fuller, 1981). The number of the appropriate lag for ADF is chosen for the absence of serial correlation using and Shwartz Information Criterion (SIC). The ADF test is based on the Ordinary Least Squares (OLS) method and requires the estimation of the following model. ΔPt=a0+δ1t+γPt−1+∑j=1qϑjΔPt−j+εt

Where, P = the price in each market; ∆ = difference parameter (i.e., ∆P1 = Pt – Pt–1, Pt–1 = Pt–1 – Pt–2 and Pn–1 = P_n–1) – P_n–2); α₀ = constant or drift; t = time trend variable; q = number of lag length and ∊_t = pure white error term.

The maximum likelihood (ML) method of cointegration is applied to check long-run wholesale prices relation between the selected markets of India (Johansen, 1988); (Johansen and Juselius, 1990). The starting point of the ML method is the vector autoregressive model of order (k) and may be written as: Pt=∑i=1kAiPt−1+μ+βt+εt(t=1,2,3…)

Where π denotes the (nx1) vector of non-stationary or integrated at order one, i.e., I (1). The procedure for estimating the cointegration vectors is based on the error correction model (ECM) as given by: ΔPt=μ+πPt−1+∑i=1k−1τiΔPt−1+βμt+εt

Where, Γi = −(I−Πi,−……,T); i=1,2,……K−1; Π = −(I−Πi, −………Πk)

Both Γi and Пi are the nxn matrices of the coefficient conveying the short and long-run information respectively, µ is a constant term, t is a trend, and ∊_t is the n-dimensional vector of the residuals that are identically and independently distributed. The vector ∆P_t is stationary means P_t is integrated at order one I (1), which will make unbalance relation as long as П matrix has a full rank of k. In this respect, the equation can be solved by inverting the matrix П–1 for P_t and as a linear combination of stationary variables (Kirchgässner, et al. 2012). The stationary linear combination of the P_t determines by the rank of Π matrix. If the rank r of the matrix П r = 0 the matrix is null and the series underlying is stationary. If the rank of the matrix П is such that 0 < rank of (П) = r < n, then there are n x r cointegrating vectors. The central point of Johansen’s procedure is simply to decompose Π into two n×r matrices such that Π = αβ’. The decomposition of Π implies that the β’P_t are r stationary linear combination.

(Johansen and Juselius, 1990) proposed two likelihood ratio test statistics (Trace and Max Eigen test statistics) to determine the number of cointegrating vectors as follows: Jtrace =−T∑i=r+1Nln⁡ln⁡(1−λ^1)Jmax=−Tln⁡(1−λ^r+1)

Where r is the number of cointegrated vectors, λ^1 is the eigenvalue and λ^r+1 is the (r + 1)^th largest squared eigenvalue obtained from the matrix П and the T is the effective number of observation. The trace statistics tested the null hypothesis of r cointegrating vector(s) against the alternative hypothesis of n cointegrating relations. The Max Eigen statistic tested the null hypothesis (r = 0) as against the alternative of r + 1.

The notion of the Granger causality is that if the two variables are integrated of order one, i.e., I(1), then the most accepted way to know the causal relationship between them is the Granger Causality proposed by (Granger 1969). The existing study also performed the Granger Causality test, which explained that the wholesale price in market A causes the price in market B if and only if the past values of market A provide additional information for the forecast of market B. The testing procedure of the Granger Causality involves three steps. In the first step, the order of integration was tested by applying the augmented Dickey-Fuller. After confirming the integration, (Johansen and Juselius, 1990) maximum likelihood approach was used to comprehend the cointegration between the markets. The Johansen cointegration test explained that if cointegration exists among the variables, then Granger causality must also exist either unidirectional or bidirectional. The Granger causality involves estimation of the simple form of vector autoregressive model (VAR) and is presented as follows: PtA=∑i=1mαiPt−1A+∑j=1mβilPt−jB+μ1tPtB=∑i=1nγiPt−iSB+∑j=1n∂iPt−jA+μ2t

Where P_t is the wholesale prices and scripts A and B indicate the two separate markets, t is the time trend, µ_Aand µ_B are the error terms of both the model.

