Uniform Confidence Bands Research Articles

Nonparametric estimation and inference under shape restrictions

Economic theory often provides shape restrictions on functions of interest in applications, such as monotonicity, convexity, non-increasing (non-decreasing) returns to scale, or the Slutsky inequality of consumer theory; but economic theory does not provide finite-dimensional parametric models. This motivates nonparametric estimation under shape restrictions. Nonparametric estimates are often very noisy. Shape restrictions stabilize nonparametric estimates without imposing arbitrary restrictions, such as additivity or a single-index structure, that may be inconsistent with economic theory and the data. This paper explains how to estimate and obtain an asymptotic uniform confidence band for a conditional mean function under possibly nonlinear shape restrictions, such as the Slutsky inequality. The results of Monte Carlo experiments illustrate the finite-sample performance of the method, and an empirical example illustrates its use in an application.

Doubly robust uniform confidence band for the conditional average treatment effect function

SummaryIn this paper, we propose a doubly robust method to estimate the heterogeneity of the average treatment effect with respect to observed covariates of interest. We consider a situation where a large number of covariates are needed for identifying the average treatment effect but the covariates of interest for analyzing heterogeneity are of much lower dimension. Our proposed estimator is doubly robust and avoids the curse of dimensionality. We propose a uniform confidence band that is easy to compute, and we illustrate its usefulness via Monte Carlo experiments and an application to the effects of smoking on birth weights.

Moving Fourier Analysis for Locally Stationary Processes with the Bootstrap in View

We introduce a moving Fourier transformation for locally stationary time series, which captures the time‐varying spectral density in a similar manner as the classical Fourier transform does for stationary time series. In particular, the resulting Fourier coefficients as well as moving local periodograms are shown to be (almost all) asymptotically uncorrelated. The moving local periodogram is obtained by thinning the local periodogram to avoid multiple information present at different but close points in time. We obtain consistent estimators for the local spectral density at each point in time by smoothing the moving local periodogram. Furthermore, the moving Fourier coefficients, respectively periodograms, are well suited to adapt stationary frequency domain bootstrap methods to the locally stationary case. For the wild time frequency toggle bootstrap, it is shown that the corresponding bootstrap covariance of a global locally stationary bootstrap samples captures the time‐varying covariance structure of the underlying locally stationary time series correctly. Furthermore, this bootstrap in addition to adaptations of other frequency domain bootstrap methods is used in a simulation study to obtain uniform confidence bands for the time‐varying autocorrelation at lag 1. Finally, this methodology is applied to a wind data set.

On copula-based conditional quantile estimators

Recently, two different copula-based approaches have been proposed to estimate the conditional quantile function of a variable Y with respect to a vector of covariates X: the first estimator is related to quantile regression weighted by the conditional copula density, while the second estimator is based on the inverse of the conditional distribution function written in terms of margins and the copula. Using empirical processes, we show that even if the two estimators look quite different, their estimation errors have the same limiting distribution. Also, we propose a bootstrap procedure for the limiting process in order to construct uniform confidence bands around the conditional quantile function.

Partial Linear Quantile Regression and Bootstrap Confidence Bands

In this paper uniform confidence bands are constructed for nonparametric quantile estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated empirical density function (EDF) for residuals. It is known that the approximation error for the uniform confidence band by the asymptotic Gumbel distribution is logarithmically slow. It is proved that the bootstrap approximation provides a substantial improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. Comparison to classic asymptotic uniform bands is presented through a simulation study. An economic application considers the labour market differential effect with respect to different education levels.

Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with non-parametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by non-parametric regression. The subjective density is estimated non-parametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

A Confidence Corridor for Expectile Functions

Let (X1; Y1), …, (Xn; Yn) be i.i.d. rvs and denote v(x) be the unknown τ - expectile regression curve of Y conditional on X. An expectile-smoother vn(x) is a localized, nonlinear estimator of v(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {vn(x) – v(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |vn(x) – v(x)|. The derived result helps in the construction of a uniform confidence band for the expectile curve v(x). This paper considers fitting a simultaneous confidence corridor (SCC) around the estimated expectile function of the conditional distribution of Y given x based on the observational data generated according to a nonparametric regression model. Moreover, we construct the simultaneous confidence corridors around the expectiles of the residuals from the temperature models to investigate the temperature risk drivers.

Optimal Sup-Norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.

