A two–stage approach to additive time series models

For nonlinear additive time series models, an appealing approach used in the literature to estimate the nonparametric additive components is the projection method. In this paper, it is demonstrated that the projection method might not be efficient in an asymptotic sense. To estimate additive components efficiently, a two–stage approach is proposed together with a local linear fitting and a new bandwidth selector based on the nonparametric version of the Akaike information criterion. It is shown that the two–stage method not only achieves efficiency but also makes bandwidth selection relatively easier. Also, the asymptotic normality of the resulting estimator is established. A small simulation study is carried out to illustrate the proposed methodology and the two–stage approach is applied to a real example from econometrics.     

A two-step approach for identifying seasonal autoregressive time series forecasting models

One of the most powerful and widely used methodologies for forecasting economic time series is the class of models known as seasonal autoregressive processes. In this article we present a new approach not only for identifying seasonal autoregressive models, but also the degree of differencing required to induce stationarity in the data. The identification method is iterative and consists in systematically fitting increasing order models to the data, and then verifying that the resulting residuals behave like white noise using a two stage autoregressive order determination criterion. Once the order of the process is determined the identified structure is tested to see if it can be simplified. The identification performance of this procedure is contrasted with other order selection procedures for models with ‘gaps.' We also illustrate the forecast performance of the identification method using monthly and quarterly economic data.     

Estimation of a model containing unobservable variables using grouped observations: An application to the permanent income hypothesis

The method of estimating relationships in a model containing unobservable variables is combined with the technique of estimation from a sample of grouped observations to analyse the permanent income hypothesis. From a sample of group means the paper measures the relative importance of such factors as wealth, occupation, age and family-size as determinants of permanent income in the United Kingdom, and also considers the mechanism by which transitory receipts influence total consumption expenditure.     

A flexible approach to parametric inference in nonlinear and time varying time series models

Many structural break and regime-switching models have been used with macroeconomic and financial data. In this paper, we develop an extremely flexible modeling approach which can accommodate virtually any of these specifications. We build on earlier work showing the relationship between flexible functional forms and random variation in parameters. Our contribution is based around the use of priors on the time variation that is developed from considering a hypothetical reordering of the data and distance between neighboring (reordered) observations. The range of priors produced in this way can accommodate a wide variety of nonlinear time series models, including those with regime-switching and structural breaks. By allowing the amount of random variation in parameters to depend on the distance between (reordered) observations, the parameters can evolve in a wide variety of ways, allowing for everything from models exhibiting abrupt change (e.g. threshold autoregressive models or standard structural break models) to those which allow for a gradual evolution of parameters (e.g. smooth transition autoregressive models or time varying parameter models). Bayesian econometric methods for inference are developed for estimating the distance function and types of hypothetical reordering. Conditional on a hypothetical reordering and distance function, a simple reordering of the actual data allows us to estimate our models with standard state space methods by a simple adjustment to the measurement equation. We use artificial data to show the advantages of our approach, before providing two empirical illustrations involving the modeling of real GDP growth.     

A unified approach to validating univariate and multivariate conditional distribution models in time series

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic $N(0,1)$ distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance.     

Risk measures for variable annuities: A hermite series expansion approach

In this study, we propose an efficient approach to the calculation of risk measures for an insurer's liability from writing a variable annuity with guaranteed benefits. Our approach is based on a novel application of the Hermite series expansions on the transition density of a diffusion process to the insurance setting. We compare our method with existing methods in the literature, including the analytical method, spectral method and Green's function method, and illustrate its substantial advantages in calculating risk measures for variable annuities with different guarantee structures. The improved efficiency makes our method flexible to practical implementation in reporting risk measures on a daily basis. We also conduct a sensitivity analysis of the risk measures with respect to key parameters.     

