----------------------------------------------------------------------------------------------------------------------------------help rlassolassopack v1.4.2 ----------------------------------------------------------------------------------------------------------------------------------Titlerlasso-- Program for lasso and sqrt-lasso estimation with data-driven penalizationSyntaxrlassodepvarregressors[weight] [ifexp] [inrange] [,sqrtpartial(varlist)pnotpen(varlist)psolver(string)norecovernoconstantfenoftoolsrobustcluster(varlist)bw(int)kernel(string)centerxdependentnumsim(int)prestdtolopt(real)tolpsi(real)tolzero(real)maxiter(int)maxpsiiter(int)maxabsxlassopsicorrnumber(int)lalternativegamma(real)maqc(real)c0(real)supscoressnumsim(int)testonlyseed(real)displayallpostallolsverbosevverbose] Note: thefeoption will take advantage of theftoolspackage (if installed) for the fixed-effects transform; the speed gains using this package can be large. See help ftools or click on ssc install ftools to install.General optionsDescription ----------------------------------------------------------------------------------------------------------------------------sqrtuse sqrt-lasso (default is standard lasso)noconstantsuppress constant from regression (cannot be used withaweightsorpweights)fefixed-effects model (requires data to bextset)noftoolsdo not use FTOOLS package for fixed-effects transform (slower; rarely used)partial(varlist)variables partialled-out prior to lasso estimation, including the constant (if present); to partial-out just the constant, specifypartial(_cons)pnotpen(varlist)variables not penalized by lassopsolver(string)override default solver used for partialling out (one of: qr, qrxx, lu, luxx, svd, svdxx, chol; default=qrxx)norecoversuppress recovery of partialled out variables after estimation.robustlasso penalty loadings account for heteroskedasticitycluster(varlist)lasso penalty loadings account for clustering; both standard (1-way) and 2-way clustering supportedbw(int)lasso penalty loadings account for autocorrelation (AC) using bandwidthint; use withrobustto account for both heteroskedasticity and autocorrelation (HAC)kernel(string)kernel used for HAC/AC penalty loadings (one of: bartlett, truncated, parzen, thann, thamm, daniell, tent, qs; default=bartlett)centercenter moments in heteroskedastic and cluster-robust loadingslassopsiuse lasso or sqrt-lasso residuals to obtain penalty loadings (psi) (default is post-lasso)corrnumber(int)number of high-correlation regressors used to obtain initial residuals; default=5; if =0, thendepvaris used in place of residualsprestdstandardize data prior to estimation (default is standardize during estimation via penalty loadings)seed(real)set Stata's random number seed prior toxdepandsupscoresimulations (default=leave state unchanged)LambdaDescription ----------------------------------------------------------------------------------------------------------------------------xdependentpenalty level is estimated depending on Xnumsim(int)number of simulations used for the X-dependent case (default=5000)lalternativealternative (less sharp) lambda0 = 2c*sqrt(N)*sqrt(2*log(2*p/gamma)) (sqrt-lasso = replace 2c with c)gamma(real)"gamma" in lambda0 function (default = 0.1/log(N); cluster-lasso = 0.1/log(N_clust))maq(HAC/AC with truncated kernel only) "gamma" in lambda0 function = 0.1/log(N/(bw+1)); mimics cluster-robustc(real)"c" in lambda0 function (default = 1.1)c0(real)(rarely used) "c" in lambda0 function in first iteration only when iterating to obtain penalty loadings (default = 1.1)OptimizationDescription ----------------------------------------------------------------------------------------------------------------------------tolopt(real)tolerance for lasso shooting algorithm (default=1e-10)tolpsi(real)tolerance for penalty loadings algorithm (default=1e-4)tolzero(real)minimum below which coeffs are rounded down to zero (default=1e-4)maxiter(int)maximum number of iterations for the lasso shooting algorithm (default=10k)maxpsiiter(int)maximum number of lasso-based iterations for penalty loadings (psi) algorithm (default=2)maxabsx(sqrt-lasso only) use max(abs(x_ij)) as initial penalty loadings as per Belloni et al. (2014)Sup-score testDescription ----------------------------------------------------------------------------------------------------------------------------supscorereport sup-score test of statistical significancetestonlyreport only sup-score test; do not estimate lasso regressionssgamma(real)test level for conservative critical value for the sup-score test (default = 0.05, i.e., 5% significance level)ssnumsim(int)number of simulations for sup-score test multiplier bootstrap (default=500; 0 => do not simulate)Display and postDescription ----------------------------------------------------------------------------------------------------------------------------displayalldisplay full coefficient vectors including unselected variables (default: display only