This is my first time using Stata for maximum likelihood estimation, and I'm encountering a problem where my ml evaluator will not converge.
I am using a parametric specification for bid distributions based on Athey, Levin, and Seira (2011). My dataset has one observation per auction with variables for the value of the highest bid, second highest bid, etc. These variables have missing values if a bid of that rank was not submitted in that auction.
The log-likelihood function for auction t is:
\[ \ln L_t\left(\rho,\lambda,\theta\right) = N_t \ln\theta + \ln\Gamma\left(\frac{1}\theta+N_t\right)-\ln\Gamma\left(\frac{1}\theta\right)+\sum_{i=1}^{N _t}\ln\left(\rho\lambda\left(\frac{b _{it}}\lambda\right)^{\rho-1}\right)+\left(\frac{1}\theta+N_t\right)\ln\left( 1+\theta\sum_{i=1}^{N_t}\left(\frac{b_{it}}\lambda \right)^\rho\right) \]
N_t is the number of bidders in auction t and gamma is the gamma function.
Here's the ml program I wrote. $ML_y1 through $ML_y12 are the top 12 bids, and $ML_y13 is the number of bidders:
When I use ml maximize I get this output:
I get similar results when I use the "difficult" option. I tried writing a smaller version of the program for auctions with only three bidders and ran it on that sample and got the same results. I used ml query to look for the problem, and found this:
Coefficient 6 corresponds to theta. I've tried everything I can think of to achieve convergence, but haven't had any luck.
One other problem I have is that I want to use the number of bidders as an independent variable in one of my parameter equations, but Stata automatically omits it because of multicollinearity. Given the form of the likelihood function I don't see how multicollinearity would be a problem, but it could be that I'm missing something. Is there a way to get around this?
I am using a parametric specification for bid distributions based on Athey, Levin, and Seira (2011). My dataset has one observation per auction with variables for the value of the highest bid, second highest bid, etc. These variables have missing values if a bid of that rank was not submitted in that auction.
The log-likelihood function for auction t is:
\[ \ln L_t\left(\rho,\lambda,\theta\right) = N_t \ln\theta + \ln\Gamma\left(\frac{1}\theta+N_t\right)-\ln\Gamma\left(\frac{1}\theta\right)+\sum_{i=1}^{N _t}\ln\left(\rho\lambda\left(\frac{b _{it}}\lambda\right)^{\rho-1}\right)+\left(\frac{1}\theta+N_t\right)\ln\left( 1+\theta\sum_{i=1}^{N_t}\left(\frac{b_{it}}\lambda \right)^\rho\right) \]
N_t is the number of bidders in auction t and gamma is the gamma function.
Here's the ml program I wrote. $ML_y1 through $ML_y12 are the top 12 bids, and $ML_y13 is the number of bidders:
Code:
program gammaweibull
args lnf lambda rho theta
tempvar ncons nmult lsumpdf sumcdf lsumcdf ///
ld1 ld2 ld3 ld4 ld5 ld6 ld7 ld8 ld9 ld10 ld11 ld12
quietly {
gen double `ncons' = $ML_y13 * ln(`theta') + lngamma(1/`theta' + $ML_y13 ) - lngamma(1/`theta')
gen double `nmult' = 1/`theta' + $ML_y13
*If bid values are missing from an auction, then the program observes ln(0)...