A non-linear filtering approach to stochastic volatility models with an application to daily stock returns

This paper develops a new model for the analysis of stochastic volatility (SV) models. Since volatility is a latent variable in SV models, it is difficult to evaluate the exact likelihood. In this paper, a non-linear filter which yields the exact likelihood of SV models is employed. Solving a series of integrals in this filter by piecewise linear approximations with randomly chosen nodes produces the likelihood, which is maximized to obtain estimates of the SV parameters. A smoothing algorithm for volatility estimation is also constructed. Monte Carlo experiments show that the method performs well with respect to both parameter estimates and volatility estimates. We illustrate our model by analysing daily stock returns on the Tokyo Stock Exchange. Since the method can be applied to more general models, the SV model is extended so that several characteristics of daily stock returns are allowed, and this more general model is also estimated. Copyright © 1999 John Wiley & Sons, Ltd.     

Shocks derived from mining windfalls and horizontal transfers: Exploring the permanent income hypothesis in Chilean municipalities from a spatial competition approach

We explore how spatial interaction affects the strategic use of municipal income when deciding between 1) an optimal long-run expenditure strategy versus 2) using the current income to finance current activities, a phenomenon known as the permanent income hypothesis. Even when this hypothesis is grounded in temporal logic, insufficient attention has been given to the impact of spatial dependence on this type of budget decision. Therefore, we present two reasons why spatial interaction adds new insight to this discussion. First, subnational governments located inside larger functional areas have lower average costs due to the population concentration, allowing for coordination between jurisdictions to achieve more power of negotiation and to potentially exploit economies of scale. Second, local government decision-making is not independent of other jurisdictions as municipalities would constantly evaluate the others’ actions regarding local tax effort, spending, and debt. While this spatial consideration remains a challenge for theoretical modeling, we offer empirical evidence to evaluate how robust the permanent income hypothesis is when geography is incorporated. Our empirical approach uses dynamic panel data with spatial dependence on debt, expenditure, and the error term. To evaluate our hypothesis, we exploit panel data from 320 Chilean municipalities between 2008 and 2020 and use two sources of income: non-matched grants via mining windfalls and horizontal fiscal transfers among cities. The evidence indicates that jurisdictions make backward-looking decisions regarding spending; that is, there are no significant differences between the short and long run. The results for debt, however, are not robust. Policy pertaining to the use of public resources should consider the spatial dependence between municipalities which should be a crucial factor in budgetary decision-making.     

Forecasting the NN5 time series with hybrid models

We propose a simple way of predicting time series with recurring seasonal periods. Missing values of the time series are estimated and interpolated in a preprocessing step. We combine several forecasting methods by taking the weighted mean of forecasts that were generated with time-domain models which were validated on left-out parts of the time series. The hybrid model is a combination of a neural network ensemble, an ensemble of nearest trajectory models and a model for the 7-day cycle. We apply this approach to the NN5 time series competition data set.     

Accelerated gaussian importance sampler with application to dynamic latent variable models

We propose a new generic and highly efficient Accelerated Gaussian Importance Sampler (AGIS) for the numerical evaluation of (very) high-dimensional density functions. A specific case of interest to us is the evaluation of likelihood functions for a broad class of dynamic latent variable models. The feasibility of our method is strikingly illustrated by means of an application to a first-order dynamic stochastic volatility model for daily stock returns, whose likelihood for an actual sample of size 2022 (!) is evaluated with high numerical accuracy by means of 10,000 Monte Carlo replications. The estimated model parsimoniously dominates ARCH and GARCH alternatives, one of which includes twelve lags.     

Testing for unit roots in time series models with non-stationary volatility

Many of the key macro-economic and financial variables in developed economies are characterized by permanent volatility shifts. It is known that conventional unit root tests are potentially unreliable in the presence of such behaviour, depending on a particular function (the variance profile) of the underlying volatility process. Somewhat surprisingly then, very little work has been undertaken to develop unit root tests which are robust to the presence of permanent volatility shifts. In this paper we fill this gap in the literature by proposing tests which are valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth-transition) volatility change processes as special cases. Our solution uses numerical methods to simulate the asymptotic null distribution of the statistics based on a consistent estimate of the variance profile which we also develop. The practitioner is not required to specify a parametric model for volatility. An empirical illustration using producer price inflation series from the Stock–Watson database is reported.     

