Comparing Possibly Misspecified Forecasts

Abstract

Recent work has emphasized the importance of evaluating estimates of a statistical functional (such as a conditional mean, quantile, or distribution) using a loss function that is consistent for the functional of interest, of which there is an infinite number. If forecasters all use correctly specified models free from estimation error, and if the information sets of competing forecasters are nested, then the ranking induced by a single consistent loss function is sufficient for the ranking by any consistent loss function. This article shows, via analytical results and realistic simulation-based analyses, that the presence of misspecified models, parameter estimation error, or nonnested information sets, leads generally to sensitivity to the choice of (consistent) loss function. Thus, rather than merely specifying the target functional, which narrows the set of relevant loss functions only to the class of loss functions consistent for that functional, forecast consumers or survey designers should specify the single specific loss function that will be used to evaluate forecasts. An application to survey forecasts of U.S. inflation illustrates the results.     

Uncertainty,information, and disagreement of economic forecasters

ABSTRACT

An information framework is proposed for studying uncertainty and disagreement of economic forecasters. This framework builds upon the mixture model of combining density forecasts through a systematic application of the information theory. The framework encompasses the measures used in the literature and leads to their generalizations. The focal measure is the Jensen–Shannon divergence of the mixture which admits Kullback–Leibler and mutual information representations. Illustrations include exploring the dynamics of the individual and aggregate uncertainty about the US inflation rate using the survey of professional forecasters (SPF). We show that the normalized entropy index corrects some of the distortions caused by changes of the design of the SPF over time. Bayesian hierarchical models are used to examine the association of the inflation uncertainty with the anticipated inflation and the dispersion of point forecasts. Implementation of the information framework based on the variance and Dirichlet model for capturing uncertainty about the probability distribution of the economic variable are briefly discussed.     

INDEX TO VOLUME 7 (1989)

Surveys of forecasters, containing respondents’ predictions of future values of key macroeconomic variables, receive a lot of attention in the financial press, from investors and from policy makers. They are apparently widely perceived to provide useful information about agents’ expectations. Nonetheless, these survey forecasts suffer from the crucial disadvantage that they are often quite stale, as they are released only infrequently. In this article, we propose MIDAS regression and Kalman filter methods for using asset price data to construct daily forecasts of upcoming survey releases. Our methods also allow us to predict actual outcomes, providing competing forecasts, and allow us to estimate what professional forecasters would predict if they were asked to make a forecast each day, making it possible to measure the effects of events and news announcements on expectations.     

INDEX TO VOLUME 5 (1987)

Real-time estimates of output gaps and inflation gaps differ from the values that are obtained using data available long after the event. Part of the problem is that the data on which the real-time estimates are based is subsequently revised. We show that vector-autoregressive models of data vintages provide forecasts of post-revision values of future observations and of already-released observations capable of improving estimates of output and inflation gaps in real time. Our findings indicate that annual revisions to output and inflation data are in part predictable based on their past vintages. This article has online supplementary materials.     

中国通货膨胀率的非线性动态行为研究——来自MS—ARFIMA模型的新证据

使用允许长记忆参数d服从区制转换的MS—ARFIMA模型对中国月度通货膨胀路径的动态行为进行新的实证研究,结果显示：中国通货膨胀不仅均值水平和不确定性存在着“低通胀”区制和“高通胀”区制,而且更为重要的是,通货膨胀序列的平稳性也表现出显著的区制转换动态。“低通胀”区制下,长记忆参数d1=0．361,说明通货膨胀是协方差平稳序列,“高通胀”区制下,长记忆参数d2=1．145,说明通货膨胀是非平稳序列。这一新的研究结论意味着中国通货膨胀冲击的持久性效应也存在相应的区制转移变化。这要求央行在管控通货膨胀过程中,既要考虑均值和不确定性的区制变化,又要兼顾平稳性和持久性的区制变化。     

