
on Econometrics 
By:  Mohitosh Kejriwal (Purdue University); Xuewen Yu (Purdue University); Pierre Perron (Boston University) 
Abstract:  This paper proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by nonstationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap supWald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regimespeciÖc bootstrap pvalues. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the finite sample improvements offered by our methods. An application to OECD ináation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures. 
Keywords:  heteroskedasticity, multiple structural changes, sequential procedure, unit root, Wald tests, wild bootstrap 
JEL:  C22 
Date:  2020–03 
URL:  http://d.repec.org/n?u=RePEc:bos:wpaper:wp2020009&r=all 
By:  Pierre Perron (Boston University); Mototsugu Shintaniz (The University of Tokyo); Tomoyoshi Yabu (Keio University) 
Abstract:  We propose a new procedure to select the unknown frequencies of a trigonometric function, a problem Örst investigated by Anderson (1971) under the assumption of serially uncorrelated noise. We extend the analysis to general linear processes without the prior knowledge of a stationary or integrated model allowing the frequencies to be unknown. We provide a consistent model selection procedure. We Örst show that if we estimate a model with fewer frequencies than those in the correct model, the estimates converge to a subset of the frequencies in the correct model. This opens the way to a consistent model selection strategy based on a speciÖc to general procedure that tests whether additional frequencies are needed. This is achieved using tests based on the feasible ísuper eÂ¢ cientî(under unit root noise) Generalized Least Squares estimator of Perron, Shintani and Yabu (2017) who assumed the frequencies to be known. We show that the limiting distributions of our test statistics are the same for both cases about the noise function. Simulation results conÖrm that our frequency selection procedure works well with sample sizes typically available in practice. We illustrate the usefulness of our method via applications to unemployment rates and global temperature series. 
Keywords:  Cyclical trends, medianunbiased estimator, nonlinear trends, supereÂ¢ cient estimator, unit root 
JEL:  C22 
Date:  2020–01 
URL:  http://d.repec.org/n?u=RePEc:bos:wpaper:wp2020012&r=all 
By:  Ting Ye; Luke Keele; Raiden Hasegawa; Dylan S. Small 
Abstract:  The method of differenceindifferences (DID) is widely used to study the causal effect of policy interventions in observational studies. DID exploits a before and after comparison of the treated and control units to remove the bias due to timeinvariant unmeasured confounders under the parallel trends assumption. Estimates from DID, however, will be biased if the outcomes for the treated and control units evolve differently in the absence of treatment, namely if the parallel trends assumption is violated. We propose a general identification strategy that leverages two groups of control units whose outcomes relative to the treated units exhibit a negative correlation, and achieves partial identification of the average treatment effect for the treated. The identified set is of a union bounds form that previously developed partial identification inference methods do not apply to. We develop a novel bootstrap method to construct valid confidence intervals for the identified set and parameter of interest when the identified set is of a union bounds form, and we establish the theoretical properties. We develop a simple falsification test and sensitivity analysis. We apply the proposed strategy for bracketing to an application on the effect of voter identification laws in Georgia and Indiana on turnout and find evidence that the laws increased turnout rates. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.02423&r=all 
By:  Rustam Ibragimov; Rasmus Pedersen; Anton Skrobotov 
Abstract:  Many key variables in finance, economics and risk management, including financial returns and foreign exchange rates, exhibit nonlinear dependence, heterogeneity and heavytailedness of some usually largely unknown type. The presence of nonlinear dependence (usually modelled using GARCHtype dynamics) and heavytailedness may make problematic the analysis of (non)efficiency, volatility clustering and predictive regressions in economic and financial markets using traditional approaches that appeal to asymptotic normality of sample autocorrelation functions (ACFs) of returns and their squares. The paper presents several new approaches to deal with the above problems. We provide the results that motivate the use of measures of market (non)efficiency, volatility clustering and nonlinear dependence based on (small) powers of absolute returns and their signed versions. The paper provides asymptotic theory for sample analogues of the above measures in the case of general time series, including GARCHtype processes. It further develops new approaches to robust inference on them in the case of general GARCHtype processes exhibiting heavytailedness properties. The approaches are based on robust inference methods exploiting conservativeness properties of tstatistics Ibragimov and Muller (2010,2016) and several new results on their applicability in the settings considered. In the approaches, estimates of parameters of interest are computed for groups of data and the inference is based on tstatistics in resulting group estimates. This results in valid robust inference under a wide range of heterogeneity and dependence assumptions satisfied in financial and economic markets. Numerical results and empirical applications confirm advantages of the new approaches over existing ones and their wide applicability. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.01212&r=all 
By:  Alessandro Casini (University of Rome Tor Vergata); Pierre Perron (Boston University) 
Abstract:  For a partial structural change in a linear regression model with a single break, we develop a continuous record asymptotic framework to build inference methods for the break date. We have T observations with a sampling frequency h over a fixed time horizon [0, N] , and let T â†’ âˆž with h â†“ 0 while keeping the time span N fixed. We impose very mild regularity conditions on an underlying continuoustime model assumed to generate the data. We consider the leastsquares estimate of the break date and establish consistency and convergence rate. We provide a limit theory for shrinking magnitudes of shifts and locally increasing variances. The asymptotic distribution corresponds to the location of the extremum of a function of the quadratic variation of the regressors and of a Gaussian centered martingale process over a certain time interval. We can account for the asymmetric informational content provided by the pre and postbreak regimes and show how the location of the break and shift magnitude are key ingredients in shaping the distribution. We consider a feasible version based on plugin estimates, which provides a very good approximation to the finite sample distribution. We use the concept of Highest Density Region to construct confidence sets. Overall, our method is reliable and delivers accurate coverage probabilities and relatively short average length of the confidence sets. Importantly, it does so irrespective of the size of the break. 