Forecasting performance of multivariate time series models with full and reduced rank: an empirical examination

Since Quenouille's influential work on multiple time series, much progress has been made towards the goal of parameter reduction and model fit. Relatively less attention has been paid to the systematic evaluation of out-of-sample forecast performance of multivariate time series models. In this paper, we update the hog data set studied by Quenouille (and other researchers who followed him). We re-estimate his model with extended observations (1867–1966), and generate recursive one- to four-steps-ahead forecasts for the period of 1967 through 2000. These forecasts are compared to forecasts from an unrestricted vector autoregression, a reduced rank regression model, an index model and a cointegration-based error correction model. The error correction model that takes into account both nonstationarity of the data and rank reduction performs best at all four forecasting horizons. However, differences among competing models are statistically insignificant in most cases. No model consistently encompasses the others at all four horizons.     

Trends and cycles in economic time series: A Bayesian approach

Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be very informative for policy makers, particularly with regard to the size and direction of the output gap and potential turning points. From the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing Markov chain Monte Carlo methods for both univariate and multivariate models. Applications to US macroeconomic series are presented.     

Estimation of some limited dependent variable models with application to housing demand

A model which extends the switching regression models and combines several different limited dependent variable models into a general framework is introduced. Methods to get consistent estimates and asymptotic efficient estimates are derived. Our estimation procedures are then used to study a housing expenditure model which takes into account the simultaneous determination of whether or not to own, and how much to spend.     

Stochastic search variable selection in vector error correction models with an application to a model of the UK macroeconomy

This paper develops methods for stochastic search variable selection (currently popular with regression and vector autoregressive models) for vector error correction models where there are many possible restrictions on the cointegration space. We show how this allows the researcher to begin with a single unrestricted model and either do model selection or model averaging in an automatic and computationally efficient manner. We apply our methods to a large UK macroeconomic model. Copyright © 2011 John Wiley & Sons, Ltd.     

An appraisal of the Box-Jenkins approach to univariate time series analysis

Summary Some comments are made on the current methodology associated with theBox-Jenkins approach. The three strokes of Identification, Estimation and Verification are considered; and a final stage, that of Interpretation, mentioned. A discussion of whether the approach is practicable is also given.     

On a criterion for the selection of models for stationary time series

Summary A general approach to model selection is described and a criterion for the selection of parametric models for stationary Gaussian time series is derived. A corresponding approximate test is also given and experiences in applying the criterion are reported.     

Estimating the codifference function of linear time series models with infinite variance

We consider the codifference and the normalized codifference function as dependence measures for stationary processes. Based on the empirical characteristic function, we propose estimators of the codifference and the normalized codifference function. We show consistency of the proposed estimators, where the underlying model is the ARMA with symmetric α-stable innovations, 0 < α ≤ 2. In addition, we derive their limiting distribution. We present a simulation study showing the dependence of the estimator on certain design parameters. Finally, we provide an empirical example using some stocks from Indonesia Stock Exchange.     

Estimating a frequency unseen: an application to ornithology

The second author is involved in a capture–mark–recapture study of some wader species. Part of his program deals with resight observations. On a particular day he visually inspects a fairly stable population to identify the ringed birds by reading their ring-number. Some ringed birds will be missed, so observations are repeated on other days. The issue of main interest is whether, after some repetitions, we can be sufficiently sure that all the ringed birds in the population have been identified or, equivalently, that the frequency of unseen birds is zero.
Most current theory is concerned with an asymptotic setting. In our 'exact' context the emphasis is on the determination of the 'probability' that the frequency of unseen birds is zero. This issue is settled by considering the more general problem of 'estimating' the frequency of the unseen birds by providing a predictive inference in the form of a probability distribution. We develop methods of inference based on the assumption of a bird-independent probability p _i of identifying a ringed bird on day i , as well as without this assumption. In Section 5 we critically examine these approaches.