MLE estimation in STATA

Notes on MLE in STATA
July 31, 2013
NOTE: I will not test you on this material, but I provide our notes here in case you need them in the future
(e.g., for a homework problem).
A good guide for using MLE in STATA has been written by Marco R. Steenbergen. I uploaded it to my website
just in case; you can find it at http://www.econ.umn.edu/~evdok003/MLE_in_stata.pdf. I will illustrate the
basic techniques with some examples, progressively more difficult.
where
We start with the simplest possible model:
The likelihood function is given by
f(y i |x i , σ 2 ) =
y i = x i β + ɛ i , e i ∼ i.i.d. N(0, σ 2 )
L = Π N i=1f(y i |x i , θ),
1
√ (y 2πσ
2 e− i −x i β)
2σ 2 == 1 σ
1
√ e − ( y i −µ i )
σ
2
2 = 1 2π σ φ(y i − µ i
).
σ
φ here is the standard normal distribution and µ i the conditional mean x i β. We can therefore write the likelihood
function in terms of φ as
L = Π N 1
i=1
σ φ(y i − µ i
).
σ
Therefore,
N∑
ln(L) = ln(φ( y i − µ i β
N∑
) − ln(σ)) = ln(f i ).
σ
The term inside the sum, ln(f i ), is programmed in STATA as follows
program define normal
version 1.0
args lnf mu sigma
quietly replace ‘lnf’=ln(normd(($ML_y1-‘mu’)/‘sigma’))-ln(‘sigma’)
end
i=1
The layout of the program is standard, so it is useful to go over it. The first line tells us that we are defining a
program called “normal.” The second line is optional and specifies the version of the program. The third line defines
the arguments of the program, which are “ln(f i )”; the parameters “ln(f i )” depends on, which are the mean (µ i ) and
the variance (σ) of the distribution of our ɛ i ; and y i . Notice that we are telling the program how y i enters into ln(f i )!
We then use the program defined above to create the full log-likelihood. This is done using the following line
ml model lf normal (y=x) (y)
Above, “lf” stands for “linear form.” This command tells STATA that ln(L) = ln(f 1 ) + ln(f 2 ) + ... + ln(f N ). This
will be true for every model we consider below. “normal” tells stata what the shape of each individual ln(f i ) is.
The two sets of opened and closed parentheses tell STATA to estimate two parameters, using dependent variable
y. The first parameter is µ i , which depends on some regressor x i (since µ i = x i β), and the second parameter is the
variance, which does not depend on any regressor.
i=1
Finally, we type
1

ml max
to maximize the log-likelihood. This will provide you the MLE estimates ˆβ MLE and ˆσ MLE that you derived analytically
in your first homework, as well as their standard errors.
We next follow Bresnahan and Reiss (1991), looking at the market for tire manufacturers as an application of
the MLE method. Our dataset (available at http:///www.econ.umn.edu/~evdok003/BR.csv) includes markets
with 0, 1, 2, 3, 4, 5 and > 5 entrants. First, we will focus on markets with 0 or 1 producers, modeling market entry
using a probit specification. Thus, {
y i = 1 if Π 1 > 0
,
0 otherwise
where Π 1 are the firm’s profits. We further assume that Π 1 = ¯Π 1 + ɛ, ¯Π 1 being the firm’s expected profits, a
nonlinear function of a number of regressors, and ɛ ∼ N(0, σ 2 ). NOTE THAT THE NONLINEAR NATURE OF
THE EXPECTED PROFIT FUNCTION MAKES STATA’S PROBIT COMMAND INAPPLICABLE. We must
therefore build the MLE estimator from scratch.
The regressors are:
• tpop - town population
• opop - nearby population
• ngrw - negative tpop growth
• pgrw - positive tpop growth
• octy - commuters out of county
• landv - value per acre of farm-land and buildings
• eld - percentage of the country population being 65 or older
• ffrac - fraction of land in farms
• pinc - per capita income
• lnhdd - log of heating degree days
The specific form of the expected profit function is assumed to be
¯Π 1 = S(Y ; λ) · V 1 (Z; α 1 , β) − F 1 (W ; γ).
Here, S(Y ; λ) is a measure of market size, which is a function of population parameters Y . V 1 (Z; α 1 , β) is a
measure of per-capita demand, which depends on demand shifters Z. F 1 (W ; γ) is a measure of costs, which depends
on cost shifters W . We assume that S, V 1 and F 1 depend on the regressors linearly in the following way:
S(Y ; λ) = tpop + λ 1 opop + λ 2 ngrw + λ 3 pgrw + λ 4 octy
V 1 (W, Z; α 1 , β) = α 1 + β 1 eld + β 2 pinc + β 3 lnhdd + β 4 ffrac
F 1 (W ; γ) = γ 1 + γ L landv
Notice that our assumptions imply that P (N = 1|Y, Z, W ) = Φ(¯Π 1 ).
The model can be estimated with STATA using the following code:
insheet using http://www.econ.umn.edu/~evdok003/BR.csv, clear
drop if tire>1
program monentry
version 1.0
2

