An Introduction to Best Empirical Models when the Parameter Space is Infinite Dimensional*

Ploberger and Phillips (Econometrica, Vol. 71, pp. 627–673, 2003) proved a result that provides a bound on how close a fitted empirical model can get to the true model when the model is represented by a parameterized probability measure on a finite dimensional parameter space. The present note extends that result to cases where the parameter space is infinite dimensional. The results have implications for model choice in infinite dimensional problems and highlight some of the difficulties, including technical difficulties, presented by models of infinite dimension. Some implications for forecasting are considered and some applications are given, including the empirically relevant case of vector autoregression (VAR) models of infinite order.     

Combining Forecasts from Nested Models*

Motivated by the common finding that linear autoregressive models often forecast better than models that incorporate additional information, this paper presents analytical, Monte Carlo and empirical evidence on the effectiveness of combining forecasts from nested models. In our analytics, the unrestricted model is true, but a subset of the coefficients is treated as being local‐to‐zero. This approach captures the practical reality that the predictive content of variables of interest is often low. We derive mean square error‐minimizing weights for combining the restricted and unrestricted forecasts. Monte Carlo and empirical analyses verify the practical effectiveness of our combination approach.     

Automated Model Selection in Finance: General‐to‐Specific Modelling of the Mean and Volatility Specifications*

General‐to‐Specific (GETS) modelling has witnessed major advances thanks to the automation of multi‐path GETS specification search. However, the estimation complexity associated with financial models constitutes an obstacle to automated multi‐path GETS modelling in finance. Making use of a recent result we provide and study simple but general and flexible methods that automate financial multi‐path GETS modelling. Starting from a general model where the mean specification can contain autoregressive terms and explanatory variables, and where the exponential volatility specification can include log‐ARCH terms, asymmetry terms, volatility proxies and other explanatory variables, the algorithm we propose returns parsimonious mean and volatility specifications.     

Testing for Heteroskedasticity and Predictive Failure in Linear Regression Models*

It is argued that, when researchers wish to carry out a Chow test of the significance of prediction errors, it is necessary to assume homoskedasticity because standard results on heteroskedasticity‐robust tests are not available. The effects of heteroskedasticity on the Chow prediction error test are examined. The implementation of tests for heteroskedasticity is discussed, with the case in which the regressors include dummy variables for prediction error tests receiving special attention. Monte Carlo results are reported.     

Specification Testing of Markov Switching Models*

This paper proposes a set of formal tests to address the goodness‐of‐fit of Markov switching models. These formal tests are constructed as tests of model consistency and of both parametric and non‐parametric encompassing. The formal tests are then combined with informal tests using simulation in combination with non‐parametric density and conditional mean estimation. The informal tests are shown to be useful in shedding light on the failure (or success) of the encompassing tests. Several examples are provided.     

Practical Considerations for Choosing Between Tobit and SCLS or CLAD Estimators for Censored Regression Models with an Application to Charitable Giving*

Practical considerations for choosing between Tobit, symmetrically censored least squares (SCLS) and censored least absolute deviations (CLAD) estimators are offered. Practical considerations deal with when a Hausman test is better than a conditional moment test for judging the severity of a misspecification, the need to bootstrap the sampling distributions of the Hausman tests, what to look for in a graphical examination of the residuals and the limited value of SCLS. The practical considerations are applied to a model of the intergenerational transmission of charitable giving using new data from the Panel Study of Income Dynamics (PSID). The paper shows how to use relative distribution methods to calculate CLAD‐based marginal effects on the observable dependent variable.     

Testing for Misspecification in Binary Response Models with Competing Distributions*

In empirical studies, the probit and logit models are often used without checks for their competing distributional specifications. It is also rare for econometric tests to be focused on this issue. Santos Silva [Journal of Applied Econometrics (2001 ), Vol. 16, pp. 577–597] is an important recent exception. By using the conditional moment test principle, we discuss a wide class of non‐nested tests that can easily be applied to detect the competing distributions for the binary response models. This class of tests includes the test of Santos Silva (2001 ) for the same task as a particular example and provides other useful alternatives. We also compare the performance of these tests by a Monte Carlo simulation.     

Dickey–Fuller Type of Tests against Nonlinear Dynamic Models*

In this paper, we introduce several test statistics testing the null hypothesis of a random walk (with or without drift) against models that accommodate a smooth nonlinear shift in the level, the dynamic structure and the trend. We derive analytical limiting distributions for all the tests. The power performance of the tests is compared with that of the unit‐root tests by Phillips and Perron [Biometrika (1988), Vol. 75, pp. 335–346], and Leybourne, Newbold and Vougas [Journal of Time Series Analysis (1998), Vol. 19, pp. 83–97]. In the presence of a gradual change in the deterministics and in the dynamics, our tests are superior in terms of power.     

A Flexible Tool for Model Building: the Relevant Transformation of the Inputs Network Approach (RETINA)*

A new method, called Relevant Transformation of the Inputs Network Approach is proposed as a tool for model building. It is designed around flexibility (with nonlinear transformations of the predictors of interest), selective search within the range of possible models, out‐of‐sample forecasting ability and computational simplicity. In tests on simulated data, it shows both a high rate of successful retrieval of the data generating process, which increases with the sample size and a good performance relative to other alternative procedures. A telephone service demand model is built to show how the procedure applies on real data.     

Simulation‐based Finite Sample Linearity Test against Smooth Transition Models*

In this paper, we use Monte Carlo (MC) testing techniques for testing linearity against smooth transition models. The MC approach allows us to introduce a new test that differs in two respects from the tests existing in the literature. First, the test is exact in the sense that the probability of rejecting the null when it is true is always less than or equal to the nominal size of the test. Secondly, the test is not based on an auxiliary regression obtained by replacing the model under the alternative by approximations based on a Taylor expansion. We also apply MC testing methods for size correcting the test proposed by Luukkonen, Saikkonen and Teräsvirta (Biometrika, Vol. 75, 1988, p. 491). The results show that the power loss implied by the auxiliary regression‐based test is non‐existent compared with a supremum‐based test but is more substantial when compared with the three other tests under consideration.     

