Showing papers on "STAR model published in 2004"

Autoregressive Latent Trajectory (ALT) Models A Synthesis of Two Traditions

Abstract: Although there are a variety of statistical methods available for the analysis of longitudinal panel data, two approaches are of particular historical importance: the autoregressive (simplex) model and the latent trajectory (curve) model. These two approaches have been portrayed as competing methodologies such that one approach is superior to the other. We argue that the autoregressive and trajectory models are special cases of a more encompassing model that we call the autoregressive latent trajectory (ALT) model. In this paper we detail the underlying statistical theory and mathematical identification of this model, and demonstrate the ALT model using two empirical data sets. The first reanalyzes a simulated repeated measures data set that was previously used to argue against the autoregressive model, and we illustrate how the ALT model can recover the true latent curve model. Second, we apply the ALT model to real family income data on N=3912 adults over a seven year period and find evidence for both a...

Small Sample Properties of Forecasts from Autoregressive Models Under Structural Breaks

Abstract: This Paper develops a theoretical framework for the analysis of small sample properties of forecasts from general autoregressive models under structural breaks. Finite-sample results for the mean-squared forecast error of one-step-ahead forecasts are derived, both conditionally and unconditionally, and numerical results for different types of break specifications are presented. It is established that forecast errors are unconditionally unbiased even in the presence of breaks in the autoregressive coefficients and/or error variances so long as the unconditional means of the process remains unchanged. Insights from the theoretical analysis are demonstrated in Monte Carlo simulations and on a range of macroeconomic time series from G7 countries. The results are used to draw practical recommendations for the choice of estimation window when forecasting from autoregressive models subject to breaks.

Vector Autoregressive and Vector Error Correction Models

Abstract: Introduction The first step in constructing a model for a specific purpose or for a particular sector of an economy is to decide on the variables to be included in the analysis. At this stage it is usually important to take into account what economic theory has to say about the relations between the variables of interest. Suppose we want to analyze the transmission mechanism of monetary policy. An important relation in that context is the money demand function, which describes the link between the real and the monetary sector of the economy. In this relationship a money stock variable depends on the transactions volume and opportunity costs for holding money. As an example we consider German M3 as the money stock variable, GNP as a proxy for the transactions volume, a long-term interest rate R as an opportunity cost variable, and the inflation rate Dp = Δp, where p denotes the log of the GNP deflator. The latter variable may be regarded as a proxy for expected inflation, which may also be considered an opportunity cost variable. Because the quantity theory suggests a log linear relation, we focus on the variables m = log M3 and gnp = log GNP. Seasonally unadjusted quarterly series for the period 1972–98 are plotted in Figure 3.1. Of course, many more variables are related to the presently considered ones and, hence, could be included in a model for the monetary sector of the economy. However, increasing the number of variables and equations does not generally lead to a better model because doing so makes it more difficult to capture the dynamic, intertemporal relations between them.

Structural Vector Autoregressive Modeling and Impulse Responses

Abstract: Introduction In the previous chapter we have seen how a model for the DGP of a set of economic time series variables can be constructed. When such a model is available, it can be used for analyzing the dynamic interactions between the variables. This kind of analysis is usually done by tracing the effect of an impulse in one of the variables through the system. In other words, an impulse response analysis is performed. Although this is technically straightforward, some problems related to impulse response analysis exist that have been the subject of considerable discussion in the literature. As argued forcefully by Cooley & LeRoy (1985), vector autoregressions have the status of “reduced form” models and therefore are merely vehicles to summarize the dynamic properties of the data. Without reference to a specific economic structure, such reduced-formVAR models are difficult to understand. For example, it is often difficult to draw any conclusion from the large number of coefficient estimates in a VAR system. As long as such parameters are not related to “deep” structural parameters characterizing preferences, technologies, and optimization behavior, the parameters do not have an economic meaning and are subject to the so-called Lucas critique. Sims (1981, 1986), Bernanke (1986), and Shapiro & Watson (1988) put forward a new class of econometric models that is now known as structural vector autoregression (SVAR) or identified VAR . Instead of identifying the (autoregressive) coefficients, identification focuses on the errors of the system, which are interpreted as (linear combinations of) exogenous shocks. In the early applications of Sargent (1978) and Sims (1980), the innovations of the VAR were orthogonalized using a Choleski decomposition of the covariance matrix.

Solution of the two-star model of a network.

Abstract: The $p$-star model or exponential random graph is among the oldest and best known of network models. Here we give an analytic solution for the particular case of the two-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry-broken phase separated from the normal phase of the model by a conventional continuous phase transition.

