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The order, (p, q), of an autoregressive-moving average sequence, y(t), may be estimated by minimizing a criterion, log σ2+(p+q)logT/T with respect to p and q, where is the maximum likelihood estimate of the variance of the innovations, ε(t). It is suggested that, instead, be estimated from a series of regressions of y(t) on y(t-1),y(t-2),...,y(t-p),e{open}(t-1),...e{open}(t-q), where the (t) are obtained by fitting a long autoregression to the data. It is shown how the sequence of regressions may, for p = q, be economically recursively calculated by embedding them in a sequence of bivariate autoregressions. Asymptotic properties of the procedure are established under very general conditions.

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... As a result, an appropriately large order is essential in Durbin's algorithm to ensure parameter accuracy and optimal forecasts, as well as improving computational complexity. The use of model selection criteria, such as the Akaike Information Criteria (AIC) Akaike (1974), has been suggested to aide in the selection of an appropriately large order AR model by Hannan and Rissanen (1982) and Broersen (1996). ...

... The steps of Durbin's Algorithm are depicted in Algorithm 1. Hannan and Rissanen (1982) extended this algorithm with trimming steps to improve the initial parameter estimates, as well as suggesting the use the Bayesian Information Criterion (BIC) as a model selection criterion. ...

... Durbin's methodology, Algorithm 1, of exploiting the asymptotic equivalence between AR(∞) and MA(q) models can similarly be used to estimate the coefficients of ARMA models. The optimal orderp can also be chosen using model selection criteria such as the BIC utilised by Hannan and Rissanen Hannan and Rissanen (1982) and the GIC utilised by Broersen (1996Broersen ( , 2000 as seen in Section 2.2. Durbin's methodology, Algorithm 1, along with the BIC and GIC criteria will be used as the estimation method for ARMA models for the remainder of this paper, and will be compared empirically in Section 4 to the new algorithm, called Rollage, developed in Section 3. ...

We develop a new method to estimate an ARMA model in the presence of big time series data. Using the concept of a rolling average, we develop a new efficient algorithm, called Rollage, to estimate the order of an AR model and subsequently fit the model. When used in conjunction with an existing methodology, specifically Durbin's algorithm, we show that our proposed method can be used as a criterion to optimally fit ARMA models. Empirical results on large-scale synthetic time series data support the theoretical results and reveal the efficacy of this new approach, especially when compared to existing methodology.

... ARMA models estimation has a very long history [1,2,5,9,12,14,15,26]. Maximum likelihood estimation is usually performed for its advantageous asymptotic properties. ...

... ARMA models are, in practice, fitted by iterative optimization algorithms that start at preliminary estimates obtained, for example, with the well-known Hannan and Rissanen (HR) method [14]. We consider this setting to carry out the experiment, in order to assess the impact of the regularization term in the common use cases. ...

... The null-hypothesis, states that all the fitting methods are equivalent and so their ranks should be equal. Table 8 reports the average of ranks over all the time series in our dataset, w.r.t. the metrics of interest (13) and (14). ...

In this paper we propose a new optimization model for maximum likelihood estimation of causal and invertible ARMA models. Through a set of numerical experiments we show how our proposed model outperforms, both in terms of quality of the fitted model as well as in the computational time, the classical estimation procedure based on Jones reparametrization. We also propose a regularization term in the model and we show how this addition improves the out of sample quality of the fitted model. This improvement is achieved thanks to an increased penalty on models close to the non causality or non invertibility boundary.

... In this article, we develop a method for estimating the parameters of ARMA model with symmetric stable noise and symmetric stable GARCH noise by modifying the Hannan-Rissanen Method (HR) [8]. For the estimation of the symmetric stable GARCH parameters, we use the model discussed in [15] as opposed to the standard GARCH model with symmetric stable noise and tempered stable noise used by Calzolari et al. [3] and Feng and Shi [5], respectively. ...

... The classical Hannan-Rissanen method [8] uses linear regression to establish estimates for the parameters and the white noise variance of an ARMA(p, q) process. However, to establish estimates for processes with infinite variance, we modify the Hannan-Rissanen method [8]. ...

... The classical Hannan-Rissanen method [8] uses linear regression to establish estimates for the parameters and the white noise variance of an ARMA(p, q) process. However, to establish estimates for processes with infinite variance, we modify the Hannan-Rissanen method [8]. Let {X t } be defined as in (2) or (4). ...

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.

... On the other hand, a three-step method was designed for ARMA model fitting, including the estimation of the autoregressive and moving average orders (p, q) (Hannan and Rissanen, 1982;Hannan and Kavalieris, 1984). The three steps are as follows: ...

... This procedure can be costly when a wide grid of (p, q) values is tested. In the scenario where p = q, an efficient algorithm that recursively computes the sequence of ARMA(p, q) regressions has been proposed (Hannan and Rissanen, 1982). ...

This paper surveys state-of-the-art methods and models dedicated to time series analysis and modeling, with the final aim of prediction. This review aims to offer a structured and comprehensive view of the full process flow, and encompasses time series decomposition, stationary tests, modeling and forecasting. Besides, to meet didactic purposes, a unified presentation has been adopted throughout this survey, to present decomposition frameworks on the one hand and linear and nonlinear time series models on the other hand. First, we decrypt the relationships between stationarity and linearity, and further examine the main classes of methods used to test for weak stationarity. Next, the main frameworks for time series decomposition are presented in a unified way: depending on the time series, a more or less complex decomposition scheme seeks to obtain nonstationary effects (the deterministic components) and a remaining stochastic component. An appropriate modeling of the latter is a critical step to guarantee prediction accuracy. We then present three popular linear models, together with two more flexible variants of the latter. A step further in model complexity, and still in a unified way, we present five major nonlinear models used for time series. Amongst nonlinear models, artificial neural networks hold a place apart as deep learning has recently gained considerable attention. A whole section is therefore dedicated to time series forecasting relying on deep learning approaches. A final section provides a list of R and Python implementations for the methods, models and tests presented throughout this review. In this document, our intention is to bring sufficient in-depth knowledge, while covering a broad range of models and forecasting methods: this compilation spans from well-established conventional approaches to more recent adaptations of deep learning to time series forecasting.

... Thanks to the equations (17)- (19), the regressors of the extended relationship (15) are fully specified (15) ...

... See [32]. For example, a number of authors: [18,19,39,45], among others, propose to identify the orders (p * , q * ) by using a sequence of linear regressions of L t on lagged variables ...