*To correct for this, manually replace additional term with 0 for missing observations
gen double `ld1' = ln(`rho'*`lambda'*($ML_y1 /`lambda')^(`rho'-1))
gen double `ld2' = `ld1' + ln(`rho'*`lambda'*($ML_y2 /`lambda')^(`rho'-1))
replace `ld2' = `ld1' if $ML_y2 ==.
gen double `ld3' = `ld2' + ln(`rho'*`lambda'*($ML_y3 /`lambda')^(`rho'-1))
replace `ld3' = `ld2' if $ML_y3 ==.
gen double `ld4' = `ld3' + ln(`rho'*`lambda'*($ML_y4 /`lambda')^(`rho'-1))
replace `ld4' = `ld3' if $ML_y4 ==.
gen double `ld5' = `ld4' + ln(`rho'*`lambda'*($ML_y5 /`lambda')^(`rho'-1))
replace `ld5' = `ld4' if $ML_y5 ==.
gen double `ld6' = `ld5' + ln(`rho'*`lambda'*($ML_y6 /`lambda')^(`rho'-1))
replace `ld6' = `ld5' if $ML_y6 ==.
gen double `ld7' = `ld6' + ln(`rho'*`lambda'*($ML_y7 /`lambda')^(`rho'-1))
replace `ld7' = `ld6' if $ML_y7 ==.
gen double `ld8' = `ld7' + ln(`rho'*`lambda'*($ML_y8 /`lambda')^(`rho'-1))
replace `ld8' = `ld7' if $ML_y8 ==.
gen double `ld9' = `ld8' + ln(`rho'*`lambda'*($ML_y9 /`lambda')^(`rho'-1))
replace `ld9' = `ld8' if $ML_y9 ==.
gen double `ld10' = `ld9' + ln(`rho'*`lambda'*($ML_y10 /`lambda')^(`rho'-1))
replace `ld10' = `ld9' if $ML_y10 ==.
gen double `ld11' = `ld10' + ln(`rho'*`lambda'*($ML_y11 /`lambda')^(`rho'-1))
replace `ld11' = `ld10' if $ML_y11 ==.
gen double `ld12' = `ld11' + ln(`rho'*`lambda'*($ML_y12 /`lambda')^(`rho'-1))
replace `ld12' = `ld11' if $ML_y12 ==.
gen `lsumpdf' = `ld12'
*Now if the bid is 0 then the additional terms will be 0, so we only need one variable
gen double `sumcdf' = ($ML_y1 / `lambda')^(`rho') ///
+ ($ML_y2 / `lambda')^(`rho') ///
+ ($ML_y3 / `lambda')^(`rho') ///
+ ($ML_y4 / `lambda')^(`rho') ///
+ ($ML_y5 / `lambda')^(`rho') ///
+ ($ML_y6 / `lambda')^(`rho') ///
+ ($ML_y7 / `lambda')^(`rho') ///
+ ($ML_y8 / `lambda')^(`rho') ///
+ ($ML_y9 / `lambda')^(`rho') ///
+ ($ML_y10 / `lambda')^(`rho') ///
+ ($ML_y11 / `lambda')^(`rho') ///
+ ($ML_y12 / `lambda')^(`rho')
gen double `lsumcdf' = ln(1+ `theta'*`sumcdf' )
replace `lnf' = `ncons'+`lsumpdf'+(`nmult'*`lsumcdf')
}
end
Code:
. ml maximize initial: log likelihood = 134279.26 rescale: log likelihood = 134279.26 rescale eq: log likelihood = 141848.78 Iteration 0: log likelihood = 141848.78 (not concave) Iteration 1: log likelihood = 141869.98 (not concave) Iteration 2: log likelihood = 141881.5 (not concave) Iteration 3: log likelihood = 141882.65 (not concave) Iteration 4: log likelihood = 141882.76 (not concave) Iteration 5: log likelihood = 141882.81 (not concave) cannot compute an improvement -- discontinuous region encountered r(430);
Code:
Current status
Coefficient values
1: -1.5495e-08
2: -.000023924
3: 265.716277
4: .000022557
5: 108.852769
6: 3.3868e+17
Function value: .
Converged: no
One other problem I have is that I want to use the number of bidders as an independent variable in one of my parameter equations, but Stata automatically omits it because of multicollinearity. Given the form of the likelihood function I don't see how multicollinearity would be a problem, but it could be that I'm missing something. Is there a way to get around this?