Lag Polynomial Research Articles

Seasonal Autoregressive Integrated Moving Average for Climate Change Time Series Forecasting

This study investigates the application of time series models, specifically ARIMA (Auto Regressive Integrated Moving Average) and SARIMAX (Seasonal Autoregressive Integrated Moving Average with eXogenous regressors), in the context of climate change. The ARIMA and SARIMAX models are mathematical methods that can be used to forecast future values of a time series related to climate change, taking into account trends and seasonality, as well as incorporating additional information through exogenous variables. The paper also delves into the mathematical foundations of the ARIMA and SARIMAX models, including the various operators used to eliminate trends, the use of lag polynomials to represent the autoregressive and moving average components of the model, and the incorporation of exogenous variables in the SARIMAX model. The study aims to provide a better understanding of the use of these models in analyzing and predicting the effects of climate change.

Estimation of time series models using residuals dependence measures

We propose new estimation methods for time series models, possibly noncausal and/or noninvertible, using serial dependence information from the characteristic function of model residuals. This allows to impose the i.i.d. or martingale difference assumptions on the model errors to identify the unknown location of the roots of the lag polynomials for ARMA models without resorting to higher order moments or distributional assumptions. We consider generalized spectral density and cumulative distribution functions to measure residuals dependence at an increasing number of lags under both assumptions and discuss robust inference to higher order dependence when only mean independence is imposed on model errors. We study the consistency and asymptotic distribution of parameter estimates and discuss efficiency when different restrictions on error dependence are used simultaneously, including serial uncorrelation. Optimal weighting of continuous moment conditions yields maximum likelihood efficiency under independence for unknown error distribution. We investigate numerical implementation and finite sample properties of the new classes of estimates.

TF-MIDAS: a transfer function based mixed-frequency model

ABSTRACT This paper tackles the mixed-frequency modelling problem from a new perspective. Instead of drawing upon the common distributed lag polynomial model, we use a transfer function representation to develop a new type of models, named TF-MIDAS. We derive the theoretical TF-MIDAS implied by the high-frequency VARMA family models for two common aggregation schemes, flow and stock. This exact correspondence leads to potential gains in terms of nowcasting and forecasting performance against the current alternatives. The estimation of the model proposed is also addressed via its state space equivalent form. A Monte Carlo simulation exercise confirms that TF-MIDAS beats U-MIDAS models (its natural competitor) in terms of out-of-sample nowcasting performance for several data generating high-frequency processes.

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TF-MIDAS: A Transfer Function Based Mixed-Frequency Model

This paper tackles the mixed-frequency modeling problem from a new perspective. Instead of drawing upon the common distributed lag polynomial model, we use a transfer function representation to develop a new type of models, named TF-MIDAS. We derive the theoretical TF-MIDAS implied by the high-frequency VARMA family models for two common aggregation schemes, flow and stock. This exact correspondence leads to potential gains in terms of now-casting and forecasting performance against the current alternatives. The estimation of the model proposed is also addressed via its state space equivalent form. A Monte Carlo simulation exercise confirms that TF-MIDAS beats U-MIDAS models (its natural competitor) in terms of out-of-sample now-casting performance for several data generating high-frequency processes.

Consistent Risk Modeling of Liquid and Illiquid Asset Returns

The authors link liquid assets like exchange-traded real estate investment trusts and public equity, which have relatively high volatilities and low autocorrelations, to illiquid assets like real estate and private equity, which have relatively low volatilities and high autocorrelations. A suitable weighted moving-average or smoothing function applied to public market proxies can yield risk estimates similar to their private market equivalents. Going in the opposite direction, the inverted lag polynomial, or desmoothing function, can transform illiquid alternative asset returns to have risk characteristics similar to those of public market proxies. Using the transform function to express liquid and illiquid views enables more consistent modeling of risk attribution and risk management for portfolios with illiquid assets.

A comparison of MIDAS and bridge equations

This paper compares two single-equation approaches from the recent nowcasting literature: mixed-data sampling (MIDAS) regressions and bridge equations. Both approaches are suitable for nowcasting low-frequency variables such as the quarterly GDP using higher-frequency business cycle indicators. Three differences between the approaches are identified: (1) MIDAS is a direct multi-step nowcasting tool, whereas bridge equations provide iterated forecasts; (2) the weighting of high-frequency predictor observations in MIDAS is based on functional lag polynomials, whereas the bridge equation weights are fixed partly by time aggregation; (3) for parameter estimation, the MIDAS equations consider current-quarter leads of high-frequency indicators, whereas bridge equations typically do not. To assist in discussing the differences between the approaches in isolation, intermediate specifications between MIDAS and bridge equations are provided. The alternative models are compared in an empirical application to nowcasting GDP growth in the Euro area, given a large set of business cycle indicators.

