Marginal Effects

Hill et al (2008) shows us that we can examine the effect of a 1 unit change in W‟ on the probability that Y1 =1 by considering the derivative

0 .0 1 .0 2 .0 3 .0 4 D e n si ty 0 20 40 60 80 100 yrs_farm 3) 2) 1)

31 ( ) ( )

Where t = and ( ) is a standard normal probability density function evaluated at . To get the result, we can use chain rule of differentiation. This result gives us the marginal effect of dependent variables on cover crop adoption.

Greene (2008) shows in more details how to calculate marginal effect for the probit model. He suggests calculating standard errors, using the linear approximation approach.

For predicted probabilities,

[ ̂] 0 ̂ ̂1 0 ̂ ̂1 where [ ̂]

As described earlier, there can be three different asymptotic covariance matrices of ̂, let ̂. Then the derivative vector is given by the following equation

0 ̂ ̂1 0 ̂ 1 0 ̂1 ̂ After combining the terms we get

[ ̂ ] ̂

which depends on the particular x variable used. This result is useful when the marginal effect is counted for a dummy variable. In this case, the estimated effect is

̂ ̂ ( ) ̂ ( ) The asymptotic variance would be

[ ̂] * ̂ _̂+ * ̂ _̂+ where 4) 5) 6) 7) 8) 9) 10)

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) 0 ̂

̂ 1 ̂ (

̅( )_{) ̂ (} ̅( )₎

The matrix of marginal effects is ̂ . ̂ ̂ / ̂ . ̂ / . ̂ / ̂ . ̂ / ̂ For probit model

so [ ̂] [ ( ) ] [ ( ) ] 3.2.1.2. Censored Data

Nitrogen use by producers is considered as censored data, because a substantial fraction of the observations on the dependent variable take a limit value, which can be equal to zero.

When the dependent variable is censored, we cannot use the least squares method to obtain regression parameters because the parameters obtained by least squares are biased and inconsistent if the data are censored. Our dependent variable is a variable with quantitative meaning, y*, and we are interested in the population regression E(y*). If y* were observed for the population we could use least squares. But the data problem arises in that y* is censored from below, when some part of the population does not use nitrogen (nitrogen use equals 0).

When a distribution is censored on the left, observations with values at or below τ are set to τy.

{

The use of τ and τy are just a generalization of having τ and τy set at 0. If a continuous

variable y has a pdf f(y) and τ is constant, then we have ( ) [ ( )] [ ( )]

12)

14)

15) 13)

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So we see that the density function of y is the same as that for y* for y > τ, and is equal to the probability of observing y* < if y* = . d is an indicator variable that equals 1 if y* > τ, so the observation is uncensored and is equal to 0 if y = τ, so the observation is biased.

The probability of the censored distribution would be,

( ) ( ) ( ) ( ) and

( ) ( ) ( ) Thus, the likelihood function can be written as

∏ * ( )+ * ( )+ 3.2.1.3. Tobit Model

So as we see, we cannot use OLS for our censored data, but we can use the Tobit model. This model recognizes that we have two types of data, the limit observations (y = 0) and non- linear observation (y > 0) (Hill at al. 2008).

Greene gives us the structural equation of the Tobit model which is:

where ( ). y* is a latent variable that is observed for values greater than and censored otherwise. The observed defined by following measurement equation

{

In the typical Tobit model, we assume that = 0, i.e. the data are censored at 0. Thus, we have { 19) 20) 21) 16) 17) 18)

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While nitrogen use by producers is considered as a left censored variable, adoption of cover crops is considered as an endogenous dummy variable. The resulting system is a LDV model defined by the amount of nitrogen use by producers, with an endogenous dummy variable that investigates whether the producer adopts cover crops. Because the censoring precludes unique or sensible solutions for the reduced forms, a condition must be imposed in a system of censored dependent variables (Heckman, 2001). The structural form of the model is given by

We assume that = is continuously observed such that

{

where Y2 represents the amount of nitrogen use by producers per acre and is censored at

zero. X‟ is a vector of independent exogenous variables, β is a vector of parameter estimates, and Y2 represents the probability of adopting cover crops. The amount of nitrogen use is dependent

on exogenous variables X and a dummy variable Y2 representing the probability of adopting a

cover crop, which is potentially endogenous.

We already showed the likelihood function for the censored normal distribution (equation 8), where is the censoring point. In the traditional Tobit model, is set to zero, and substitute µ for Xiβ. This gives us the likelihood function for the Tobit model.

∏ * ( )+ * ( )+ The log likelihood function for the Tobit model is

∑ , ( ( )) ( ) ( ( ))- 22)

23)

24)

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The overall log-likelihood is made up of two parts. The first corresponds to the classical regression for the uncensored observations, while the second part corresponds to the relevant probabilities that an observation is censored.

3.2.1.3.1. Marginal Effects

There are potentially three conditional mean fractions to consider, depending on the purpose of the study (Greene). As there are three expected values, there are three marginal effects as well; marginal effect on the latent dependent variable, y*, marginal effect on the expected value for y for uncensored observations, and marginal effect on the expected value for y (censored and uncensored). We can calculate the marginal effect on the latent dependent variable

y* by the following equation;

[ ]

Greene (2000) suggests that McDonald and Moffitt‟s (1980) decomposition is very useful tool for calculating marginal effects for Tobit model. Marginal effect on the expected value for y (censored and uncensored observations);

[ ]

( ) which can be rewritten as the following,

[ ] ( ) [ ] ) [ ]) ( )

Thus, the reported Tobit coefficients indicate how a one unit change in an independent variable xk alters the latent variable.

3.2.1.4. Endogeneity

As we already mentioned above, the structural form of the tobit model is given by 26)

27)

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The amount of nitrogen use is dependent on exogenous variables X and a dummy variable Y2 representing the probability of adopting a cover crop, which is potentially

endogenous. Probability of adoption of cover crops is dependent on Z variables which are uncorrelated with ui. Endogeneity tests of acres of GM corn planted and hours worked off the farm are considered. We use the Smith Blundell test to determine exogeneity as proposed by Baum (1999), who computes a test for exogeneity based on the Smith and Blundell‟s test (1989) where, under the null hypothesis, the models are appropriately specified with all explanatory variables as exogenous. Under the alternative hypothesis, the suspected endogenous variables are expressed as linear projections of a set of instruments, and the residuals from the first stage regressions are added to the model.

Considerable literature has evolved in the use of limited dependent variable model with endogenous dummy variable. Amemiya (1974) considers a model in which all endogenous variables are truncated to zero, revealing certain necessary restrictions on the model and suggesting a method of estimation using the indirect least squares method. Nelson and Olson (1978) proposed a two-stage least squares procedure for Tobit analysis proving that the estimates are asymptotically normal. More recent studies have applied these models for specifying effects on adoption of technologies including Blundell and Smith (1989) who compared estimates of marginal and marginal and new conditional maximum likelihood procedures. Goodwin and Mishra (2004) used the simultaneous equation framework to determine multiple job holdings and resulting effects on farming efficiency. A more detailed discussion on use of LDV with dummy endogenous model is presented by Angrist, J.D. (2001).

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CHAPTER 4: RESULTS