This document discusses time series forecasting and summarizes four illustrations of time series analysis and forecasting:
1. A multivariate model is used to analyze the European business cycle based on trends, common cycles, and leads/lags between economic indicators like GDP, industrial production, and confidence.
2. A bivariate unobserved components model is applied to daily Nordpool electricity spot prices and consumption data. The model decomposes the data into trends, seasons, cycles and residuals. Forecasting results show the bivariate model outperforms the univariate.
3. A periodic dynamic factor model is jointly modeled to 24 hours of French electricity load data. The model accounts for long-term trends, various seasonal patterns,
This document describes a collapsed dynamic factor analysis model for macroeconomic forecasting. It summarizes that multivariate time series models can more accurately capture relationships between economic variables compared to univariate models. The document then presents a collapsed dynamic factor model that relates a target time series (yt) to unobserved dynamic factors (Ft) estimated from related macroeconomic data (gt). Out-of-sample forecasting experiments on US personal income and industrial production data demonstrate the model achieves more accurate point forecasts than univariate benchmarks like random walk or AR(2) models.
This document discusses state space methods for time series analysis and forecasting. It begins by introducing the basic state space model framework, which represents a time series using unobserved states that evolve over time according to a state equation and generate observations according to an observation equation. The document then provides examples of how various time series models, such as regression models with time-varying coefficients, ARMA models, and univariate component models can be expressed as state space models. Finally, it introduces the Kalman filter algorithm, which provides a recursive means of estimating the unobserved states from the observations.
This document provides an overview of a course on forecasting time series using state space methods and unobserved components models. The course covers introduction to univariate component models, state space methods, forecasting different time series components, and exercises for practical forecasting applications with examples. Key topics include white noise processes, random walk processes, the local level model, and simulated data from a local level model.
Prévision de consommation électrique avec adaptive GAMCdiscount
The document discusses generalized additive models (GAM) for short-term electricity load forecasting. GAMs are smooth additive models that decompose a response variable into additive components like trends, cyclic patterns, and nonlinear effects. They summarize how GAMs can model various drivers of electricity consumption, including temperature effects, day-of-week patterns, and lagged load values. Big additive models (BAM) allow applying GAMs to large electricity load datasets. BAMs use QR decomposition and online updating to efficiently estimate high-dimensional additive models.
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...Eesti Pank
This document presents a Markov-switching DSGE model for Estonia that allows for time-varying risk premia. The model includes domestic consumption, inflation, and interest rates that are linked to foreign variables through a fixed exchange rate. Shocks follow Markov-switching processes where volatility varies between states. The model is solved using a forward-looking approach and estimated via Bayesian methods. Estimated parameters provide evidence of regime-dependent behavior and time-varying volatility consistent with financial crises in Estonia.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 14.
More info at http://summerschool.ssa.org.ua
The document summarizes a simulation of news and insider influences on stock market price dynamics using a non-linear model. It describes two types of trader strategies (F and N), how aggregate excess demand is calculated, dynamic price adjustment, and how the share of each trader type changes over time based on past relative returns. It also discusses using the model to simulate dollar/ruble exchange rates and analyze patterns in the time series data.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
This document describes a collapsed dynamic factor analysis model for macroeconomic forecasting. It summarizes that multivariate time series models can more accurately capture relationships between economic variables compared to univariate models. The document then presents a collapsed dynamic factor model that relates a target time series (yt) to unobserved dynamic factors (Ft) estimated from related macroeconomic data (gt). Out-of-sample forecasting experiments on US personal income and industrial production data demonstrate the model achieves more accurate point forecasts than univariate benchmarks like random walk or AR(2) models.
This document discusses state space methods for time series analysis and forecasting. It begins by introducing the basic state space model framework, which represents a time series using unobserved states that evolve over time according to a state equation and generate observations according to an observation equation. The document then provides examples of how various time series models, such as regression models with time-varying coefficients, ARMA models, and univariate component models can be expressed as state space models. Finally, it introduces the Kalman filter algorithm, which provides a recursive means of estimating the unobserved states from the observations.
This document provides an overview of a course on forecasting time series using state space methods and unobserved components models. The course covers introduction to univariate component models, state space methods, forecasting different time series components, and exercises for practical forecasting applications with examples. Key topics include white noise processes, random walk processes, the local level model, and simulated data from a local level model.
Prévision de consommation électrique avec adaptive GAMCdiscount
The document discusses generalized additive models (GAM) for short-term electricity load forecasting. GAMs are smooth additive models that decompose a response variable into additive components like trends, cyclic patterns, and nonlinear effects. They summarize how GAMs can model various drivers of electricity consumption, including temperature effects, day-of-week patterns, and lagged load values. Big additive models (BAM) allow applying GAMs to large electricity load datasets. BAMs use QR decomposition and online updating to efficiently estimate high-dimensional additive models.
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...Eesti Pank
This document presents a Markov-switching DSGE model for Estonia that allows for time-varying risk premia. The model includes domestic consumption, inflation, and interest rates that are linked to foreign variables through a fixed exchange rate. Shocks follow Markov-switching processes where volatility varies between states. The model is solved using a forward-looking approach and estimated via Bayesian methods. Estimated parameters provide evidence of regime-dependent behavior and time-varying volatility consistent with financial crises in Estonia.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 14.
More info at http://summerschool.ssa.org.ua
The document summarizes a simulation of news and insider influences on stock market price dynamics using a non-linear model. It describes two types of trader strategies (F and N), how aggregate excess demand is calculated, dynamic price adjustment, and how the share of each trader type changes over time based on past relative returns. It also discusses using the model to simulate dollar/ruble exchange rates and analyze patterns in the time series data.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
1) The document outlines the nonlinear equilibrium conditions of a simple New Keynesian model without capital. It discusses formulating the model's nonlinear equations to study optimal monetary policy and higher-order solutions.
2) It presents the key components of the model, including household and firm behavior assumptions. Households maximize utility from consumption and labor. Firms set prices according to Calvo pricing and maximize profits.
3) The document derives the nonlinear equilibrium conditions that characterize household and firm optimization, including the household's intertemporal FOC and the intermediate firm's price-setting problem. It expresses the model's equilibrium objects like marginal costs and the price index.
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...SYRTO Project
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...SSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 8.
More info at http://summerschool.ssa.org.ua
This document outlines a lecture on linear factor models and event studies in financial econometrics. It discusses the motivation for linear factor pricing models and outlines two main econometric approaches: time-series regression based tests that focus on intercepts when factors are excess returns, and cross-sectional regression based residual tests when factors are not excess returns. It also provides details on specific methods for time-series approach including the joint intercept test of Black, Jensen and Scholes (1972) and extensions by Gibbons, Ross and Shanken (1989) and MacKinlay and Richardson (1991).
This document provides an outline and overview of linear factor models and event studies. For linear factor models, it discusses the time-series approach of regressing returns on factors to test pricing restrictions, the cross-sectional approach of regressing average returns on factor betas, and compares the two approaches. It also discusses formulating linear factor models and tests in a generalized method of moments framework. For event studies, it briefly outlines the motivation and basic methodology.