selected, unpenalized and partialled-out)postallpost full coefficient vector including unselected variables in e(b) (default: e(b) has only selected, unpenalized and partialled-out)olspost OLS coefs using lasso-selected variables in e(b) (default is lasso coefs)verboseshow additional outputvverboseshow even more outputdotsshow dots corresponding to repetitions in simulations (xdepandsupscore) ---------------------------------------------------------------------------------------------------------------------------- Postestimation:predict[type]newvar[if] [in] [,xbueuexburesidlassonoisilyols]predictis not currently supported after fixed-effects estimation.OptionsDescription ----------------------------------------------------------------------------------------------------------------------------xbgenerate fitted values (default)residualsgenerate residualsegenerate overall error component e(it). Only afterfe.uegenerate combined residuals, i.e., u(i) + e(it). Only afterfe.xbuprediction including fixed effect, i.e., a + xb + u(i). Only afterfe.ufixed effect, i.e., u(i). Only afterfe.noisilydisplays beta used for prediction.lassouse lasso coefficients for prediction (default is posted e(b) matrix)olsuse OLS coefficients based on lasso-selected variables for prediction (default is posted e(b) matrix) ---------------------------------------------------------------------------------------------------------------------------- Replay:rlasso[,displayall]OptionsDescription ----------------------------------------------------------------------------------------------------------------------------displayalldisplay full coefficient vectors including unselected variables (default: display only selected, unpenalized and partialled-out) ----------------------------------------------------------------------------------------------------------------------------rlassomay be used with time-series or panel data, in which case the data must be tsset or xtset first; see helptssetorxtset.aweightsandpweightsare supported; see helpweights.pweightsis equivalent toaweights+robust. All varlists may contain time-series operators or factor variables; see helpvarlist.Description Estimation methods Penalty loadings Sup-score test of joint significance Computational notes Miscellaneous Version notes Examples of usage Saved results References Website Installation Acknowledgements Citation of lassopackContentsDescriptionrlassois a routine for estimating the coefficients of a lasso or square-root lasso (sqrt-lasso) regression where the lasso penalization is data-dependent and where the number of regressors p may be large and possibly greater than the number of observations. The lasso (Least Absolute Shrinkage and Selection Operator, Tibshirani1996) is a regression method that uses regularization and the L1 norm.rlassoimplements a version of the lasso that allows for heteroskedastic and clustered errors; see Belloni et al. (2012,2013,2014,2016). For an overview ofrlassoand the theory behind it, see Ahrens et al. (2020) The default estimator implemented byrlassois the lasso. An alternative that does not involve estimating the error variance is the square-root-lasso (sqrt-lasso) of Belloni et al. (2011,2014), available with thesqrtoption. The lasso and sqrt-lasso estimators achieve sparse solutions: of the full set of p predictors, typically most will have coefficients set to zero and only s<<p will be non-zero. The "post-lasso" estimator is OLS applied to the variables with non-zero lasso or sqrt-lasso coefficients, i.e., OLS using the variables selected by the lasso or sqrt-lasso. The lasso/sqrt-lasso and post-lasso coefficients are stored ine(beta)ande(betaOLS), respectively. By default,rlassoposts the lasso or sqrt-lasso coefficients ine(b). To post ine(b)the OLS coefficients based on lasso- or sqrt-lasso-selected variables, use theolsoption.Estimation methodsrlassosolves the following problem min 1/N RSS + lambda/N*||Psi*beta||_1, where RSS = sum(y(i)-x(i)'beta)^2 denotes the residual sum of squares, beta is a p-dimensional parameter vector, lambda is the overall penalty level, ||.||_1 denotes the L1-norm, i.e., sum_i(abs(a[i])); Psi is a p by p diagonal matrix of predictor-specific penalty loadings. Note thatrlassotreats Psi as a row vector. N number of observations If the optionsqrtis specified,rlassoestimates the sqrt-lasso estimator, which is defined as the solution to: min sqrt(1/N*RSS) + lambda/N*||Psi*beta||_1. Note: the above lambda differs from the definition used in parts of the lasso and elastic net literature; see for example the R packageglmnetby Friedman et al. (2010). The objective functions here follow the format of Belloni et al. (2011,2012). Specifically,lambda(r)=2*N*lambda(GN)wherelambda(r)is the penalty level used byrlassoandlambda(GN)is the penalty level used byglmnet.rlassoobtains the solutions to the lasso sqrt-lasso using coordinate descent algorithms. The algorithm was first proposed by Fu (1998) for the lasso (then referred