We are faced with more and more data, often with many, and poorly described or understood, variables. We can even have more variables than we do data. Classical techniques break down when applied to such data.
The lasso is designed to sift through this kind of data and extract features that have the ability to predict outcomes.
Stata gives you the tools to use lasso for predicton and for characterizing the groups and patterns in your data (model selection). Use the lasso itself to select the variables that have real information about your response variable. Use split-sampling and goodness of fit to be sure the features you find generalize outside of your training (estimation) sample.
With the lasso command, you specify potential covariates, and it selects the covariates to appear in the model. The fitted model is suitable for making out-of-sample predictions but not directly applicable for statistical inference. If inference is your interest, see our description of Lasso for inference.
There are lots of lasso commands. Here are the most important ones for prediction.
You have an outcome y and variables x1-x1000. Among them might be a subset good for predicting y. Lasso attempts to find them. Type
. lasso linear y x1-x1000
To see the variables selected, type
To make predictions with new data, type
. use newdata . predict yhat
To see the fit in the new data, type
Lasso fits logit, probit, Poisson and Cox proportional hazards models too.
. lasso logit z x1-x1000 . lasso probit z x1-x1000 . lasso poisson c x1-x1000 . lasso cox x1-x1000
And it fits elastic-net models.
. elasticnet linear y x1-x1000 . elasticnet logit z x1-x1000 . elasticnet probit z x1-x1000 . elasticnet poisson c x1-x1000 . elasticnet cox x1-x1000
Because ridge regression is a special case of elastic net, it fits ridge regressions too.
Square-root lasso is a variant of lasso for linear models.
. sqrtlasso y x1-x1000
You can force the selection of variables such as x1-x4.
. lasso linear y (x1-x4) x5-x1000
After fitting a lasso, you can use the postlasso commands.
. lassoknots table of estimated models by lambda . lassocoef selected variables . lassogof goodness of fit . lassoselect lambda = 0.1 select model for another lambda . coefpath plot coefficient path . cvplot plot cross-validation function
And then there are features that will make it easier to do all the above. Need to split your data into training and testing samples? Type
. splitsample, generate(sample) nsplit(2)
Need to manage large variable lists? You do. We typed x1-x1000 above, but your variables will have real names, and you do not want to type them all. Use the vl commands to create lists of variables:
. vl set // creates vlcontinuous, vlcategorical, ... . vl create myconts = vlcontinuous . vl modify myconts = myconts - (kl srh srd polyt) . vl create myfactors = vlcategorical . vl substitute myvarlist = i.myfactors myconts i.myfactors#c.myconts
The # sign creates interactions.
We just created myvarlist, which is ready for use in a lasso command such as
. lasso linear y $myvarlist
We are going to show you three examples.
And then we are going to compare them.
Rather than typing
. lasso linear q104 (i.gender i.q3 i.q4 i.q5) i.q2 i.q6 i.q7 i.q8 i.q9 i.q10 i.q11 i.q13 i.q14 i.q16 i.q17 i.q19 i.q25 i.q26 i.q29 i.q30 i.q32 i.q33 i.q34 i.q36 i.q37 i.q38 i.q40 i.q41 i.q42 i.q43 i.q44 i.q46 i.q47 i.q48 i.q49 i.q50 i.q51 i.q55 i.q56 i.q57 i.q58 i.q59 i.q61 i.q64 i.q65 i.q67 i.q68 i.q69 i.q71 i.q72 i.q73 i.q74 i.q75 i.q77 i.q78 i.q79 i.q82 i.q83 i.q84 i.q85 i.q86 i.q88 i.q89 i.q90 i.q91 i.q94 i.q95 i.q96 i.q97 i.q98 i.q100 i.q101 i.q102 i.q105 i.q108 i.q109 i.q110 i.q113 i.q114 i.q115 i.q116 i.q117 i.q118 i.q122 i.q123 i.q125 i.q126 i.q128 i.q130 i.q133 i.q134 i.q136 i.q137 i.q138 i.q140 i.q142 i.q143 i.q144 i.q145 i.q146 i.q147 i.q148 i.q149 i.q150 i.q151 i.q152 i.q153 i.q154 i.q155 i.q156 i.q158 i.q159 i.q160 i.q161 age q1 q15 q18 q20 q21 q22 q24 q31 q35 q45 q52 q53 q62 q63 q70 q76 q87 q93 q103 q111 q112 q120 q121 q129 q131 q132 q139 q157
we have used vl behind the scenes, so that we can type
. lasso linear q104 ($idemographics) $ifactors $vlcontinuous
And so that we can compare the out-of-sample predictions for the three models, we have already split our sample in two by typing
. splitsample, generate(sample) nsplit(2) rseed(1234)
We will fit all three models on sample==1 and later compare predictions using sample==2.