Inference on Breakdown Frontiers

A breakdown frontier is the boundary between the set of assumptions which lead to a specific conclusion and those which do not. In a potential outcomes model with a binary treatment, we consider two conclusions: First, that ATE is at least a specific value (e.g., nonnegative) and second that the proportion of units who benefit from treatment is at least a specific value (e.g., at least 50%). For these conclusions, we derive the breakdown frontier for two kinds of assumptions: one which indexes deviations from random assignment of treatment, and one which indexes deviations from rank invariance. These classes of assumptions nest both the point identifying assumptions of random assignment and rank invariance and the opposite end of no constraints on treatment selection or the dependence structure between potential outcomes. This frontier provides a quantitative measure of robustness of conclusions to deviations from the point identifying assumptions. We derive \sqrt{N}-consistent sample analog estimators for these frontiers. We then provide two asymptotically valid bootstrap procedures for constructing lower uniform confidence bands for the breakdown frontier. As a measure of robustness, estimated breakdown frontiers and their corresponding confidence bands can be presented alongside traditional point estimates and confidence intervals obtained under point identifying assumptions. We illustrate this approach in an empirical application to the effect of child soldiering on wages. We find that the conclusions we consider are fairly robust to failure of rank invariance, when random assignment holds, but these conclusions are much more sensitive to both assumptions for small deviations from random assignment.

Inference under Shape Restrictions

We propose a uniformly valid inference method for an unknown function or parameter vector satisfying certain shape restrictions. The method applies very generally, namely to a wide range of finite dimensional and nonparametric problems, such as regressions or instrumental variable estimation, to both kernel or series estimators, and to many different shape restrictions. A major application of our inference method is to construct uniform confidence bands for an unknown function of interest. Our confidence bands are asymptotically equivalent to standard unrestricted confidence bands if the true function strictly satisfies all shape restrictions, but they can be much smaller if some of the shape restrictions are binding or close to binding. We illustrate these sizable width gains as well as the wide applicability of our method in Monte Carlo simulations and in an empirical application.

Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide

This article reviews recent advances in estimation and inference for nonparametric and semiparametric models with endogeneity. It first describes methods of sieves and penalization for estimating unknown functions identified via conditional moment restrictions. Examples include nonparametric instrumental variables (NPIV) regression, nonparametric quantile IV regression, and many more semi/nonparametric structural models. Asymptotic properties of the sieve estimators and the sieve Wald, quasi-likelihood ratio hypothesis tests of functionals with nonparametric endogeneity are presented. For sieve NPIV estimation, the rate-adaptive data-driven choices of sieve regularization parameters and the sieve score bootstrap uniform confidence bands are described. Finally, simple sieve variance estimation and overidentification tests for the semiparametric two-step generalized method of moments are reviewed. Monte Carlo examples are also included.

Inference for Monotone Functions Under Short- and Long-Range Dependence: Confidence Intervals and New Universal Limits

ABSTRACTWe introduce new point-wise confidence interval estimates for monotone functions observed with additive, dependent noise. Our methodology applies to both short- and long-range dependence regimes for the errors. The interval estimates are obtained via the method of inversion of certain discrepancy statistics. This approach avoids the estimation of nuisance parameters such as the derivative of the unknown function, which previous methods are forced to deal with. The resulting estimates are therefore more accurate, stable, and widely applicable in practice under minimal assumptions on the trend and error structure. The dependence of the errors especially long-range dependence leads to new phenomena, where new universal limits based on convex minorant functionals of drifted fractional Brownian motion emerge. Some extensions to uniform confidence bands are also developed. Supplementary materials for this article are available online.

Leveraged ETF Options Implied Volatility Paradox: A Statistical Study

In this paper, we study the statistical properties of the moneyness scaling transformation by Leung and Sircar (2015). This transformation adjusts the moneyness coordinate of the implied volatility in an attempt to remove the discrepancy between the IV smiles for levered and unlevered ETF options. We construct bootstrap uniform confidence bands which indicate that there remains a possibility that the implied volatility smiles are still not the same, even after moneyness scaling has been performed. This presents possible arbitrage opportunities on the (L)ETF market which can be exploited by traders. An empirical data application shows that there are indeed such opportunities in the market which result in risk-free gains for the investor. A dynamic trade-with-the-smile strategy based on a dynamic semiparametric factor model is presented. This strategy utilizes the dynamic structure of implied volatility surface allowing out-of-sample forecasting and information on unleveraged ETF options to construct theoretical one-step-ahead implied volatility surfaces. Additionally, we propose a semi-analytic and a simulation-based estimation approach for incorporating stochastic volatility into the moneyness scaling method. This approach allows to infer the expected integrated variance smile from the data.

