
on Econometrics 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Aman Ullah (UCR); He Wang (UCR) 
Abstract:  The higherorder asymptotic properties provide better approximation of the bias for a class of estimators. The firstorder asymptotic properties of the asymmetric least squares (ALS) estimator have been investigated by Newey and Powell (1987). This paper develops the secondorder asymptotic properties (bias and mean squared error) of the ALS estimator, extending the secondorder asymptotic results for the symmetric least squares (LS) estimators of Rilstone, Srivastava and Ullah (1996). The LS gives the mean regression function while the ALS gives the "expectile" regression function, a generalization of the usual regression function. The secondorder bias result enables an improved bias correction and thus an improved ALS estimation in finite sample. In particular, we show that the secondorder bias is much larger as the asymmetry is stronger, and therefore the benefit of the secondorder bias correction is greater when we are interested in extreme expectiles which are used as a risk measure in financial economics. The higherorder MSE result for the ALS estimation also enables us to better understand the sources of estimation uncertainty. The Monte Carlo simulation confirms the benefits of the secondorder asymptotic theory and indicates that the secondorder bias is larger at the extreme low and high expectiles. 
Keywords:  asymmetric least squares, expectile, delta function, secondorder bias, Monte Carlo. 
JEL:  C13 C33 C52 
Date:  2018–12 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201910&r=all 
By:  Marc Hallin; Davide La Vecchia; H Liu 
Abstract:  We propose a new class of estimators for semiparametric VARMA models with the innovation density playing the role of nuisance parameter. Our estimators are Restimators based on the multivariate concepts of centeroutward ranks and signs recently proposed by Hallin~(2017). We show how these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield a robust yet flexible and powerful class of estimation procedures for multivariate time series. We develop the relevant asymptotic theory of our Restimators, establishing their rootn consistency and asymptotic normality under a broad class of innovation densities including, e.g. multimodal mixtures of Gaussians or and multivariate skewt distributions. An implementation algorithm is provided in the supplementary material, available online. A Monte Carlo study compares our Restimators with the routinelyapplied Gaussian quasilikelihood ones; the latter appear to be quite significantly outperformed away from elliptical innovations. Numerical results also provide evidence of considerable robustness gains. Two real data examples conclude the paper. 
Keywords:  Multivariate ranks, Distributionfreeness, Local asymptotic normality, Measure transportation, Quasi likelihood estimation, Skew innovation density 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:eca:wpaper:2013/294809&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Bai Huang (CUFE); Aman Ullah (UCR) 
Abstract:  This paper examines the asymptotic properties of the Steintype shrinkage combined (averaging) estimation of panel data models. We introduce a combined estimation when the fixed effects (FE) estimator is inconsistent due to endogeneity arising from the correlated common effects in the regression error and regressors. In this case the FE estimator and the CCEP estimator of Pesaran (2006) are combined. This can be viewed as the panel data model version of the shrinkage to combine the OLS and 2SLS estimators as the CCEP estimator is a 2SLS or control function estimator that controls for the endogeneity arising from the correlated common effects. The asymptotic theory, Monte Carlo simulation, and empirical applications are presented. According to our calculation of the asymptotic risk, the Steinlike shrinkage estimator is more efficient estimation than the CCEP estimator. 
Keywords:  Endogeneity, Panel data, Fixed effect, Common correlated effects, Shrinkage, Model averaging, Local asymptotics, Hausman test. 
JEL:  C13 C33 C52 
Date:  2018–09 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201905&r=all 
By:  Aman Ullah (Department of Economics, University of California Riverside); Shujie Ma (Department of Statistics, University of California Riverside); Jeffrey Racine (Department of Economics, McMaster University) 
Abstract:  We consider a Bspline regression approach towards nonparametric modelling of a random effects (error component) model. We focus our attention on the estimation of marginal effects (derivatives) and their asymptotic properties. Theoretical underpinnings are provided, finitesample performance is evaluated via Monte Carlo simulation, and an application that examines the contribution of different types of public infrastructure on private production is investigated using panel data comprising the 48 contiguous states in the US over the period 19701986. 
