
on Econometrics 
By:  Zongwu Cai (Department of Economics, The University of Kansas, Lawrence, KS 66045, USA); Ted Juhl (School of Business, The University of Kansas, Lawrence, KS 66045, USA) 
Abstract:  We find the asymptotic distribution for rolling linear regression models various window widths. The limiting distribution depends on using the width of the rolling window, and on a Òbias processÓ that is typically ignored in practice. Based on the distribution, we tabulate critical values used to find uniform confidence intervals for the average values of regression parameters over the windows. We propose a corrected rolling regression technique that removes the bias process by rolling over smoothed parameter estimates. The procedure is illustrated using a series of Monte Carlo experiments. The paper includes an empirical example to show the how the confidence bands suggest alternative conclusions about the persistence of inflation. 
Keywords:  Asymptotic distribution; Bias correction; Nonparametric estimation; Rolling regressions; Timevarying parameters. 
JEL:  C14 C22 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:kan:wpaper:202013&r=all 
By:  Anastasios Panagiotelis; Puwasala Gamakumara; George Athanasopoulos; Rob J Hyndman 
Abstract:  We develop a framework for prediction of multivariate data that follow some known linear constraints, such as the example where some variables are aggregates of others. This is particularly common when forecasting time series (predicting the future), but also arises in other types of prediction. For point prediction, an increasingly popular technique is reconciliation, whereby predictions are made for all series (socalled `base' predictions) and subsequently adjusted to ensure coherence with the constraints. This paper extends reconciliation from the setting of point prediction to probabilistic prediction. A novel definition of reconciliation is developed and used to construct densities and draw samples from a reconciled probabilistic prediction. In the elliptical case, it is proven that the true predictive distribution can be recovered from reconciliation even when the location and scale matrix of the base prediction are chosen arbitrarily. To find reconciliation weights, an objective function based on scoring rules is optimised. The energy and variogram scores are considered since the log score is improper in the context of comparing unreconciled to reconciled predictions, a result also proved in this paper. To account for the stochastic nature of the energy and variogram scores, optimisation is achieved using stochastic gradient descent. This method is shown to improve base predictions in simulation studies and in an empirical application, particularly when the base prediction models are severely misspecified. When misspecification is not too severe, extending popular reconciliation methods for point prediction can result in a similar performance to score optimisation via stochastic gradient descent. The methods described here are implemented in the ProbReco package for R. 
Keywords:  scoring rules, probabilistic forecasting, hierarchical time series, stochastic gradient descent 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:msh:ebswps:202026&r=all 
By:  Ruben LoaizaMaya; Didier Nibbering 
Abstract:  The multinomial probit model is a popular tool for analyzing choice behaviour as it allows for correlation between choice alternatives. Because current model specifications employ a full covariance matrix of the latent utilities for the choice alternatives, they are not scalable to a large number of choice alternatives. This paper proposes a factor structure on the covariance matrix, which makes the model scalable to large choice sets. The main challenge in estimating this structure is that the model parameters require identifying restrictions. We identify the parameters by a tracerestriction on the covariance matrix, which is imposed through a reparamatrization of the factor structure. We specify interpretable prior distributions on the model parameters and develop an MCMC sampler for parameter estimation. The proposed approach substantially improves performance in large choice sets relative to existing multinomial probit specifications. Applications to purchase data show the economic importance of including a large number of choice alternatives in consumer choice analysis. 
Keywords:  multinomial probit model, factor analysis, parameter identification, spherical coordinates 
JEL:  C11 C25 C35 C38 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:msh:ebswps:202025&r=all 
By:  Silvia Goncalves; Ana María Herrera; Lutz Kilian; Elena Pesavento 
Abstract:  We study the construction of nonlinear impulse responses in structural dynamic models that include nonlinearly transformed regressors. Such models have played an important role in recent years in capturing asymmetries, thresholds and other nonlinearities in the responses of macroeconomic variables to exogenous shocks. The conventional approach to estimating nonlinear responses is by Monte Carlo integration. We show that the population impulse responses in this class of models may instead be derived analytically from the structural model. We use this insight to study under what conditions linear projection (LP) estimators may be used to recover the population impulse responses. We find that, unlike in vector autoregressive models, the asymptotic equivalence between estimators based on the structural model and LP estimators breaks down. Only in one important special case can the population impulse response be consistently estimated by LP methods. The construction of this LP estimator, however, differs from the LP approach currently used in the literature. Simulation evidence suggests that the modified LP estimator is less accurate in finite samples than estimators based on the structural model, when both are valid. 