To fit a lasso with the default cross-validation selection method, we type
. lasso linear q104 ($idemographics) $ifactors $vlcontinuous if sample == 1 10-fold cross-validation with 100 lambdas ... Grid value 1: lambda = .9109571 no. of nonzero coef. = 4 Folds: 1...5....10 CVF = 16.93341 output omitted Grid value 23: lambda = .1176546 no. of nonzero coef. = 74 Folds: 1...5....10 CVF = 12.17933 ... cross-validation complete ... minimum found Lasso linear model No. of obs = 458 No. of covariates = 273 Selection: Cross-validation No. of CV folds = 10
|No. of Out-of- CV mean|
|nonzero sample prediction|
|ID||Description lambda coef. R-squared error|
|1||first lambda .9109571 4 0.0147 16.93341|
|18||lambda before .1873395 42 0.2953 12.10991|
|* 19||selected lambda .1706967 49 0.2968 12.08516|
|20||lambda after .1555325 55 0.2964 12.09189|
|23||last lambda .1176546 74 0.2913 12.17933|
Lambda (λ) is lasso's penalty parameter. Lasso fits a range of models, from models with no covariates to models with lots, corresponding to models with large λ to models with small λ.
Lasso then selected a model. Because we did not specify otherwise, it used its default, cross-validation (CV) to choose model ID=19, which has λ=0.171. The model has 49 covariates.
Cross-validation chooses the model that minimizes the cross-validation function. Here is a graph of it.
We plan on comparing this model with two other models, so we will store these estimates. We will store them under the name cv.
. estimates store cv
We can select the model corresponding to any λ we wish after fitting the lasso. Picking the λ that has the minimum Bayes information criterion (BIC) gives good predictions under certain conditions.
To fit a lasso with minimum BIC, we use the same command and specify the additional option selection(bic):
. lasso linear q104 ($idemographics) $ifactors $vlcontinuous if sample == 1, selection(bic)
|Evaluating up to 100 lambdas in grid ...|
|Grid value 1: lambda = .9109571 no. of nonzero coef. = 4|
|BIC = 2618.642|
|Grid value 2: lambda = .8300302 no. of nonzero coef. = 7|
|BIC = 2630.961|
|Grid value 3: lambda = .7562926 no. of nonzero coef. = 8|
|BIC = 2626.254|
|Grid value 4: lambda = .6891057 no. of nonzero coef. = 9|
|BIC = 2619.727|
|Grid value 5: lambda = .6278874 no. of nonzero coef. = 10|
|BIC = 2611.577|
|Grid value 6: lambda = .5721076 no. of nonzero coef. = 13|
|BIC = 2614.155|
|Grid value 7: lambda = .5212832 no. of nonzero coef. = 13|
|BIC = 2597.164|
|Grid value 8: lambda = .4749738 no. of nonzero coef. = 14|
|BIC = 2588.189|
|Grid value 9: lambda = .4327784 no. of nonzero coef. = 16|
|BIC = 2584.638|
|Grid value 10: lambda = .3943316 no. of nonzero coef. = 18|
|BIC = 2580.891|
|Grid value 11: lambda = .3593003 no. of nonzero coef. = 22|
|BIC = 2588.984|
|Grid value 12: lambda = .327381 no. of nonzero coef. = 26|
|BIC = 2596.792|
|Grid value 13: lambda = .2982974 no. of nonzero coef. = 27|
|BIC = 2586.521|
|Grid value 14: lambda = .2717975 no. of nonzero coef. = 28|
|BIC = 2578.211|
|Grid value 15: lambda = .2476517 no. of nonzero coef. = 32|
|BIC = 2589.632|
|Grid value 16: lambda = .225651 no. of nonzero coef. = 35|
|BIC = 2593.753|
|Grid value 17: lambda = .2056048 no. of nonzero coef. = 37|
|BIC = 2592.923|
|Grid value 18: lambda = .1873395 no. of nonzero coef. = 42|
|BIC = 2609.975|
|Grid value 19: lambda = .1706967 no. of nonzero coef. = 49|
|BIC = 2639.437|
|... selection BIC complete ... minimum found|
|ID||Description lambda coef. R-squared BIC|
|1||first lambda .9109571 4 0.0308 2618.642|
|13||lambda before .2982974 27 0.3357 2586.521|
|* 14||selected lambda .2717975 28 0.3563 2578.211|
|15||lambda after .2476517 32 0.3745 2589.632|
|19||last lambda .1706967 49 0.4445 2639.437|
We can draw the BIC function plot:
We will store these results by minBIC.