This paper deals especially with a two-stage approach to forecasting hourly electricity demand by using a linear regression model with serially correlated residuals. Firstly, ordinary least squares are applied to estimate a linear regression model based on purely deterministic predictors (essentially, polynomials in time and calendar dummy variables). In the case wherein the regression residuals are not a white noise series, a SARMA (seasonal autoregressive moving average) process is applied to the estimated regression residuals. After examining a vast set of potential representations, the stationary and invertible process associated with the smaller Akaike information criterion and the smaller Ljung–Box statistic is selected. Secondly, two sets of instrumental predictors are added to the current model: the estimated residuals of the first regression model plus the estimated errors of the chosen SARMA process. The new regression model is estimated by again using ordinary least squares, but taking advantage of the fact that the new regressors eliminate serial correlation. Practical issues in points and interval forecasting are illustrated with reference to nine-day ahead prediction performance for short-term electric loads in Italy.

... The general task of obtaining initial estimates that are accurate and fast to compute for a nonlinear optimization algorithm is referred to as the initialization problem. Hannan and Rissanen (1982) proposed first estimating time series residuals via a long VAR, and regressing the data upon the lagged data, and estimated residuals to get VARMA estimates. [The extension of the original univariate method to the VARMA case was first studied in Kavalieris (1984, 1986).] ...

... [Although attributed to Hannan and Rissanen (1982), examination of Durbin (1960, equation (25)) shows the same methodology.] It is very fast to compute, only requiring two ordinary least squares (OLS) multivariate regressions. ...

A new method for the estimation of a vector moving average (VMA) process is presented. The technique uses Kullback–Leibler discrepancy with inverse spectra, and yields a Yule–Walker system of equations in inverse autocovariances for the VMA coefficients. This provides a direct formula for the coefficients, which always results in a stable matrix polynomial. The paper provides asymptotic results, as well as an analysis of the method's performance, in terms of speed, bias, and precision. Applications to preliminary estimation of VMA models are discussed, and the method is illustrated on retail data.

... For shorter lengths of time series data, it is prudent to use a hierarchical time series model. It is claimed that the hierarchical times series model can detect outbreaks faster than the lab based exceedance system [29]. ...

... Augmented Dickey Fuller test was conducted to establish that Russian series was stationary in level while Brazil was integrated in the first order and the remaining three series namely India, Spain and US were integrated in second order. The model specification determined by Hannan Rissanen algorithm [29] was India (4,2,4), Brazil (3,1,2), Russia (3,0,0), Spain (4,2,4) and US (1,2,1) respectively. The residuals of the ARIMA series were plotted and found to be stationary. ...

Background and aims
In a little over six months, the Corona virus epidemic has affected over ten million and killed over half a million people worldwide as on June 30, 2020. With no vaccine in sight, the spread of the virus is likely to continue unabated. This article aims to analyze the time series data for top five countries affected by the COVID-19 for forecasting the spread of the epidemic.
Material and methods
Daily time series data from 15th February to June 30, 2020 of total infected cases from the top five countries namely US, Brazil, India, Russia and Spain were collected from the online database. ARIMA model specifications were estimated using Hannan and Rissanen algorithm. Out of sample forecast for the next 77 days was computed using the ARIMA models.
Results
Forecast for the first 18 days of July was compared with the actual data and the forecast accuracy was using MAD and MAPE were found within acceptable agreement. The graphic plots of forecast data suggest that While Russia and Spain have reached the inflexion point in the spread of epidemic, the US, Brazil and India are still experiencing an exponential curve.
Conclusion
Our analysis shows that India and Brazil will hit 1.38 million and 2.47 million mark while the US will reach the 4.29 million mark by 31st July. With no effective cure available at the moment, this forecast will help the governments to be better prepared to combat the epidemic by ramping up their healthcare facilities.

... In this article, we develop a method for estimating the parameters of ARMA model with symmetric stable noise and symmetric stable GARCH noise by modifying the Hannan-Rissanen Method [11]. For the estimation of the GARCH parameters, we propose the modified empirical characteristic function method which is based on the method discussed in [6]. ...

... We propose two methods for the estimation of the parameters φ, θ, c, a, b, a G and b G for the processes defined in (2), (3) and (4). For estimation of φ and θ, we suitably modify the Hannen-Rissanen method [11] and for estimating c, a, b, a G and b G we modify the empirical characteristic function method discussed in [6]. ...

In this article, we first propose the modified Hannan-Rissanen Method for estimating the parameters of au-toregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods through Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.

... In this article, we develop a method for estimating the parameters of ARMA model with symmetric stable noise and symmetric stable GARCH noise by modifying the Hannan-Rissanen Method [11]. For the estimation of the GARCH parameters, we propose the modified empirical characteristic function method which is based on the method discussed in [6]. ...

... We propose two methods for the estimation of the parameters φ, θ, c, a, b, a G and b G for the processes defined in (2), (3) and (4). For estimation of φ and θ, we suitably modify the Hannen-Rissanen method [11] and for estimating c, a, b, a G and b G we modify the empirical characteristic function method discussed in [6]. ...

In this article, we first propose the modified Hannan-Rissanen Method for estimating the parameters of au-toregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods through Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.

... To design upgraded and enhanced functions of the system for the early warning of adverse disease events, we survey many suitable algorithms for syndromic surveillance, and analyze and evaluate some algorithms. According to our survey, the CUmulative SUM (CUSUM) [6,7] and the Early Aberration Reporting System (EARS) [7] statistical detection approaches and the autoregressive integrated moving average (ARIMA) [8] and Holt-Winters [9,10] time series forecasting approaches have been used in a large number of papers and applications. ...

Early detection of infectious disease outbreaks is one of the important and significant issues in syndromic surveillance systems. It helps to provide a rapid epidemiological response and reduce morbidity and mortality. In order to upgrade the current system at the Korea Centers for Disease Control and Prevention (KCDC), a comparative study of state-of-the-art techniques is required. We compared four different temporal outbreak detection algorithms: the CUmulative SUM (CUSUM), the Early Aberration Reporting System (EARS), the autoregressive integrated moving average (ARIMA), and the Holt-Winters algorithm. The comparison was performed based on not only 42 different time series generated taking into account trends, seasonality, and randomly occurring outbreaks, but also real-world daily and weekly data related to diarrhea infection. The algorithms were evaluated using different metrics. These were namely, sensitivity, specificity, positive predictive value (PPV), negative predictive value (NPV), F1 score, symmetric mean absolute percent error (sMAPE), root-mean-square error (RMSE), and mean absolute deviation (MAD). Although the comparison results showed better performance for the EARS C3 method with respect to the other algorithms, despite the characteristics of the underlying time series data, Holt⁻Winters showed better performance when the baseline frequency and the dispersion parameter values were both less than 1.5 and 2, respectively.