to as "shooting"). For further details of how the lasso and sqrt-lasso solutions are obtained, seelasso2.rlassofirst estimates the lasso penalty level and then uses the coordinate descent algorithm to obtain the lasso coefficients. For the homoskedastic case, a single penalty level lambda is applied; in the heteroskedastic and cluster cases, the penalty loadings vary across regressors. The methods are discussed in detail in Belloni et al. (2012,2013,2014,2016) and are described only briefly here. For a detailed discussion of an R implementation ofrlasso, see Spindler et al. (2016). For compatibility with the wider lasso literature, the documentation here uses "lambda" to refer to the penalty level that, combined with the possibly regressor-specific penalty loadings, is used with the estimation algorithm to obtain the lasso coefficients. "lambda0" refers to the component of the overall lasso penalty level that does not depend on the error variance. Note that this terminology differs from that in the R implementation ofrlassoby Spindler et al. (2016). The default lambda0 for the lasso is 2c*sqrt(N)*invnormal(1-gamma/(2p)), where p is the number of penalized regressors and c and gamma are constants with default values of 1.1 and 0.1/log(N), respectively. In the cluster-lasso (Belloni et al.2016) the default gamma is 0.1/log(N_clust), where N_clust is the number of clusters (saved ine(N_clust)). The default lambda0s for the sqrt-lasso are the same except replace 2c with c. The constant c>1.0 is a slack parameter; gamma controls the confidence level. The alternative formula lambda0 = 2c*sqrt(N)*sqrt(2*log(2p/gamma)) is available with thelaltoption. The constants c and gamma can be set using thec(real)andgamma(real)options. Thexdepoption is another alternative that implements an "X-dependent" penalty level lambda0; see Belloni and Chernozhukov (2011) and Belloni et al. (2013) for discussion. The default lambda for the lasso in the i.i.d. case is lambda0*rmse, where rmse is an estimate of the standard deviation of the error variance. The sqrt-lasso differs from the standard lasso in that the penalty term lambda is pivotal in the homoskedastic case and does not depend on the error variance. The default for the sqrt-lasso in the i.i.d. case is lambda=lambda0=c*sqrt(N)*invnormal(1-gamma/(2*p)) (note the absence of the factor of "2" vs. the lasso lambda).As is standard in the lasso literature, regressors are standardized to have unit variance. By default, standardization is achieved by incorporating the standard deviations of the regressors into the penalty loadings. In the default homoskedastic case, the penalty loadings are the vector of standard deviations of the regressors. The normalized penalty loadings are the penalty loadings normalized by the SDs of the regressors. In the homoskedastic case the normalized penalty loadings are a vector of 1s.Penalty loadingsrlassosaves the vector of penalty loadings, the vector of normalized penalty loadings, and the vector of SDs of the regressors X ine(.)macros. Penalty loadings are constructed after the partialling-out of unpenalized regressors and/or the FE (fixed-effects) transformation, if applicable. A alternative to partialling-out unpenalized regressors with thepartial(varlist)option is to give them penalty loadings of zero with thepnotpen(varlist)option. By the Frisch-Waugh-Lovell Theorem for the lasso (Yamada2017), the estimated lasso coefficients are the same in theory (but seebelow) whether the unpenalized regressors are partialled-out or given zero penalty loadings, so long as the same penalty loadings are used for the penalized regressors in both cases. Note that the calculation of the penalty loadings in both thepartial(.)andpnotpen(.)cases involves adjustments for the partialled-out variables. This is different from thelasso2handling of unpenalized variables specified in thelasso2optionnotpen(.), where no such adjustment of the penalty loadings is made (and is why the two no-penalization options are named differently). Regressor-specific penalty loadings for the heteroskedastic and clustered cases are derived following the methods described in Belloni et al. (2012,2013,2014,2015,2016). The penalty loadings for the heteroskedastic-robust case have elements of the form sqrt[avg(x^2e^2)]/sqrt[avg(e^2)] where x is a (demeaned) regressor, e is the residual, and sqrt[avg(e^2)] is the root mean squared error; the normalized penalty loadings have elements sqrt[avg(x^2e^2)]/(sqrt[avg(x^2)]sqrt[avg(e^2)]) where the sqrt(avg(x^2) in the denominator is SD(x), the standard deviation of x. This corresponds to the presentation of penalty loadings in Belloni et al. (2014; see Algorithm 1 but note that in their presentation, the predictors x are assumed already to be standardized). NB: in the presentation we use here, the penalty loadings for the lasso and sqrt-lasso are the same; what differs is the overall penalty term lambda. The cluster-robust case is similar to the