Task $1.$

a$.$ You are given an article on the Dagum regression by Domma, Condino, and Giordano $(2018)($search for public articule on the web "A new formulation of the Dagum distribution in terms of income inequality and poverty measures"$).$ In it$,$ they propose $3$ variants of a Dagum regression to simultaneously analyze inequality, income median, and poverty.

b$.$ Find or create a dataset in R where the dependent variable is income and the independent variable is a binary variable. Name this variable $($for example, gender: $1=$male, $0=$female; or sector: $1=$Primary, $0=$Secondary and tertiary$)(10$ points$).($It is recommended not to have a very large sample $(500$ is ok$)).$

c$.$ Manually $($no using the fuctions in R already for Dagum regression$)$ estimate the parameters beta, lambda, and delta $($and their standard errors$)$ of the likelihood function presented on page $113($below equation $38)$ and equations $27$ and $28$ using R$.$ Do this using the mle or mle$2$ function. Note that this estimation should be performed only with the dependent variable, without considering the predictor. $P=F_{Da}(t$;$,,)=(1+t^{-}{)}^{-}$

$me={}^{\frac{1}{}}(2^{\frac{1}{}}-1{)}^{-\frac{1}{}}$

${}_1=\frac{1}{}.$

After algebra, we can express the parameters $,$ and $$ as function of ${}_1,$ me and $P,$ i$.$e$.l($;$y,W)={\displaystyle \underset{i=1}{\overset{n}{}}}\{log{}_i^{}+log{}_i^{}+log{}_i^{}-[{}_i^{}+1]log(y_i)-[{}_i^{}+1]log(1+{}_i^{}{y}_i^{-{}_i^{}})\}$

Consider that $p_j$ is the number of determinants of $I_j,j=1,2,3,$ so the overall likelihood equations are $(p_1+p_2+p_3),$

calculated for the general formulation as follows

$\frac{\text{d}\text{e}\text{l}\text{l}(\text{;}y,W)}{\text{d}\text{e}\text{l}{}_{j,r_j}}={\displaystyle \underset{i=1}{\overset{n}{}}}\{\frac{{}_{j,r_j,i}^{}}{{}_i^{}}+\frac{{}_{j,r_j,i}^{}}{{}_i^{}}+\frac{{}_{j,r_j,i}^{}}{{}_i^{}}\}-{\displaystyle \underset{i=1}{\overset{n}{}}}{}_{j,r_j,i}^{}log(y_i)+$

$-{\displaystyle \underset{i=1}{\overset{n}{}}}\{{}_{j,r_j,i}^{}log[1+{}_i^{}{y}_i^{-{}_i}]-[{}_i^{}+1]y_i^{-{}_i}\frac{{}_{j,r_j,i}^{}-{}_i^{}{}_{j,r_j,i}^{}log(y_i)}{1+{}_i^{}{y}_i^{-{}_i}}\}=0$

where

${}_{j,r_j,i}^{}=\frac{\text{d}\text{e}\text{l}{}_i^{}}{\text{d}\text{e}\text{l}{}_{j,r_j}}=\frac{\text{d}\text{e}\text{l}(I_{1,i},I_{2,i},I_{3,i})}{del{I}_{j,i}}\frac{del{I}_{j,i}}{\text{d}\text{e}\text{l}{}_{j,r_j}},$

${}_{j,r_j,i}^{}=\frac{\text{d}\text{e}\text{l}{}_i^{}}{\text{d}\text{e}\text{l}{}_{j,r_j}}=\frac{\text{d}\text{e}\text{l}(I_{1,i},I_{2,i},I_{3,i})}{del{I}_{j,i}}\frac{del{I}_{j,i}}{\text{d}\text{e}\text{l}{}_{j,r_j}}$

and

${}_{j,r_j,i}^{}=\frac{\text{d}\text{e}\text{l}{}_i^{}}{\text{d}\text{e}\text{l}{}_{j,r_j}}=\frac{\text{d}\text{e}\text{l}(I_{1,i},I_{2,i},I_{3,i})}{del{I}_{j,i}}\frac{del{I}_{j,i}}{\text{d}\text{e}\text{l}{}_{j,r_j}}$

d$.$ Based on the article, try to interpret these parameters $(10$ points$).$

e$.$ Based on the likelihood function from point $($c$)$ and the first formulation $($equations $39),$ estimate the parameters gamma $($and their standard errors$)$ for the $3$ simultaneous equations $($for beta, median, and delta$)$ using mle and mle$2($Hint: Use the results obtained in part $($c$)$ for the initial values of the estimation.$)$

$--$

First formulation

The indicators $I_1={}_1,I_2=me$ and $I_3={}_1$ which govern the first formulation of the Dagum model $(21)$ are all positive,

hence appropriate link functions can be$,$ for example, the following

${}_{1,i}=e^{w_{1,i}^{\text{'}}}$

$u{?}_{(}()()1),m{e}_i=e^{w_{2,i}^{\text{'}}{}_2},{}_{1,i}=e^{w_{3,i}^{\text{'}}{}_3},$

f$.$ Interpret the gamma coefficients according to the article $($results given for Italy$)$

I NEED THE CODE OF THE ANSWERS NOT THE EXPLANATION