TU4.L09 - FOUR-COMPONENT SCATTERING POWER DECOMPOSITION WITH ROTATION OF COHE...grssieee
The document proposes a four-component scattering power decomposition method with rotation of the coherency matrix. This improves upon existing decomposition methods by minimizing the HV component through rotation, resulting in better separation of surface, double bounce, volume, and helix scattering mechanisms. The new method is applied to fully polarimetric SAR data sets to provide improved classification results.
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
This document analyzes regional agglomeration in Portugal using linear models from New Economic Geography that emphasize spatial factors like distance and transportation costs. It estimates equations to examine real wage determinants and clustering. The results show some concentration in Lisboa e Vale do Tejo, indicating regional divergence. Productivity significantly improves models' ability to explain regional clustering in Portugal. In conclusion, agglomeration processes exhibit concentration in Lisboa e Vale do Tejo, evidencing divergence, and productivity notably enhances explaining regional clustering despite being ignored in New Economic Geography models.
The document discusses a stochastic volatility model that incorporates jumps in volatility and the possibility of default. It describes the dynamics of the model and how it can be used to price volatility and credit derivatives. Analytical and numerical methods are presented for solving the pricing problem. As an example application, the model is fit to data on General Motors to analyze the implications.
This document contains the homework assignment for Dr. Ashu Sabharwal's ELEC 430 class at Rice University due on February 19, 2009. It includes 3 exercises on topics related to signaling and detection:
1. The likelihood ratio test is derived for a binary communication system with additive white Gaussian noise. Conditional probability density functions are found and used to determine the probability of error as a function of threshold.
2. Matched filters are discussed for an antipodal signaling system. The impulse response and output of the matched filter are sketched. Expressions are derived for the noise variance and probability of error.
3. Orthogonal signal properties are explored for a set of signals that are modified by subtracting
I conti economici trimestrali: avanzamenti metodologici e prospettive di innovazione
Seminario
Roma, 21 aprile 2016
Istat, Aula Magna
Via Cesare Balbo, 14
This document summarizes a mathematical economics model of stock prices using differential equations. It begins by modeling stock prices based on dividends and interest rates, assuming stock prices converge to the fundamental price over time. It then models stock prices under rational expectations, finding stock prices instantly jump to the fundamental price. Finally, it briefly introduces a dynamic IS-LM model to analyze output and interest rates over time.
This document discusses Bayesian reliability analysis of systems when component lifetimes follow a geometric distribution. It provides the probability mass function of the geometric distribution and defines the prior and posterior distributions of the distribution parameter θ. It then derives the Bayesian reliability estimators for k-out-of-n, series, parallel and cold standby systems. Simulation studies are presented to analyze the Bayes risk of the estimators for different values of the model parameters.
The document discusses a stochastic differential equation (SDE) with Markovian switching that is used to model financial market behavior subject to random regime shifts. Key results shown include:
1) The SDE model exhibits the same long-run growth and deviation properties as classical geometric Brownian motion, obeying the law of the iterated logarithm.
2) Mathematical analysis is presented of the SDE with Markovian switching.
3) The results are applied to financial market models to account for random regime shifts between bullish and bearish states and changes in market sentiment.
This document discusses the analysis and formation of an optimal questionnaire to monitor the connection between re-engineering economic parameters in small and medium enterprises. It presents an algorithm for transforming an initial questionnaire into an optimal questionnaire through arranging the questions and associated weights and values. It then applies this methodology to analyze the connection between economic parameters for a re-engineering model, presenting an initial questionnaire with groups of parameters, events, and associated weights and values.
101 Tips for a Successful Automation Career Appendix FISA Interchange
This document discusses three types of first principle process dynamics:
1) Self-regulating processes, where negative feedback causes the process output to decelerate to a new steady state. Over 90% of processes are self-regulating.
2) Integrating processes, where there is no feedback, so the process output will continually ramp up or down. Batch and level processes typically behave this way.
3) Runaway processes, where positive feedback causes the process output to accelerate until limits are reached. These are rare but important for safety in some chemical processes.
The document then provides equations to model the time constants and gains of back-mixed and plug flow volumes, which are important for control system design.
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
This document describes a collapsed dynamic factor analysis model for macroeconomic forecasting. It summarizes that multivariate time series models can more accurately capture relationships between economic variables compared to univariate models. The document then presents a collapsed dynamic factor model that relates a target time series (yt) to unobserved dynamic factors (Ft) estimated from related macroeconomic data (gt). Out-of-sample forecasting experiments on US personal income and industrial production data demonstrate the model achieves more accurate point forecasts than univariate benchmarks like random walk or AR(2) models.
1) The document outlines the nonlinear equilibrium conditions of a simple New Keynesian model without capital. It discusses formulating the model's nonlinear equations to study optimal monetary policy and higher-order solutions.
2) It presents the key components of the model, including household and firm behavior assumptions. Households maximize utility from consumption and labor. Firms set prices according to Calvo pricing and maximize profits.
3) The document derives the nonlinear equilibrium conditions that characterize household and firm optimization, including the household's intertemporal FOC and the intermediate firm's price-setting problem. It expresses the model's equilibrium objects like marginal costs and the price index.
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...SYRTO Project
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.
Parameter Estimation in Stochastic Differential Equations by Continuous Optim...SSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 8.
More info at http://summerschool.ssa.org.ua
This document outlines a lecture on linear factor models and event studies in financial econometrics. It discusses the motivation for linear factor pricing models and outlines two main econometric approaches: time-series regression based tests that focus on intercepts when factors are excess returns, and cross-sectional regression based residual tests when factors are not excess returns. It also provides details on specific methods for time-series approach including the joint intercept test of Black, Jensen and Scholes (1972) and extensions by Gibbons, Ross and Shanken (1989) and MacKinlay and Richardson (1991).
This document provides an outline and overview of linear factor models and event studies. For linear factor models, it discusses the time-series approach of regressing returns on factors to test pricing restrictions, the cross-sectional approach of regressing average returns on factor betas, and compares the two approaches. It also discusses formulating linear factor models and tests in a generalized method of moments framework. For event studies, it briefly outlines the motivation and basic methodology.
TU4.L09 - FOUR-COMPONENT SCATTERING POWER DECOMPOSITION WITH ROTATION OF COHE...grssieee
The document proposes a four-component scattering power decomposition method with rotation of the coherency matrix. This improves upon existing decomposition methods by minimizing the HV component through rotation, resulting in better separation of surface, double bounce, volume, and helix scattering mechanisms. The new method is applied to fully polarimetric SAR data sets to provide improved classification results.
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
This document analyzes regional agglomeration in Portugal using linear models from New Economic Geography that emphasize spatial factors like distance and transportation costs. It estimates equations to examine real wage determinants and clustering. The results show some concentration in Lisboa e Vale do Tejo, indicating regional divergence. Productivity significantly improves models' ability to explain regional clustering in Portugal. In conclusion, agglomeration processes exhibit concentration in Lisboa e Vale do Tejo, evidencing divergence, and productivity notably enhances explaining regional clustering despite being ignored in New Economic Geography models.
The document discusses a stochastic volatility model that incorporates jumps in volatility and the possibility of default. It describes the dynamics of the model and how it can be used to price volatility and credit derivatives. Analytical and numerical methods are presented for solving the pricing problem. As an example application, the model is fit to data on General Motors to analyze the implications.