Inference and Forecasting Based on the Phillips Curve

In this paper, we conduct uniform inference of two widely used versions of the Phillips curve, specifically the random-walk Phillips curve and the New-Keynesian Phillips curve (NKPC). For both specifications, we propose a potentially time-varying natural unemployment (NAIRU) to address the uncertainty surrounding the inflation-unemployment trade-off. The inference is conducted through the construction of what is known as the uniform confidence band (UCB). The proposed methodology is then applied to point-ahead inflation forecasting for the Korean economy. This paper finds that the forecasts can benefit from conducting UCB-based inference and that the inference results have important policy implications.

Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide

This paper reviews recent advances in estimation and inference for nonparametric and semiparametric models with endogeneity. It first describes methods of sieves and penalization for estimating unknown functions identified via conditional moment restrictions. Examples include nonparametric instrumental variables regression (NPIV), nonparametric quantile IV regression and many more semi-nonparametric structural models. Asymptotic properties of the sieve estimators and the sieve Wald, quasi-likelihood ratio (QLR) hypothesis tests of functionals with nonparametric endogeneity are presented. For sieve NPIV estimation, the rate-adaptive data-driven choices of sieve regularization parameters and the sieve score bootstrap uniform confidence bands are described. Finally, simple sieve variance estimation and over-identification test for semiparametric two-step GMM are reviewed. Monte Carlo examples are included.

Capital asset pricing model: A time-varying volatility approach

In this paper, we propose a methodology to conduct uniform inference of volatility in the capital asset pricing model (CAPM). To that end, relevant theory is employed to construct the uniform confidence band of the volatility in the CAPM. The methodology is applied to the U.S. stock return data. The empirical results show strong evidence of co-movement among the volatility estimates for six U.S. stocks of large market capitalization. The hypothesis of constant volatility for the CAPM is rejected unanimously, mainly due to the surge in volatility in the early 2000s and during the 2008 financial crisis.

Doubly Robust Uniform Confidence Band for the Conditional Average Treatment Effect Function

In this paper, we propose a doubly robust method to present the heterogeneity of the average treatment effect with respect to observed covariates of interest. We consider a situation where a large number of covariates are needed for identifying the average treatment effect but the covariates of interest for analyzing heterogeneity are of much lower dimension. Our proposed estimator is doubly robust and avoids the curse of dimensionality. We propose a uniform confidence band that is easy to compute, and we illustrate its usefulness via Monte Carlo experiments and an application to the effects of smoking on birth weights.

Confidence band for expectation dependence with applications

Motivated by the applications of the concept of expectation dependence in economics and finance, we propose a method to construct uniform confidence band for expectation dependence. It is derived based on Hoeffding’s inequality. Our proposed confidence band can be explicitly expressed and thus it is very easy to implement. Our method has applications to demand for a risky asset and first-order risk aversion problems. Simulations suggest our proposed confidence interval can control the coverage probabilities very well, and the average lengths are very short. Two empirical applications are presented to illustrate the usefulness of the constructed confidence band of expectation dependence.

Was it risk? Or was it fundamentals? Explaining excess currency returns with kernel smoothed regressions

This paper uses recently developed kernel smoothing regression procedures and uniform confidence bounds to investigate the forward premium anomaly. These new statistical methods estimate the local time-varying slope coefficient of the regression of spot returns on the lagged interest rate differential. Uniform confidence bands are used to test when uncovered interest parity is violated. The estimated betas in the forward premium smoothed regression are found to vary substantially over time and to be partially explicable in terms of lagged fundamentals and money growth volatilities arising from risk premium. Frequentist model averaging procedures indicate the relative importance of these variables in terms of explaining movements in the betas and hence the apparent causes of regimes where UIP fails.

Some new asymptotic theory for least squares series: Pointwise and uniform results

In econometric applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements over a pre-specified functional form, especially if they nest some successful parametric economically-motivated forms. Series method offers exactly that by approximating the unknown function based on k basis functions, where k is allowed to grow with the sample size n to balance the trade off between variance and bias. In this work we consider series estimators for the conditional mean in light of four new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the L∞ and L2-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge at some rate, and (iv) strong approximations to series-type processes.These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken considerably the condition on the number k of approximating functions used in series estimation from the typical k2/n→0 to k/n→0, up to log factors, which was available only for spline series before. Second, under the same weak conditions we derive L2 rates and pointwise central limit theorems results when the approximation error vanishes. Under an incorrectly specified model, i.e. when the approximation error does not vanish, analogous results are also shown. Third, under stronger conditions we derive uniform rates and functional central limit theorems that hold if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the nonparametric function.Finally and most importantly, from a point of view of practice, we derive uniform rates, Gaussian approximations, and uniform confidence bands for a wide collection of linear functionals of the conditional expectation function, for example, the function itself, the partial derivative function, the conditional average partial derivative function, and other similar quantities. All of these results are new.