heteroskedastic case except that numerator sqrt[avg(x^2e^2)] in the heteroskedastic case is replaced by sqrt[avg(u_i^2)], where (using the notation of the Stata manual's discussion of the _robust command) u_i is the sum of x_ij*e_ij over the j members of cluster i; see Belloni et al. (2016). Again in the presentation used here, the cluster-lasso and cluster-sqrt-lasso penalty loadings are the same. The unit vector is again the benchmark for the standardized penalty loadings. NB: also following_robust, the denominator of avg(u_i^2) and Tbar is (N_clust-1).cluster(varname1 varname2)implements two-way cluster-robust penalty loadings (Cameron et al.2011; Thompson2011). "Two-way cluster-robust" means the penalty loadings accommodate arbitrary within-group correlation in two distinct non-nested categories defined byvarname1andvarname2. Note that the asymptotic justification for the two-way cluster-robust approach requires both dimensions to be "large" (go off to infinity). Autocorrelation-consistent (AC) and heteroskedastic and autocorrelation-consistent (HAC) penalty loadings can be obtained by using thebw(int)option on its own (AC) or in combination with therobustoption (HAC), whereintspecifies the bandwidth; see Chernozhukov et al. (2018, 2020) and Ahrens et al. (2020). Syntax and usage follows that used byivreg2; see theivreg2help file for details. The default is to use the Bartlett kernel; this can be changed using thekerneloption. The full list of kernels available is (abbreviations in parentheses): Bartlett (bar); Truncated (tru); Parzen (par); Tukey-Hanning (thann); Tukey-Hamming (thamm); Daniell (dan); Tent (ten); and Quadratic-Spectral (qua or qs). AC and HAC penalty loadings can also be used for (large T) panel data; this requires the dataset to bextset. Note that for some kernels it is possible in finite samples to obtain negative variances and hence undefined penalty loadings; the same is true of two-way cluster-robust. Intutively, this arises because the covariance term in a calculation like var+var-2cov is "too big". When this happens,rlassoissues a warning and (arbitrarily) replaces 2cov with cov. Thecenteroption centers the x_ij*e_ij terms (or in the cluster-lasso case, the u_i terms) prior to calculating the penalty loadings.Sup-score test of joint significancerlassowith thesupscoreoption reports a test of the null hypothesis H0: beta_1 = ... = beta_p = 0. i.e., a test of the joint significance of the regressors (or, alternatively, a test that H0: s=0; of the full set of p regressors, none is in the true model). The test follows Chernozhukov et al. (2013, Appendix M); see also Belloni et al. (2012,2013). (The variables are assumed to be rescaled to be centered and with unit variance.) If the null hypothesis is correct and the rest of the model is well-specified (including the assumption that the regressors are orthogonal to the disturbance e), then E(e*x_j) = E((y-beta_0)*x_j) = 0, j=1...p where beta_0 is the intercept. The sup-score statistic is S=sqrt(N)*max_j(abs(avg((y-b_0)*x_j))/(sqrt(avg(((y-b_0)*x_j)^2)))), where: (a) the numerator abs(avg((y-b_0)*x_j)) is the absolute value of the average score for regressor x_j and b_0 is sample mean of y; (b) the denominator sqrt(avg(((y-b_0)*x_j)^2)) is the sample standard deviation of the score; (c) the statistic is sqrt(N) times the maximum across the p regressors of the ratio of (a) to (b). The p-value for the sup-score test is obtained by a multiplier bootstrap procedure simulating the statistic W, defined as W=sqrt(N)*max_j(abs(avg((y-b_0)*x_j*u))/(sqrt(avg(((y-b_0)*x_j)^2)))) where u is an iid standard normal variate independent of the data. Thessnumsim(int)option controls the number of simulated draws (default=500);ssnumsim(0)requests that the sup-score statistic is reported without a simulation-based p-value.rlassoalso reports a conservative critical value (asymptotic bound) as per Belloni et al. (2012,2013), defined as c*invnormal(1-gamma/(2p)); this can be set by the optionssgamma(int)(default = 0.05).A computational alternative to the default of standardizing "on the fly" (i.e., incorporating the standardization into the lasso penalty loadings) is to standardize all variables to have unit variance prior to computing the lasso coefficients. This can be done using theComputational notesprestdoption. The results are equivalent in theory. Theprestdoption can lead to improved numerical precision or more stable results in the case of difficult problems; the cost is (a typically small) computation time required to standardize the data. Either thepartial(varlist)option or thepnotpen(varlist)option can be used for variables that should not be penalized by the lasso. The options are equivalent in theory (see above), but numerical results can differ in practice because of the different calculation methods used. Partialling-out variables can lead to improved numerical precision or more stable results in the case of difficult problems vs. specifying the