Forecasting Inflation Rates Using Daily Data: A Nonparametric MIDAS Approach

In this paper a nonparametric approach for estimating mixed‐frequency forecast equations is proposed. In contrast to the popular MIDAS approach that employs an (exponential) Almon or Beta lag distribution, we adopt a penalized least‐squares estimator that imposes some degree of smoothness to the lag distribution. This estimator is related to nonparametric estimation procedures based on cubic splines and resembles the popular Hodrick–Prescott filtering technique for estimating a smooth trend function. Monte Carlo experiments suggest that the nonparametric estimator may provide more reliable and flexible approximations to the actual lag distribution than the conventional parametric MIDAS approach based on exponential lag polynomials. Parametric and nonparametric methods are applied to assess the predictive power of various daily indicators for forecasting monthly inflation rates. It turns out that the commodity price index is a useful predictor for inflations rates 20–30 days ahead with a hump‐shaped lag distribution. Copyright © 2015 John Wiley & Sons, Ltd.

Matrix Powers in Spatial Econometric Models

This paper demonstrates several implications of the Cayley-Hamilton Theorem for specification, estimation, and identification of spatial autoregressive models. First, the minimal polynomial of the spatial weights matrix implies an upper bound on the admissible lag order in spatial autoregressive models when powers of the weights matrix are used to form higher-order spatial lags. Second, knowledge of the minimal polynomial facilitates the derivation of closed-form solutions of the marginal effects. Compared to the common practice of numerically inverting the spatial lag polynomial, the use of closed-form solutions enhances the interpretation of the structural model parameters and can be computationally attractive in large samples. Moreover, I demonstrate that the numerical inversion approach may result in biased finite-sample estimates of the marginal effects due to the sample size dependence of the spatial weights matrix even if all coefficient estimates are unbiased. The closed-form solutions can be used to obtain an exact expression of the bias and to avoid it in the first place. Third, I show that the matrix exponential spatial specification has a restricted spatial autoregressive model representation. Finally, I highlight that the order of the minimal polynomial of the spatial weights (or social interaction) matrix is related to the identification of endogenous and exogenous interaction effects in social networks.

Estimating and forecasting Bahrain quarterly GDP growth using simple regression and factor-based methods

In this paper, simple regression estimates and factor-based models are utilised to produce forecasts for Bahrain quarterly gross domestic product growth. Using simulated out-of-sample experiments, we assess and compare the performance of the simple regression estimates, which exploit the available information on selected indicator variables, with factor-based estimates. These estimates use up to 65 variables to obtain new factors that embody most of the potential information and handle it in a systematic way following the Stock–Watson approach. Additionally, we compare the performance of the nowcast factor MIDAS with the quarterly factor (static-SW) models based on time-aggregated data, which neglect the most recent information. Our empirical findings can be summarised as follows. First, using more information does not help to produce more accurate results. Preselected indicator variables can clearly improve the forecast performance in comparison with the use of large dataset. Second, quarterly factor models are in general outperformed by the nowcast factor models that directly relate low-frequency data to those of high frequency. Third, the best forecasting performance can be reached using simple regression estimates with a handful of variables. However, it fails in density forecast evaluation tests. Thus, the alternative factor-MIDAS model is considered as an optimal model that passes all the performance evaluation tests. Fourth, concerning the difference between MIDAS projection methods, the results indicate that MIDAS with exponential distributed lag functions outperforms the MIDAS with unrestricted lag polynomials. The best performing projection based on the number of factors is the model with three factors. They can pick up the rapid switch in the utility of the indicators automatically. Finally, although the most accurate Flash estimates are obtained at 84 days, nowcasting using industrial production into a bridge equation witnessed insignificant loss in accuracy at 54 days only.

Real-Time Forecasting with a MIDAS VAR

This paper presents a MIDAS type mixed frequency VAR forecasting model. First, we propose a general and compact mixed frequency VAR framework using a stacked vector approach. Second, we integrate the mixed frequency VAR with a MIDAS type Almon lag polynomial scheme which is designed to reduce the parameter space while keeping models fexible. We show how to recast the resulting non-linear MIDAS type mixed frequency VAR into a linear equation system that can be easily estimated. A pseudo out-of-sample forecasting exercise with US real-time data yields that the mixed frequency VAR substantially improves predictive accuracy upon a standard VAR for dierent VAR specications. Forecast errors for, e.g., GDP growth decrease by 30 to 60 percent for forecast horizons up to six months and by around 20 percent for a forecast horizon of one year.