... Based on a subjective judgement, these plots could be very useful [4]. Pattern identification tools could include the smallest canonical correlation table (SCAN) [24], the extended sample autocorrelation function table (ESACF) [23] and the minimum information criterion table (MINIC) [12]. ...

Time series model selection has been widely studied in recent years. It is of importance to select the best model among candidate models proposed for a series in terms of explaining the procedure that governs the series and providing the most accurate forecast for the future observations. In this study, it is aimed to create an algorithm for order selection in Box–Jenkins models that combines penalized natural logarithm of mutual information among the original series and predictions coming from each candidate. The penalization is achieved by subtracting the number of parameters in each candidate and empirical information the data provide.Simulation studies under various scenarios and applications on real data sets imply that our algorithm offers a promising and satisfactory alternative to its counterparts.

... ARIMA (Hannan and Rissanen 1982): We use ARIMA model with Kalman filter which is widely used in time series forecasting. ...

Recent improvement and availability of remote satellite and IoT data offers interesting and diverse applications of artificial intelligence in precision agriculture. Soil moisture is an important component of multiple agricultural and food supply chain practices. It measures the amount of water stored in various depth of soil. Existing data driven approaches for soil moisture prediction use conventional models which fail to capture the dynamic dependency of soil moisture values in near-by locations over time. In this work, we propose to convert the problem of soil moisture prediction as a semi-supervised learning on temporal graphs. We propose a dynamic graph neural network which can use the dependency of related locations over a region to predict soil moisture. However, unlike social or information networks, graph structure is not explicitly given for soil moisture prediction. Hence, we incorporate the problem of graph structure learning in the framework of dynamic GNN. Our algorithm, referred as DGLR, provides an end-to-end learning which can predict soil moisture over multiple locations in a region over time and also update the graph structure in between. Our solution achieves state-of-the-art results on real-world soil moisture datasets compared to existing machine learning approaches.

... In Stan, an incorrect order selection might be considered an ill model, producing multiple divergent transitions. Several procedures for automatic order selection have been proposed Tsay (2010), Hannan and Rissanen (1982) and Gomez (1998). A Bayesian version of Hyndman, Athanasopoulos, Bergmeir, Caceres, Chhay, O'Hara-Wild, Petropoulos, Razbash, Wang, and Yasmeen (2020) algorithm implemented in their forecast (Hyndman and Khandakar 2008) package is proposed. ...

varstan is an R package (R Core Team 2017) for Bayesian analysis of time series models using Stan (Stan, Development. Team 2017). The package offers a dynamic way to choose a model, define priors in a wide range of distributions, check model's fit, and forecast with the m-steps ahead predictive distribution. The users can widely choose between implemented models such as multiplicative seasonal ARIMA, dynamic regression, random walks, GARCH, dynamic harmonic regressions,VARMA, stochastic Volatility Models , and generalized t-student with unknown degree freedom GARCH models. Every model constructor in varstan defines weakly informative priors, but prior specifications can be changed in a dynamic and flexible way, so the prior distributions reflect the parameter's initial beliefs. For model selection, the package offers the classical information criteria: AIC, AICc, BIC, DIC, Bayes factor. And more recent criteria such as Widely-applicable information criteria (WAIC), and the Bayesian leave one out cross-validation (loo). In addition, a Bayesian version for automatic order selection in seasonal ARIMA and dynamic regression models can be used as an initial step for the time series analysis.

... Reported in [29], it suggested to use forgetting factor that is slightly less than one (0 < (l) ≤ 1) . In addition, [15] suggested that the estimation of this model is solely age weighted algorithm where the data on the next period of time, t, has a weight . For the quarterly data in macroeconomics, however, [20] mentioned that the forgetting factor should be set to = 0.99 in which technically implies that the observations five years ago receive approximately 80% as much weight from the last period's observation. ...

Economic and financial data is extremely volatile relative to the others especially in the time series data. Foreign Exchange market or forex data is one among the others. Despite the fact that it is still impossible to guarantee the profits from trading using advanced mathematical model for this time series, but the predictive performance via the mean squares forecasting error and mean absolute forecasting error obtained from new techniques recently is very much improved. In this paper, we apply vector autoregression (VAR) to forecast most traded forex pairs according to dailyfx.com. The parameters obtained in our work are extraordinary where we use the algorithm so-called dynamic model averaging (DMA) and dynamic model selection (DMS) to deal with the model uncertainty. This algorithm is based closely on Kalman Filtering which has a huge advantage when compared with Markov chain Monte Carlo method. We are able to perform up to 27 variables in VAR and set the lag of 14 periods. We forecast EUR-USD, GBP-USD and EUR-JPY with the number of horizon from \(h=1\) to \(h=14\) or one-day-ahead through fourteen-day-ahead prediction where each predicted value is obtained via iterated forecasting method. The findings in this work are: first, DMA algorithm, the Large-VAR with time-varying parameters perform well in predicting GBP-USD. EUR-USD and EUR-JPY. In addition, DMS, an algorithm that selects the highest model probability in Bayesian model averaging obviously outperforms DMA method. Secondly, by having algorithm which is able to track the degree of changes in each dimension for VAR via forgetting factors, the larger matrix in data usage results in more time variation degree in VAR parameters. Furthermore, we also illustrate the dynamic Minnesota prior for each point in time. We finally found that the proposed method delivered excellent predictive performance for large time-varying parameter VAR up to 27 variables.

... 1. Identify the VARMA model with an iterative likelihood-ratio test procedure (see Gómez [12] for an analytical description). 2. Carry out a preliminary estimation of the parameters by the Hannan-Rissanen method (see Gómez [12] and Hannan et al. [22]). 3. Refine the estimation using the conditional method described by Lütkepohl ( see [23]), Reinsel [24]) and Gómez (see [12,14] for the implementation). ...

Normally, econometric models that forecast the Italian Industrial Production Index do not exploit information already available at time t + 1 for their own main industry groupings. The new strategy proposed here uses state–space models and aggregates the estimates to obtain improved results. The performance of disaggregated models is compared at the same time with a popular benchmark model, a univariate model tailored on the whole index, with persistent not formally registered holidays, a vector autoregressive moving average model exploiting all information published on the web for main industry groupings. Tests for superior predictive ability confirm the supremacy of the aggregated forecasts over three steps horizon using absolute forecast error and quadratic forecast error as a loss function. The datasets are available online.