JEL:  C14 C23 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201920&r=all 
By:  Sarojini Hirshleifer (Department of Economics, University of California Riverside); Dalia Ghanem (UC Davis); Karen OrtizBecerra (UC Davis) 
Abstract:  We approach attrition in field experiments with baseline outcome data as an identification problem in a panel model. A systematic review of the literature indicates that there is no consensus on how to test for attrition bias. We establish identifying assumptions for treatment effects for both the respondent subpopulation and the study population. We then derive their sharp testable implications on the baseline outcome distribution and propose randomization procedures to test them. We demonstrate that the most commonly used test does not control size in general when internal validity holds. Simulations and applications illustrate the empirical relevance of our analysis. 
Keywords:  attrition, field experiments, randomized experiments, randomized controlled trials, internal validity, KolmogorovSmirnov, CramervonMises, randomization tests 
JEL:  C12 C21 C33 C93 
Date:  2019–08 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201919&r=all 
By:  Xuan Liang; Jiti Gao; Xiaodong Gong 
Abstract:  This paper develops a timevarying coefficient spatial autoregressive panel data model with the individual fixed effects to capture the nonlinear effects of the regressors, which vary over the time. To effectively estimate the model, we propose a method that incorporates the nonparametric local linear method and the concentrated quasimaximum likelihood estimation method to obtain consistent estimators for the spatial coefficient and the timevarying coefficient function. The asymptotic properties of these estimators are derived as well, showing the regular sqrt(NT)rate of convergence for the parametric parameters and the common sqrt(NTh)rate of convergence for the nonparametric component, respectively. Monte Carlo simulations are conducted to illustrate the finite sample performance of our proposed method. Meanwhile, we apply our method to study the Chinese labor productivity to identify the spatial influences and the timevarying spillover effects among 185 Chinese cities with comparison to the results on a subregion East China. 
Keywords:  concentrated quasimaximum likelihood estimation, local linear estimation, timeâ€“varying coefficient. 
JEL:  C21 C23 
Date:  2019 
URL:  http://d.repec.org/n?u=RePEc:msh:ebswps:201926&r=all 
By:  Jaeheon Jung 
Abstract:  This paper deals with the timevarying high dimensional covariance matrix estimation. We propose two covariance matrix estimators corresponding with a timevarying approximate factor model and a timevarying approximate characteristicbased factor model, respectively. The models allow the factor loadings, factor covariance matrix, and error covariance matrix to change smoothly over time. We study the rate of convergence of each estimator. Our simulation and empirical study indicate that timevarying covariance matrix estimators generally perform better than timeinvariant covariance matrix estimators. Also, if characteristics are available that genuinely explain true loadings, the characteristics can be used to estimate loadings more precisely in finite samples; their helpfulness increases when loadings rapidly change. 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1910.11965&r=all 
By:  Sven Husmann; Antoniya Shivarova; Rick Steinert 
Abstract:  The global minimumvariance portfolio is a typical choice for investors because of its simplicity and broad applicability. Although it requires only one input, namely the covariance matrix of asset returns, estimating the optimal solution remains a challenge. In the presence of highdimensionality in the data, the sample estimator becomes illconditioned, which negates the positive effect of diversification in an outofsample setting. To address this issue, we review recent covariance matrix estimators and extend the literature by suggesting a multifold crossvalidation technique. In detail, conducting an extensive empirical analysis with four datasets based on the S\&P 500, we evaluate how the datadriven choice of specific tuning parameters within the proposed crossvalidation approach affects the outofsample performance of the global minimumvariance portfolio. In particular, for cases in which the efficiency of a covariance estimator is strongly influenced by the choice of a tuning parameter, we detect a clear relationship between the optimality criterion for its selection within the crossvalidation and the evaluated performance measure. Finally, we show that using crossvalidation can improve the performance of highly efficient estimators even when the datadriven covariance parameter deviates from its theoretically optimal value. 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1910.13960&r=all 
By:  Aman Ullah (Department of Economics, University of California Riverside); Yoonseok Lee (Syracuse University); Debasri Mukherjee (Western Michigan University) 
Abstract:  This paper considers multivariate local linear least squares estimation of panel data models when fixed effects present. One step estimation of the local marginal effect is of the main interest. A withingroup type nonparametric estimator is developed, where the fixed effects are eliminated by subtracting individualspecific locally weighted time average (i.e., using the local within transformation). It is shown that the localwithintransformationbased estimator satisfies the standard properties of the local linear estimator. In comparison, the nonparametric estimators based on the conventional (i.e., global) within transformation or first difference result in biased estimators, where the bias does not degenerate even with large samples. The new estimator is used to examine the nonlinear relationship between income and nitrogenoxide level (i.e., the environmental Kuznets curve) based on the U.S. statelevel panel data. 