Keywords:  local projection; structural model; censored regressor; nonlinear transformation; nonlinear responses; Monte Carlo integration 
JEL:  C22 C32 C51 
Date:  2020–06–30 
URL:  http://d.repec.org/n?u=RePEc:fip:feddwp:88270&r=all 
By:  Robert Adamek; Stephan Smeekes; Ines Wilms 
Abstract:  The desparsified lasso is a highdimensional estimation method which provides uniformly valid inference. We extend this method to a time series setting under NearEpoch Dependence (NED) assumptions allowing for nonGaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an oracle inequality for the (regular) lasso, relaxing the commonly made exact sparsity assumption to a weaker alternative, which permits many small but nonzero parameters. The weak sparsity coupled with the NED assumption means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions. Additionally, we show consistency of a longrun variance estimator, thus providing a complete set of tools for performing inference in highdimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.10952&r=all 
By:  Riani, Marco; Atkinson, Anthony C.; Corbellini, Aldo; Perrotta, Domenico 
Abstract:  Minimum density power divergence estimation provides a general framework for robust statistics, depending on a parameter a, which determines the robustness properties of the method. The usual estimation method is numerical minimization of the power divergence. The paper considers the special case of linear regression. We developed an alternative estimation procedure using the methods of Sestimation. The rho function so obtained is proportional to one minus a suitably scaled normal density raised to the power a. We used the theory of Sestimation to determine the asymptotic efficiency and breakdown point for this new form of Sestimation. Two sets of comparisons were made. In one, S power divergence is compared with other Sestimators using four distinct rho functions. Plots of efficiency against breakdown point show that the properties of S power divergence are close to those of Tukey's biweight. The second set of comparisons is between S power divergence estimation and numerical minimization. Monitoring these two procedures in terms of breakdown point shows that the numerical minimization yields a procedure with larger robust residuals and a lower empirical breakdown point, thus providing an estimate of a leading to more efficient parameter estimates. 
Keywords:  Estimation of a; Monitoring; Numerical minimization; Sestimation; Tukey's biweight 
JEL:  C1 
Date:  2020–03–31 
URL:  http://d.repec.org/n?u=RePEc:ehl:lserod:103931&r=all 
By:  Jesper R.V. Soerensen (Department of Economics, University of Copenhagen, Denmark) 
Abstract:  This paper proposes a new test for a class of conditional moment restrictions whose parameterization involves unknown, unrestricted conditional expectation functions. Examples of such conditional moment restrictions are conditional mean independence (leading to a nonparametric significance test) and conditional homoskedasticity (with an otherwise unrestricted conditional mean) and also arise from models of singleagent discrete choice under uncertainty and static games of incomplete information. The proposed test may be viewed as a semi/nonparametric extension of the Bierens (1982) goodnessoffit test of a parametric model for the conditional mean. Estimating conditional expectations using series methods and employing a Gaussian multiplier bootstrap to obtain critical values, the resulting test is shown to be asymptotically correctly sized and consistent. A simulation study applies the procedure to test the specification of a twoplayer, binaryaction static game of incomplete information, treating equilibrium beliefs as nonparametric conditional expectations. 
Keywords:  Omnibus specification testing; Semiparametric; Conditional moment restrictions; Conditional expectation; Series estimation; Bootstrap; Cramervon Mises distance. 
JEL:  C01 C14 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:kud:kuiedp:2004&r=all 
By:  Christian Schluter (AMSE  AixMarseille Sciences Economiques  EHESS  École des hautes études en sciences sociales  ECM  École Centrale de Marseille  CNRS  Centre National de la Recherche Scientifique  AMU  Aix Marseille Université) 
Abstract:  City size distributions are not strictly Pareto, but upper tails are rather Pareto like (i.e. tails are regularly varying). We examine the properties of the tail exponent estimator obtained from ordinary least squares (OLS) rank size regressions (Zipf regressions for short), the most popular empirical strategy among urban economists. The estimator is then biased towards Zipf's law in the leading class of distributions. The Pareto quantilequantile plot is shown to offer a simple diagnostic device to detect such distortions and should be used in conjunction with the regression residuals to select the anchor point of the OLS regression in a datadependent manner. Applying these updated methods to some wellknown data sets for the largest cities, Zipf's law is now rejected in several cases. 