variables as unpenalized, but may be slower in terms of computation time. Both thepartial(varlist)andpnotpen(varlist)options use least squares. This is implemented in Mata using one of Mata's solvers. In cases where the variables to be partialled out are collinear or nearly so, different solvers may generate different results. Users may wish to check the stability of their results in such cases. Thepsolver(.)option can be used to specify the Mata solver used. The default behavior ofrlassoto solve AX=B for X is to use the QR decomposition applied to (A'A) and (A'B), i.e., qrsolve((A'A),(A'B)), abbreviated qrxx. Available options are qr, qrxx, lu, luxx, svd, svdxx, where, e.g., svd indicates using svsolve(A,B) and svdxx indicates using svsolve((A'A),(A'B)).rlassowill warn if collinear variables are dropped when partialling out. By default the constant (if present) is not penalized if there are no regressors being partialled out; this is equivalent to mean-centering prior to estimation. The exception to this is ifaweightsoraweightsare specified, in which case the constant is partialled-out. Thepartial(varlist)option will automatically also partial out the constant (if present); to partial out just the constant, specifypartial(_cons). The within transformation implemented by thefeoption automatically mean-centers the data; thenoconsoption is redundant in this case and may not be specified with this option. Theprestdandpnotpen(varlist)vs.partial(varlist)options can be used as simple checks for numerical stability by comparing results that should be equivalent in theory. If the results differ, the values of the minimized objective functions (e(pmse)ore(prmse)) can be compared. Thefefixed-effects option is equivalent to (but computationally faster and more accurate than) specifying unpenalized panel-specific dummies. The fixed-effects ("within") transformation also removes the constant as well as the fixed effects. The panel variable used by thefeoption is the panel variable set byxtset. To use weights with fixed effects, the ftools must be installed.By defaultMiscellaneousrlassoreports only the set of selected variables and their lasso and post-lasso coefficients; the omitted coefficients are not reported in the regression output. Thepostallanddisplayalloptions allow the full coefficient vector (with coefficients of unselected variables set to zero) to be either posted ine(b)or displayed as output.rlasso, like the lasso in general, accommodates possibly perfectly-collinear sets of regressors. Stata'sfactor variablesare supported byrlasso(as well as bylasso2). Users therefore have the option of specifying as regressors one or more complete sets of factor variables or interactions with no base levels using theibnprefix. This can be interpreted as allowingrlassoto choose the members of the base category. The choice of whether to usepartial(varlist)orpnotpen(varlist)will depend on the circumstances faced by the user. Thepartial(varlist)option can be helpful in dealing with data that have scaling problems or collinearity issues; in these cases it can be more accurate and/or achieve convergence faster than thepnotpen(varlist)option. Thepnotpen(varlist)option will sometimes be faster because it avoids using the pre-estimation transformation employed bypartial(varlist). The two options can be used simultaneously (but not for the same variables). The treatment of standardization, penalization and partialling-out inrlassodiffers from that oflasso2. In therlassotreatment, standardization incorporates the partialling-out of regressors listed in thepnotpen(varlist)list as well as those in thepartial(varlist)list. This is in order to maintain the equivalence of the lasso estimator irrespective of which option is used for unpenalized variables (see the discussion of the Frisch-Waugh-Lovell Theorem for the lasso above). In thelasso2treatment, standardization takes place after the partialling-out of only the regressors listed in thenotpen(varlist)option. In other words,rlassoadjusts the penalty loadings for any unpenalized variables;lasso2does not. For further details, seelasso2. The initial overhead for fixed-effects estimation and/or partialling out and/or pre-estimation standardization (creating temporary variables and then transforming the data) can be noticable for large datasets. For problems that involve looping over data, users may wish to first transform the data by hand. If a small number of correlations is set using thecorrnum(int)option, users may want to increase the number of penalty loadings iterations from the default of 2 to something higher using themaxpsiiter(int)option. The sup-score p-value is obtained by simulation, which can be time-consuming for large datasets. To skip this and use only the conservative (asymptotic bound) critical value, set the number of simulations to zero with thessnumsim(0)option.Detailed version notes can be found inside the ado files rlasso.ado and lassoutils.ado. Noteworthy changes