Interaction Between Autocorrelation and Conditional Heteroscedasticity: A Random-Coefficient Approach

In applied econometrics, we tend to tackle specification problems one at a time rather than considering them jointly. This has serious consequences for statistical inference. One example of this is considering autocorrelation and autoregressive conditional heteroscedasticity (ARCH) separately. In this article we consider a linear regression model with random coefficient autoregressive disturbances that provides a convenient framework to analyze autocorrelation and ARCH simultaneously. Our stationarity conditions and testing results reveal the strong interaction between ARCH and autocorrelation. An empirical example of testing the unbiasedness of experts' expectations of inflation demonstrates that neglecting conditional heteroscedasticity or misspecifying the autocorrelation structure might result in unreliable inference.     

经济周期中产出波动与通货膨胀不对称动态关系研究

建立不对称动态菲利普斯曲线理论研究经济周期中产出波动与通货膨胀不对称动态关系。该理论蕴含了经济扩张与收缩期中通胀持续性、产出波动对通胀的长短期影响差异特征及相关检验方法。运用该理论对中国相关季度数据进行了实证分析,结果表明:产出波动对通货膨胀短期中具有"顺周期"的正相关性,充当了"晴雨器"作用;长期中具有"逆周期"的负相关性,充当了"稳定器"作用。统计检验表明,经济周期中通货膨胀持续性及产出波动对通货膨胀的长短期影响具有显著不对称性,这种不对称性是中国经济转型期经济运行质量的历史检验,对现阶段追求经济增长质量具有深刻的政策启示。     

我国通货膨胀运动阶段的非线性平滑转换

本文分别构建了两、三以及四机制C-STAR模型来研究我国通货膨胀的非线性运动特征。实证结果表明：我国通胀率是平稳的非线性均值回归过程,依据三机制模型划分的通缩、通缩-通胀中间态以及通胀的三阶段能很好地刻画我国通货膨胀的运动特点;但是把我国通胀率划分为通缩、通胀适中、温和通胀和高通胀的四阶段能进一步提高模型的解释与拟合能力。我们详细阐述了通货膨胀在不同阶段的转换特点以及持久性与不确定性特征,发现通货膨胀的持久性与不确定性成反向非线性关系,与水平值的大小没有必然联系。在温和通胀阶段,其持久性最强,不确定性最小;在通胀适中阶段,其持久性最弱,不确定性最大。另外,温和通胀阶段也是最优通胀目标区间,为了保持经济的稳定增长,央行应把通胀率控制在该区间内。最后,本文给出了实证结果所蕴含的政策涵义。     

Effects of a single outlier on arma identification

Fox (1972), Box and Tiao (1975), and Abraham and Box (1979) have proposed methods for detecting outliers in time series whose ARMA form is known (or identified). We show that the existence of a single aberrant observation, innovation, or intervention causes an ARMA model to be misidentified using unadjusted autocorrelation (acf) and partial autocorrelation estimates. The magnitude, location, type of outlier, and in some cases the ARMA's parameters, affect the identification outcome. We use variance inflation, signal-to-noise ratios, and acf critical values to determine an ARMA model's susceptibility to misidentifi-cation. Numerical and simulation examples suggest how to iteratively use the outlier detection methods in practice.     

A New Model of Trend Inflation

This article introduces a new model of trend inflation. In contrast to many earlier approaches, which allow for trend inflation to evolve according to a random walk, ours is a bounded model which ensures that trend inflation is constrained to lie in an interval. The bounds of this interval can either be fixed or estimated from the data. Our model also allows for a time-varying degree of persistence in the transitory component of inflation. In an empirical exercise with CPI inflation, we find the model to work well, yielding more sensible measures of trend inflation and forecasting better than popular alternatives such as the unobserved components stochastic volatility model. This article has supplementary materials online.     