This document contains the homework assignment for Dr. Ashu Sabharwal's ELEC 430 class at Rice University due on February 19, 2009. It includes 3 exercises on topics related to signaling and detection:
1. The likelihood ratio test is derived for a binary communication system with additive white Gaussian noise. Conditional probability density functions are found and used to determine the probability of error as a function of threshold.
2. Matched filters are discussed for an antipodal signaling system. The impulse response and output of the matched filter are sketched. Expressions are derived for the noise variance and probability of error.
3. Orthogonal signal properties are explored for a set of signals that are modified by subtracting
I conti economici trimestrali: avanzamenti metodologici e prospettive di innovazione
Seminario
Roma, 21 aprile 2016
Istat, Aula Magna
Via Cesare Balbo, 14
This document summarizes a mathematical economics model of stock prices using differential equations. It begins by modeling stock prices based on dividends and interest rates, assuming stock prices converge to the fundamental price over time. It then models stock prices under rational expectations, finding stock prices instantly jump to the fundamental price. Finally, it briefly introduces a dynamic IS-LM model to analyze output and interest rates over time.
This document discusses Bayesian reliability analysis of systems when component lifetimes follow a geometric distribution. It provides the probability mass function of the geometric distribution and defines the prior and posterior distributions of the distribution parameter θ. It then derives the Bayesian reliability estimators for k-out-of-n, series, parallel and cold standby systems. Simulation studies are presented to analyze the Bayes risk of the estimators for different values of the model parameters.
The document discusses a stochastic differential equation (SDE) with Markovian switching that is used to model financial market behavior subject to random regime shifts. Key results shown include:
1) The SDE model exhibits the same long-run growth and deviation properties as classical geometric Brownian motion, obeying the law of the iterated logarithm.
2) Mathematical analysis is presented of the SDE with Markovian switching.
3) The results are applied to financial market models to account for random regime shifts between bullish and bearish states and changes in market sentiment.
This document discusses the analysis and formation of an optimal questionnaire to monitor the connection between re-engineering economic parameters in small and medium enterprises. It presents an algorithm for transforming an initial questionnaire into an optimal questionnaire through arranging the questions and associated weights and values. It then applies this methodology to analyze the connection between economic parameters for a re-engineering model, presenting an initial questionnaire with groups of parameters, events, and associated weights and values.
101 Tips for a Successful Automation Career Appendix FISA Interchange
This document discusses three types of first principle process dynamics:
1) Self-regulating processes, where negative feedback causes the process output to decelerate to a new steady state. Over 90% of processes are self-regulating.
2) Integrating processes, where there is no feedback, so the process output will continually ramp up or down. Batch and level processes typically behave this way.
3) Runaway processes, where positive feedback causes the process output to accelerate until limits are reached. These are rare but important for safety in some chemical processes.
The document then provides equations to model the time constants and gains of back-mixed and plug flow volumes, which are important for control system design.
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
This document describes a collapsed dynamic factor analysis model for macroeconomic forecasting. It summarizes that multivariate time series models can more accurately capture relationships between economic variables compared to univariate models. The document then presents a collapsed dynamic factor model that relates a target time series (yt) to unobserved dynamic factors (Ft) estimated from related macroeconomic data (gt). Out-of-sample forecasting experiments on US personal income and industrial production data demonstrate the model achieves more accurate point forecasts than univariate benchmarks like random walk or AR(2) models.
Présentation Olivier Biau Random forests et conjonctureCdiscount
1. The document proposes using random forest (RF), a machine learning technique, to forecast euro area GDP based on large survey datasets.
2. An out-of-sample exercise from 2004-2009 finds that a pure RF model outperforms an AR model but not official European Commission forecasts. However, during the financial crisis, RF performs poorly due to non-negative values in the training data.
3. A modified two-step RF-linear model approach selects important variables from RF and chooses the best linear model, improving crisis period performance and comparing well to official forecasts overall.
Prévision consommation électrique par processus à valeurs fonctionnellesCdiscount
The document discusses predicting functional time series. It introduces functional data as slices of a continuous stochastic process over time. The goal is to predict future values of this process given past observations. An autoregressive Hilbertian process of order 1 is proposed, where the current value is a linear function of the past value plus noise. Under certain conditions, this defines a strictly stationary process whose best predictor is the past value multiplied by the linear operator. The document outlines clustering functional data using wavelets and applying these methods to predict electricity demand.
This document proposes a framework for predicting links in dynamic graph sequences. It formulates the problem as a convex optimization that minimizes three terms: (1) how well feature vectors of past graphs predict future feature vectors, (2) how well predicted features match predicted graph features, and (3) a penalty on the predicted graph to encourage simplicity. The framework assumes graph features change gradually over time and the predicted graph is low rank. It aims to leverage trade-offs between these terms to select predictive graph features.
- The document discusses strategies for analyzing large datasets that are too big to fit into memory, including using cloud computing, the ff and rsqlite packages in R, and sampling with the data.sample package.
- The ff and rsqlite packages allow working with data beyond RAM limits but require rewriting code, while data.sample provides sampling without rewriting code but introduces sampling error.
- Cloud computing avoids rewriting code and has no memory limits but requires setup, and sampling is good for analysis but not reporting exact values.
The document discusses random forests, which are ensemble learning methods for classification and regression. It notes that Leo Breiman promoted random forests using tree averaging to obtain good rules, with the base trees being simple and randomized. Breiman's ideas were influenced by previous work on geometric feature selection, random subspace methods, and random split selection. Random forests have emerged as serious competitors to state-of-the-art methods due to their speed, accuracy, ability to handle many variables without overfitting, and general effectiveness. However, their mathematical properties remain largely unknown.
This document provides an overview of time-series analysis techniques. It defines the components of a time series as trend, seasonal variation, and cyclical variation. Smoothing techniques like moving averages and exponential smoothing are described to reduce random fluctuations and expose patterns. Methods for separating out the trend, seasonal, and cyclical components are outlined, including linear regression for trends and calculating seasonal factors. The goal of time-series analysis is described as describing historical patterns and making forecasts by projecting patterns into the future.
INTRODUCTION TO TIME SERIES REGRESSION AND FORCASTINGSPICEGODDESS
What Is Time Series Regression? Time series regression is a statistical method for predicting a future response based on the response history (known as autoregressive dynamics) and the transfer of dynamics from relevant predictors.
Plenary session 5 5. osberg polarization of time and income a multidimensio...IARIW 2014
This document summarizes research on polarization of time and income in Germany between 1991/92 and 2001/02 using a multidimensional approach. The researchers developed methods to identify multidimensional poverty, affluence, and polarization using time and income data. They found that while multidimensional polarization headcount ratios decreased slightly, the mean minimum multidimensional polarization gap increased significantly, indicating greater polarization. Certain groups like the self-employed and families with children experienced higher multidimensional polarization. The study provides insights but leaves questions around how to define time poverty and affluence and how results generalize across all age groups.
Is the Macroeconomy Locally Unstable and Why Should We Care?ADEMU_Project
The document discusses evidence that macroeconomic dynamics may exhibit locally unstable behavior like limit cycles rather than stable convergence to a steady state. It presents a reduced-form model allowing for nonlinearities, estimated on total hours and other labor market variables. The estimates provide intriguing evidence of limit cycle dynamics, with simulations showing attracting cycles. This suggests macroeconomic fluctuations could be endogenous rather than just responses to shocks, with implications for stabilization policy.