appear below. In versions ofVersion noteslassoutilsprior to 1.1.01 (8 Nov 2018), the very first iteration to obtain penalty loadings set the constant c=0.55. This was dropped in version 1.1.01, and the constant c is unchanged in all iterations. To replicate the previous behavior ofrlasso, use thec0(real)option. For example, with the default value of c=1.1, to replicate the earlier behavior usec0(0.55). In versions oflassoutilsprior to 1.1.01 (8 Nov 2018), the sup-score test statistic S was N*max_j rather than sqrt(N)*max_j as in Chernozhukov et al. (2013), and similarly for the simulated statistic W.Examples using prostate cancer data from Hastie et al. (2009Load prostate cancer data. . clear . insheet using https://web.stanford.edu/~hastie/ElemStatLearn/datasets/prostate.data, tab Estimate lasso using data-driven lambda penalty; default homoskedasticity case. . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45 Use square-root lasso instead. . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45, sqrt Illustrate relationships between lambda, lambda0 and penalty loadings: Basic usage: homoskedastic case, lasso . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45 lambda=lambda0*SD is lasso penalty; incorporates the estimate of the error variance default lambda0 is 2c*sqrt(N)*invnormal(1-gamma/(2*p)) . di e(lambda) . di e(lambda0) In the homoskedastic case, penalty loadings are the vector of SDs of penalized regressors . mat list e(ePsi) ...and the standardized penalty loadings are a vector of 1s. . mat list e(sPsi) Heteroskedastic case, lasso . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45, robust lambda and lambda0 are the same as for the homoskedastic case . di e(lambda) . di e(lambda0) Penalty loadings account for heteroskedasticity as well as incorporating SD(x) . mat list e(ePsi) ...and the standardized penalty loadings are not a vector of 1s. . mat list e(sPsi) Homoskedastic case, sqrt-lasso . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45, sqrt with the sqrt-lasso, the default lambda=lambda0=c*sqrt(N)*invnormal(1-gamma/(2*p)); note the difference by a factor of 2 vs. the standard lasso lambda0 . di e(lambda) . di e(lambda0))rlassovs.lasso2(if installed) . rlasso lpsa lcavol lweight age lbph svi lcp gleason pgg45 lambda=lambda0*SD is lasso penalty; incorporates the estimate of the error variance default lambda0 is 2c*sqrt(N)*invnormal(1-gamma/(2*p)) . di %8.5f e(lambda) Replicaterlassoestimates usingrlassolambda andlasso2. lasso2 lpsa lcavol lweight age lbph svi lcp gleason pgg45, lambda(44.34953)Examples using data from Acemoglu-Johnson-Robinson (2001Load and reorder AJR data for Table 6 and Table 8 (datasets need to be in current directory). . clear . (click to download maketable6.zip from economics.mit.edu) . unzipfile maketable6 . (click to download maketable8.zip from economics.mit.edu) . unzipfile maketable8 . use maketable6 . merge 1:1 shortnam using maketable8 . keep if baseco==1 . order shortnam logpgp95 avexpr lat_abst logem4 edes1975 avelf, first . order indtime euro1900 democ1 cons1 democ00a cons00a, last Alternatively, load AJR data from our website (no manual download required): . clear . use https://statalasso.github.io/dta/AJR.dta Basic usage: . rlasso logpgp95 lat_abst edes1975 avelf temp* humid* steplow-oilres Heteroskedastic-robust penalty loadings: . rlasso logpgp95 lat_abst edes1975 avelf temp* humid* steplow-oilres, robust Partialling-out vs. non-penalization: . rlasso logpgp95 lat_abst edes1975 avelf temp* humid* steplow-oilres, partial(lat_abst) . rlasso logpgp95 lat_abst edes1975 avelf temp* humid* steplow-oilres, pnotpen(lat_abst) Request sup-score test (H0: all betas=0): . rlasso logpgp95 lat_abst edes1975 avelf temp* humid* steplow-oilres, supscore)Examples using data from Angrist-Krueger (1991Load AK data and rename variables (dataset needs to be in current directory). NB: this is a large dataset (330k observations) and estimations may take some time to run on some installations. . clear . (click to download asciiqob.zip from economics.mit.edu) . unzipfile asciiqob.zip . infix lnwage 1-9 edu 10-20 yob 21-31 qob 32-42 pob 43-53 using asciiqob.txt Alternatively, get data from our website source (no unzipping needed): . use https://statalasso.github.io/dta/AK91.dta xtset data by place of birth (state): . xtset pob State (place of birth) fixed effects; regressors are year of birth, quarter of birth and QOBxYOB. . rlasso edu i.yob# #i.qob, fe As above but explicit penalized state dummies and all categories (no base category) for all factor vars. Note that the (unpenalized) constant is reported. . rlasso edu ibn.yob# #ibn.qob ibn.pob State fixed effects; regressors are YOB, QOB and QOBxYOB; cluster on state. . rlasso edu i.yob# #i.qob, fe cluster(pob))Example using data from Belloni et al. (2015Load dataset on eminent domain (available at journal website). . clear . import excel using CSExampleData.xlsx, first Settings used in Belloni et al. ()2015) - results