# 经济代写|计量经济学代写Econometrics代考|A more mathematical approach

## 经济代写|计量经济学代写Econometrics代考|A more mathematical approach

Suppose we want to model:
$$D_i=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}+u_i$$
where $D_i$ is a dichotomous dummy variable as in the problem of labour force participation discussed earlier. To motivate the probit model, assume that the decision to join the work force or not depends on an unobserved variable (also known as a latent variable) $Z_i$ that is determined by other observable variables (say, level of family income as in our previous example) such as:
$$Z_i=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}$$
and
$$P_i=F\left(Z_i\right)$$

If we assume normal distribution, then the $F\left(Z_i\right)$ comes from the normal cumulative density function given by:
$$F\left(Z_i\right)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{Z_i} e^{-\frac{Z^2}{2}} d Z$$
Expressing $Z$ as the inverse of the normal cumulative density function we have:
$$Z_i=F^{-1}\left(P_i\right)=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}$$
which is the expression for the probit model.
The model is estimated by applying the maximum-likelihood method to Equation (12.36), but the results obtained from the use of any statistical software are given in the form of Equation (12.37).

The interpretation of the marginal effect is obtained by differentiation in order to calculate $\partial P / \partial X_i$ which in this case is:
$$\frac{\partial P}{\partial X_i}=\frac{\partial P}{\partial Z} \frac{\partial Z}{\partial X_i}=F^{\prime}(Z) \beta_i$$
Since $F\left(Z_i\right)$ is the standard normal cumulative distribution, $F^{\prime}\left(Z_i\right)$ is just the standard normal distribution itself given by:
$$F^{\prime}\left(Z_i\right)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{Z^2}{2}}$$
In order to obtain a statistic for the marginal effect, we first calculate $Z$ for the mean values of the explanatory variables, then calculate $F^{\prime}\left(Z_i\right)$ from Equation (12.37) and then multiply this result by $\beta_i$ to get the final result, as in Equation (12.38).
The overall goodness of fit is examined as for the logit model.

## 经济代写|计量经济学代写Econometrics代考|Multinomial logit and probit models

Multinomial logit and probit models are multi-equation models. A dummy dependent variable with $k$ categories will create $k-1$ equations (and cases to examine). This is easy to see if we consider that for the dichotomous dummy $D=(1,0)$ we have only one logit/probit equation to capture the probability that the one or the other will be chosen. Therefore, if we have a trichotomous (with three different choices) variable we shall need two equations, and for a $k$ categories variable, $k-1$ equations.

Consider the example given before. We have a firm that is planning to make a takeover bid by means of (a) cash, (b) shares or (c) a mixture. Therefore, we have a response variable with three levels. We can define these variable levels as follows:
\begin{aligned} D_S & = \begin{cases}1 & \text { if the takeover is by shares } \ 0 & \text { if otherwise }\end{cases} \ D_C & = \begin{cases}1 & \text { if the takeover is by cash } \ 0 & \text { if otherwise }\end{cases} \ D_M & = \begin{cases}1 & \text { if the takeover is by a mixture } \ 0 & \text { if otherwise }\end{cases} \end{aligned}
Note that we need only two of the three dummies presented here, because one dummy will be reserved as the reference point. Therefore, we have two equations:
\begin{aligned} & D_S=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}+u_i \ & D_C=a_1+a_2 X_{2 i}+a_3 X_{3 i}+\cdots+a_k X_{k i}+v_i \end{aligned}
which can be estimated by either the logit or the probit method, based on the assumption to be made for the distribution of the disturbances.

The fitted values of the two equations can be interpreted as the probabilities of using the method of takeover described by each equation. Since all three alternatives should add to one, by subtracting the two obtained probabilities from unity we can derive the probability for the takeover by using a mixed strategy.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|A more mathematical approach

$$D_i=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}+u_i$$

$$Z_i=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}$$

$$P_i=F\left(Z_i\right)$$

$$F\left(Z_i\right)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{Z_i} e^{-\frac{Z^2}{2}} d Z$$

$$Z_i=F^{-1}\left(P_i\right)=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}$$

$$\frac{\partial P}{\partial X_i}=\frac{\partial P}{\partial Z} \frac{\partial Z}{\partial X_i}=F^{\prime}(Z) \beta_i$$

$$F^{\prime}\left(Z_i\right)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{Z^2}{2}}$$

## 经济代写|计量经济学代写Econometrics代考|Multinomial logit and probit models

$D_S=\left{\begin{array}{ll}1 & \text { if the takeover is by shares } 0\end{array}\right.$ if otherwise $D_C= \begin{cases}1 & \text { if the }\end{cases}$

$$D_S=\beta_1+\beta_2 X_{2 i}+\beta_3 X_{3 i}+\cdots+\beta_k X_{k i}+u_i \quad D_C=a_1+a_2 X_{2 i}+a_3 X_{3 i}+$$

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