Keywords:  Asymptotic distribution, break date, changepoint, highest density region, semimartingale 
JEL:  C10 C12 C22 
Date:  2020–03 
URL:  http://d.repec.org/n?u=RePEc:bos:wpaper:wp2020013&r=all 
By:  Sander Barendse; Andrew J. Patton 
Abstract:  We develop tests for outofsample forecast comparisons based on loss functions that contain shape parameters. Examples include comparisons using average utility across a range of values for the level of risk aversion, comparisons of forecast accuracy using characteristics of a portfolio return across a range of values for the portfolio weight vector, and comparisons using a recentlyproposed â€œMurphy diagramsâ€ for classes of consistent scoring rules. An extensive Monte Carlo study verifies that our tests have good size and power properties in realistic sample sizes, particularly when compared with existing methods which break down when then number of values considered for the shape parameter grows. We present three empirical illustrations of the new test. 
Keywords:  Forecasting, model selection, outofsample testing, nuisance parameters 
JEL:  C53 C52 C12 
Date:  2020–05–27 
URL:  http://d.repec.org/n?u=RePEc:oxf:wpaper:909&r=all 
By:  Mohitosh Kejriwal (Purdue University); Pierre Perron (Boston University); Xuewen Yu (Purdue University) 
Abstract:  Kejriwal and Perron (2010, KP) provided a comprehensive treatment for the problem of testing multiple structural changes in cointegrated regression models. A variety of models were considered depending on whether all regression coefficients are allowed to change (pure structural change) or a subset of the coefficients is held Öxed (partial structural change). In this note, we Örst show that the limit distributions of the test statistics in the latter case are not invariant to changes in the coeÂ¢ cients not being tested; in fact, they diverge as the sample size increases. To address this issue, we propose a simple two step procedure to test for partial parameter stability. The Örst entails the application of a joint test of stability for all coeÂ¢ cients as in KP. Upon a rejection, the second conducts a stability test on the subset of coeÂ¢ cients of interest while allowing the other coeÂ¢ cients to change at the estimated breakpoints. Its limit distribution is standard chisquare. The relevant asymptotic theory is provided along with simulations that illustrates the usefulness of the procedure in finite samples. 
Keywords:  cointegration, partial structural change, break date, supWald tests, joint hypothesis testing 
JEL:  C22 
Date:  2020–02 
URL:  http://d.repec.org/n?u=RePEc:bos:wpaper:wp2020011&r=all 
By:  S. Broda (Department of Economics and Econometrics, University of Amsterdam); Juan Carlos ArismendiZambrano (Department of Economics, Finance and Accounting, Maynooth University & ICMA Centre, Henley Business School, University of Reading) 
Abstract:  Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closedform expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic random vectors. This ﬂexible distribution nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the ﬁrst partial moment is also obtained, which plays a vital role in ﬁnancial risk management. The proof involves a generalization of the classic inversion formula due to GilPelaez (1951). Two numerical applications are considered: ﬁrst, the finitesample distribution of the two stage least squares estimator of a structural parameter. Second, the Value at Risk and expected shortfall of a quadratic portfolio with heavytailed risk factors. An empirical application is examined, in which a portfolio of Dow Jones Industrial Index stock options is optimized with respect to its expected shortfall. The results demonstrate the beneﬁts of the analytical expression. 
Keywords:  Characteristic Function; Conditional Value at Risk; Expected Shortfall; Transform Inversion; Two Stage Least Squares. 
JEL:  C10 C13 C14 C15 C18 C63 C65 G32 
Date:  2020 
URL:  http://d.repec.org/n?u=RePEc:may:mayecw:n30220.pdf&r=all 
By:  Tenglong Li; Kenneth A. Frank 
Abstract:  The internal validity of observational study is often subject to debate. In this study, we define the unobserved sample based on the counterfactuals and formalize its relationship with the null hypothesis statistical testing (NHST) for regression models. The probability of a robust inference for internal validity, i.e., the PIV, is the probability of rejecting the null hypothesis again based on the ideal sample which is defined as the combination of the observed and unobserved samples, provided the same null hypothesis has already been rejected for the observed sample. When the unconfoundedness assumption is dubious, one can bound the PIV of an inference based on bounded belief about the mean counterfactual outcomes, which is often needed in this case. Essentially, the PIV is statistical power of the NHST that is thought to be built on the ideal sample. We summarize the process of evaluating internal validity with the PIV into a sixstep procedure and illustrate it with an empirical example (i.e., Hong and Raudenbush (2005)). 