特有冲击、共同冲击与通货膨胀持续性

中国分类价格指数的通货膨胀持续性有何特征？如何理解分类价格指数和加总价格指数通货膨胀持续性之间的关系？我们对2001年1月至2011年12月的数据进行实证分析,结果表明分类价格指数比较明显的表现出行业间异质的通货膨胀波动性和通货膨胀持续性,而与加总价格指数相比,它的通货膨胀波动性更大而通货膨胀持续性更低。通货膨胀持续性的这些现象可以从特有冲击和共同冲击的角度加以理解。共同冲击的相对重要性存在差异解释了分类价格指数的通货膨胀持续性呈现行业间异质性;特有冲击的重要性在数据加总时被削弱,解释了分类价格指数具有比加总价格指数更低的通货膨胀持续性。     

Who Forecasts Better?

This article suggests that the emphasis on forecast evaluations should be redirected from searching for “best” forecasters to finding those who are “better.” A new methodology is developed, and the results indicate that it is possible to identify “better” forecasters.     

系统抽样、脉冲响应与通货膨胀惯性

利用时间序列模型研究经济变量的动态特征时,模型估计结果通常随样本观测频率改变而变化。本文针对系统抽样问题,研究样本观测频率变化对脉冲响应函数和累积脉冲响应的影响,探讨不同观测频率下脉冲响应函数的理论联系,以及对变量持久性特征更加稳健的度量方法,在此基础上考察中国通货膨胀的动态特征。结果表明,累积脉冲响应和惯性系数随样本观测频率提高而增加;系统抽样对脉冲响应函数的影响源于外生冲击界定的不同,脉冲响应的数值意义有限,但可用其考察冲击影响的持续时间;外生冲击对中国通货膨胀的影响持续2年左右,且90%在1年内实现,央行要对通胀走势有很强的预判能力,政策滞后将增加反通胀成本。     

我国通货膨胀率非线性特征研究

 本文应用带单位根的门限自回归模型，对我国1990年来以来通货膨胀率的动态路径进行了模拟分析。估计和检验发现了我国通货膨胀具有明显的非线性特征，模型较好地拟合了通货膨胀的动态调整过程。我国通货膨胀调整存在减速通货膨胀状态、适中通货膨胀状态和加速通货膨胀状态三个区制。适中通货膨胀状态是一个平稳的自回归过程，减速通货膨胀状态、加速通货膨胀状态则是具有单位根的自回归过程，具有自我加速的作用。在不同的区制下，通货膨胀率均有较高的持久性，但中间状态的持久性明显低于其他两种状态。     

Comparing Greenbook and Reduced Form Forecasts Using a Large Realtime Dataset

Many recent articles have found that atheoretical forecasting methods using many predictors give better predictions for key macroeconomic variables than various small-model methods. The practical relevance of these results is open to question, however, because these articles generally use ex post revised data not available to forecasters and because no comparison is made to best actual practice. We provide some evidence on both of these points using a new large dataset of vintage data synchronized with the Fed’s Greenbook forecast. This dataset consist of a large number of variables as observed at the time of each Greenbook forecast since 1979. We compare realtime, large dataset predictions to both simple univariate methods and to the Greenbook forecast. For inflation we find that univariate methods are dominated by the best atheoretical large dataset methods and that these, in turn, are dominated by Greenbook. For GDP growth, in contrast, we find that once one takes account of Greenbook’s advantage in evaluating the current state of the economy, neither large dataset methods, nor the Greenbook process offers much advantage over a univariate autoregressive forecast.     