... where α i = 0 for i > r and β i = 0 for i > q, and the innovation ν t = y 2 t − σ 2 t is a white noise (not i.i.d. in general) and identically distributed under the strict stationary assumption of y t . According to Hannan and Rissanen (1982), the LS estimators for an ARMA model are obtained as follows: (a) Fit a high order autoregressive model of order m, AR(m), with m > max( p, q), to the data by Yule-Walker method to obtain ...

In this paper, we adapt sufficient and ordered non-overlapping block bootsrap methods into jackknife-after-bootstrap (JaB) algorithm to estimate the standard error of a statistic where observations form a stationary sequence. We also extend the JaB algorithm to obtain prediction intervals for future returns and volatilities of GARCH processes. The finite sample properties of the proposed methods are illustrated by an extensive simulation study and they are applied to S&P 500 stock index data. Our findings reveal that the proposed algorithm often exhibits improved performance and, is computationally more efficient compared to conventional JaB method.

... According to [18], the LS estimators of an ARMA model are obtained as follows: (a) First, a high order autoregressive model of order m, AR(m), with m > max(p, q), is fitted to the data by Yule-Walker method to obtain ν t , where m is determined from the data by using Akaike information criteria or Bayesian information criteria. (b) Then a linear regression of y 2 t onto y 2 t−1 , . . . ...

In this paper, we propose a new resampling algorithm based on block bootstrap to
obtain prediction intervals for future returns and volatilities of GARCH processes.
The finite sample properties of the proposed methods are illustrated by an extensive
simulation study and they are applied to Japan Yen (JPY) / U.S. dollar (USD) daily
exchange rate data. Our results indicate that: (i) the proposed algorithm is a good
competitor or even better and (ii) computationally more efficient than traditional
method(s).

... Pentru procese pure AR, o bunǎ estimaţie preliminarǎ a parametrilor modelului poate fi obţinutǎ cu algoritmul Whittle, sau cu versiunea multivariabilǎ a algoritmului Burg (Jones, 1978). Alţi algoritmi utili pot fi gǎsiţiîn lucrarea lui Lütkepohl (1991),în particular o metodǎ de tipul celor mai mici pǎtrate condiţionaleşi metoda propusǎ de Hannanşi Rissanen (1982), ultima fiind foarte utilǎ pentru estimarea preliminarǎ a parametrilor modelelor ARMA(p, q). Metodele spectrale de estimare a parametrilor proceselor multivariabile ARMA sunt, de asemenea, frecvent utilizateîn acest scop (Anderson, 1980). ...

... Time series forecasting has been always challenging task for every researchers and forecast experts due to its uncertainty property. In order to solve the problem of time series prediction, the models such as ARIMA model (Hannan and Rissanen, 1982), exponential smoothing method, state space model and Kalman filter model have been used earlier (Morrison and Pike, 1977). However these models hardly fit nonlinear data. ...

ABSTRACT
The values of some time series in the real world usually change randomly but they may contain information from history. In these cases, today value can depend not only on yes- terday value but also on further values in history. Hence, a forecast model which takes the information from two or three days ago to predict today value can give a more accurate prediction. This paper presents a novel higher Markov model for time series forecasting where the state space of the Markov chain was contructed from different levels of changes of the time series. Once the transition matrix has been calculated based on the fuzzy sets, the mean of the levels of changes along with transition probabilities allow caculating the data for forecast values. The experiment with different data shows a significantly improved accuracy compared to other previous models such as ARIMA, ANN , HMM-based models and combined HMM-Fuzzy models.

... Following the procedure suggested by Hannan and Rissanen (1982) to identify a suitable ARIMA model for the regression residuals, we arrive to two structural ARIMA (1,1,1) models 1 . The first, based on equation (1), with two explanatory variables. ...

In recent decades there has been a growing literature dealing with the empirical estimation of the rate of profit and other Marxian variables in several countries. Nonetheless, there has been a paucity of econometric research about the impact of those Marxian variables on the growth rate in developing countries. This paper seeks to evaluate the rate of profit and the rate of accumulation as determinants of the growth rate in Colombia during 1967-2019, using a VAR model. We find that both variables are statistically significant and, in concordance with Marxian theory predictions, affect positively the growth rate. We also identify direct impacts of growth rate over the profit rate and the accumulation rate as well as an inverse relationship between these last variables.

... In Stan, an incorrect order selection might be considered an ill model, producing multiple divergent transitions. Several procedures for automatic order selection have been proposed Tsay (2010), Hannan and Rissanen (1982) and Gomez (1998). A Bayesian version of Hyndman, Athanasopoulos, Bergmeir, Caceres, Chhay, O'Hara-Wild, Petropoulos, Razbash, Wang, and Yasmeen (2020) algorithm implemented in their forecast (Hyndman and Khandakar 2008) package is proposed. ...

Normality is the main assumption for analyzing dependent data in several time series models, and tests of normality have been widely studied in the literature, however, the implementations of these tests are limited. The \textbf{nortsTest} package performs the tests of \textit{Lobato and Velasco, Epps, Psaradakis and Vavra} and \textit{random projection} for normality of stationary processes. In addition, the package offers visual diagnostics for checking stationarity and normality assumptions for the most used time series models in several \R packages. The aim of this work is to show the functionality of the package, presenting each test performance with simulated examples, and the package utility for model diagnostic in time series analysis.

... The algorithm used is a simplified version of Hyndman and Khandakar (2008) consisting on searching exclusively on stationary and non-seasonal models, since both non-stationarity and seasonality are assumed already captured by the trend and seasonal components. The procedure uses standard regression to increase speed by taking advantage of the linear approximation proposed by Hannan and Rissanen (1982), by which an initial noise estimation is obtained by fitting a long autoregressive model to the innovations identified by BIC. ...

Automatic identification of time series models is a necessity once the big data era has come and is staying among us. This has become obvious for many companies and public entities that have passed from a crafted analysis of each individual problem to handle a tsunami of information that has to be processed efficiently, online and in record time. Automatic identification tools have never been tried out on Unobserved Components models (UC). This chapter shows how information criteria, such as Akaike’s or Schwarz’s, are rather useful for model selection within the UC family. The difficulty lies, however, on choosing an appropriate and as general as possible set of models to search in. A set too narrow would render poor forecast accuracy, while a set too wide would be highly time consuming. The forecasting results suggest that UC models are powerful potential forecasting competitors to other well-known methods. Though there are several pieces of software available for UC modeling, this is the first implementation of an automatic algorithm for this class of models, to the best of the author’s knowledge.