. estimates store minBIC
Adaptive lasso is another selection technique that tends to select fewer covariates. It also uses cross-validation but runs multiple lassos. By default, it runs two.
To fit an adaptive lasso, we use the same command and specify the additional option selection(adaptive):
. lasso linear q104 ($idemographics) $ifactors $vlcontinuous if sample == 1, selection(adaptive) Lasso step 1 of 2: 10-fold cross-validation with 100 lambdas ... Grid value 1: lambda = .9109571 no. of nonzero coef. = 4 Folds: 1...5....10 CVF = 17.012 Grid value 2: lambda = .8300302 no. of nonzero coef. = 7 [output omitted] Grid value 24: lambda = .1072025 no. of nonzero coef. = 78 Folds: 1...5....10 CVF = 12.40012 ... cross-validation complete ... minimum found Lasso step 2 of 2: Evaluating up to 100 lambdas in grid ... Grid value 1: lambda = 51.68486 no. of nonzero coef. = 4 [output omitted] Grid value 100: lambda = .0051685 no. of nonzero coef. = 59 10-fold cross-validation with 100 lambdas ... Fold 1 of 10: 10....20....30....40....50....60....70....80....90....100 [output omitted] Fold 10 of 10: 10....20....30....40....50....60....70....80....90....100 ... cross-validation complete Lasso linear model No. of obs = 458 No. of covariates = 273 Selection: Adaptive No. of lasso steps = 2 Final adaptive step results
|No. of Out-of- CV mean|
|nonzero sample prediction|
|ID||Description lambda coef. R-squared error|
|25||first lambda 51.68486 4 0.0101 17.01083|
|77||lambda before .4095937 46 0.3985 10.33691|
|* 78||selected lambda .3732065 46 0.3987 10.33306|
|79||lambda after .3400519 47 0.3985 10.33653|
|124||last lambda .0051685 59 0.3677 10.86697|
Adaptive lasso selected a model with 46 covariates instead of the 49 selected by ordinary lasso.
We will store these results as adaptive.
. estimates store adaptive
We have three sets of results.
cv contains the model selected by CV.
minBIC contains the model selected by us that corresponds to the minimum BIC.
adaptive contains the model selected by adaptive lasso.
First, let's compare the variables each selected. The lassocoef command does this. We specify sort(coef, standardized) so that the variables with the largest absolute values of their coefficients are listed first.
. lassocoef cv minBIC adaptive, sort(coef, standardized) nofvlabel
|cv minBIC adaptive|
|0.q19||x x x|
|0.q85||x x x|
|1.q5||x x x|
|3.q156||x x x|
|0.q101||x x x|
|0.q88||x x x|
|0.q48||x x x|
|q22||x x x|
|4||x x x|
|q139||x x x|
|0.q56||x x x|
|q31||x x x|
|0.q73||x x x|
|0.q96||x x x|
|1.gender||x x x|
|0.q50||x x x|
|1.q3||x x x|
|3.q16||x x x|
|2.q84||x x x|
|0.q43||x x x|
|0.q149||x x x|
|0.q159||x x x|
|3.q134||x x x|
|0.q115||x x x|
|0.q108||x x x|
|3||x x x|
|0.q160||x x x|
|_cons||x x x|
Start at the top and look down, and you will see that all three approaches selected the first 23 variables listed in the table, the variables with the largest coefficients.
Which model produces the best predictions? Let's do out-of-sample prediction to find out. We split our data into two samples at the outset for just this purpose. We fit the models on sample 1. We can compare predictions for sample 2.
The lassogof command reports goodness-of-fit statistics. We specify option postselection to compare predictions based on the postselection coefficients instead of the penalized coefficients. We specify option over(sample) so that lassogof calculates fit statistics for each sample separately.
. lassogof cv minBIC adaptive, over(sample) postselection Penalized coefficients
|Name sample||MSE R-squared Obs|
|1||8.652771 0.5065 503|
|2||14.58354 0.2658 493|
|1||9.740229 0.4421 508|
|2||13.44496 0.3168 503|
|1||8.637575 0.5057 504|
|2||14.70756 0.2595 494|
We compare MSE and R-squared for sample 2. minBIC did best by both measures.
Learn more about Stata's lasso features.
See [D] splitsample for more about the splitsample command.
See [D] vl for more about the vl command for constructing long variable lists.
Also see Bayesian lasso.