Title
blp -- Berry Levinsohn Pakes random coefficients logit estimator
Syntax
blp sharevar [varlist] [if] [in], endog(endogvars=inst) stochastic(svar1=dvars1,svar2=dvars2,..) markets(marketvar) [draws(#) demofile(demodata) tolin(#) tolout(#) elast(evar,emarket) Robust Twostep Noisily Nocons]
Description
blp estimates the random parameters logit demand model from product market shares, using the algorithm proposed by Berry Levinsohn and Pakes(1995). This allows for endogenous prices, and individual specific coefficients, that can be partially explained by observed variation in demographic variables. The heterogeneity parameters are estimated using the modified Newton-Raphson method with analytic gradient.
Options for blp
endog(endogvars=inst) identifies endogvars as the endogenous variables, and inst and the instruments, excluding those in varlist. Instruments can still be listed if all variables are exogenous.
stochastic(svar1=dvars1,svar2=dvars2,..) is required and identifies svar1, svar2 etc as variables with random coefficients, and dvars1,dvars2 as the demographic variables (optional). Coefficient equations can contains different demographic variables.
markets(marketvar) is required and identifies marketvar as the market variable in the data, and in the demographic data if used.
draws(#) specifies the number of simulation draws used to approximate the intgral over the random coefficients. The defualt is 100.
demofile(demodata) identifies demodata as the path to the file that contains the random draws of the demographic variables by market.
tolin(#) specifies the tolerance level used to define convergence of the contraction mapping algorithm. The default is 1E-14.
tolout(#) specifies the tolerance level used to define convergence of the GMM estimator. The defualt is 10E-12.
elast(evar,emarket) computes the matrix of demand elasticities for variable evar in market emarket. Note that these can only be computed for a variable in svar1,svar2 etc.
Robust computes an estimate of the standard errors that are robust to heteroskedasticity across markets, and correlation across products. The default assumes that the errors are iid.
Twostep computes the two-step GMM estimator.
Noisily displays the iterations during estimation. This indicates convergence of the contraction mapping by market, heterogeneity parameters and associated analytical gradients.
Nocons estimates the model without the constant term.
Remarks
blp requires the data to be in long form, where each market contains observations on product shares and characteristics. There is no requirement for products to be the same across markets.
The number of draws # in draws(#) is set to 100, but the user should test the stability of the estimates for larger values.
If demographic variables are used, it is necessary for demofile to contain an equal number of draws across markets (in long-form), and with market identifier marketvar. This number will override #. As Nevo(2000) points out, identification of the coefficients, requires variation in the distribution of demographic variables over several markets.
Examples
Example 1: No demographic variables
In this example, consumers can select from 6 alternatives including the outside good. Choice data is generated for 50,000 individuals over 25 markets, and aggregated to form shares. The observed characteristics (excluding the constant) are: x1, x2 which are exogenous, and price p which is endogenous. The coefficients vary over individuals, with means (1,1,-1) and standard deviations (1,1,0.5) respectively. Price is instrumented by an suply side variable z, and to identify the standard deviations, a further set of instruments z2=z^2, x12=x1^2, x22=x2^2, expx1=exp(x1), and expx2=exp(x2) are created. The option elast(p,1) is included to post the price elasticities for market 1.