... In the last three decades several attempts have been made to automate the modeling of ARIMA processes. Hannan and Rissanen (1982) proposed a method to identify the order of an ARIMA model for a series of stationary time. In his method, innovations can be obtained by adjusting a long autoregressive model for the data, and later the likelihood of the potential models is calculated through a series of standard regressions. ...

The present research aims to provide a relevant algorithm for the proper identification of a data generating process (economic and financial variables) by using the "auto.arima" command, belonging to the forecast library (R statistical software), to identify the optimal parameters of the ARIMA model. Considering the methodological framework of Box and Jenkins process (1970), it seeks to achieve, by solving a stochastic difference equation, the correct calibration and obtaining an optimal forecast for the Mexican Monetary Base, one of the main economic variabes.

... The vectors are, n = (Φ n , Θ n ) = ( X , … , p , X , … , q ) = ( X , … , pWq ) (5) where is the order of model, is the parameter and is the parameter. Basing on the n and [18], we define a multivariate equation ( n ) as, ...

The Covid-19 pandemic is still spreading around the world and seriously imperils humankind's health. This swift spread has caused the public to panic and look to scientists for answers. Fortunately, these scientists already have a wealth of datathe Covid-19 reports that each country releases, reports with valuable spatial-temporal properties. These data point toward some key actions that humans can take in their fight against Covid-19. Technically, the Covid-19 records can be described as sequences, which represent spatial-temporal linkages among the data elements with graph structure. Therefore, we propose a novel framework, the Interaction-Temporal Graph Convolution Network (IT-GCN), to analyze pandemic data. Specifically, IT-GCN introduces ARIMA into GCN to model the data which originate on nodes in a graph, indicating the severity of the pandemic in different cities. Instead of regular spatial topology, we construct the graph nodes with the vectors via ARIMA parameterization to find out the interaction topology underlying in the pandemic data. Experimental results show that IT-GCN is able to capture the comprehensive interaction-temporal topology and achieve well-performed short-term prediction of the Covid-19 daily infected cases in the United States. Our framework outperforms state-of-art baselines in terms of MAE, RMSE and MAPE. We believe that IT-GCN is a valid and reasonable method to forecast the Covid-19 daily infected cases and other related time-series. Moreover, the prediction can assist in improving containment policies.

... Hannan and Rissanen [11] suggested a method to recognise the order of an ARMA model for a stationary series by fitting an extended autoregressive model and the likelihood of potential models is computed via a series of standard regressions. The Hannan-Rissanen identification method was further extended by Gomez [8] to include multiplicative seasonal ARIMA model identification. ...

Automated forecasting is essential to business operations that handle scores of univariate time series. Practitioners have to deal with thousands of time series with a periodicity ranging from seconds to monthly. The sheer velocity and volume of time series make it challenging for human labour to manually identify the order of the time series to forecast the results. An automated forecasting algorithm or framework is essential to complete the task. The approach must be robust in the identification of the order of the time series, and readily applicable to scores of time series without manual intervention. The most modern automated forecasting algorithms are derived from exponential smoothing or ARIMA models. In this paper, the authors proposed a new heuristics approach to identify the initial starting point for a neighbourhood search to obtain the most appropriate model. The results of this method are used to compare against the methods proposed in the literature.

... In this section we show that there is virtually no loss incurred in using our supLM tests when no previous information on the order is available, provided a proper model selection procedure is adopted. We advocate the use of the consistent ARMA order selection proposed in Hannan and Rissanen (1982) (see also Choi (1992)). ...

We present supremum Lagrange Multiplier tests to compare a linear ARMA specification against its threshold ARMA extension. We derive the asymptotic distribution of the test statistics both under the null hypothesis and contiguous local alternatives. Moreover, we prove the consistency of the tests. The Monte Carlo study shows that the tests enjoy good finite-sample properties, are robust against model mis-specification and their performance is not affected if the order of the model is unknown. The tests present a low computational burden and do not suffer from some of the drawbacks that affect the quasi-likelihood ratio setting. Lastly, we apply our tests to a time series of standardized tree-ring growth indexes and this can lead to new research in climate studies.

... Applying the procedure of Hannan-Rissanen (1982) will provide an opportunity for the information criteria to choose a plausible model. models with less parameters, so it eventually selects specification with 5 autoregressive terms. ...

The multiplier as a macroeconomic variable represents a linking device between the fiscal impulses and the real economy. The dynamics of the Keynesian expenditure multiplier in Bulgaria confirms that no single value of the multiplier exists. Its magnitude depends on various factors which impedes the forecasting of the exact multiplier value at a given moment as well as the fiscal policy effects on the economy. In line with this aim the econometric evaluation of the multiplier time series might reveal some of its properties leading to the description of its future dynamics.

... Decades ago, universal coding has evolved into the so-called universal modeling, which is no longer restricted to how to encode data but rather to pursue optimal models, above all an optimal universal model. Distill these thinkings, a universal modeling principle, the MDL for statistical inference, then, generalizes the older idea of parameter estimator in statistics [10,24,27], and it incorporates the model complexity which affects all aspects of model performance into its coding scheme [29]. ...

Unified Granger causality analysis (uGCA) alters conventional two-stage Granger causality analysis into a unified code-length guided framework. We have presented several forms of uGCA methods to investigate causal connectivities, and different forms of uGCA have their own characteristics, which capable of approaching the ground truth networks well in their suitable contexts. In this paper, we considered comparing these several forms of uGCA in detail, then recommend a relatively more robust uGCA method among them, uGCA-NML, to reply to more general scenarios. Then, we clarified the distinguished advantages of uGCA-NML in a synthetic 6-node network. Moreover, uGCA-NML presented its good robustness in mental arithmetic experiments, which identified a stable similarity among causal networks under visual/auditory stimulus. Whereas, due to its commendable stability and accuracy, uGCA-NML will be a prior choice in this unified causal investigation paradigm.

... Neural stochastic differential equations Neural stochastic differential equations offer a shift in this paradigm. By parameterising the drift and diffusion of an SDE as neural networks, then modelling (Generative) time series models are of classical interest, with forecasting models such as Holt-Winters [17,18], ARMA [19] and so on. It has also attracted much recent interest with (besides Neural SDEs) the development of models such as Time Series GAN [20], Latent ODEs [21], GRU-ODE-Bayes [22], ODE 2 VAE [23], CTFPs [24], Neural ODE Processes [25] and Neural Jump ODEs [26]. ...