Are Capital Controls Effective in Preventing Financial CrisesALing Huang
This document summarizes research on whether capital controls are effective in preventing financial crises. It finds that most currency crash events occurred under tight capital control settings, and loosening controls is linked to a reduced likelihood of crises after accounting for other factors. Theoretically, capital controls can distort markets and facilitate corruption, lowering productivity. Financial openness may also exert "discipline effects" that encourage sound macroeconomic policies.
Innovations in technology has revolutionized financial services to an extent that large financial institutions like Goldman Sachs are claiming to be technology companies! It is no secret that technological innovations like Data science and AI are changing fundamentally how financial products are created, tested and delivered. While it is exciting to learn about technologies themselves, there is very little guidance available to companies and financial professionals should retool and gear themselves towards the upcoming revolution.
In this master class, we will discuss key innovations in Data Science and AI and connect applications of these novel fields in forecasting and optimization. Through case studies and examples, we will demonstrate why now is the time you should invest to learn about the topics that will reshape the financial services industry of the future!
Topics in Econometrics
This document summarizes the results of estimating equations for taxes, consumption, and money supply (M2) using a structural macroeconomic model. The model contains 11 behavioral equations estimated using two-stage least squares, with some recursive equations estimated using ordinary least squares. Historical simulations from 1960-1993 show close fits between actual and predicted values for taxes and consumption. Forecasts for 1994-1995 also closely match actual tax and consumption values.
The impact of business cycle fluctuations on aggregate endogenous growth ratesGRAPE
This document summarizes Marc Bielecki's research on modeling the impact of business cycle fluctuations on endogenous growth rates. The research aims to develop a single framework to analyze both business cycle and growth phenomena. Key contributions include a microfounded aggregate R&D intensity function and modeling of innovating, heterogeneous firms hit by aggregate and idiosyncratic shocks. Empirical evidence shows entry, expansions and contractions are procyclical. The model can replicate these features and shows that temporary productivity shocks can have permanent effects by shifting the long-run growth rate.
The impact of business cycle fluctuations on aggregate endogenous growth ratesGRAPE
This document summarizes Marc Bielecki's research on modeling the impact of business cycle fluctuations on endogenous growth rates. The research aims to develop a single framework to analyze both business cycle and growth phenomena. Key contributions include a microfounded aggregate R&D intensity function and modeling of innovating, heterogeneous firms hit by aggregate and idiosyncratic shocks. Empirical evidence shows entry, expansions and contractions are procyclical. The model can replicate these features and shows that temporary productivity shocks can have permanent effects by shifting the long-run growth rate.
1) The document analyzes the effect of changes in oil prices on stock price indices and industrial production in the US, UK, and Japan from 1986 to 2008 using a VAR model.
2) It finds that oil price shocks explain 9.51% of the variance in the US stock market, 7.51% in the UK market, and 4.4% in the Japanese market over a 24 month period.
3) The response of stock markets to oil price changes was stronger in the first half of the sample period compared to the second half.
1) The document analyzes the effect of oil price shocks on stock markets in the US, UK, and Japan using a VAR model.
2) It finds that oil price changes explain 9.51% of the variance in the US stock market index, 7.51% in the UK index, and 4.4% in the Japanese index.
3) The results indicate that oil price shocks negatively impact industrial production but do not appear to significantly affect stock market performance over the long run.
1) The document analyzes the effect of changes in oil prices on stock price indices and industrial production in the US, UK, and Japan from 1986 to 2008 using a VAR model.
2) The results show that oil price shocks explain 9.51% of the variance in the US stock market, 7.51% in the UK market, and 4.4% in the Japanese market over a 24 month period.
3) The response of stock markets to oil price changes was asymmetric, with a larger effect from negative price changes compared to positive changes.
The document discusses risk premia in credit markets and their implications. It finds that:
1) Credit spread components like expected losses and risk premia vary significantly over time and with credit quality.
2) Systematic risk, not just expected losses, influences the pricing of credit default swaps and stock options relative to their underlyings.
3) Ratings alone are insufficient for credit assessments - risk-adjusted ratings may be needed to account for varying risk premia.
This document discusses quantitative forecasting methods, including time series and causal models. It covers key time series components like trend, seasonality, and cycles. Three main time series methods are described: smoothing, trend projection, and trend projection adjusted for seasonal influence. Moving averages and exponential smoothing are explained as common techniques for forecasting stationary time series. The document also covers decomposing a time series into trend, seasonal, and irregular components. Regression methods are mentioned as another approach when a trend is present in the data.
L-14 Modeling Strategies and Policy Analysis - NR.pptxRiyadhJack
This document outlines a training on modeling inflation in Australia using a markup model and vector error correction model (VECM). It begins with introducing the markup model which relates consumer prices to unit labor costs, import prices, petrol prices, and a markup term. Preliminary data analysis shows the variables are integrated of order one. A VECM is then estimated showing one cointegrating relationship consistent with the markup model. Further reductions to the VECM support the variables being weakly exogenous and the model satisfying long-run homogeneity. The document concludes by reducing the VECM to a single-equation autoregressive distributed lag model for inflation consistent with the economic theory.
The document discusses time series decomposition and smoothing. It introduces the components of time series patterns including trends, cycles, and seasonality. Time series can be decomposed into trend, seasonal, and remainder components using either additive or multiplicative models. Simple exponential smoothing is presented as a method to smooth time series data by applying weighted averages with weights that decrease exponentially. Autocorrelation and differencing methods are discussed to make time series stationary for ARIMA modeling.
This document summarizes the results of an econometrics analysis examining the relationship between macroeconomic variables in the US and Italy. It tests for unit roots and cointegration, estimates vector autoregression models in levels and first differences, and analyzes impulse response functions and variance decompositions. The key findings are: 1) some variables are stationary while others have unit roots; 2) there are two cointegrating relationships; 3) monetary shocks have a significant positive effect on GDP for several quarters in the levels model; 4) variance decompositions show monetary shocks do not explain significant portions of GDP variance.
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1. Time Series Forecasting
Siem Jan Koopman
http://personal.vu.nl/s.j.koopman
Department of Econometrics
VU University Amsterdam
Tinbergen Institute
2012
2. Unobserved components: decomposing time series
A basic model for representing a time series is the additive model
yt = µt + γt + εt , t = 1, . . . , n,
also known as the classical decomposition.
yt = observation,
µt = slowly changing component (trend),
γt = periodic component (seasonal),
εt = irregular component (disturbance).
In a Structural Time Series Model (STSM)
or a Unobserved Components Model (UCM),
the components are modelled explicitly as stochastic processes.
Basic example is the local level model.
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3. Illustrations
We present various illustrations of time series analysis and
forecasting:
1. European business cycle
2. Bivariate analysis: decomposing and forecasting of Nordpool
daily (average) of spot prices and consumption.
3. Periodic dynamic factor analysis: joint modeling of 24 hours
in a daily panel of electricity loads.
4. Modelling house prices in Europe.
5. Modelling the U.S. Yield Curve.
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4. Illustration 1: European business cycle
Azevedo, Koopman and Rua (JBES, 2006) consider European
business cycle based on
• a multivariate model consisting of generalised components for
trend and cycle with band-pass filter properties;
• data-set includes nine time series (quarterly, monthly) where
individual series that may be leading/lagging GDP;
• a model where all equations have individual trends but share
one common “business cycle” component.