Keywords:  regular variation,city size distributions,Zipf's law,rank size regression,extreme value index,heavy tails 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:hal:journl:hal02880544&r=all 
By:  Zhenzhong Wang; Zhengyuan Zhu; Cindy Yu 
Abstract:  In the datarich environment, using many economic predictors to forecast a few key variables has become a new trend in econometrics. The commonly used approach is factor augment (FA) approach. In this paper, we pursue another direction, variable selection (VS) approach, to handle highdimensional predictors. VS is an active topic in statistics and computer science. However, it does not receive as much attention as FA in economics. This paper introduces several cuttingedge VS methods to economic forecasting, which includes: (1) classical greedy procedures; (2) l1 regularization; (3) gradient descent with sparsification and (4) metaheuristic algorithms. Comprehensive simulation studies are conducted to compare their variable selection accuracy and prediction performance under different scenarios. Among the reviewed methods, a metaheuristic algorithm called sequential Monte Carlo algorithm performs the best. Surprisingly the classical forward selection is comparable to it and better than other more sophisticated algorithms. In addition, we apply these VS methods on economic forecasting and compare with the popular FA approach. It turns out for employment rate and CPI inflation, some VS methods can achieve considerable improvement over FA, and the selected predictors can be well explained by economic theories. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.10160&r=all 
By:  Jushan Bai; Serena Ng 
Abstract:  Estimates of the approximate factor model are increasingly used in empirical work. Their theoretical properties, studied some twenty years ago, also laid the ground work for analysis on large dimensional panel data models with crosssection dependence. This paper presents simplified proofs for the estimates by using alternative rotation matrices, exploiting properties of low rank matrices, as well as the singular value decomposition of the data in addition to its covariance structure. These simplifications facilitate interpretation of results and provide a more friendly introduction to researchers new to the field. New results are provided to allow linear restrictions to be imposed on factor models. 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2008.00254&r=all 
By:  Nishihat, Masaya; Otsu, Taisuke 
Abstract:  This paper studies finite sample performances of the conditional GMM es timators for a particular conditional moment restriction model, which is commonly ap plied in economic analysis using gravity models of international trade. We consider the GMM estimator with growing moments and Dominguez and Lobato’s (2004) process based GMM estimator. Under the simulation designs by Santos Silva and Tenreyro (2006, 2011), we find that Dominguez and Lobato’s (2004) estimator is favorably com parable with the Poisson pseudo maximum likelihood estimator, and outperforms other estimators. 
JEL:  J1 
Date:  2020–04–29 
URL:  http://d.repec.org/n?u=RePEc:ehl:lserod:105083&r=all 
By:  Nicolas Van de Sipe; Frank Windmeijer 
Abstract:  We show that the Likelihood Ratio (LR) statistic for testing the value of the coefficient $\beta$ in a linear instrumental variables model with a single endogenous variable is identical to the $t_0(\hat{\beta}_L)^2$ statistic as proposed by Mills, Moreira, and Vilela (2014), where $\hat{\beta}_L$ is the LIML estimator. This implies the equivalence of their conditional versions that are robust to weak instruments. From this result, properties of the power of the Conditional LR (CLR) test can be understood; in particular the asymmetric nature of the power curve as a function of the true value of $\beta$ when testing H0: $\beta =\beta_0$ for fixed $\beta_0$, when the instruments are weak and the variance matrix of the structural and firststage errors is held constant. Power curves of the CLR and related tests have often been presented for a design where instead the variance matrix of the reducedform and firststage errors has been held constant. This latter design changes the endogeneity features at each value of $\beta$ and results in a power curve that is close to the points with maximum power in the design with fixed variance of the structural and firststage errors. As the results for the design with fixed variance of the structural and firststage errors are informative for the behaviour of the testbased confidence intervals, it seems more natural to consider this design. We find that LIML and Fullerbased conditional Wald and conditional $t_0({\hat{\beta}_{Full}})^2$ tests, which are not unbiased tests, are more powerful than the CLR test when the degree of endogeneity is low to moderate. JEL codes: C12, C26 
Date:  2020–08–04 
URL:  http://d.repec.org/n?u=RePEc:nuf:econwp:2009&r=all 
By:  Gilles Zumbach 
Abstract:  A new test for measuring the accuracy of financial market risk estimations is introduced. It is based on the probability integral transform (PIT) of the ex post realized returns using the ex ante probability distributions underlying the risk estimation. If the forecast is correct, the result of the PIT, that we called probtile, should be an iid random variable with a uniform distribution. The new test measures the variance of the number of probtiles in a tiling over the whole sample. Using different tilings allow to check the dynamic and the distributional aspect of risk methodologies. The new test is very powerful, and new benchmarks need to be introduced to take into account subtle mean reversion effects induced by some risk estimations. The test is applied on 2 data sets for risk horizons of 1 and 10 days. The results show unambiguously the importance of capturing correctly the dynamic of the financial market, and exclude some broadly used risk methodologies. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.12431&r=all 
By:  Gandy, Axel; Veraart, Luitgard A. M. 