Maximum studentized score tests for the detection of outliers in time series regression models

Efficient score tests exist among others, for testing the presence of additive and/or innovative outliers that are the result of the shifted mean of the error process under the regression model. A sample influence function of autocorrelation-based diagnostic technique also exists for the detection of outliers that are the result of the shifted autocorrelations. The later diagnostic technique is however not useful if the outlying observation does not affect the autocorrelation structure but is generated due to an inflation in the variance of the error process under the regression model. In this paper, we develop a unified maximum studentized type test which is applicable for testing the additive and innovative outliers as well as variance shifted outliers that may or may not affect the autocorrelation structure of the outlier free time series observations. Since the computation of the p-values for the maximum studentized type test is not easy in general, we propose a Satterthwaite type approximation based on suitable doubly non-central F-distributions for finding such p-values [F.E. Satterthwaite, An approximate distribution of estimates of variance components, Biometrics 2 (1946), pp. 110–114]. The approximations are evaluated through a simulation study, for example, for the detection of additive and innovative outliers as well as variance shifted outliers that do not affect the autocorrelation structure of the outlier free time series observations. Some simulation results on model misspecification effects on outlier detection are also provided.     

Best linear corrector of classification estimates of proportions of objects in several unknown classes

It has been recognized that counting the objects allocated by a rule of classification to several unknown classes often does not provide good estimates of the true class proportions of the objects to be classified. We propose a linear transformation of these classification estimates, which minimizes the mean squared error of the transformed estimates over all possible sets of true proportions. This so-called best-linear-corrector (BLC) transformation is a function of the confusion (classification-error) matrix and of the first and second moments of the prior distribution of the vector of proportions. When the number of objects to be classified increases, the BLC tends to the inverse of the confusion matrix. The estimates that are obtained directly by this inverse-confusion corrector (ICC) are also the maximum-likelihood unbiased estimates of the probabilities that the objects originate from one or the other class, had the objects been preselected with those probabilities. But for estimating the actual proportions, the ICC estimates behave less well than the raw classification estimates for some collections. In that situation, the BLC is substantially superior to the ICC even for some large collections of objects and is always substantially superior to the raw estimates. The statistical model is applied concretely to the measure of forest covers in remote sensing.     

中国通货膨胀预期的结构和持久性

本文通过改进的预期通货膨胀率模型,将城镇居民调查问卷中未来物价变动预期的定性数据,转换为有效的预期通货膨胀率。同时,本文对通货膨胀预期的性质进行了结构性分析,结果表明,核心通货膨胀内涵下的通货膨胀预期变动符合理性预期假说,食品价格内涵下的通货膨胀预期变动更具备适应性预期的特点。通货膨胀预期的持久性较差,实际核心通货膨胀对通货膨胀预期不具备持续影响,而来自食品因素的实际通货膨胀对通货膨胀预期有持久的影响,但通货膨胀预期对实际通货膨胀的反馈效果并不稳定。因此,在充分重视核心通货膨胀的变动的基础上,货币政策的持续性和稳定性显得尤为重要。     

Zero-inflated compound Poisson distributions in integer-valued GARCH models

In this paper we introduce a wide class of integer-valued stochastic processes that allows to take into consideration, simultaneously, relevant characteristics observed in count data namely zero inflation, overdispersion and conditional heteroscedasticity. This class includes, in particular, the compound Poisson, the zero-inflated Poisson and the zero-inflated negative binomial INGARCH models, recently proposed in literature. The main probabilistic analysis of this class of processes is here developed. Precisely, first- and second-order stationarity conditions are derived, the autocorrelation function is deduced and the strict stationarity is established in a large subclass. We also analyse in a particular model the existence of higher-order moments and deduce the explicit form for the first four cumulants, as well as its skewness and kurtosis.     

A New Test for Short Memory in Long Memory Time Series

This article considers short memory characteristics in a long memory process. We derive new asymptotic results for the sample autocorrelation difference ratios. We used these results to develop a new portmanteau test that determines if short memory parameters are statistically significant. In simulations, the new test can detect short memory components more often than the Ljung-Box test when these short memory components are in fact within a long memory process. Interestingly, our test finds short memory autocorrelations in U.S. inflation rate data, whereas the Ljung-Box test fails to find these autocorrelations. Modeling these short memory autocorrelations of the inflation rate data leads to improved model accuracy and more precise prediction.