Instrumental variable estimation of a spatial autoregressive model with autoregressive disturbances: large and small sample results

Abstract: The purpose of this paper is two-fold. First, on a theoretical level we introduce a series-type instrumental variable (IV) estimator of the parameters of a spatial first order autoregressive model with first order autoregressive disturbances. We demonstrate that our estimator is asymptotically efficient within the class of IV estimators, and has a lower computational count than an efficient IV estimator that was introduced by Lee (2003). Second, via Monte Carlo techniques we give small sample results relating to our suggested estimator, the maximum likelihood (ML) estimator, and other IV estimators suggested in the literature. Among other things we find that the ML estimator, both of the asymptotically efficient IV estimators, as well as an IV estimator introduced in Kelejian and Prucha (1998), have quite similar small sample properties. Our results also suggest the use of iterated versions of the IV estimators.

Applied Time Series Econometrics: Smooth Transition Regression Modeling

Abstract: Introduction Nonlinear models have gained a foothold in both macroeconomic and financial modeling. Linear approximations to nonlinear economic phenomena have served macroeconomic modelers well, but in many cases nonlinear specifications have turned out to be useful. Nonlinear econometric models can be divided in two broad categories. The first one contains the models that do not nest a linear model as a special case. Disequilibrium models [e.g., Fair & Jaffee (1972)] are a case in point. The second category embraces several popular models that do nest a linear model. The switching regression model, various Markov-switching models, and the smooth transition regression model are examples of models that belong to this class. Researchers interested in applying them can then choose a linear model as their starting-point and consider nonlinear extensions should they turn out to be necessary. In this chapter, the discussion is centered on modeling of economic time series using the family of smooth transition regression models as a tool. This chapter is organized as follows. The smooth transition regression model is presented in Section 6.2. The modeling cycle, consisting of specification, estimation and evaluation stages, is the topic of Section 6.3. In Section 6.4, the modeling strategy and its application using JMu⌉Ti is illustrated by two empirical examples. Section 6.5 presents some final remarks. The Model The smooth transition regression (STR) model is a nonlinear regression model that may be viewed as a further development of the switching regression model that Quandt (1958) introduced. The univariate version of the switching regression model has long been known as the threshold autoregressive model; for a thorough review, see Tong (1990).

Weak Dependence: Models and Applications to Econometrics

Abstract: In this paper we discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH(∞) processes. For Volterra processes, mixing properties obtain given additional regularity assumptions on the distribution of the innovations.We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.We first thank the editor for the huge amount of additional editorial work provided for this review paper. The efficiency of the numerous referees was especially useful. The error pointed out in Hall and Horowitz (1996) was the origin of the present paper, and we thank the referees for asking for a more detailed treatment of a correct proof for this paper in Section 2.3. Also we thank Marc Henry and Rafal Wojakowski for a very careful rereading of the paper. An anonymous referee has been particularly helpful in the process of revision of the paper. The authors thank him for his numerous suggestions of improvement, including important results on negatively associated sequences and a thorough update in standard English.

Tracking Articulated Motion using a Mixture of Autoregressive Models

Abstract: We present a novel approach to modelling the non-linear and time-varying dynamics of human motion, using statistical methods to capture the characteristic motion patterns that exist in typical human activities. Our method is based on automatically clustering the body pose space into connected regions exhibiting similar dynamical characteristics, modelling the dynamics in each region as a Gaussian autoregressive process. Activities that would require large numbers of exemplars in example based methods are covered by comparatively few motion models. Different regions correspond roughly to different action-fragments and our class inference scheme allows for smooth transitions between these, thus making it useful for activity recognition tasks. The method is used to track activities including walking, running, etc., using a planar 2D body model. Its effectiveness is demonstrated by its success in tracking complicated motions like turns, without any key frames or 3D information.

Spatio-temporal autoregressive models defined over brain manifolds.

Abstract: Multivariate Autoregressive time series models (MAR) are an increasingly used tool for exploring functional connectivity in Neuroimaging. They provide the framework for analyzing the Granger Causality of a given brain region on others. In this article, we shall limit our attention to linear MAR models, in which a set of matrices of autoregressive coefficients Ak (k = 1,...,p) describe the dependence of present values of the image on lagged values of its past. Methods for estimating the Ak and determining which elements that are zero are well-known and are the basis for directed measures of influence. However, to date, MAR models are limited in the number of time series they can handle, forcing the a priori selection of a (small) number of voxels or regions of interest for analysis. This ignores the full spatio-temporal nature of functional brain data which are, in fact, collections of time series sampled over an underlying continuous spatial manifold the brain. A fully spatio-temporal MAR model (ST-MAR) is developed within the framework of functional data analysis. For spatial data, each row of a matrix Ak is the influence field of a given voxel. A Bayesian ST-MAR model is specified in which the influence fields for all voxels are required to vary smoothly over space. This requirement is enforced by penalizing the spatial roughness of the influence fields. This roughness is calculated with a discrete version of the spatial Laplacian operator. A massive reduction in dimensionality of computations is achieved via the singular value decomposition, making an interactive exploration of the model feasible. Use of the model is illustrated with an fMRI time series that was gathered concurrently with EEG in order to analyze the origin of resting brain rhythms.