Forecasting GDP growth using mixed-frequency models with switching regimes

For modelling mixed-frequency data with a business cycle pattern, we introduce the Markov-switching Mixed Data Sampling model with unrestricted lag polynomial (MS-U-MIDAS). Usually, models of the MIDAS-class use lag polynomials of a specific function which impose some structure on the weights of the regressors included in the model. This may lead to a deterioration in the predictive power of the model if the structure imposed differs from the data generating process. When the difference between the available data frequencies is small and there is no risk of parameter proliferation, using an unrestricted lag polynomial might not only simplify the model estimation, but also improve its forecasting performance. We allow the parameters of the MIDAS model with an unrestricted lag polynomial to change according to a Markov-switching scheme in order to account for the business cycle pattern observed in many macroeconomic variables. Thus, we combine the unrestricted MIDAS with a Markov-switching approach and propose a new Markov-switching MIDAS model with unrestricted lag polynomial (MS-U-MIDAS). We apply this model to a large dataset with the help of factor analysis. Monte Carlo experiments and an empirical forecasting comparison carried out for the U.S. GDP growth show that the models of the MS-U-MIDAS class exhibit nowcasting and forecasting performances which are similar to or better than those of their counterparts with restricted lag polynomials.

Midas and Bridge Equations

This paper compares two single-equation approaches from the recent nowcast literature: Mixed-data sampling (MIDAS) regressions and bridge equations. Both approach are used to nowcast a low-frequency variable such as quarterly GDP growth by higher-frequency business cycle indicators. Three differences between the approaches are discussed: 1) MIDAS is a direct multi-step nowcasting tool, whereas bridge equations are based on iterated forecasts; 2) MIDAS equations employ empirical weighting of high-frequency predictor observations with functional lag polynomials, whereas the weights of indicator observations in bridge equations are partly fixed stemming from time aggregation. 3) MIDAS equations can consider current-quarter leads of high-frequency indicators in the regression, whereas bridge equations typically do not. However, the conditioning set for nowcasting includes the most recent indicator observations in both approaches. To discuss the differences between the approaches in isolation, intermediate specifications between MIDAS and bridge equations are provided. The alternative models are compared in an empirical application to nowcasting GDP growth in the Euro area given a large set of business cycle indicators.

Unrestricted Mixed Data Sampling (MIDAS): MIDAS Regressions with Unrestricted Lag Polynomials

SummaryMixed data sampling (MIDAS) regressions allow us to estimate dynamic equations that explain a low frequency variable by high frequency variables and their lags. When the difference in sampling frequencies between the regressand and the regressors is large, distributed lag functions are typically employed to model dynamics avoiding parameter proliferation. In macroeconomic applications, however, differences in sampling frequencies are often small. In such a case, it might not be necessary to employ distributed lag functions. We discuss the pros and cons of unrestricted lag polynomials in MIDAS regressions. We derive unrestricted-MIDAS (U-MIDAS) regressions from linear high frequency models, discuss identification issues and show that their parameters can be estimated by ordinary least squares. In Monte Carlo experiments, we compare U-MIDAS with MIDAS with functional distributed lags estimated by non-linear least squares. We show that U-MIDAS performs better than MIDAS for small differences in sampling frequencies. However, with large differing sampling frequencies, distributed lag functions outperform unrestricted polynomials. The good performance of U-MIDAS for small differences in frequency is confirmed in empirical applications on nowcasting and short-term forecasting euro area and US gross domestic product growth by using monthly indicators.

News shocks, nonfundamentalness and volatility

Rational expectations models with news shocks may generate moving average representation that are nonfundamental. The nonfundamentalness typically arises from the lag polynomial associated with news shocks. This paper provides an exact solution formula for this special type of polynomial and discusses its main properties. In the presence of news shocks, the solutions may be used to convert a nonfundamental moving average representation into a fundamental one and vice versa. From the properties of these solutions, we conclude that the destabilizing effects of news shocks are exclusively due to its anticipation characteristic.

