Switching Regression

Consider the following selection model that describes farmers' choices of joining the contract and their performance with and without the contract:

If _γZ_i + u_i > 0, farmer i chooses to join the contract, which is described by = I_i = 1;

If _γZ_i + u_i ≤ 0, farmer i chooses not to join the contract, which is described by = I_i = 0;

Farmer i's profitability with the contract (I_i = 1) is y_1i = β₁X_1i + ε_1i;

Farmer i's profitability with the contract (I_i = 0) is y_0i = β₀X_0i + ε_0i;

In the model, Z_i is a vector of farm characteristics that affect farmers’ decision to join the contract; X_1i and X_0i are two vectors of farm characteristics that affect farmers’ performance under the contract and without the contract; and y_1i and y_0i are dependent variables measuring farmers’ profitability. γ, β₁ and β₀ are vectors of parameters to be estimated. u_i, ε_1i, and ε_0i are three random error terms that follow a trivariate normal distribution.

After the parameters are estimated, we can calculate:

xb_1i = E(y_1i|x_1i) = x_1iβ_1i

xb_0i = E(y_0i|x_0i) = x_0iβ_0i

yc_{1_1i} = E(y_1i|I_i = 1, x_1i) = x_1iβ₁ + σ₁ρ₁f(γZ_i) | F(γZ_i)

yc_{0_1i} = E(y_0i|I_i = 1, x_1i) = x_1iβ₀ + σ₀ρ₀f(γZ_i) | F(γZ_i)

yc_{0_0i} = E(y_0i|I_i = 0, x_0i) = x_0iβ₀ - σ₀ρ₀f(γZ_i) | [ 1 - F(γZ_i)]

yc_{1_0i} = E(y_1i|I_i = 0, x_0i) = x_0iβ₁ - σ₁ρ₁f(γZ_i) | [ 1 - F(γZ_i)]

xb_1i represents the unconditional expectation of farmers' performance under the contract; xb_0i represents the unconditional expectation of farmers' performance without the contract; yc_{1_1i} represents the conditional expectation of contract farmers' performance under the contract; yc_{0_1i} represents the conditional expectation of contract farmers' performance without the contract; yc_{0_0i} represents the conditional expectation of non-contract farmers' performance without the contract; and yc1_0i represents the conditional expectation of noncontract farmers' performance under the contract. σ₁ and σ₀ are the standard errors of ε_1i, and ε_1i; ρ₁ is the correlation coefficient between ε_1i and u_i; ρ₀ is the correlation coefficient between ε_0i and μ_i; f(.) is the normal density function; and F[.] is the cumulative normal distribution.

Indicators for premiums of joining the contract

yc_{1_1i} and yc_{0_1i} represent, respectively, the average of contract farmers’ actual performance under the contract and the average of their counterfactual performance without the contract. The difference Π₁ = yc_{1_1i} − yc_{0_1i} provides a measure of the impact of contract farming on the performance of farmers who actually chose to join the contract. Π₁ > 0 (or Π₁ < 0 ) would indicate a positive (or negative) impact of contract farming. Similarly, Π₀ = yc_{1_0i} − yc_{0_0i} provides a measure of the impact of contract farming on the performance of farmers who actually chose not to join the contract.

Indicators for selection bias

The estimated correlation coefficients, ρ₀ and ρ₁ , provide interesting insights of the sampled farms in choosing the contractual arrangement. For example, ρ > 0 would indicate that farms that actually chose to enter the contractual arrangement have above average performance under the contract. The average performance in this case is defined as x₁β₁, assuming all farms in the sample were subjected to the contractual arrangement. In other words, a positive ρ₁ implies “positive selection” into choosing the contract.

Furthermore, if non-contract farms had in fact chosen to join the contract, their performance would be worse than those farms that actually chose to enter the contract. On the other hand, ρ₁ < 0 implies “negative selection” into choosing the contract, or farms that actually chose to enter the contractual arrangement have below-average performance under the contract. In this case, if the non-contract farms had in fact chosen to join the contract, their performance would have been above that of the contracted farms.

Conversely, ρ₀ > 0 implies “negative selection” into not choosing the contract for the noncontract farms. In other words, non-contract farms have below-average performance, and if the contract farms had in fact chosen not to join the contract, their performance would have been better than that of the non-contract farms.