Some Nonlinear Threshold Autoregressive Time Series Models for Actuarial Use

Abstract: This paper introduces nonlinear threshold time series modeling techniques that actuaries can use in pricing insurance products, analyzing the results of experience studies, and forecasting actuarial assumptions. Basic “self-exciting” threshold autoregressive (SETAR) models, as well as heteroscedastic and multivariate SETAR processes, are discussed. Modeling techniques for each class of models are illustrated through actuarial examples. The methods that are described in this paper have the advantage of being direct and transparent. The sequential and iterative steps of tentative specification, estimation, and diagnostic checking parallel those of the orthodox Box-Jenkins approach for univariate time series analysis.

Linearity tests and stationarity

Abstract: Summary This paper shows that the asymptotic distributions of LM-type linearity tests against Smooth Transition Autoregressive (STAR) models, in the presence of a unit root, are non-standard and using standard χ2 critical values may lead to incorrect inference as the tails of the distribution of tests will be thicker than the χ2. This finding also indicates that one needs to test for stationarity prior to applying linearity tests.

Critical Values of the Empirical F-Distribution for Threshold Autoregressive and Momentum Threshold Autoregressive Models

Abstract: This paper provides exact (finite-sample) test critical values for carrying out tests of no cointegration versus some forms of nonlinear (threshold autoregressive) cointegration. The nonlinear models, which include threshold autoregressive and momentum threshold autoregressive behavior of deviations from long-run equilibrium, are easier to evaluate with the aid of the reported critical values. The results cover a variety of practical situations, with varying sample sizes, lag lengths, and number of time series. JEL Classification Codes: C12, C3, C32

Detecting Mean Reversion in Real Exchange Rates from a Multiple Regime STAR Model

Abstract: Recent studies on general equilibrium models with transaction costs show that the dynamics of the real exchange rate are necessarily nonlinear. Our contribution to the literature on nonlinear price adjustment mechanisms is threefold. First, we model the real exchange rate by a Multi-Regime Logistic Smooth Transition AutoRegression (MR-LSTAR), allowing for both ESTAR-type and SETAR-type dynamics. This choice is motivated by the fact that even the theoretical models, which predict a smooth behavior for the real exchange rate, do not rule out the possibility of a discontinuous adjustment as a limit case. Second, we propose two classes of unit-root tests against this MR-LSTAR alternative, based respectively on the likelihood and on an auxiliary model. Their asymptotic distributions are derived analytically. Third, when applied to 28 bilateral real exchange rates, our tests reject the null hypothesis of a unit root for eleven series bringing evidence in favor of the purchasing power parity.

The live method for generalized additive volatility models

Abstract: We investigate a new separable nonparametric model for time series, which includes many autoregressive conditional heteroskedastic (ARCH) models and autoregressive (AR) models already discussed in the literature We also propose a new estimation procedure called LIVE, or local instrumental variable estimation, that is based on a localization of the classical instrumental variable method Our method has considerable computational advantages over the competing marginal integration or projection method We also consider a more efficient two-step likelihood-based procedure and show that this yields both asymptotic and finite-sample performance gains

An autoregressive model for analysis of ice sheet elevation change time series

Abstract: We present an autoregressive (AR) model that can effectively characterize both seasonal and interannual variations in ice sheet elevation change time series constructed from satellite radar or laser altimeter data. The AR model can be used in conjunction with weighted least squares regression to accurately estimate any longer term linear trend present in the cyclically varying elevation change time series. This approach is robust in that it can account for seasonal and interannual elevation change variations, missing points in the time series, signal aperiodicity, time series heteroscedasticity, and time series with a noninteger number of yearly cycles. In addition, we derive a theoretically valid estimate of the uncertainty (standard error) in the long-term linear trend. Monte Carlo simulations were conducted that closely emulated actual characteristics of five-year elevation change time series from Antarctica. The Monte Carlo results indicate that the autoregressive approach yields long-term linear trends that are less biased than two other approaches that have been recently used for analysis of ice sheet elevation change time series. In addition, the simulation results demonstrate that the variability (uncertainty) of the long-term linear trend estimates from the AR approach is in very good agreement with the derived theoretical standard error estimates.