• a common cycle that is allowed to shift for individual time
series using techniques developed by R¨nstler (2002).
u
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5. Shifted cycles
0.2
0.0
−0.2
estimated cycles
gdp (red) versus
cons confidence (blue)
−0.4
1980 1985 1990 1995
0.2
0.0
−0.2
estimated cycles
gdp (red) versus
shifted cons confidence (blue)
−0.4
1980 1985 1990 1995
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6. Shifted cycles
In standard case, cycle ψt is generated by
ψt+1 cos λ sin λ ψt κt
+ =φ + +
ψt+1 − sin λ cos λ ψt κ+
t
The cycle
+
cos(ξλ)ψt + sin(ξλ)ψt ,
is shifted ξ time periods to the right (when ξ > 0) or to the left
(when ξ < 0).
Here, − 1 π < ξ0 λ < 2 π (shift is wrt ψt ).
2
1
More details in R¨nstler (2002) for idea of shifting cycles in
u
multivariate unobserved components time series model of
Harvey and Koopman (1997).
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7. The basic multivariate model
Panel of N economic time series, yit ,
(k) (m) +(m)
yit = µit + λi cos(ξi λ)ψt + sin(ξi λ)ψt + εit ,
where
• time series have mixed frequencies: quarterly and monthly;
(k)
• generalised individual trend µit for each equation;
(m) +(m)
• generalised common cycle based on ψt and ψt ;
• irregular εit .
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8. Business cycle
Stock and Watson (1999) states that fluctuations in aggregate
output are at the core of the business cycle so the cyclical
component of real GDP is a useful proxy for the overall business
cycle and therefore we impose a unit common cycle loading and
zero phase shift for Euro area real GDP.
Time series 1986 – 2002:
quarterly GDP
industrial production
unemployment (countercyclical, lagging)
industrial confidence
construction confidence
retail trade confidence
consumer confidence
retail sales
interest rate spread (leading)
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10. Details of model, estimation
• we have set m = 2 and k = 6 for generalised components
• leads to estimated trend/cycle estimates with band-pass
properties, Baxter and King (1999).
• frequency cycle is fixed at λ = 0.06545 (96 months, 8 years),
see Stock and Watson (1999) for the U.S. and ECB (2001)
for the Euro area
• shifts ξi are estimated
• number of parameters for each equation is four (σi2,ζ , λi , ξi ,
σi2,ε ) and for the common cycle is two (φ and σκ )
2
• total number is 4N = 4 × 9 = 36
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11. Decomposition of real GDP
14.2 0.003
14.1
0.002
14.0
0.001
GDP Euro Area Trend slope
13.9
1990 1995 2000 1990 1995 2000
0.01 0.0050
0.0025
0.00
0.0000
−0.0025
−0.01
−0.0050
Cycle irregular
1990 1995 2000 1990 1995 2000
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12. The business cycle coincident indicator
Selected estimation results
series load shift R2d
gdp −− −− 0.31
indutrial prod 1.18 6.85 0.67
Unemployment −0.42 −15.9 0.78
industriual c 2.46 7.84 0.47
construction c 0.77 1.86 0.51
retail sales c 0.26 −0.22 0.67
consumer c 1.12 3.76 0.33
retail sales 0.11 −4.70 0.86
int rate spr 0.57 16.8 0.22
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13. Coincident indicator for Euro area business cycle
0.010
0.005
0.000
−0.005
−0.010
−0.015
1990 1995 2000
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14. Coincident indicator for growth
• tracking economic activity growth is done by growth indicator
• we compare it with EuroCOIN indicator
• EuroCOIN is based on generalised dynamic factor model of
Forni, Hallin, Lippi and Reichlin (2000, 2004)
• it resorts to a dataset of almost thousand series referring to
six major Euro area countries
• we were able to get a quite similar outcome with a less
involved approach by any standard
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16. Illustration 2: Nord Pool data
• we consider Norwegian electricity prices and consumption
from Nord Pool.
• mostly hydroelectric power stations; supply depends on
weather.
• Norway’s yearly hydro power plant capacity is 115 Tw hours.
• Nord Pool is day ahead market: daily trades for next day
delivery.
• daily series of average of 24 hourly price and consumption.
• spot prices measured in Norwegian Kroner (8 NOK ≈ 1 Euro).
• sample: Jan 4, 1993 to April 10, 2005; 640 weeks or 4480
days.
• data are subject to yearly cycles, weekly patterns, level
changes, and jumps.
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17. Bivariate analysis: daily spot prices and consumption
Our unobserved components model is given by
2
yt = µt + γt + ψt + xt′ λ + εt , εt ∼ NID(0, σε ),
where
• yt is bivariate: electricity spot price and load consumption;
• µt is long term level;
• γt is seasonal effect with S = 7 (day of week effect);
• ψt is yearly cycle changes (summer/winter effects);
′
• xt λ has regression effects, mainly dummies for special days;
• εt is the irregular noise.
Parameter estimation and forecasting of observations have been
carried out by the STAMP 8 program of Koopman, Harvey,
Doornik and Shephard (2008, stamp-software.com):
user-friendly but still flexible, also for multivariate models.
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18. Daily spot electricity prices from the Nord Pool
(i) (ii)
0.1
6
0.0
5
4 −0.1
3
−0.2
100 200 300 400 500 600 100 200 300 400 500 600
(iii) (iv)
0.50
1
0.25
0
0.00
−1
−0.25
100 200 300 400 500 600 100 200 300 400 500 600
Univariate decomposition of Nord Pool daily prices January 4, 1993 to April 10, 2005:
(i) data and estimated trend plus regression; (ii) seasonal component (S = 7, the day-of-week effect); (iii) yearly
cycle; (iv) irregular.
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19. Joint decomposition of electricity prices & consumption
7
(i−a) 0.50
(i−b)
6 0.25
0.00
5 −0.25
450 500 550 600 450 500 550 600
(ii−a) 0.010
(ii−b)
0.05 0.005
0.000
−0.05
−0.005
450 500 550 600 450 500 550 600
(iii−a) 0.50
(iii−b)
1 0.25
0.00
0
−0.25
450 500 550 600 450 500 550 600
(iv−a) (iv−b)
0.2 0.025
0.0
−0.025
−0.2
450 500 550 600 450 500 550 600
Bivariate decomposition of prices and consumption: Feb 19, 2001 to April 10, 2005:
(ia,b) data and estimated trend plus regression; (iia,b) seasonal component (S = 7, the day-of-week effect); (iiia,b)
yearly cycle; (iva,b) irregular.
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20. Forecasting results
We present MAPE for forecasting of one- to seven-days ahead prices for both uni- and bivariate models. The one-
to seven-days ahead forecasts are for the next seven days. The first forecast is for Monday, March 14, 2005 in
Week 637. The last forecast is for Sunday, April 10, 2005 in Week 640. The weeks 638 and 639 contain calendar
effects for Maundy Thursday (March 24, 2005) and the days until Easter Monday (March 28, 2005).
week 637 week 638 week 639 week 640
uni biv uni biv uni biv uni biv
horizon
1M 0.83 1.11 0.15 0.07 0.83 1.01 0.92 0.27
2T 0.86 0.94 0.51 0.53 1.20 1.36 0.74 0.20
3W 1.43 1.55 0.67 0.79 1.40 1.52 0.62 0.16
4T 1.94 2.09 0.64 0.88 1.71 1.75 0.60 0.14
5F 1.69 1.93 0.65 0.72 2.01 2.00 0.60 0.30
6S 1.62 1.95 0.58 0.69 2.26 2.17 0.67 0.43
7S 1.61 2.05 0.68 0.90 2.44 2.27 0.79 0.56
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21. Illustration 3: periodic dynamic factor analysis
Aim: the joint modeling of 24 hours in a daily panel of electricity
loads for EDF.