Date:  2020–05 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2005.12784&r=all 
By:  Marianne Bl\'ehaut; Xavier D'Haultfoeuille; J\'er\'emy L'Hour; Alexandre B. Tsybakov 
Abstract:  The synthetic control method is a an econometric tool to evaluate causal effects when only one unit is treated. While initially aimed at evaluating the effect of largescale macroeconomic changes with very few available control units, it has increasingly been used in place of more wellknown microeconometric tools in a broad range of applications, but its properties in this context are unknown. This paper introduces an alternative to the synthetic control method, which is developed both in the usual asymptotic framework and in the highdimensional scenario. We propose an estimator of average treatment effect that is doubly robust, consistent and asymptotically normal. It is also immunized against firststep selection mistakes. We illustrate these properties using Monte Carlo simulations and applications to both standard and potentially highdimensional settings, and offer a comparison with the synthetic control method. 
Date:  2020–05 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2005.12225&r=all 
By:  Calypso Herrera; Florian Krach; Josef Teichmann 
Abstract:  Continuous stochastic processes are widely used to model time series that exhibit a random behaviour. Predictions of the stochastic process can be computed by the conditional expectation given the current information. To this end, we introduce the controlled ODERNN that provides a datadriven approach to learn the conditional expectation of a stochastic process. Our approach extends the ODERNN framework which models the latent state of a recurrent neural network (RNN) between two observations with a neural ordinary differential equation (neural ODE). We show that controlled ODEs provide a general framework which can in particular describe the ODERNN, combining in a single equation the continuous neural ODE part with the jumps introduced by RNN. We demonstrate the predictive capabilities of this model by proving that, under some regularities assumptions, the output process converges to the conditional expectation process. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.04727&r=all 
By:  Alessandra Amendola; Vincenzo Candila; Fabrizio Cipollini; Giampiero M. Gallo 
Abstract:  We suggest the Doubly Multiplicative Error class of models (DMEM) for modeling and forecasting realized volatility, which combines two components accommodating low, respectively, highfrequency features in the data. We derive the theoretical properties of the Maximum Likelihood and Generalized Method of Moments estimators. Two such models are then proposed, the ComponentMEM, which uses daily data for both components, and the MEMMIDAS, which exploits the logic of MIxedDAta Sampling (MIDAS). The empirical application involves the S&P 500, NASDAQ, FTSE 100 and Hang Seng indices: irrespective of the market, both DMEM's outperform the HAR and other relevant GARCHtype models. 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:2006.03458&r=all 
By:  Jesús FernándezVillaverde (University of Pennsylvania, NBER and CEPR); Samuel Hurtado (Banco de España); Galo Nuño (Banco de España) 
Abstract:  We postulate a nonlinear DSGE model with a financial sector and heterogeneous households. In our model, the interaction between the supply of bonds by the financial sector and the precautionary demand for bonds by households produces significant endogenous aggregate risk. This risk induces an endogenous regimeswitching process for output, the riskfree rate, excess returns, debt, and leverage. The regimeswitching generates i) multimodal distributions of the variables above; ii) timevarying levels of volatility and skewness for the same variables; and iii) supercycles of borrowing and deleveraging. All of these are important properties of the data. In comparison, the representative household version of the model cannot generate any of these features. Methodologically, we discuss how nonlinear DSGE models with heterogeneous agents can be efficiently computed using machine learning and how they can be estimated with a likelihood function, using inference with diffusions. 
Keywords:  heterogeneous agents, wealth distribution, financial frictions, continuoustime, machine learning, neural networks, structural estimation, likelihood function 
JEL:  C45 C63 E32 E44 G01 G11 
Date:  2020–06 
URL:  http://d.repec.org/n?u=RePEc:bde:wpaper:2013&r=all 
By:  Xie, Tian (Shanghai University of Finance and Economics); Yu, Jun (School of Economics, Singapore Management University); Zeng, Tao (Zhejiang University) 
Abstract:  The data market has been growing at an exceptional pace. Consequently, more sophisticated strategies to conduct economic forecasts have been introduced with machine learning techniques. Does machine learning pose a threat to conventional econometric methods in terms of forecasting? Moreover, does machine learning present great opportunities to crossfertilize the field of econometric forecasting? In this report, we develop a pedagogical framework that identifies complementarity and bridges between the two strands of literature. Existing econometric methods and machine learning techniques for economic forecasting are reviewed and compared. The advantages and disadvantages of these two classes of methods are discussed. A class of hybrid methods that combine conventional econometrics and machine learning are introduced. New directions for integrating the above two are suggested. The outofsample performance of alternatives is compared when they are employed to forecast the Chicago Board Options Exchange Volatility Index and the harmonized index of consumer prices for the euro area. In the first exercise, econometric methods seem to work better, whereas machine learning methods generally dominate in the second empirical application. 
Date:  2020–05–30 
URL:  http://d.repec.org/n?u=RePEc:ris:smuesw:2020_016&r=all 