Hi,
I was trying to estimate the following regression model about divorce rate, which is a simple regression model but with a unknown kinked cost function :
divrate_st = b0 + b1* F_st + b2*X2 + b3*X3 + ... + epsilon_st
where F_st = F (w_st, w*) = w_st + (w* - w_st) * I(w_st > w*) = w_st if w_st <= w*
= w* if w_st > w* ,
here I( ) is the indicator function which equals 1 is ( ) is satisfied and quals 0 otherwise, w_st is the wait time (cost) in state s at time t to legally divorce.
This model is no longer linear since the "kink" w* is unknown, therefore I need to estimate w* along with other parameters b0, b1, ....
For some reasons I don't think the non-linear least squares problem would work well for my model, one of the reasons is that the partial derivatives of the max value with respect to w* are messy. I think if we want to use NLLS, we need to have nice, not messy partial derivatives wrt each parameter.
A statistician Muggeo (2003) provides an approach to estimate this type of model in the "kink" w* direction. Specifically, he expands
b * (w* - w_st) * I(w_st > w*) = b* (w* - w) * I(w > w*_r) + c * I(w>w*_r), where w*_r is the r-th candidate value for true w* and we estimate c and iterate to get
w*_(r+1) = w*_r + c/b until c/b is very small.
I am wondering if there is anything that has been done in Stata to handle this type of "kink" problem. I am sorry that this probably is confusing and a little too technical, but I will be happy to know whatever information that is related, and any thoughts you have.
Thank you so much!!
Jeff Wang
I was trying to estimate the following regression model about divorce rate, which is a simple regression model but with a unknown kinked cost function :
divrate_st = b0 + b1* F_st + b2*X2 + b3*X3 + ... + epsilon_st
where F_st = F (w_st, w*) = w_st + (w* - w_st) * I(w_st > w*) = w_st if w_st <= w*
= w* if w_st > w* ,
here I( ) is the indicator function which equals 1 is ( ) is satisfied and quals 0 otherwise, w_st is the wait time (cost) in state s at time t to legally divorce.
This model is no longer linear since the "kink" w* is unknown, therefore I need to estimate w* along with other parameters b0, b1, ....
For some reasons I don't think the non-linear least squares problem would work well for my model, one of the reasons is that the partial derivatives of the max value with respect to w* are messy. I think if we want to use NLLS, we need to have nice, not messy partial derivatives wrt each parameter.
A statistician Muggeo (2003) provides an approach to estimate this type of model in the "kink" w* direction. Specifically, he expands
b * (w* - w_st) * I(w_st > w*) = b* (w* - w) * I(w > w*_r) + c * I(w>w*_r), where w*_r is the r-th candidate value for true w* and we estimate c and iterate to get
w*_(r+1) = w*_r + c/b until c/b is very small.
I am wondering if there is anything that has been done in Stata to handle this type of "kink" problem. I am sorry that this probably is confusing and a little too technical, but I will be happy to know whatever information that is related, and any thoughts you have.
Thank you so much!!
Jeff Wang
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