Keywords:  Nonparametric estimation, panel data, fixed effects, multivariate, local linear least squares, local within transformation, environmental Kuznets curve. 
JEL:  C14 C23 
Date:  2018–09 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201901&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Bai Huang (CUFE); Aman Ullah (UCR) 
Abstract:  When some of the regressors in a panel data model are correlated with the random individual effects, the random effect (RE) estimator becomes inconsistent while the fixed effect (FE) estimator is consistent. Depending on the various degree of such correlation, we can combine the RE estimator and FE estimator to form a combined estimator which can be better than each of the FE and RE estimators. In this paper, we are interested in whether the combined estimator may be used to form a combined forecast to improve upon the RE forecast (forecast made using the RE estimator) and the FE forecast (forecast using the FE estimator) in outofsample forecasting. Our simulation experiment shows that the combined forecast does dominate the FE forecast for all degrees of endogeneity in terms of mean squared forecast errors (MSFE), demonstrating that the theoretical results of the risk dominance for the insample estimation carry over to the outofsample forecasting. It also shows that the combined forecast can reduce MSFE relative to the RE forecast for moderate to large degrees of endogeneity and for large degrees of heterogeneity in individual effects. 
Keywords:  Endogeneity, Panel Data, Fixed Effect, Random Effect, Hausman test, Combined Estimator, Combined Forecast. 
JEL:  C13 C33 C52 
Date:  2018–12 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201906&r=all 
By:  Adrian Pagan (School of Economics, University of Sydney, CAMA, Australian National University); Tim Robinson (Melbourne Institute: Applied Economic & Social Research, The University of Melbourne) 
Abstract:  Implications of partial information for applied macroeconomic modelling along four dimensions are shown, and analysis provided on how they can be addressed. First, when permanent shocks are present a Vector ErrorCorrection Model including latent, as well as observed, variables is required to capture macroeconomic dynamics. Second, the assumption in Dynamic Stochastic General Equilibrium models that shocks are autocorrelated provides identifying information usable in Structural Vector AutoRegressions. Third, estimating models with more shocks than observed variables must yield correlated estimated structural shocks. Fourth, including measurement error, as commonly specified, implies a lack of cointegration between variables, even when actually present 
Keywords:  SVAR; Partial Information; Identification; Measurement Error; DSGE. 
JEL:  E37 C51 C52 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:iae:iaewps:wp2019n12&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Bai Huang (Central University of Finance and Economics); Aman Ullah (University of California Riverside) 
Abstract:  The combined estimation for the semiparametric panel data models is proposed. The properties of estimators for the semiparametric panel data models with random effects (RE) and fixed effects (FE) are examined. When the RE estimator suffers from endogeneity due to the individual effects correlated with the regressors, the semiparametric RE and FE estimators may be adaptively combined, with the combining weights depending on the degree of endogeneity. The asymptotic distributions of these three estimators (RE, FE, and combined estimators) for the semiparametric panel data models are derived using a local asymptotic framework. These three estimators are then compared in asymptotic risk. The semiparametric combined estimator has strictly smaller asymptotic risk than the semiparametric fixed effect estimator. The Monte Carlo study shows that the semiparametric combined estimator outperforms semiparametric FE and RE estimators except when the degrees of endogeneity and heterogeneity of the individual effects are very small. Also presented is an empirical application where the effect of public sector capital in the private economy production function is examined using the US state level panel data. 