Focus: modelling and short-term forecasting of hourly electricity
loads, from one day ahead to one week ahead.
• EDF provides a long time series: 9 years of hourly loads
• We can establish a long-term trend component but also
• different levels of seasonality (yearly, weekly, daily)
• special day effects (EJP)
• weather dependence (temperature, cloud cover)
• We look at the intra-year as well as the long-run dynamics by
using these different components.
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22. Periodic dynamic factor model specification
The adopted methodology builds on Dordonnat, et al (2008, IJF):
• Model is for high-frequency data, for hourly data);
• Hours are in the cross-section (yt is 24 × 1 vector);
• The model dynamics are formulated for days: a multivariate
daily time series model;
• In effect, we adopt a periodic approach to time series
modelling;
• The right-hand side of the model is set-up as a multiple
regression model;
• We let the regression parameters evolve over time (days);
• We have a time-varying regression model, written in
state-space form;
• The 24-dimensional time-varying parameters are subject to
common dynamics (random walks);
• Novelty: dynamic factors in the time-varying parameters.
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23. Daily National Electricity Load, 1995-2004
80000
80000
National Load (MegaWatts)
National Load (MegaWatts)
60000
60000
40000
40000
2000 2005 2002 2003
Year (a) Date (b)
80000
50000
National Load (MegaWatts)
National Load (MegaWatts)
60000
40000
40000
30000
0 5 10 15 20 −5 0 5 10 15 20 25 30
Days elapsed since August 8th,2004 (c) National Temperature (°C) (d)
Time series and temperature effects at 9 AM
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25. Multivariate Time Series Model
A periodic approach: from a univariate hourly to a daily 24 × 1
vector:
yt = (y1,t . . . yS,t )′ , S = 24 hours per day, t = 1, . . . , T days.
Our multivariate time-varying parameter regression model is given
by:
K
yt = µ t + Btk xtk + εt , εt ∼ IIN (0, Σε ) , t = 1, . . . , T ,
k=1
• Trends: µt = (µ1,t . . . µS,t )′
k k k
• Daily vectors of explanatory variables xt = (x1,t . . . xS,t )′ ,
k = 1, . . . , K , depending only on the day or on the hour of the
day.
k k k
• Regression coefficients βt = (β1,t . . . βS,t )′ , k = 1, . . . , K . In
matrix form: Bt k = diag (β k ), k = 1, . . . , K
t
• Irregular Gaussian white noise εt = (ε1,t . . . εS,t )′ .
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26. Time-varying regressions and dynamic factors
K
yt = µ t + Btk xtk + εt , εt ∼ IIN (0, Σε ) , t = 1, . . . , T ,
k=1
where the time-varying regression parameters are given by
µt = c 0 + Λ0 ft0 ,
k
βt = c k + Λk ftk , k = 1, . . . , K , t = 1, . . . , T ,
j j
with constant c j = (c1 . . . cS )′ , S × R j factor loading matrices Λj
and R j dynamic factors ftj = (f1,t . . . fR j ,t )′ , for j = 0, . . . , K and
j j
0 ≤ R j ≤ S.
• Factor structure requires 0 < R j < S;
• Constant parameter component for R j = 0;
• Model has unrestricted component when R j = S;
• Identification restrictions apply.
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27. Dynamic factor specfications
Local linear trend model for factors in trends µt :
ft+1 = ftj
j
+ gtj + v jt , v jt ∼ IIN(0, Σj )
v
j
gt+1 = gtj + w jt , w jt ∼ IIN(0, Σj )
w
• vector of dynamic factors ftj ,
• slope or gradient vector gtj ,
• level disturbance v jt and slope disturbance w jt .
Random walk model for factors in the regression coefficients:
ft+1 = ftj + e jt ,
j
e jt ∼ IIN(0, Σe ),
j
j = 1, . . . , K , t = 1, . . . , T ,
with regression coefficient disturbance e jt .
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28. Empirical application to French national hourly Loads
• French national hourly electricity loads from Sept-95 until
Aug-04
• Estimation of trivariate models for neighbouring hours
• Smooth trends
• Intentional missing values for special days (EJP) and turn of
the year. No problem for state space models.
• Yearly pattern regressors: sine/cosine functions of time are
used (2 frequencies)
• Day-of-the-week effects: day-type dummy regressors
• Weather dependence: heating degrees, smoothed-heating
degrees and cloud cover
• Heating degrees beneath treshold temperature of 15 C
• Exponentially smoothed temperature
• Cooling degrees above treshold temperature of 18 C
• Implemention: SsfPack 3 of Koopman, Shephard and
Doornik (2008, ssfpack.com) for Ox 6 (2008, doornik.com)
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29. Yearly pattern estimates per hour
4 ˆk k
µs,t +
ˆ k=1 βs,t xs,t for hours (a) s = 0, 1, 2 ; (b) s = 3, 4, 5 ; (c) s = 6, 7, 8 (with extra component), etc.
Estimation: Jan 1997 - Aug 2003, Graph: Jan 1998 - Aug 2003.
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33. Conclusions
• A general, flexible and insightful methodology is developed.
• Many dynamic features of load and price data can be
captured.
• We can detect many interesting signals which are not
discovered before.
• Decent forecasts.
• Decent diagnostics.
• Many possible extensions.
• Remaining challenge: a full multivariate unobserved
components model for all 24 hours to capture evolutions of
complete intradaily load pattern.
• More work is required !
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34. Short Bibliography
• “Multivariate structural time series models” by Harvey and
Koopman (1997), Chapter in Heij et al. (1997) Wiley.
• “Time-series analysis by state-space methods” by Durbin and
Koopman (Oxford, 2001)
• “Periodic Seasonal Reg-ARFIMA-GARCH Models for Daily
Electricity Spot Prices” by Koopman, Ooms and Carnero
(JASA, 2007).
• “An hourly periodic state-space for modelling French national
electricity load” by Dordonnat, et.al. (International Journal of
Forecasting, 2008)
• “Forecasting economic time series using unobserved
components time series models” by Koopman and Ooms
(2011), Chapter in Clements and Hendry, OUP Handbook of
Forecasting.
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35. Illustration 4: The macroeconomy in the euro area
Quarterly time series, 1981 – 2008, GDP in volumes,
for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.
(i) (ii)
13.2
12.8
13.0
12.6
12.8
12.4
1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010
12.7 (iii) 12.25 (iv)
12.6
12.00
12.5
11.75
12.4
12.3 11.50
1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010
35 / 91
36. Illustration 4: The housing market in the euro area
Quarterly time series, 1981 – 2008, real house prices (HP),
for countries (i) France, (ii) Germany, (iii) Italy and (iv) Spain.