The above-mentioned two equations with respect to market A and B can be jointly tested using OLS and then conduct an F-test for the three different expressions.

Expression 1: [δ₁₁, δ₁₂ …δ_n] ≠ 0 and [∂₂₁, ∂₂₂,….. ∂_n] = 0

Expression 1 indicates the unidirectional causality from PtB to PtA denoted as PtB⟶PtA.

Expression 2: [δ₁₁, δ₁₂ …δ_n] = 0 and [∂₂₁ ,∂₂₂,….. ∂_n] ≠ 0

Expression 2 indicates the unidirectional causality from PtA to PtB denoted as PtA⟶PtB.

Expression 3: [δ₁₁, δ₁₂ …δ_n] ≠ 0 and [∂₂₁ ,∂₂₂,…..δ_n] ≠ 0

Expression 3 indicates the bidirectional causality between PtA to PtB denoted as PtA↔PtB.

When the sets of market A and market B coefficients are statistically significant, it is said to be Feedback, or bilateral causality (Gujarati, 2003). Unidirectional causality from market A to market B is indicated if the estimated coefficient on the lagged of market B is statistically different from zero and vice versa.

Granger causality test provides only the direction of causality for the selected time span. However, it fails to demonstrate the effect of shock on future values. The impulse response function shows a specific point of time t₀, that a shock originates from one equation and proceeds through the system (Kirchgässner et al. 2012). Generalized impulse response function was initially developed by (Koop et al. 1996), and since then, many have added for the development of both the theory and application of it. The existing study also applied the generalized impulse response as given below: IRFt+k=(μ,Pt,Pt−1…)=E|Pt+k|Pt=pt+μ,Pt−1=[pt−1…]−E[Pt+kPt=pt,Pt−1=Pt−1]

Where, IRF_t+k = Impulse Response Function, lower case letters, i.e., p represent realized values, and u is the impulse shock P_t–1 is the history.

The market integration among the selected apple markets was analyzed using Johansen’s cointegration method, which necessitates that the time series should be integrated at order one, i.e., I (1). Therefore, the standard Augmented Dickey-Fuller unit root test (ADF) was applied to determine the order of integration, and results have been presented in Table 2. The empirical evidence suggests that price series had unit root problems at their level form. The null hypothesis of the unit root at level form cannot be rejected for all the price series as absolute values of ADF statistics are well below the 5 percent critical values of test statistics. Thus, it is concluded that all the price series are nonstationary at their level forms. In order to test the level or number of unit roots in the data, a unit root test of first difference was conducted, which showed the number of unit roots to be equal to one, since the data became stationary after first differencing as absolute values of ADF statistics were greater than 5 percent critical values of the test statistic. Thus, the selected price series were integrated at order one, i.e., I (1), and the number of lag lengths was chosen as suggested by Schwartz Information Criterion (SIC).

The results of Johansen’s maximum likelihood approach (maximum eigen value and trace test) are given in Table 3. The Johansen’s procedure for the apple markets of India was applied by following three steps, firstly appropriate lag length was chosen as suggested by SIC criteria, secondly, the order of integration was confirmed by using ADF test; in the third step, two tests i.e., Trace and Max Eigen tests of Johansen’s approach based on the VAR (vector autoregressive model) were put into application to analyse the cointegrating vectors between selected apple markets. The first null hypothesis of Max Eigen statistics and Trace test, tests no cointegration (r = 0) against alternative hypothesis (r ≥ 1) of at least one cointegrated equation prevailed in the VAR system. Both these tests rejected the null hypothesis of no cointegration. Maximum Eigen values and Trace test statistic values were found higher than 5 per cent critical values and accepted the alternative of one or more cointegrating vectors. Similarly, the null hypothesis from r ≤ 1 to r ≤ 3, for Trace statistic were rejected against the alternative hypothesis from r ≥ 1 to r ≥ 3, as their critical values are less than the test statistic and the corresponding probability values were also less than 0.05. This implies that Trace statistic gives four cointegrating equation in overall cointegration of apple markets, while, Max Eigen value Statistic gives only three cointegrating equations. Therefore, keeping in view both the tests, we considered three cointegrating equations out of five cointegrating equations indicating that they are well integrated, and price signals are transferred from one market to another to ensure efficiency. Thus, Johansen’s cointegration tests have shown that even though the selected apple markets are geographically isolated and spatially segmented, they are well connected in terms of prices of apple, demonstrating that the apple markets have long-run price linkages across them. Kar et al. (2014) and Baeg and Singla (2014) have reported similar findings.