Neural SDEs combine many of the best qualities of both RNNs and SDEs, and as such are a natural choice for modelling many types of temporal dynamics. They offer memory efficiency, high-capacity function approximation, and strong priors on model space. Neural SDEs may be trained as VAEs or as GANs; in either case it is necessary to backpropagate through the SDE solve. In particular this may be done by constructing a backwards-in-time SDE whose solution is the desired parameter gradients. However, this has previously suffered from severe speed and accuracy issues, due to high computational complexity, numerical errors in the SDE solve, and the cost of reconstructing Brownian motion. Here, we make several technical innovations to overcome these issues. First, we introduce the reversible Heun method: a new SDE solver that is algebraically reversible -- which reduces numerical gradient errors to almost zero, improving several test metrics by substantial margins over state-of-the-art. Moreover it requires half as many function evaluations as comparable solvers, giving up to a $1.98\times$ speedup. Next, we introduce the Brownian interval. This is a new and computationally efficient way of exactly sampling and reconstructing Brownian motion; this is in contrast to previous reconstruction techniques that are both approximate and relatively slow. This gives up to a $10.6\times$ speed improvement over previous techniques. After that, when specifically training Neural SDEs as GANs (Kidger et al. 2021), we demonstrate how SDE-GANs may be trained through careful weight clipping and choice of activation function. This reduces computational cost (giving up to a $1.87\times$ speedup), and removes the truncation errors of the double adjoint required for gradient penalty, substantially improving several test metrics. Altogether these techniques offer substantial improvements over the state-of-the-art.

... Decades ago, universal coding has evolved into the socalled universal modeling, which is no longer restricted to how to encode data but rather to pursue optimal models, above all an optimal universal model. Distill these thinkings, a universal modeling principle, the MDL for statistical inference, then, generalizes the older idea of parameter estimator in statistics [12][13][14], and it incorporates the model complexity which affects all aspects of model performance into its coding scheme [8]. ...

Unified Granger causality analysis (uGCA) alters conventional two-stage Granger causality analysis into a unified code-length guided framework. We have presented several forms of uGCA methods to investigate causal connectivities, and different forms of uGCA have their own characteristics, which capable of approaching the ground truth networks well in their suitable contexts. In this paper, we considered comparing these several forms of uGCA in detail, then recommend a relatively more robust uGCA method among them, uGCA-NML, to reply to more general scenarios. Then, we clarified the distinguished advantages of uGCA-NML in a synthetic 6-node network. Moreover, uGCA-NML presented its good robustness in mental arithmetic experiments, which identified a stable similarity among causal networks under visual/auditory stimulus. Whereas, due to its commendable stability and accuracy, uGCA-NML will be a prior choice in this unified causal investigation paradigm.

... The authors obtained different optimal ARIMA models for each country: (4,2,4), (3,1,2), (3,0,0), (4,2,4), and (1,2,1), respectively. Model specifications were estimated using Hannan and Rissanen algorithm [6]. The data for the study were taken from February 15 to June 30, 2020, for training and July 1 to July 18, 2020, for testing. ...

Forecasting the spread of COVID-19 infection is an important aspect of public health management. In this paper, we propose an approach to forecasting the spread of the pandemic based on the vector autoregressive model. Concretely, we combine the time series for the number of new cases and the number of new deaths to obtain a joint forecasting model. We apply the proposed model to forecast the number of new cases and deaths in the UAE, Saudi Arabia, and Kuwait. Test results based on out-of-sample forecast show that the proposed model achieves a high level of accuracy that is superior to many existing methods. Concretely, our model achieves mean absolute percentage error (MAPE) of 0.35%, 2.03%, and 3.75% in predicting the number of daily new cases for the three countries, respectively. Furthermore, interpolating our predictions to forecast the cumulative number of cases, we obtain MAPE of 0.0017%, 0.002%, and 0.024%, respectively. The strong performance of the proposed approach indicates that it could be a valuable tool in managing the pandemic.

... For noises other than Gaussian noises, the calculation of the MLE is generally more complicated in practice and requires the use of algorithms such as MCMC, EM, Stochastic Approximation of EM (SAEM), SMC, or alternative estimation strategies (see [1], [10] and [48]), which require a longer computation time. In the case of Laplace's noise, we use the R package tseries [54] to fit an ARMA(1,1) model to the Y i observations by a conditional least squares method [30]. Moreover, it is important to note that even in the most favourable case for calculating the MLE, our approach is faster than the Kalman filter as it only requires the minimization of an explicitly known contrast function as opposed to the MLE where the Kalman filter is used to construct the likelihood of the model to be maximized. ...

This paper develops a simple and computationally efficient parametric approach to the estimation of general hidden Markov models (HMMs). For non-Gaussian HMMs, the computation of the maximum likelihood estimator (MLE) involves a high-dimensional integral that has no analytical solution and can be difficult to approach accurately. We develop a new alternative method based on the theory of estimating functions and a deconvolution strategy. Our procedure requires the same assumptions as the MLE and deconvolution estimators. We provide theoretical guarantees about the performance of the resulting estimator; its consistency and asymptotic normality are established. This leads to the construction of confidence intervals. Monte Carlo experiments are investigated and compared with the MLE. Finally, we illustrate our approach using real data for ex-ante interest rate forecasts.

... As such, we may treat neural SDEs as generative time series models.(Generative) time series models are of classical interest, with forecasting models such as Holt-Winters [Hol57; Win60], ARMA[HR82], ARCH[Eng82], GARCH[Bol86] and so on.It has also attracted much recent interest with, besides neural SDEs, the development of ODE-based models like latent ODEs (Section 2.2.4) 3 ; discrete-time models like Time Series GAN[YJS19]; non-ODE continuous-time models like CTFPs [Den+20; Den+21] and Copula Processes[WG10]. ...

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

... We initialize with a crude but fast estimate of Φ and Θ, and if the estimates do not satisfy the constraints they are shrunk towards the constrained space following the algorithm [SHRINK] given in the supplementary material. A fast consistent estimator is provided by the Hannan and Rissanen (1982) algorithm. ...

We present a reparameterization of vector autoregressive moving average (VARMA) models that allows parameter estimation under the constraints of causality and invertibility. This reparameterization is accomplished via a bijection from the complicated causal-invertible parameter space to Euclidean space. The bijection facilitates computation of maximum likelihood estimators (MLE) via unconstrained optimization, as well as computation of Bayesian estimates via prior specification on the constrained space. The proposed parameterization is connected to the Schur-stability of polynomials and the associated Stein transformation, which are often used in dynamical systems; we establish a fundamental characterization of Schur stable polynomials via a novel characterization of positive definite block Toeplitz matrices. Our results also generalize some classical results in dynamical systems.