(i) 0.3 (ii)
5.0
0.2
4.5 0.1
4.0 0.0
1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010
(iii) (iv)
3.0
0.25
0.00 2.5
−0.25 2.0
1980 1985 1990 1995 2000 2005 2010 1980 1985 1990 1995 2000 2005 2010
36 / 91
38. The basic multivariate model
Multiple set of M economic time series, yit , is collected in
yt = (y1t , . . . , yMt )′ and model is given by
(1) (2)
yt = µt + ψt + ψt + εt ,
where the disturbance driving each vector component is a vector
too, with a variance matrix. The structure of the variance matrix
determines the dynamic interrelationships between the M time
series.
For example, if trend component µt follows the random walk,
µt+1 = µt + ηt with disturbance vector ηt , with variance matrix
Ση :
• diagonal Ση , independent trends;
• rank(Ση ) = p < M, common trends (cointegration);
• rank(Ση ) = 1, single underlying trend;
• Ση is zero matrix, constant.
Similar considerations apply to other components.
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39. Dynamic factor representations
We can formulate the multivariate unobserved components model
also by
(1) (2)
yt = µ∗ + Aη µt + A(1) ψt
κ + A(2) ψt
κ + Aε εt ,
where, for the trend component, for example, the loading matrix
Aη is such that
′
Ση = Aη Aη ,
and, similarly, loading matrices are defined for the other variance
matrices of disturbances that drive the components.
(1) (2)
Here the dynamic factors or unobserved components µt , ψt , ψt
and εt are ”normalised”.
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40. STAMP
Model is effectively a state space model: Kalman filter methods
can be applied for maximum likelihood estimation of parameters
(such as the loading matrices).
Kalman filter methods are employed for the evaluation of the
likelihood function and score vector.
Kalman filter and smoothing methods are employed for signal
extraction or the estimation of the unobserved components.
User-friendly software is available for state space analysis.
We have used S T A M P for this research project: a
multi-platform, user-friendly software: econometrics, time series
and forecasting by clicking.
It can treat multivariate unobserved components time series
models...
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41. Motivation of our study
• Evidence of any relationship between housing prices and GDP
in the euro area.
• Focus on more recent developments...
• We prefer to model the time series jointly and establish
interrelationships between the time series
• Focus on cyclical dynamics, long-term and short-term
• Emphasis on real housing prices: relevant for the monetary
policy
• We also like to discuss synchronisation of housing markets in
euro area
Empirical results are based on our data-set with two variables:
GDP and real house prices (HP); and for four euro area countries:
France, Germany, Italy and Spain.
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42. Relevant literature
• Unobserved components model: Harvey (1989)
• State space methods: Durbin and Koopman (2001)
• Multivariate unobserved components: Harvey and Koopman
(1997), Azevedo, Koopman and Rua (2006);
• Parametric approaches for house prices:
• Probit regressions: Borio and McGuire, 2004, van den Noord,
2006;
• Dynamic Factor models: Terrones, 2004, DelNegro and Otrok,
2007, Stock and Watson, 2008;
• VAR: Vargas-Silva, 2008, Goodheart and Hofmann, 2008.
42 / 91
43. Univariate analysis
Objectives:
• Verify the trend-cycle decomposition for each series
• Verify whether possible restrictions are realistic
Results for GDP:
• two short cycles in France and Italy are detected (¡6 years);
• Germany and Spain contain both a short cycle (5.42 and 3.62
years, resp.) and a long cycle (13.5 and 9.11 years)
• Various cycles are deterministic (fixed sine-cosine wave)
Results for HP:
• Results are quite different for each series
• Two cycles for Germany (5.4 and 13.5 years)
• Two short cycles for Italy (3.0 and 5.5 years) and France (3.1
and 5.8 years)
• For Spain a cycle reduces to an AR(1) process
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44. Univariate results for GDP
France Germany Italy Spain
GDP R R R R
Trend var 0.65 0.03 0.01 0.03 0.48 0.03 0.10 0.03
Cycle 1 var 0.81 0.17 0.00 0.15 3.85 5.75 0.07 0.00
Cycle 1 ρ 0.94 0.90 1.0 0.90 0.87 0.90 0.95 0.90
Cycle 1 p 3.04 5 5.42 5 2.97 5 3.62 5
Cycle 2 var 0.00 1 1.81 2.86 0.00 7.79 0.00 2.31
Cycle 2 ρ 1.0 0.95 0.95 0.95 1.00 0.95 1.00 0.95
Cycle 2 p 5.8 12 13.5 12 5.50 12 9.11 12
Irreg var 1 0.0 1 1 1 1 1 1
N 7.2 11.4 3.23 5.23 6.58 11.1 27.1 34.9
Q 14.5 24.9 15.1 14.6 9.26 13.3 22.1 24.8
R2 0.31 0.24 0.11 0.02 0.23 0.12 0.22 0.12
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45. Univariate results for HP
France Germany Italy Spain
RHP R R R R
Trend var 0.59 0.03 0.34 0.03 0.00 0.03 0.39 0.03
Cycle 1 var 0.00 0.01 0.31 1.51 0.04 0.02 1 0.01
Cycle 1 ρ 1.0 0.90 0.97 0.90 0.96 0.90 0.34 0.90
Cycle 1 p 6.34 5 4.48 5 1.11 5 – 5
Cycle 2 var 0.00 2.19 1 19.9 1 49.4 0.00 39.5
Cycle 2 ρ 1.0 0.95 0.61 0.95 0.99 0.95 0.99 0.95
Cycle 2 p 8.37 12 2.82 12 13.3 12 – 12
Irreg var 1 1 0 1 0 1 0 1
N 23.8 0.59 5.89 9.95 7.03 8.32 36.1 11.9
Q 10.6 187 55.5 111 13.7 68.4 29.3 127
R2 0.61 0.25 0.35 0.15 0.56 0.22 0.47 0.28
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46. Cycle correlations from univariate analysis
(1) (2)
Correlations for combined cycles (ψt + ψt ):
• Strong correlations between GDP of four countries
(correlations range from 0.52 to 0.94)
• The correlations with German GDP are the lowest
• Correlations between HP of four countries range from 0.42 to
0.94
• The highest correlation is between Spain and France HP’s
• Correlation on combined cycle are mostly due to long-term
cycle, not to the short-term cycle
• Correlations between GDP and HP for each country range
from 0.06 for Germany to 0.76 for Spain
• Overall low cross-correlations between GDP of one country
and HP of another country
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47. Correlations between combined cycles for eight series
(1) (2)
Combined cycle (ψt + ψt )
F GDP F HP G GDP G HP I GDP I HP S GDP S HP
F GDP 1.00 0.51 0.52 0.23 0.83 0.15 0.89 0.61
F HP 0.51 1.00 0.44 0.44 0.52 0.68 0.68 0.94
G GDP 0.52 0.44 1.00 0.50 0.54 0.47 0.61 0.44
G HP 0.23 0.44 0.50 1.00 0.08 0.80 0.22 0.42
I GDP 0.83 0.52 0.54 0.08 1.00 0.06 0.84 0.64
I HP 0.15 0.68 0.47 0.80 0.06 1.00 0.29 0.64
S GDP 0.89 0.68 0.61 0.22 0.84 0.29 1.00 0.76
S HP 0.61 0.94 0.44 0.42 0.64 0.64 0.76 1.00
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48. Correlations between short cycle for eight series
(1)
Short cycle ψt
F GDP F HP G GDP G HP I GDP I HP S GDP S HP
F GDP 1.00 0.46 0.40 0.24 0.64 -0.46 0.57 0.42
F HP 0.46 1.00 0.29 0.62 0.33 -0.51 0.35 0.39
G GDP 0.40 0.29 1.00 0.32 0.75 -0.16 0.67 0.58
G HP 0.24 0.62 0.32 1.00 0.18 -0.52 0.06 0.13
I GDP 0.64 0.33 0.75 0.18 1.00 -0.13 0.61 0.65
I HP -0.46 -0.51 -0.16 -0.52 -0.13 1.00 -0.25 -0.19
S GDP 0.57 0.35 0.67 0.06 0.61 -0.25 1.00 0.75
S HP 0.42 0.39 0.58 0.13 0.65 -0.19 0.75 1.00
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49. Correlations between long cycle for eight series
(2)
Long cycle ψt
F GDP F HP G GDP G HP I GDP I HP S GDP S HP
F GDP 1.00 0.51 0.53 0.23 0.89 0.16 0.90 0.63
F HP 0.51 1.00 0.46 0.44 0.58 0.68 0.68 0.94
G GDP 0.53 0.46 1.00 0.52 0.44 0.49 0.62 0.46
G HP 0.23 0.44 0.52 1.00 0.07 0.82 0.22 0.43
I GDP 0.89 0.58 0.44 0.07 1.00 0.08 0.90 0.72
I HP 0.16 0.68 0.49 0.82 0.08 1.00 0.29 0.64
S GDP 0.90 0.68 0.62 0.22 0.90 0.29 1.00 0.76
S HP 0.63 0.94 0.46 0.43 0.72 0.64 0.76 1.00
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50. Bivariate analysis
For each country, we carry out a bivariate analysis between GDP
and RHP:
(1) (2)
yt = µt + ψt + ψt + εt ,
where yt is a 2 × 1 vector for two series: GDP and HP.
