?===================================================================

? Sample selection with multinomial logit selection model          =

? Requires extensive modification by user for application.         =

? Note lines below marked by *                                     =

?                                                                  =

? User must define the data matrix with the following line:        =

?                                                                  =

?        NAMELIST         ; V = RHS in logit model                 =

?                                                                  =

? User's dependent variable is Y.  Change the following line       =

?                                                                  =

?        CREATE           ; Y = the dependent variable $           =

?                                                                  =

? Logit dependent variable is Z.  Change the following line        =

?                                                                  =

?        CREATE           ; Z = the logit dependent variable $     =

?===================================================================

?

? Estimate the multinomial logit model.  User supplies the name of z.

?

       LOGIT          ; LHS = z  ; RHS = V $ 

?

? Pick off the subvectors of à in blocks of K2.  Where the values taken by

? z are 0,1,...,J, there are J blocks of K2 coefficients. User must modify

? this to put the right values in the PART commands. We also recover the

? covariance matrix, SG.

?

(*)    MATRIX         ; a1 =  part(b,1,K2) ; a2 = part(b,...)  ; ...

                      ; SG  =  varb $ 

?

? Compute the predicted probabilities for the logit model.

?

       CREATE  ; p1 = exp(dot(V,a1))

               ; p2 = exp(dot(V,a2))

(*)            ; ... (do this J times)

               ; p0 = 1 / (1 + p1 + p2 + ...)

               ; p1 = p1 * p0 ; p2 = p2 * p0 ; ... $

?

? Do the sample selection.  

?

(*)    INCLUDE ; NEW ;  z = ...  $ (your value to select on)

?

? Compute lambda and delta

?

(*)    CREATE  ; H = inp(...)    ? (insert pj = the prob. of your choice)

               ; lambda = n01(H) / phi(H)

               ; delta  = lambda * (H + lambda) $

?

? Define the regressor vector for the primary equation.  User supplies the set

? of names and the name of y in the regression command to follow.

?

(*)    NAMELIST ; W = lambda,rhs in regression .... $

?

? The next command gets the right coefficients, but the wrong standard errors.

?

       REGRESS  ; lhs = y ; rhs = W $

?

? We now obtain the standard errors.  First, we need estimates of rhoj and

? sigmaj.  We use the correction to the OLS estimate given by Heckman.

?

       CALC     ; thetajsq   = b(1) ^ 2

                ; sigmajsq   = sumsqdev / n  + thetajsq * xbr(delta)

                ; rhojsq     = thetajsq / sigmajsq $

?

? The q variables in the next create statement must be set up by the user. If

? you have selected on z=0, then qj = -p0 * pj for the J outcomes. If you have

? selected on some other value, then for the J-1 outcomes not including 0 and not ? equal to the

selected one, qk = -pj * pk.  For the j'th one, qj = pj - pj^2.

? Then, each one must be multiplied by delta when we compute F. Example: suppose

? there are four choices, j=0,1,2,3.  We select on j=2.  Then, q1 = -delta*p1*p2,

? q2 = delta*p2*(1 - p2), q3 = -delta*p2*p3. User supplies the format for these.

? Variable t is used to compute W'(I-rhosq D)W simply as a weighted sum of

? squares and cross products.

?

       CREATE         ; t = 1 - rhojsq * delta 

(*)                   ; q1 = ... ; q2 = ... ... $

?

? The next set of matrix commands computes the corrected asymptotic covariance

? matrix then displays the results using STAT.

?

(*)    MATRIX         ; F1   = xdot(W,V,q1) ; F2=xdot(W,V,q2) ; ...

                      ; F    = F1,F2,...

                      ; PSI  = sigmajsq * xdot(W,W,t) 

                               & thetajsq * F | SG | F' 

                      ; C    = xpxi(W) | PSI | xpxi(W)

                      ; stat(B,C) $