. use blp_nodemo,clear
. set seed 2001862510
. blp s x1 x2, stochastic(x1,x2,p) endog(p=z z2 x12 x22 expx1 expx2) markets(mkt) draws(200) elast(p,1)
Iteration 0: f(p) = 3.5424634 (not concave) Iteration 1: f(p) = 1.3188697 Iteration 2: f(p) = .97161122 Iteration 3: f(p) = .8951762 Iteration 4: f(p) = .89338484 Iteration 5: f(p) = .89338385 Iteration 6: f(p) = .89338385
One step estimates Number of obs= 125 Number of markets= 25 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Mean Utility | cons | 5.065248 .909824 5.57 0.000 3.282026 6.84847 x1 | 1.057797 .124657 8.49 0.000 .813474 1.30212 x2 | 1.066779 .1448797 7.36 0.000 .7828198 1.350738 p | -.9908965 .1233748 -8.03 0.000 -1.232707 -.7490864 -------------+---------------------------------------------------------------- x1 | SD | 1.081303 .2331608 4.64 0.000 .6243167 1.53829 -------------+---------------------------------------------------------------- x2 | SD | 1.06941 .234606 4.56 0.000 .6095908 1.52923 -------------+---------------------------------------------------------------- p | SD | .5490221 .1436066 3.82 0.000 .2675583 .8304858 ------------------------------------------------------------------------------
All parameters are significant at the 5% level, and the estimates of the standard-deviations (denoted SD in the above) are very close to their true values. To display the price elasticity matrix type:
. matrix list e(elast)
e(elast)[5,5] c1 c2 c3 c4 c5 r1 -1.8574321 .23411686 1.1157211 .08263752 .27073708 r2 1.7334231 -4.0139038 1.0088303 .21048866 .3898307 r3 2.1954042 .26810508 -3.3967697 .07055121 .26821809 r4 1.1356809 .3906932 .49274782 -3.3028753 .57820293 r5 1.8106098 .35211236 .91160432 .28137038 -4.0760402
The ij-elements of the elasticity matrix represent (approximately) the %-change in the demand for product i, following a 1% increase in the price of product j.
Example 2: Demographic data
This example extends the DGP, to allow the coefficient on x1 to be a linear function of demographic variable D, which has a true coefficient of 1. The demographic variable is generated from a log-normal distribution, and to assist in parameter identification, the variance of D differs over markets. Furthermore, the number of available alternatives is increased to 10. File demodata.dta contains 2000 draws of D, and following Romeo(2010), the mean demographics for each market mu_D is constructed and used an an additional instrument.
. use blp_demo,clear
. set seed 2001862510
. blp s x1 x2, stochastic(x1=D,x2,p) endog(p=z z2 x12 x22 mu_D) markets(mkt) demofile(demodata)
number of draws set to number in demographic data file draws per market is: 2000 Demographic variables appearing in random coefficients equation
D x1 1 x2 0 p 0 Do you wish to continue?: 1=yes, 0=no?. 1 estimation continuing Iteration 0: f(p) = 15.625377 (not concave) Iteration 1: f(p) = 9.3971018 (not concave) Iteration 2: f(p) = 4.2458365 Iteration 3: f(p) = .27343605 Iteration 4: f(p) = .00312217 Iteration 5: f(p) = 9.873e-07 Iteration 6: f(p) = 3.970e-13 Iteration 7: f(p) = 9.301e-20
One step estimates Number of obs= 225 Number of markets= 25 ------------------------------------------------------------------------------ | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Mean Utility | cons | -5.221238 .5138024 -10.16 0.000 -6.228272 -4.214203 x1 | .9782134 .2445123 4.00 0.000 .4989781 1.457449 x2 | .9997277 .1193572 8.38 0.000 .7657918 1.233664 p | -1.048863 .2948107 -3.56 0.000 -1.626682 -.4710451 -------------+---------------------------------------------------------------- x1 | D | 1.110372 .0919074 12.08 0.000 .9302363 1.290507 SD | .9767599 .1012825 9.64 0.000 .7782499 1.17527 -------------+---------------------------------------------------------------- x2 | SD | 1.012472 .1785324 5.67 0.000 .662555 1.362389 -------------+---------------------------------------------------------------- p | SD | .5880409 .2002728 2.94 0.003 .1955133 .9805684 ------------------------------------------------------------------------------
Hence from the above, the coefficient on x1 given D, has an estimated mean of 0.978+1.11*D and a standard deviation of 0.977.
Author
David Vincent Deloitte Economic Consulting LLP Athene Place, 66 Shoe Lane, London, EC4A 3BQ davivincent@deloitte.co.uk
References
Berry, S., J. Levinsohn, and A. Pakes, (1995), “Automobile Prices in Market Equilibrium,” Econometrica, 63, 4, 841-90.
Nevo, Aviv, (2000), “A Practitioner's Guide to Estimation of Random-Coefficients Logit Models of Demand,” Journal of Economics & Management Strategy.
Romeo, Charles J., (2010), “Filling Out the Instrument Set in Mixed Logit Demand Systems for Aggregate Data”, US Department of Justice,600 E Street, NW–Suite 10000 Washington, DC 20530. .