Volatility forecast comparison using imperfect volatility proxies

The use of a conditionally unbiased, but imperfect, volatility proxy can lead to undesirable outcomes in standard methods for comparing conditional variance forecasts. We motivate our study with analytical results on the distortions caused by some widely used loss functions, when used with standard volatility proxies such as squared returns, the intra-daily range or realised volatility. We then derive necessary and sufficient conditions on the functional form of the loss function for the ranking of competing volatility forecasts to be robust to the presence of noise in the volatility proxy, and derive some useful special cases of this class of “robust” loss functions. The methods are illustrated with an application to the volatility of returns on IBM over the period 1993 to 2003.     

Testing Steady‐State Restrictions of Linear Rational Expectations Models when Data are Highly Persistent*

Steady‐state restrictions are commonly imposed on highly persistent variables to achieve stationarity prior to confronting rational expectations models with data. However, the resulting steady‐state deviations are often surprisingly persistent indicating that some aspects of the underlying theory may be empirically problematic. This paper discusses how to formulate steady‐state restrictions in rational expectations models with latent forcing variables and test their validity using cointegration techniques. The approach is illustrated by testing steady‐state restrictions for alternative specifications of the New Keynesian model and shown to be able to discriminate between different assumptions on the sources of the permanent shocks.     

The Likelihood Ratio Test for the Rank of a Cointegration Submatrix*

This paper proposes a likelihood ratio test for rank deficiency of a submatrix of the cointegrating matrix. Special cases of the test include the one of invalid normalization in systems of cointegrating equations, the feasibility of permanent–transitory decompositions and of subhypotheses related to neutrality and long‐run Granger noncausality. The proposed test has a chi‐squared limit distribution and indicates the validity of the normalization with probability one in the limit, for valid normalizations. The asymptotic properties of several derived estimators of the rank are also discussed. It is found that a testing procedure that starts from the hypothesis of minimal rank is preferable.     

Regression Models with Variables of Different Frequencies: The Case of a Fixed Frequency Ratio*

An increasing variety of data frequencies available in economics, finance, etc. gives rise to a question how to build and estimate a regression model with variables observed at different frequencies. In a unifying framework of (m,d)‐aggregation we consider various approaches by discussing some potential and limitations. A Monte Carlo experiment and an empirical example illustrate that the traditional fixed aggregation approach, widely used in applied economics, might be inconsistent with data and highly inferior in terms of model precision.     

Examining macroeconomic models through the lens of asset pricing

We develop new methods for representing the asset-pricing implications of stochastic general equilibrium models. We provide asset-pricing counterparts to impulse response functions and the resulting dynamic value decompositions (DVDs). These methods quantify the exposures of macroeconomic cash flows to shocks over alternative investment horizons and the corresponding prices or investors’ compensations. We extend the continuous-time methods developed in Hansen and Scheinkman (2012) and Borovi?ka et al. (2011) by constructing discrete-time, state-dependent, shock-exposure and shock-price elasticities as functions of the investment horizon. Our methods are applicable to economic models that are nonlinear, including models with stochastic volatility.     

The Effects of Additive Outliers and Measurement Errors when Testing for Structural Breaks in Variance*

This article discusses the asymptotic and finite‐sample properties of the CUSUM tests for detecting structural breaks in volatility when the data are perturbed with (additive) outliers and/or measurement errors. The special focus is on the parametric and non‐parametric tests in Inclán and Tiao (1994) and Kokoszka and Leipus (2000) . Whereas the asymptotic distribution of the former can be largely affected, the distribution of the latter remains invariant and renders consistent break‐point estimates. In small samples, however, large additive outliers are able to generate sizeable distortions in both tests, which explains some of the contradictory findings in previous literature.     

Panel Stationarity Test with Structural Breaks*

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.     

Selecting a Nonlinear Time Series Model using Weighted Tests of Equal Forecast Accuracy*

Nonlinear time series models have become fashionable tools to describe and forecast a variety of economic time series. A closer look at reported empirical studies, however, reveals that these models apparently fit well in‐sample, but rarely show a substantial improvement in out‐of‐sample forecasts, at least over linear models. One of the many possible reasons for this finding is the use of inappropriate model selection criteria and forecast evaluation criteria. In this paper we therefore propose a novel criterion, which we believe does more justice to the very nature of nonlinear models. Simulations show that this criterion outperforms those criteria currently in use, in the sense that the true nonlinear model is more often found to perform better in out‐of‐sample forecasting than a benchmark linear model. An empirical illustration for US GDP emphasizes its relevance.     

Controlling the Overall Significance Level of a Battery of Least Squares Diagnostic Tests*

Double bootstrap methods are used to control the overall significance level of a battery of diagnostic tests applied to a regression model estimated by ordinary least squares. Monte Carlo evidence on the finite sample performance of the bootstrap methods is reported and discussed.     

Disproving Causal Relationships Using Observational Data*

Economic theory is replete with causal hypotheses that are scarcely tested because economists are generally constrained to work with observational data. We describe a method for testing a hypothesis that one observed random variable causes another. Contingent on a sufficiently strong correspondence between the two variables, an appropriately related third variable can be employed for the test. The logic of the procedure naturally suggests strong and weak grounds for rejecting the causal hypothesis. Monte Carlo results suggest that weakly grounded rejections are unreliable for small samples, but reasonably reliable for large samples. Strongly grounded rejections are highly reliable, even for small samples.