Abstract:  We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships be tween the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for the model parameters. We test the new modelling framework on two types of financial networks: a network of financial institutions in which the edge weights represent exposures from trading Credit Default Swaps and a network of countries in which the edge weights represent crossborder lending. The compound Poisson Gamma distributions with regression fit the data well in both situations. We illustrate how this modelling framework can be used for predicting unobserved edges and their weights in an only partially observed network. This is for example relevant for assessing systemic risk in financial networks. 
Keywords:  eighted directed networks; compound Poisson distribution; regression; subnetwork prediction; financial networks; systemic risk 
JEL:  C02 C46 C53 D85 G32 
Date:  2020–05–29 
URL:  http://d.repec.org/n?u=RePEc:ehl:lserod:104185&r=all 
By:  NengChieh Chang 
Abstract:  Mean, median, and mode are three essential measures of the centrality of probability distributions. In program evaluation, the average treatment effect (mean) and the quantile treatment effect (median) have been intensively studied in the past decades. The mode treatment effect, however, has long been neglected in program evaluation. This paper fills the gap by discussing both the estimation and inference of the mode treatment effect. I propose both traditional kernel and machine learning methods to estimate the mode treatment effect. I also derive the asymptotic properties of the proposed estimators and find that both estimators follow the asymptotic normality but with the rate of convergence slower than the regular rate $\sqrt{N}$, which is different from the rates of the classical average and quantile treatment effect estimators. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.11606&r=all 
By:  Denis Belomestny; Ekaterina Krymova; Andrey Polbin 
Abstract:  The main goal of this paper is to develop a methodology for estimating time varying parameter vector autoregression (TVPVAR) models with a timeinvariant longrun relationship between endogenous variables and changes in exogenous variables. We propose a Gibbs sampling scheme for estimation of model parameters as well as timeinvariant longrun multiplier parameters. Further we demonstrate the applicability of the proposed method by analyzing examples of the Norwegian and Russian economies based on the data on real GDP, real exchange rate and real oil prices. Our results show that incorporating the time invariance constraint on the longrun multipliers in TVPVAR model helps to significantly improve the forecasting performance. 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2008.00718&r=all 
By:  Paolo Andreini; Cosimo Izzo; Giovanni Ricco 
Abstract:  We propose a novel deep neural net framework  that we refer to as Deep Dynamic Factor Model (D2FM) , to encode the information available, from hundreds of macroeconomic and financial timeseries into a handful of unobserved latent states. While similar in spirit to traditional dynamic factor models (DFMs), differently from those, this new class of models allows for nonlinearities between factors and observables due to the deep neural net structure. However, by design, the latent states of the model can still be interpreted as in a standard factor model. In an empirical application to the forecast and nowcast of economic conditions in the US, we show the potential of this framework in dealing with high dimensional, mixed frequencies and asynchronously published time series data. In a fully realtime outofsample exercise with US data, the D2FM improves over the performances of a stateoftheart DFM. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.11887&r=all 
By:  Chris Muris; Pedro Raposo; Sotiris Vandoros 
Abstract:  We study a fixedT panel data logit model for ordered outcomes that accommodates fixed effects and state dependence. We provide identification results for the autoregressive parameter, regression coefficients, and the threshold parameters in this model. Our results require only four observations on the outcome variable. We provide conditions under which a composite conditional maximum likelihood estimator is consistent and asymptotically normal. We use our estimator to explore the determinants of selfreported health in a panel of European countries over the period 20032016. We find that: (i) that the autoregressive parameter is positive and analogous to a linear AR(1) coefficient of about 0.25, indicating persistence in health status; (ii) that the association between income and health becomes insignicant once we control for unobserved heterogeneity and persistence. 