Keywords:  Endogeneity, Panel Data, Semiparametric FE estimator, Semiparametric RE estimator, Semiparametric Combined Estimator, Local Asymptotics, Hausman Test. 
JEL:  C13 C33 C52 
Date:  2018–07 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201915&r=all 
By:  Domenico Delli Gatti; Jakob Grazzini 
Abstract:  We propose two novel methods to “bring ABMs to the data”. First, we put forward a new Bayesian procedure to estimate the numerical values of ABM parameters that takes into account the time structure of simulated and observed time series. Second, we propose a method to forecast aggregate time series using data obtained from the simulation of an ABM. We apply our methodological contributions to a mediumscale macro agentbased model. We show that the estimated model is capable of reproducing features of observed data and of forecasting oneperiod ahead outputgap and investment with a remarkable degree of accuracy. 
Keywords:  agentbased models, estimation, forecasting 
JEL:  C11 C13 C53 C63 
Date:  2019 
URL:  http://d.repec.org/n?u=RePEc:ces:ceswps:_7894&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Jianghao Chu (UCR); Aman Ullah (UCR) 
Abstract:  In this paper we consider the "Regularization of Derivative Expectation Operator" (Rodeo) of Lafferty and Wasserman (2008) and propose a modified Rodeo algorithm for semiparametric single index models in big data environment with many regressors. The method assumes sparsity that many of the regressors are irrelevant. It uses a greedy algorithm, in that, to estimate the semiparametric single index model (SIM) of Ichimura (1993), all coefficients of the regressors are initially set to start from near zero, then we test iteratively if the derivative of the regression function estimator with respect to each coefficient is significantly different from zero. The basic idea of the modified Rodeo algorithm for SIM (to be called SIMRodeo) is to view the local bandwidth selection as a variable selection scheme which amplifies the coefficients for relevant variables while keeping the coefficients of irrelevant variables relatively small or at the initial starting values near zero. For sparse semiparametric single index models, the SIMRodeo algorithm is shown to attain consistency in variable selection. In addition, the algorithm is fast to finish the greedy steps. We compare SIMRodeo with SIMLasso method in Zeng et al. (2012). Our simulation results demonstrate that the proposed SIMRodeo method is consistent for variable selection and show that it has smaller integrated mean squared errors than SIMLasso. 
Keywords:  Single index model (SIM), Variable selection, Rodeo, SIMRodeo, Lasso, SIMLasso. 
JEL:  C25 C44 C53 C55 
Date:  2018–09 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201908&r=all 
By:  Ruoyao Shi (Department of Economics, University of California Riverside); Cheng Chou (University of Leicester) 
Abstract:  It has been widely acknowledged that the measurement of labor supply in the Current Population Survey (CPS) and other conventional microeconomic surveys has nonclassical measurement error, which will bias the estimates of crucial parameters in labor economics, such as labor supply elasticity. Time diary studies, such as the American Time Use Survey (ATUS), only have accurate measurement of hours worked on a single day, hence the weekly hours worked are unobserved. Despite the missing data problem, we provide several consistent estimators of the parameters in weekly labor supply equation using the information in the time use surveys. The consistency of our estimators does not require more conditions beyond those for a usual two stage least square (2SLS) estimator when the true weekly hours worked are observed. We also show that it is impossible to recover the weekly number of hours worked or its distribution function from time use surveys like the ATUS. In our empirical application we find considerable evidence of nonclassical measurement error in the hours worked in the CPS, and illustrate the consequences of using mismeasured weekly hours worked in empirical studies. 