The live method for generalized additive volatility models

Abstract: We investigate a new separable nonparametric model for time series, which includes many autoregressive conditional heteroskedastic (ARCH) models and autoregressive (AR) models already discussed in the literature. We also propose a new estimation procedure called LIVE, or local instrumental variable estimation, that is based on a localization of the classical instrumental variable method. Our method has considerable computational advantages over the competing marginal integration or projection method. We also consider a more efficient two-step likelihood-based procedure and show that this yields both asymptotic and finite-sample performance gains. This paper is based on Chapter 2 of the first author's Ph.D. dissertation from Yale University. We thank Wolfgang Hardle, Joel Horowitz, Peter Phillips, and Dag Tjostheim for helpful discussions. We are also grateful to Donald Andrews and two anonymous referees for valuable comments. The second author thanks the National Science Foundation and the ESRC for financial support.

Tools for non-linear time series forecasting in economics – an empirical comparison of regime switching vector autoregressive models and recurrent neural networks

Abstract: The purpose of this study is to contrast the forecasting performance of two non-linear models, a regime-switching vector autoregressive model (RS-VAR) and a recurrent neural network (RNN), to that of a linear benchmark VAR model. Our specific forecasting experiment is U.K. inflation and we utilize monthly data from 1969 to 2003. The RS-VAR and the RNN perform approximately on par over both monthly and annual forecast horizons. Both non-linear models perform significantly better than the VAR model.

Karhunen-Loeve Representation of Periodic Second-Order Autoregressive Processes

Abstract: In dynamic data driven applications modeling accurately the uncertainty of various inputs is a key step of the process. In this paper, we first review the basics of the Karhunen-Loeve decomposition as a means for representing stochastic inputs. Then, we derive explicit expressions of one-dimensional covariance kernels associated with periodic spatial second-order autoregressive processes. We also construct numerically those kernels by employing the Karhunen-Loeve expansion and making use of Fourier representation in order to solve efficiently the associated eigenvalue problem. Convergence and accuracy of the numerical procedure are checked by comparing the covariance kernels obtained from the Karhunen-Loeve expansions against theoretical solutions.

Bayesian analysis of random coefficient autoregressive models

Abstract: Random Coefficient AutoRegressive (RCAR) models are obtained by introducing random coefficients to an AR or more generally ARMA model. These models have second order properties similar to that of ARCH and GARCH models. In this article, a Bayesian approach to estimate the first order RCAR models is considered. A couple of Bayesian testing criteria for the unit-root hypothesis are proposed: one is based on the Posterior Interval, and the other one is based on Bayes Factor. In the end, two real life examples involving the daily stock volume transaction data are presented to show the applicability of the proposed methods.

Estimating threshold subset autoregressive moving-average models by genetic algorithms

Abstract: Summary -A genetic algorithm is proposed to estimate the parameters of a selfexciting threshold subset autoregressive moving-average model The threshold model is composed of several linear autoregressive moving-average models Each one of these models applies according to a “switch mechanism” that is based on the comparison between the delayed observation and some “threshold” values Our procedure incorporates the identification in each “regime” of a “subset” model Subset models are useful as they allow the number of parameters to be reduced so that only those really needed are included in the model The proposed procedure is used for modeling the well-known Canadian lynx data

PLP$^2$: Autoregressive modeling of auditory-like 2-D spectro-temporal patterns

Abstract: The temporal trajectories of the spectral energy in auditory critical bands over 250 ms segments are approximated by an all-pole model, the time-domain dual of conventional linear prediction. This quarter-second auditory spectro-temporal pattern is further smoothed by iterative alternation of spectral and temporal all-pole modeling. Just as Perceptual Linear Prediction (PLP) uses an autoregressive model in the frequency domain to estimate peaks in an auditory-like short-term spectral slice, PLP$^2$ uses all-pole modeling in both time and frequency domains to estimate peaks of a two-dimensional spectro-temporal pattern, motivated by considerations of the auditory system.

Linear models, smooth transition autoregressions, and neural networks for forecasting macroeconomic time series: A re-examination

Abstract: In this paper we examine the forecast accuracy of linear autoregressive, smooth transition autoregressive (STAR), and neural network (NN) time series models for 47 monthly macroeconomic variables of the G7 economies. Unlike previous studies that typically consider multiple but fixed model specifications, we use a single but dynamic specification for each model class. The point forecast results indicate that the STAR model generally outperforms linear autoregressive models. It also improves upon several fixed STAR models, demonstrating that careful specification of nonlinear time series models is of crucial importance. The results for neural network models are mixed in the sense that at long forecast horizons, an NN model obtained using Bayesian regularization produces more accurate forecasts than a corresponding model specified using the specific-to-general approach. Reasons for this outcome are discussed.