If ρ₀ < 0 , there is “positive selection” into not choosing the contract for the non-contract farms, or farms that actually chose not to enter the contract have above average performance without the contract. In this case, if the contract farms had in fact chosen to not join the contract, their performance would have been worse than that of the non-contract farms.

Following Maddala (1983) and Hamilton and Nickerson (2003) but using the correlation coefficients instead of the covariances, four interesting cases can be discerned from the two correlation coefficients.

Case 1: ρ₀ < 0 and ρ₁ > 0

In this case, farms that chose to enter the contractual agreement have above average performance under the contract, while farms that chose to stay outside the contract have above average performance without the contract.³ In other words, both contract and noncontract farms chose the correct or appropriate tactics by which they have relative advantage. This case may be characterized as a situation where both contract and noncontract farms are in fact capturing their “comparative advantage.”

Case 2: ρ₀ > 0 and ρ₁ > 0

In this case, farms that actually chose to enter the contract (i.e., the contract farms) would have above-average performance whether they are under the contract or without the contract. In other words, contract farms have an “absolute advantage” in the sense that they have above-average performance with or without the contract. Conversely, non-contract farms in general have below-average performance whether they are under the contract or without the contract.

Case 3: ρ₀ < 0 and ρ₁ < 0

In contrast to case 2, non-contract farms in this case have an “absolute advantage” in the sense that they tend to have below-average performance both under the contract and without the contract, while contract farms have below-average performance both under the contract and without the contract.

Case 4: ρ₀ > 0 and ρ₁ < 0

In this case, contract farms would in general have below-average performance under the contract but above-average performance without the contract, while non-contract farms would have above-average performance under the contract but below-average performance without the contract. In this sense, farms chose the tactics that provide them “comparative disadvantage.” This would not happen most of the time except when there are factors that may force farms to adopt less-desirable tactics.

Comparison of contract farmers’ and non-contract farmers’ profitability in commercial rice farming

Based on the above switching regression model, we use “movestay” module (Lokshin and Sajaia, 2004) in the STATA program to evaluate factors that affect farmers’ decisions to join the contract and their performance with or without the contract. We measure farmers’ performance by their profits per hectare in their commercial operations.

The selection model includes the following variables: household characteristics, including family size and ratio of females in the household; and farm characteristics, including farm size, value of production assets, value of consumption assets, value of transportation assets, the distance of the farm to the market and the distance of the farm to the highway. The profit functions⁴ include farm size, family size, and the value of consumption assets. The estimated results of the selection model and profit functions are presented in Appendix Table A.1 [ PDF 20.5KB | 1 page ] and Table A.2 [ PDF 20.5KB | 1 page ], respectively. The overall model is significant at the 10% level as indicated by Wald’s X².

Using the indicators described above, the premiums from joining the contract and their selection bias indicators are calculated. Figure 2 depicts the distribution of contract and noncontract farmers’ profits under contract and without the contract.

The counterfactual analysis indicates that both contract and non-contract farmers tend to increase their profitability by joining the contract. The contract farmers’ profits under contract bottom left graph) are on average higher than their counterfactual profits without the contract (top left graph). Joining the contract is estimated to have increased the profits of contract farmers by 4.63 million kip. In the case of non-contract farmers, the counterfactual profits under contract (bottom right graph) are on average higher than the actual profits outside the contract (top right graph). In other words, the profits of non-contract farmers would have increased by 3.21 million kip had they joined the contract.

As shown in Appendix Table A.1, the estimated ρ₀ and ρ₁ are both negative, although ρ₁ is not statistically significant. This pattern is described above as case 3, indicating that contract farmers have below-average performance both under contract and without the contract. In other words, contract farmers are less profitable than non-contract farmers, both under contract and without the contract. This suggests that the observed higher profitability of contract farming is not due to contract farming attracting more profitable farms; rather, contract farming tends to be more attractive and more beneficial to farmers with relatively low performance.

Figure 3: Profitability Comparison of Contract and Non-Contract Farmers [ PDF 24.4KB | 1 page ]

Download this Discussion Paper [ PDF 127.3KB| 24 pages ].

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