The causal relation between the selected price series of apple markets was examined through Granger causality technique. Granger’s causality shows the direction of price transmission between two markets and related spatial arbitrage, i.e., physical movement of a commodity to adjust the price differences (Gafoor et al. 2009). The results of Granger’s causality are shown in Table 4, which shows that all the four F- statistics for the causality tests of wholesale prices in the Shimla market are statistically significant. The null hypothesis of no Granger’s causality was rejected in the case of Shimla. Besides, Chandigarh has three, while Delhi and Bengaluru have two each, and Mumbai has one F-statistics statistically significant on other market prices.

The results of Granger’s causality revealed that unidirectional causality was found between market pairs; Shimla-Chandigarh, Delhi-Chandigarh wholesale markets, meaning that a price change in the former market in each pair Granger cause price change in the latter market and same is not feedbacked by the price change in the former market in each pair. There exists bidirectional causality between Mumbai-Shimla, Delhi-Shimla, Bengaluru-Shimla, and Bengaluru-Chandigarh. In these cases, the former market in each pair Granger causes the wholesale price formation in the latter market, which in turn provides the feedback to the former market as well. Further, four market pairs, Delhi-Mumbai, Chandigarh- Mumbai, Bengaluru-Mumbai, and Bengaluru- Delhi, were found to have no direct causality between them.

Therefore, it was further concluded from the table that since all the four F-statistics for the causality tests of wholesale prices in the Shimla market are statistically significant. Therefore, the Shimla market is holding a key position in price determination in other markets.

Impulse response function was used to determine the relative strength of causality effect beyond the selected time span, as causality tests are inappropriate because these tests are unable to show how much feedback exists from one variable to the other beyond the selected sample period (Rehman and Shahbaz, 2013). The best way to interpret the implications of the models for the patterns of price transmission, causality, and adjustments is to consider the time paths of prices after exogenous shock i.e., impulse response. The impulse response function explicates the responsiveness of one of the endogenous variables due to the shock on the current and future values of all the other endogenous variables in the VAR system. The shock affects the variable itself and is transmitted to the rest of the explanatory variables (Bhanumurthy et al. 2012). The results of impulse response function analysis are presented in Fig. 4.11(a) to 4.11(e). Figure 4.11(a) presents the response of Shimla to a standard deviation shock given to Mumbai, Chandigarh, Delhi and Bengaluru apple prices. The Shimla apple prices reacted to it immediately, plummeting initially for 4 to 5 months and then stabilized for the remaining period.

The results of impulse response function in Chandigarh market were almost similar to those of Shimla market, i.e., the prices dropped immediately during the past 4 months but stabilized thereafter. In Delhi, market price shock of one-unit standard deviation resulted in an immediate decline in prices which then stabilized after 4 to 5 months. The wholesale prices in the Delhi market were observed to be inversely related to those of the Mumbai market for the initial 5 months. In the Bengaluru market, the results of the impulse response function revealed a sharp decline in prices. Similarly, the wholesale prices in the Mumbai market were observed to be inversely related to Bengaluru up to the first 3 months. In the Mumbai market, prices reacted immediately by going down and then stabilized after 4 to 5 months. The overall results of the impulse response function explicate that the responses exhibit large magnitudes over 2600 unit standard deviations. Moreover, the price information process is quick for all the selected apple markets as they respond immediately to a shock that seems to fade away in 4 to 5 months.