Keywords:  measurement error, missing data, instrumental variable, asymptotic efficiency, labor supply 
JEL:  C13 C21 C26 C81 J22 
Date:  2019–01 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201912&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Aman Ullah (University of California, Riverside); Ran Wang (University of California, Riverside) 
Abstract:  Bootstrap Aggregating (Bagging) is an ensemble technique for improving the robustness of forecasts. Random Forest is a successful method based on Bagging and Decision Trees. In this chapter, we explore Bagging, Random Forest, and their variants in various aspects of theory and practice. We also discuss applications based on these methods in economic forecasting and inference. 
Keywords:  bagging, decision trees, random forests, forecasting 
JEL:  C2 C3 C4 C5 
Date:  2019–07 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201918&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Jianghao Chu (UCR); Aman Ullah (UCR) 
Abstract:  Freund and Schapire (1997) introduced "Discrete AdaBoost" (DAB) which has been mysteriously effective for the highdimensional binary classi cation or binary prediction. In an effort to understand the myth, Friedman, Hastie and Tibshirani (FHT, 2000) show that DAB can be understood as statistical learning which builds an additive logistic regression model via Newtonlike updating minimization of the exponential loss. From this statistical point of view, FHT proposed three modi fications of DAB, namely, Real AdaBoost (RAB), LogitBoost (LB), and Gentle AdaBoost (GAB). All of DAB, RAB, LB, GAB solve for the logistic regression via different algorithmic designs and different objective functions. The RAB algorithm uses class probability estimates to construct realvalued contributions of the weak learner, LB is an adaptive Newton algorithm by stagewise optimization of the Bernoulli likelihood, and GAB is an adaptive Newton algorithm via stagewise optimization of the exponential loss. The same authors of FHT published an influential textbook, The Elements of Statistical Learn ing (ESL, 2001 and 2008). A companion book An Introduction to Statistical Learning (ISL) by James et al. (2013) was published with applications in R. However, both ESL and ISL (e.g., sections 4.5 and 4.6) do not cover these four AdaBoost algorithms while FHT provided some simulation and empirical studies to compare these methods. Given numerous potential applications, we believe it would be useful to collect the R libraries of these AdaBoost algorithms, as well as more recently developed extensions to Ad aBoost for probability prediction with examples and illustrations. Therefore, the goal of this chapter is to do just that, i.e., (i) to provide a user guide of these alternative AdaBoost algorithms with stepbystep tutorial of using R (in a way similar to ISL, e.g., Section 4.6), (ii) to compare AdaBoost with alternative machine learning classi fication tools such as the deep neural network (DNN), logistic regression with LASSO and SIMRODEO, and (iii) to demonstrate the empirical applications in economics, such as prediction of business cycle turning points and directional prediction of stock price indexes. We revisit Ng (2014) who used DAB for prediction of the business cycle turning points by comparing the results from RAB, LB, GAB, DNN, logistic regression and SIMRODEO. 
Keywords:  AdaBoost, R, Binary classi cation, Logistic regression, DAB, RAB, LB, GAB, DNN 
Date:  2018–07 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201907&r=all 
By:  Halbleib, Roxana; Dimitriadis, Timo 
JEL:  C1 C4 C5 
Date:  2019 
URL:  http://d.repec.org/n?u=RePEc:zbw:vfsc19:203669&r=all 
By:  Krikamol Muandet; Arash Mehrjou; Si Kai Lee; Anant Raj 
Abstract:  We present a novel singlestage procedure for instrumental variable (IV) regression called DualIV which simplifies traditional twostage regression via a dual formulation. We show that the common twostage procedure can alternatively be solved via generalized least squares. Our formulation circumvents the firststage regression which can be a bottleneck in modern twostage procedures for IV regression. We also show that our framework is closely related to the generalized method of moments (GMM) with specific assumptions. This highlights the fundamental connection between GMM and twostage procedures in IV literature. Using the proposed framework, we develop a simple kernelbased algorithm with consistency guarantees. Lastly, we give empirical results illustrating the advantages of our method over the existing twostage algorithms. 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1910.12358&r=all 
By:  Sergei Seleznev (Bank of Russia, Russian Federation) 
Abstract:  We construct priors for the tempered hierarchical Dirichlet process vector autoregression model (tHDPVAR) that in practice do not lead to explosive forecasting dynamics. Additionally, we show that tHDPVAR and its variational Bayesian approximation with heuristics demonstrate competitive or even better forecasting performance on US and Russian datasets. 