as in text discussion (p=147): . rlasso NumProCase Z* BA BL DF, robust lalt corrnum(0) maxpsiiter(100) c0(0.55) . di e(p) Settings used in Belloni et al. (2015) - results as in journal replication file (p=144): . rlasso NumProCase Z*, robust lalt corrnum(0) maxpsiiter(100) c0(0.55) . di e(p). use http://fmwww.bc.edu/ec-p/data/wooldridge/phillips.dta . tsset year, yearly Autocorrelation-consistent (AC) penalty loadings; bandwidth=3; default kernel is Bartlett. . rlasso cinf L(0/10).unem, bw(3) Heteroskedastic- and autocorrelation-consistent (HAC) penalty loadings; bandwidth=5; kernel is quadratic-spectral. . rlasso cinf L(0/10).unem, bw(5) rob kernel(qs)Examples illustrating AC/HAC penalty loadingssSaved resultsrlassosaves the following ine(): scalarse(N)sample sizee(N_clust)number of clusters in cluster-robust estimation; in the case of 2-way cluster-robust,e(N_clust)=min(e(N_clust1),e(N_clust2))e(N_g)number of groups in fixed-effects modele(p)number of penalized regressors in modele(s)number of selected regressorse(s0)number of selected and unpenalized regressors including constant (if present)e(lambda0)penalty level excluding rmse (default = 2c*sqrt(N)*invnormal(1-gamma/(2*p)))e(lambda)lasso: penalty level including rmse (=lambda0*rmse); sqrt-lasso: lambda=lambda0e(slambda)standardized lambda; equiv to lambda used on standardized data; lasso: slambda=lambda/SD(depvar); sqrt-lasso: slambda=lambda0e(c)parameter in penalty level lambdae(gamma)parameter in penalty level lambdae(niter)number of iterations for shooting algorithme(maxiter)max number of iterations for shooting algorithme(npsiiter)number of iterations for loadings algorithme(maxpsiiter)max iterations for loadings algorithme(r2)R-sq for lasso estimatione(rmse)rmse using lasso resdualse(rmseOLS)rmse using post-lasso residualse(pmse)minimized objective function (penalized mse, standard lasso only)e(prmse)minimized objective function (penalized rmse, sqrt-lasso only)e(cons)=1 if constant in model, =0 otherwisee(fe)=1 if fixed-effects model, =0 otherwisee(center)=1 if moments have been centerede(bw)(HAC/AC only) bandwidth usede(supscore)sup-score statistice(supscore_p)sup-score p-valuee(supscore_cv)sup-score critical value (asymptotic bound) macrose(cmd)rlassoe(cmdline)command linee(depvar)name of dependent variablee(varX)all regressorse(varXmodel)penalized regressorse(pnotpen)unpenalized regressorse(partial)partialled-out regressorse(selected)selected and penalized regressorse(selected0)all selected regressors including unpenalized and constant (if present)e(method)lasso or sqrt-lassoe(estimator)lasso, sqrt-lasso or post-lasso ols posted in e(b)e(robust)heteroskedastic-robust penalty loadingse(clustvar)variable defining clusters for cluster-robust penalty loadings; if two-way clustering is used, the variables are ine(clustvar1)ande(clustvar2)e(kernel)(HAC/AC only) kernel usede(ivar)variable defining groups for fixed-effects model matricese(b)posted coefficient vectore(beta)lasso or sqrt-lasso coefficient vectore(betaOLS)post-lasso coefficient vectore(betaAll)full lasso or sqrt-lasso coefficient vector including omitted, factor base variables, etc.e(betaAllOLS)full post-lasso coefficient vector including omitted, factor base variables, etc.e(ePsi)estimated penalty loadingse(sPsi)standardized penalty loadings (vector of 1s in homoskedastic case functionse(sample)estimation sampleAcemoglu, D., Johnson, S. and Robinson, J.A. 2001. The colonial origins of comparative development: An empirical investigation.ReferencesAmerican Economic Review, 91(5):1369-1401. https://economics.mit.edu/files/4123 Ahrens, A., Aitkens, C., Dizten, J., Ersoy, E., Kohns, D. and M.E. Schaffer. 2020. A Theory-based Lasso for Time-Series Data. Invited paper for the International Conference of Econometrics of Vietnam, January 2020. Forthcoming inStudiesin Computational Intelligence(Springer). Ahrens, A., Hansen, C.B. and M.E. Schaffer. 2020. lassopack: model selection and prediction with regularized regression in Stata.The Stata Journal, 20(1):176-235. https://journals.sagepub.com/doi/abs/10.1177/1536867X20909697. Working paper version: https://arxiv.org/abs/1901.05397. Angrist, J. and Kruger, A. 1991. Does compulsory school attendance affect schooling and earnings?Quarterly Journal ofEconomics106(4):979-1014. http://www.jstor.org/stable/2937954 Belloni, A. and Chernozhukov, V. 2011. High-dimensional sparse econometric models: An introduction. In Alquier, P., Gautier E., and Stoltz, G. (eds.), Inverse problems and high-dimensional estimation. Lecture notes in statistics, vol. 203. Springer, Berlin, Heidelberg. https://arxiv.org/pdf/1106.5242.pdf Belloni, A., Chernozhukov, V. and Wang, L. 2011. Square-root lasso: Pivotal recovery of sparse signals via conic programming.Biometrika98:791-806. https://doi.org/10.1214/14-AOS1204 Belloni, A., Chen, D., Chernozhukov, V. and Hansen, C. 2012. Sparse models and methods for optimal instruments with an application to eminent