Dynamic Censored Regression and the Open Market Desk Reaction Function

The censored regression model and the Tobit model are standard tools in econometrics. This paper provides a formal asymptotic theory for dynamic time series censored regression when lags of the dependent variable have been included among the regressors. The central analytical challenge is to prove that the dynamic censored regression model satisfies stationarity and weak dependence properties if a condition on the lag polynomial holds. We show the formal asymptotic correctness of conditional maximum likelihood estimation of the dynamic Tobit model, and the correctness of Powell’s least absolute deviations procedure for the estimation of the dynamic censored regression model. The paper is concluded with an application of the dynamic censored regression methodology to temporary purchases of the Open Market Desk. This article has supplementary material online.

U-Midas: Midas Regressions with Unrestricted Lag Polynomials

Mixed-data sampling (MIDAS) regressions allow to estimate dynamic equations that explain a low-frequency variable by high-frequency variables and their lags. When the difference in sampling frequencies between the regressand and the regressors is large, distributed lag functions are typically employed to model dynamics avoiding parameter proliferation. In macroeconomic applications, however, differences in sampling frequencies are often small. In such a case, it might not be necessary to employ distributed lag functions. In this paper, we discuss the pros and cons of unrestricted lag polynomials in MIDAS regressions. We derive unrestricted MIDAS regressions (U-MIDAS) from linear high-frequency models, discuss identification issues, and show that their parameters can be estimated by OLS. In Monte Carlo experiments, we compare U-MIDAS to MIDAS with functional distributed lags estimated by NLS. We show that U-MIDAS generally performs better than MIDAS when mixing quarterly and monthly data. On the other hand, with larger differences in sampling frequencies, distributed lag-functions outperform unrestricted polynomials. In an empirical application on out-of-sample nowcasting GDP in the US and the Euro area using monthly predictors, we find a good performance of U-MIDAS for a number of indicators, albeit the results depend on the evaluation sample. We suggest to consider U-MIDAS as a potential alternative to the existing MIDAS approach in particular for mixing monthly and quarterly variables. In practice, the choice between the two approaches should be made on a case-by-case basis, depending on their relative performance.

MIDAS vs. mixed-frequency VAR: Nowcasting GDP in the euro area

Abstract This paper compares the mixed-data sampling (MIDAS) and mixed-frequency VAR (MF-VAR) approaches to model specification in the presence of mixed-frequency data, e.g. monthly and quarterly series. MIDAS leads to parsimonious models which are based on exponential lag polynomials for the coefficients, whereas MF-VAR does not restrict the dynamics and can therefore suffer from the curse of dimensionality. However, if the restrictions imposed by MIDAS are too stringent, the MF-VAR can perform better. Hence, it is difficult to rank MIDAS and MF-VAR a priori, and their relative rankings are better evaluated empirically. In this paper, we compare their performances in a case which is relevant for policy making, namely nowcasting and forecasting quarterly GDP growth in the euro area on a monthly basis, using a set of about 20 monthly indicators. It turns out that the two approaches are more complements than substitutes, since MIDAS tends to perform better for horizons up to four to five months, whereas MF-VAR performs better for longer horizons, up to nine months.

Midas Versus Mixed-Frequency VAR: Nowcasting GDP in the Euro Area

This paper compares the mixed-data sampling (MIDAS) and mixed-frequency VAR (MF-VAR) approaches to model specification in the presence of mixed-frequency data, e.g., monthly and quarterly series. MIDAS leads to parsimonious models based on exponential lag polynomials for the coefficients, whereas MF-VAR does not restrict the dynamics and therefore can suffer from the curse of dimensionality. But if the restrictions imposed by MIDAS are too stringent, the MF-VAR can perform better. Hence, it is difficult to rank MIDAS and MF-VAR a priori, and their relative ranking is better evaluated empirically. In this paper, we compare their performance in a relevant case for policy making, i.e., nowcasting and forecasting quarterly GDP growth in the euro area, on a monthly basis and using a set of 20 monthly indicators. It turns out that the two approaches are more complementary than substitutes, since MF-VAR tends to perform better for longer horizons, whereas MIDAS for shorter horizons.

Assessing the Effects of Terrorism on Tourism by Use of Time Series Methods

This paper presents an assessment of the effects of terrorism on tourism by using time series methods, namely the ARMAX (autoregressive moving average with explanatory variables) model. This is a single-equation approach, which has the ability to provide impact analysis easily. The use of the ARMAX model allows for the general shape of the lag distribution of the impacts of the explanatory variables based on the ratio of lag polynomials for the independent and dependent variables. The ARMAX models, like the ARIMA models, provide for a short-term assessment of terrorist incidents on tourism.