Autocorrelation and partial autocorrelation, which provide a mathematical tool to understand repeating patterns in time series data, are often used to facilitate the identification of model orders of time series models (e.g., moving average and autoregressive models). Asymptotic methods for testing autocorrelation and partial autocorrelation such as the 1/T approximation method and the Bartlett's formula method may fail in finite samples and are vulnerable to non-normality. Resampling techniques such as the moving block bootstrap and the surrogate data method are competitive alternatives. In this study, we use a Monte Carlo simulation study and a real data example to compare asymptotic methods with the aforementioned resampling techniques. For each resampling technique, we consider both the percentile method and the bias-corrected and accelerated method for interval construction. Simulation results show that the surrogate data method with percentile intervals yields better performance than the other methods. An R package pautocorr is used to carry out tests evaluated in this study.

We address the issue of modelling and forecasting macroeconomic variables using rich datasets by adopting the class of Vector Autoregressive Moving Average (VARMA) models. We overcome the estimation issue that arises with this class of models by implementing an iterative ordinary least squares (IOLS) estimator. We establish the consistency and asymptotic distribution of the estimator for weak and strong VARMA(p,q) models. Monte Carlo results show that IOLS is consistent and feasible for large systems, outperforming the MLE and other linear regression based efficient estimators under alternative scenarios. Our empirical application shows that VARMA models are feasible alternatives when forecasting with many predictors. We show that VARMA models outperform the AR(1), ARMA(1,1), Bayesian VAR, and factor models, considering different model dimensions.

A nonparametric procedure for identifying the differencing operator of a non-stationary time series is presented and tested. Any proposed differencing operator is first applied to the time series, and the spectral density is tested for zeroes corresponding to the polynomial roots of the operator. A nonparametric tapered spectral density estimator is used, and the subsampling methodology is applied to obtain critical values. Simulations explore the effectiveness of the procedure under a variety of scenarios involving non-stationary processes.

This paper proposes an approach to the simultaneous identification and estimation of transfer function models. An estimation procedure is advanced consisting of a sequence of four ordinary least squares calculations and the asymptotic properties of each step are presented, consistency and asymptotic efficiency being shown. A strategy for the identification of the model structure based on a selection criterion that arises naturally from the last step of the estimation algorithm is also investigated. Consideration is given to the numerical implementation of the techniques discussed, and finally a practical example is employed to illustrate the methods.

Statistically speaking, a time seriesy is a finite set of values {y1…, yn} taken by certain k-dimensional random vectors {Y1…, Yn}. The proper framework in which to study time series is that of stochastic processes.

In a quarterly unbalanced panel of 24 developed and developing countries, direct survey measures of capacity utilisation rates are stationary, positively correlated with growth in the short run and uncorrelated with growth in the long run. We show how these stylised facts are related to the 'convergence debate', i.e. the inability of actual capacity utilisation to converge to its normal or desired value in the long-run: In the baseline Neo-Kaleckian model, while trend capacity utilisation is not restricted, it should be positively correlated with growth in the long-run; in contrast, the Sraffian Supermultiplier where capacity utilisation converges to its long-run exogenous value implies utilisation is stationary and uncorrelated with growth in the long-run. Although both models' empirical predictions in the short-run are confirmed, our results reject the baseline Neo-Kaleckian model in favor of the Sraffian Supermultiplier in the long-run.
JEL classification: E11, C22

This dissertation is devoted to the study of some properties of the random coefficient autoregressive
process. This process is commonly referred to as a sequence {Xt, t 2 Z} fully
described by its past values multiplied by random coefficients and disturbed by white noise.
We treat some problems of the study of such processes: stationarity, conditions of existence
of a stationary solution and unknown parameters estimation. We end this work with simulations carried out by the R language.

This work is a comparative study of different univariate and multivariate time series predictive models as applied to Bitcoin, other cryptocurrencies, and other related financial time series data. ARIMA models, long regarded as the gold standard of univariate financial time series prediction due to both its flexibility and simplicity, are used a baseline for prediction. Given the highly correlative nature amongst different cryptocurrencies, this work aims to show the benefit of forecasting with multivariate time series models—primarily focusing on a novel parameter optimization of VARIMA models outlined in this paper. These models are trained on 3 years of historical data, aggregated from different cryptocurrency exchanges by Coinmarketcap.com, which includes: daily average prices and trading volume. Historical time series data of traditional market data, including the stock Nvidia, the de facto leading manufacture of gaming GPU’s, is also analyzed in conjunction with cryptocurrency prices, as gaming GPU’s have played a significant role in solving the profitable SHA256 hashing problems associated with cryptocurrency mining and have seen equivalently correlated investor attention as a result. Models are trained on this historical data using moving window subsets, with window lengths of 100, 200, and 300 days and forecasting 1 day into the future. Validation of this prediction against the actually price from that day are done with following metrics: Directional Forecasting (DF), Mean Absolute Error (MAE), and Mean Squared Error (MSE).

This chapter examines methods for modeling and forecasting univariate time series. The models most commonly used for stationary time series are the autoregressive moving‐average (ARMA) models, that have proved to be useful in many scientific fields. An alternative way to model nonstationary time series is to treat it as an additive model of trend, seasonality, and irregular components. This approach is called the structural time series modelling. An important property, that distinguishes integrated processes from stationary ones, is the way by which the serial dependence behaves. Stationary time series are useful and widely used in practice, and many nonstationary time series can be transformed to stationary ones via differencing. An alternative approach to analyzing stationary time series is to explore the cyclical properties of the data. Estimation of a specified ARIMA model is often carried out via the conditional Maximum Likelihood method under the normality assumption.