domain.Econometrica80(6):2369-2429. http://onlinelibrary.wiley.com/doi/10.3982/ECTA9626/abstract Belloni, A., Chernozhukov, V. and Hansen, C. 2013. Inference for high-dimensional sparse econometric models. InAdvancesin Economics and Econometrics: 10th World Congress, Vol. 3: Econometrics, Cambridge University Press: Cambridge, 245-295. http://arxiv.org/abs/1201.0220 Belloni, A., Chernozhukov, V. and Hansen, C. 2014. Inference on treatment effects after selection among high-dimensional controls.Review of Economic Studies81:608-650. https://doi.org/10.1093/restud/rdt044 Belloni, A., Chernozhukov, V. and Hansen, C. 2015. High-dimensional methods and inference on structural and treatment effects.Journal of Economic Perspectives28(2):29-50. http://www.aeaweb.org/articles.php?doi=10.1257/jep.28.2.29 Belloni, A., Chernozhukov, V., Hansen, C. and Kozbur, D. 2016. Inference in high dimensional panel models with an application to gun control.Journal of Business and Economic Statistics34(4):590-605. http://amstat.tandfonline.com/doi/full/10.1080/07350015.2015.1102733 Belloni, A., Chernozhukov, V. and Wang, L. 2014. Pivotal estimation via square-root-lasso in nonparametric regression.Annals of Statistics42(2):757-788. https://doi.org/10.1214/14-AOS1204 Chernozhukov, V., Chetverikov, D. and Kato, K. 2013. Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors.Annals of Statistics41(6):2786-2819. https://projecteuclid.org/euclid.aos/1387313390 Cameron, A.C., Gelbach, J.B. and D.L. Miller. Robust Inference with Multiway Clustering.Journal of Business & EconomicStatistics29(2):238-249. https://www.jstor.org/stable/25800796. Working paper version: NBER Technical Working Paper 327, http://www.nber.org/papers/t0327. Chernozhukov, V., Hardle, W.K., Huang, C. and W. Wang. 2018 (rev 2020). LASSO-driven inference in time and space.Workingpaper. https://arxiv.org/abs/1806.05081 Correia, S. 2016. FTOOLS: Stata module to provide alternatives to common Stata commands optimized for large datasets. https://ideas.repec.org/c/boc/bocode/s458213.html Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent.Journal of Statistical Software33(1), 1\9622. https://doi.org/10.18637/jss.v033.i01 Fu, W.J. 1998. Penalized regressions: The bridge versus the lasso.Journal of Computational and Graphical Statistics7(3):397-416. http://www.tandfonline.com/doi/abs/10.1080/10618600.1998.10474784 Hastie, T., Tibshirani, R. and Friedman, J. 2009.The elements of statistical learning(2nd ed.). New York: Springer-Verlag. https://web.stanford.edu/~hastie/ElemStatLearn/ Spindler, M., Chernozhukov, V. and Hansen, C. 2016. High-dimensional metrics. https://cran.r-project.org/package=hdm. Thompson, S.B. 2011. Simple formulas for standard errors that cluster by both firm and time.Journal of FinancialEconomics99(1):1-10. Working paper version: http://ssrn.com/abstract=914002. Tibshirani, R. 1996. Regression shrinkage and selection via the lasso.Journal of the Royal Statistical Society. Series B(Methodological)58(1):267-288. https://doi.org/10.2307/2346178 Yamada, H. 2017. The Frisch-Waugh-Lovell Theorem for the lasso and the ridge regression.Communications in Statistics -Theory and Methods46(21):10897-10902. http://dx.doi.org/10.1080/03610926.2016.1252403Please check our website https://statalasso.github.io/ for more information.WebsiteInstallationrlassois part of thelassopackpackage. To get the latest stable version oflassopackfrom our website, check the installation instructions at https://statalasso.github.io/installation/. We update the stable website version more frequently than the SSC version. Earlier versions of lassopack are also available from the website. To verify thatlassopackis correctly installed, click on or type whichpkg lassopack (which requireswhichpkgto be installed; ssc install whichpkg).Thanks to Alexandre Belloni for providing Matlab code for the square-root-lasso and to Sergio Correia for advice on the use of the FTOOLS package.AcknowledgementsCitation of rlassorlassois not an official Stata command. It is a free contribution to the research community, like a paper. Please cite it as such: Ahrens, A., Hansen, C.B., Schaffer, M.E. 2018 (updated 2020). LASSOPACK: Stata module for lasso, square-root lasso, elastic net, ridge, adaptive lasso estimation and cross-validation http://ideas.repec.org/c/boc/bocode/s458458.html Ahrens, A., Hansen, C.B. and M.E. Schaffer. 2020. lassopack: model selection and prediction with regularized regression in Stata.The Stata Journal, 20(1):176-235. https://journals.sagepub.com/doi/abs/10.1177/1536867X20909697. Working paper version: https://arxiv.org/abs/1901.05397.Achim Ahrens, Public Policy Group, ETH Zurich, Switzerland achim.ahrens@gess.ethz.ch Christian B. Hansen, University of Chicago, USA Christian.Hansen@chicagobooth.edu Mark E. Schaffer, Heriot-Watt University, UK m.e.schaffer@hw.ac.ukAuthorsHelp:Also seelasso2,cvlasso,lassologit, pdslasso, ivlasso (if installed)

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