Keywords:  panel data; ordered choice; health satisfaction 
JEL:  C23 C25 I14 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:mcm:deptwp:202014&r=all 
By:  Philippe Goulet Coulombe; Maxime Leroux; Dalibor Stevanovic; St\'ephane Surprenant 
Abstract:  From a purely predictive standpoint, rotating the predictors' matrix in a lowdimensional linear regression setup does not alter predictions. However, when the forecasting technology either uses shrinkage or is nonlinear, it does. This is precisely the fabric of the machine learning (ML) macroeconomic forecasting environment. Preprocessing of the data translates to an alteration of the regularization  explicit or implicit  embedded in ML algorithms. We review old transformations and propose new ones, then empirically evaluate their merits in a substantial pseudooutsample exercise. It is found that traditional factors should almost always be included in the feature matrix and moving average rotations of the data can provide important gains for various forecasting targets. 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2008.01714&r=all 
By:  ShaoPeng Hong 
Abstract:  Due to the skessed distribution, high peak and thick tail and asymmetry of financial return data, it is difficult to describe the traditional distribution. In recent years, generalized autoregressive score (GAS) has been used in many fields and achieved good results. In this paper, under the framework of generalized autoregressive score (GAS), the asymmetric Laplace distribution (ALD) is improved, and the GASALD model is proposed, which has the characteristics of timevarying parameters, can describe the peak thick tail, biased and asymmetric distribution. The model is used to study the Shanghai index, Shenzhen index and SME board index. It is found that: 1) the distribution parameters and moments of the three indexes have obvious timevarying characteristics and aggregation characteristics. 2) Compared with the commonly used models for calculating VaR and ES, the GASALD model has a high prediction effect. 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2008.01277&r=all 
By:  Jose Blanchet; Henry Lam; Yang Liu; Ruodu Wang 
Abstract:  Quantile aggregation with dependence uncertainty has a long history in probability theory with wide applications in problems in finance, risk management, statistics, and operations research. Using a recent result on infconvolution of RangeValueatRisk, which includes ValueatRisk and Expected Shortfall as special cases, we establish new analytical bounds which we call convolution bounds. These bounds are easy to compute, and we show that they are sharp in many relevant cases. We pay a special attention to the problem of quantile aggregation, and the convolution bounds help us to identify approximations for the extremal dependence structure. The convolution bound enjoys several advantages, including interpretability, tractability and theoretical properties. To the best of our knowledge, there is no other theoretical result on quantile aggregation which is not covered by the convolution bounds, and thus the convolution bounds are genuinely the best one available. The results can be applied to compute bounds on the distribution of the sum of random variables. Some applications to operations research are discussed. 
Date:  2020–07 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2007.09320&r=all 
By:  Tam\'as Krisztin; Philipp Piribauer; Michael W\"ogerer 
Abstract:  The paper proposes a Bayesian multinomial logit model to analyse spatial patterns of urban expansion. The specification assumes that the logodds of each class follow a spatial autoregressive process. Using recent advances in Bayesian computing, our model allows for a computationally efficient treatment of the spatial multinomial logit model. This allows us to assess spillovers between regions and across land use classes. In a series of Monte Carlo studies, we benchmark our model against other competing specifications. The paper also showcases the performance of the proposed specification using European regional data. Our results indicate that spatial dependence plays a key role in land sealing process of cropland and grassland. Moreover, we uncover land sealing spillovers across multiple classes of arable land. 
Date:  2020–08 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2008.00673&r=all 