Keywords:  Bayesian nonparametrics, forecasting, hierarchical Dirichlet process, infinite hidden Markov model. 
JEL:  C11 C32 C53 E37 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:bkr:wpaper:wps47&r=all 
By:  TaeHwy Lee (Department of Economics, University of California Riverside); Jianghao Chu (University of California, Riverside); Aman Ullah (University of California, Riverside); Ran Wang (University of California, Riverside) 
Abstract:  In the era of Big Data, selecting relevant variables from a potentially large pool of candidate variables becomes a newly emerged concern in macroeconomic researches, especially when the data available is highdimensional, i.e. the number of explanatory variables (p) is greater than the length of the sample size (n). Common approaches include factor models, the principal component analysis and regularized regressions. However, these methods require additional assumptions that are hard to verify and/or introduce biases or aggregated factors which complicate the interpretation of the estimated outputs. This chapter reviews an alternative solution, namely Boosting, which is able to estimate the variables of interest consistently under fairly general conditions given a large set of explanatory variables. Boosting is fast and easy to implement which makes it one of the most popular machine learning algorithms in academia and industry. 
Keywords:  Boosting, AdaBoost, Gradient Boosting, Functional Gradient Descent, Decision Tree, Shrinkage 
JEL:  C2 C3 C4 C5 
Date:  2019–05 
URL:  http://d.repec.org/n?u=RePEc:ucr:wpaper:201917&r=all 
By:  Timo Dimitriadis; Andrew J. Patton; Patrick Schmidt 
Abstract:  Rational respondents to economic surveys may report as a point forecast any measure of the central tendency of their (possibly latent) predictive distribution, for example the mean, median, mode, or any convex combination thereof. We propose tests of forecast rationality when the measure of central tendency used by the respondent is unknown. These tests require us to overcome an identification problem when the measures of central tendency are equal or in a local neighborhood of each other, as is the case for (exactly or nearly) symmetric and unimodal distributions. As a building block, we also present novel tests for the rationality of mode forecasts. We apply our tests to survey forecasts of individual income, Greenbook forecasts of U.S. GDP, and random walk forecasts for exchange rates. We find that the Greenbook and random walk forecasts are best rationalized as mean, or nearmean forecasts, while the income survey forecasts are best rationalized as mode forecasts. 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1910.12545&r=all 
By:  Sven Husmann; Antoniya Shivarova; Rick Steinert 
Abstract:  The popularity of modern portfolio theory has decreased among practitioners because of its unfavorable outofsample performance. Estimation errors tend to affect the optimal weight calculation noticeably, especially when a large number of assets is considered. To overcome these issues, many methods have been proposed in recent years, although most only address a small set of practically relevant questions related to portfolio allocation. This study therefore sheds light on different covariance estimation techniques, combines them with sparse model approaches, and includes a turnover constraint that induces stability. We use two datasets  comprising 319 and 100 companies of the S&P 500, respectively  to create a realistic and reproducible data foundation for our empirical study. To the best of our knowledge, this study is the first to show that it is possible to maintain the lowrisk profile of efficient estimation methods while automatically selecting only a subset of assets and further inducing low portfolio turnover. Moreover, we provide evidence that using the LASSO as the sparsitygenerating model is insufficient to lower turnover when the involved tuning parameter can change over time. 
Date:  2019–10 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1910.11840&r=all 