args lnf s v f
quietly replace ‘lnf’=ln(normal(‘s’*‘v’-‘f’)) if $ML_y1==1
quietly replace ‘lnf’=ln(1-normal(‘s’*‘v’-‘f’)) if $ML_y1==0
end
ml model lf monentry (lambda:tire=opop ngrw pgrw octy,nocons offset(tpop)) \\\ (beta:tire=eld pinc lnhdd ff
ml search lambda -1 1 beta 0 1 gammaL -1 1
ml max
The results are
Table 1: Estimation results (probit model)
Variable Coefficient (Std. Err.)
opop (λ 1 ) 7.404 (21.593)
ngrw (λ 2 ) -12.980 (44.674)
pgrw (λ 3 ) 21.061 (65.999)
octy (λ 4 ) -3.349 (15.151)
α 1 -0.163 (1.173)
eld (β 1 ) 0.431 (0.977)
pinc (β 2 ) 0.031 (0.083)
lnhdd (β 3 ) -0.007 (0.139)
ffrac (β 4 ) 0.182 (0.397)
landv (γ L ) -0.725 (0.727)
γ 1 0.802 ∗ (0.366)
We will now look at the whole dataset and model the number of entrants using ordered probit. Following
standard economic theory, we assume that profits are highest in a monopoly, and that entry of additional firms
drives them down. Thus ¯Π 1 > ¯Π 2 > ¯Π 3 > ¯Π 4 > ¯Π 5 . Following the standard ordered logit argument,
P (N = 0|Y, Z, W ) = 1 − Φ( ¯Π 1 )
P (N = J|Y, Z, W ) = Φ(¯Π J ) − Φ( ¯Π J+1 ) ∀J = 1, 2, 3, 4
P (N ≥ 5|Y, Z, W ) = Φ(¯Π 5 )
where ¯Π N = S(Y ; λ) · V N (W, Z; α, β) − F N (W ; γ).
The specific forms of S, V N and F N are similar to that of F 1 , V 1 , and F 1 that we worked with above. Bresnahan
and Reiss assume, however, that α 4 = 0, so we will impose the same restriction.
Again, the method of Maximum Likelihood can be used to estimate the parameters of the model. The STATA code
is provided below:
insheet using http://www.econ.umn.edu/~evdok003/BR.csv, clear
program firmentry
version 1.0
args lnf s v f alpha2 alpha3 alpha4 alpha5 gamma2 gamma3 gamma4 gamma5
tempvar p2 p3 p4 p5
qui gen double ‘p2’=normal(‘s’*(‘v’-‘alpha2’)-‘f’-‘gamma2’)
qui gen double ‘p3’=normal(‘s’*(‘v’-‘alpha2’-‘alpha3’)-‘f’-‘gamma2’-‘gamma3’)
qui gen double ‘p4’=normal(‘s’*(‘v’-‘alpha2’-‘alpha3’-‘alpha4’)-‘f’-‘gamma2’-‘gamma3’-‘gamma4’)
qui gen double ‘p5’=normal(‘s’*(‘v’-‘alpha2’-‘alpha3’-‘alpha4’-‘alpha5’)-‘f’-‘gamma2’-‘gamma3’\\\
-‘gamma4’-‘gamma5’)
quietly replace ‘lnf’=ln(1-normal(‘s’*(‘v’)-‘f’)) if $ML_y1==0
quietly replace ‘lnf’=ln(normal(‘s’*(‘v’)-‘f’)-‘p2’) if $ML_y1==1
quietly replace ‘lnf’=ln(‘p2’-‘p3’) if $ML_y1==2
quietly replace ‘lnf’=ln(‘p3’-‘p4’) if $ML_y1==3
3

quietly replace ‘lnf’=ln(‘p4’-‘p5’) if $ML_y1==4
quietly replace ‘lnf’=ln(‘p5’) if $ML_y1>=5
end
constraint 1 [alpha4]_cons=0
ml model lf firmentry (lambda:tire=opop ngrw pgrw octy,nocons offset(tpop)) \\\
(beta:tire=eld pinc lnhdd ffrac) (gammaL:tire=landv) (alpha2:tire=) \\\
(alpha3:tire=) (alpha4:tire=) (alpha5:tire=) (gamma2:tire=) (gamma3:tire=) \\\
(gamma4:tire=) (gamma5:tire=), constraint(1)
ml search lambda 0 50 beta 0 1 alpha2 0 1 alpha3 0 1 alpha4 0 1 alpha5 0 1 \\\
gamma2 0 1 gamma3 0 1 gamma4 0 1 gamma5 0 1 gammaL -1 1
ml max
Table 2: Estimation results (ordered probit)
Variable Coefficient (Std. Err.)
opop (λ 1 ) -0.532 (0.404)
ngrw (λ 2 ) 2.253 ∗ (0.976)
pgrw (λ 3 ) 0.343 (0.612)
octy (λ 4 ) 0.227 (0.408)
eld (β 1 ) -0.488 (0.626)
pinc (β 2 ) -0.031 (0.029)
lnhdd (β 3 ) 0.004 (0.056)
ffrac (β 4 ) -0.021 (0.077)
α 1 0.863 † (0.464)
α 2 0.035 (0.116)
α 3 0.150 (0.093)
α 4 0.000 (0.000)
α 5 0.081 (0.050)
landv (γ L ) -0.737 † (0.403)
γ 1 0.529 ∗ (0.220)
γ 2 0.756 ∗∗ (0.186)
γ 3 0.465 ∗ (0.196)
γ 4 0.598 ∗∗ (0.113)
γ 5 0.120 (0.174)
Notice that this replicates the last column of Table 4 in the paper (p. 994).
4