Dans cette thèse nous nous intéressons principalement à la validation des modèles ARMA saisonniers et/ou périodiques (SARMA pour Seasonal AutoRegressive Moving-Average, PARMA pour Periodic ARMA et SPARMA Seasonal PARMA) en considérant des termes d’erreur non corrélés mais qui peuvent contenir des dépendances non linéaires. Ces modèles sont appelés des ARMA saisonniers et/ou périodiques faibles et permettent de traiter des processus qui peuvent avoir des dynamiques non linéaires très générales. Par opposition nous appelons modèles ARMA saisonniers et/ou périodiques forts quand le terme d'erreur est indépendant et identiquement distribué (iid). Relâcher l'hypothèse d'indépendance sur le terme d'erreur (une hypothèse habituellement imposée dans la littérature) permet aux modèles SARMA, PARMA et SPARMA faibles d'élargir considérablement leurs champs d'application. Nous étudions les propriétés asymptotiques de l’estimateur du quasi-maximum de vraisemblance et de l’estimateur des moindres carrés généralisés et quasi-généralisés des modèles SARMA faibles et SPARMA faibles. Ensuite nous nous intéressons aux tests fondés sur les résidus, qui ont pour objet de vérifier que les résidus des modèles estimés sont bien des estimations de bruits blancs. Plus particulièrement, nous nous intéressons aux tests portmanteau, aussi appelés tests d’autocorrélation. Nous montrons que les autocorrélations résiduellesde ces modèles ont une distribution asymptotique normal de matrice de covariance différente de celle obtenue dans le cas standard (c’est-à-dire sous les hypothèses iid sur le bruit). Cela nous permet d'avoir le comportement asymptotique des statistiques de tests portmanteau et de proposer ainsi des versions modifiées de ces tests. La distribution asymptotique des tests portmanteau est approximée par un chi-deux lorsque le terme d'erreur est supposé iid. Dans le cas général, nous montrons que cette distribution asymptotique est celle d'une somme pondérée de chi-deux, elle peut être très différente de l'approximation chi-deux usuelle du cas fort (iid). Nous proposons également une approche qui nous permet d’éviter l’estimation de la matrice de covariance asymptotique des autocorrélations empiriques dite d’auto-normalisation.

The purpose of this paper was to analyze the impacts of the global pandemic of COVID – 19 on the price volatility of three important commodities exported by Brazil: soybean, corn and cotton. Therefore, the historical volatility of the respective daily prices was observed and the univariate time series models of the generalized conditional heteroscedasticity (GARCH) with Threshold (TGARCH) were analyzed. Historical volatility estimation showed that the volatility of soybean, corn and cotton prices increased with the world pandemic, that is, the price risk for these commodities increased with the pandemic. In addition, the price of corn had an ARCH effect after the global pandemic announcement made by the World Health Organization (WHO).

The Neo‐Kaleckian model predicts that actual capacity utilisation is endogenous to demand shocks and positively correlated with growth in the short and long run. Competing macroeconomic theories predict that such correlation does not exist in the long run and demand shocks have transitory effects on capacity utilisation. Using a quarterly unbalanced panel of 21 developed and developing countries, we show that taking into account direct survey measures, capacity utilisation is stationary, positively correlated with growth in the short run and uncorrelated with growth in the long run. These results are inconsistent with the long‐run behaviour of the Neo‐Kaleckian model.

Although multivariate stochastic volatility models usually produce more accurate forecasts compared to the MGARCH models, their estimation techniques such as Bayesian MCMC typically suffer from the curse of dimensionality. We propose a fast and efficient estimation approach for MSV based on a penalized OLS framework. Specifying the MSV model as a multivariate state space model, we carry out a two‐step penalized procedure. We provide the asymptotic properties of the two‐step estimator and the oracle property of the first‐step estimator when the number of parameters diverges. The performances of our method are illustrated through simulations and financial data.

The paper consists of three parts. In part one we give a consistency proof for subspace methods, in part two we show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. In part three a simulation study comparing two subspace methods and the maximum likelihood method is performed.

This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models.

Le but de cette communication est un examen des méthodes d'estimation efficaces des paramètres dans quelques modèles employés dans l'analyse des séries temporelles. On considère les modèles suivants: Le modèle autorégressif: ut+α 1ut-1+....+α kut-k=ε t. (1) Régression avec variables x indépendentes et y retardés: yt+α 1yt-1+....+α pyt-p=β 1x1t+....β qxqt+ε t. (2) Régression avec variables x indépendentes et perturbations autorégressives: yt=β tx1t+....+β qxqt+ut (3) avec ut+α 1ut-1+....+α put-p=ε t. Modèle défini par des moyennes mobiles: ut=ε t+β 1ε t-1+....+β hε t-h. (4) Modèle autorégressif et erreurs définies par moyennes mobiles: ut+γ 1ut-1+....+γ put-p=ε t+δ 1ε t-1+....+δ qε t-q. (5) Dans chaque cas, il est supposé que {ε t} définit une série de variables aléatoires indépendentes avec la même distribution. Pour les modèles (1) et (2) les qualités de l'estimation par les moindres carrés sont examinées. Pour le modèle (3) on considère le cas simple yt=β xt+ut avec ut+α ut-1=ε t. Soient a, b, c les coefficients de régression (par les moindres carrés) de yt sur -yt-1′xt′xt-1. Il est démontré que β̂=(1+ar)b+(a+r)c/1+2ar+a2, lorsque r=Σ xtxt-1/Σ xt2, est un estimateur asymptotiquement efficace (à variance minimum) de β. Une extension au cas général est indiquée. Le traitement des modèles (4) et (5) est basé sur l'idée d'un ajustement aux données par la méthode des moindres carrés d'un modèle autorégressif d'ordre k, lorsque k est grand. En examinant les distributions multi-dimensionnelles des coefficients obtenus, on trouve que les estimations efficaces de β 1,...,β h dans (4) sont les solutions d'une série d'équations linéaires. Cette méthode permet d'obtenir des estimations de δ 1,...,δ q dans (5), en supposant que les valeurs γ 1,...,γ p sont connues. En utilisant ces deux méthodes alternativement, on obtient une méthode itérative pour l'ajustement du modèle (5). Une méthode d'estimation plus simple mais moins efficace, qui pourrait être utilisée pour obtenir des valeurs de départ pour l'itération est indiquée.

Under general conditions strong consistency of certain estimates of the maximum lags of an autoregressive moving average process is established. A theorem on weak consistency is also proved and in certain cases where consistency does not hold the probability of over-estimation of a maximum lag is evaluated.

The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.

Standard real business cycle models must rely on total factor productivity (TFP) shocks to explain the observed comovement of consumption, investment, and hours worked. This paper shows that a neoclassical model consistent with observed heterogeneity in labor supply and consumption can generate comovement in the absence of TFP shocks. Intertemporal substitution of goods and leisure induces comovement over the business cycle through heterogeneity in the consumption behavior of employed and unemployed workers. This result owes to two model features introduced to capture important characteristics of U.S. labor market data. First, individual consumption is affected by the number of hours worked: Employed agents consume more on average than the unemployed do. Second, changes in the employment rate, a central factor explaining variation in total hours, affect aggregate consumption. Demand shocks--such as shifts in the marginal efficiency of investment, as well as government spending shocks and news shocks--are shown to generate economic fluctuations consistent with observed business cycles.

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.