We can conclude that
• highest correlation is found for cycle components (except
Italy)
• for France, high correlation for medium-term cycle (8 years)
but no dependence for long-term cycle (15.6 years)
• for Spain, strong correlations for both medium-term (8.2
years) and long-term (14.4 years)
• for Germany, correlations for both cycles, but with low periods
(4.3 and 7 years)
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51. Bivariate results for GDP and HP
GDP RHP corr per ρ diag GDP RHP
FRA
trend 0.0 0.0 0.0 – – N 3.25 13.4
cyc 1 3.0 3.3 0.88 8.0 0.98 Q 17.0 17.4
cyc 2 1.0 126 0.07 15.6 0.99 R2 0.38 0.63
irreg 0.6 1.6 -0.19 – –
GER
trend 0.0 0.003 0.0 – – N 8.52 1.08
cyc 1 2.5 5.4 -0.6 4.3 0.90 Q 6.86 42.1
cyc 2 3.1 0.5 1.0 7.0 0.98 R2 0.39 0.29
irreg 4.3 1.1 0.58 – –
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52. Bivariate results for GDP and HP
GDP RHP corr per ρ diag GDP RHP
ITA
trend 0.1 0.9 -0.15 – – N 4.19 4.57
cyc 1 4.3 16.2 -0.08 6.0 0.92 Q 10.1 8.60
cyc 2 0.0 8.4 0.0 1.1 0.94 R2 0.14 0.47
irreg 0.8 1.2 0.96 – –
SPN
trend 0.0 0.0 0.0 – – N 9.05 21.7
cyc 1 3.3 11.9 0.95 8.2 0.98 Q 17.5 43.0
cyc 2 0.0 83.3 0.82 14.4 0.99 R2 0.45 0.73
irreg 3.9 7.7 -0.35 – –
52 / 91
53. Four-variate cross-country analysis of GDP and RHP
Now we incorporate earlier findings and impose a strict short- and
long-term cycle decomposition for our analysis.
In particular, we have
• an independent trend µt (i.e. diagonal variance matrix Ση for
disturbance vectors of µt+1 = µy + ηt )
• similarly, an independent irregular component εt (i.e. diagonal
variance matrix Σε )
• a two-cycle parametrization with restricted periods of 5 and
12 years
• the rank of the 4 × 4 cycle variance matrices Σκ is 2:
common cyles ...
• we load the two ”independent” cycles on France and Germany,
i.e. cyclical dynamics of Spain and Italy are obtained as linear
functions of the two times two (short and long) cyclical factors
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55. Four-variate results for cross-country: GDP
Fra Ger Ita Spn Fra Ger
Cycle short (cov ×10−6 ) factor loadings
Fra 4.11 0.25 ∗ 0.77 ∗ -0.40 ∗ 1 0
Ger 1.77 11.8 0.81 ∗ 0.78 ∗ 0 1
Ita 5.65 10.1 13.1 0.27 ∗ 1.08 0.69
Spn -1.04 3.50 1.27 1.65 -0.41 0.35
Cycle long (cov ×10−6 )
Fra 8.08 0.79 ∗ 0.48 ∗ 0.98 ∗ 1 0
Ger 7.94 12.5 -0.16 ∗ 0.64 ∗ 0 1
Ita 3.43 -1.39 6.28 0.66 ∗ 1.42 -1.02
Spn 11.2 9.11 6.73 16.4 1.79 -0.41
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56. Four-variate results for cross-country: GDP
• Diagnostic statistics are satisfactory
• Strong correlation France-Germany for long-term cycle
• Business cycles for Italy and Spain are closely connected with
the one for France (however, negative ??? marginal
correlation Fra-Spa for short-term cycle)
• German cycles strongly affect business cycles in Italy and
Spain (however, their marginal correlations for longer cycle are
negative)
56 / 91
58. Four-variate results for cross-country: HP
Fra Ger Ita Spn Fra Ger
Cycle short (cov ×10−6 ) factor loadings
Fra 15.5 0.37 ∗ -0.89 ∗ 0.05 ∗ 1 0
Ger 4.73 10.8 0.10 ∗ -0.91 ∗ 0 1
Ita -21.0 1.97 36.2 -0.50 ∗ -1.64 0.90
Spn 0.89 -14.6 -14.6 23.8 0.55 -1.60
Cycle long (cov ×10−6 )
Fra 44.5 0.38 ∗ 0.70 ∗ 0.93 ∗ 1 0
Ger 4.43 3.13 -0.40 ∗ 0.69 ∗ 0 1
Ita 66.9 -10.3 207.1 0.38 ∗ 2.13 -6.30
Spn 100.4 19.9 88.3 262.8 1.89 3.69
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59. Four-variate results for cross-country: HP
• Overall, these results seem to indicate that there is less
evidence of common (cyclical) dynamics in HP series
• Low correlations between France and Germany
• Strong negative correlations for the 5-year cycle between
Fra-Ita and Ger-Spa
• However, more commonalities for the 12-year cycle (Fra-Spa,
Fra-Ita, Ger-Spa)
• Similarities between correlation matrices for the 12-year HP
and GDP cycles, except that relationship Fra-Ger is stronger
for GDP (0.79 against 0.38 for HP)
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60. Eight-variate results: HP and GDP for four countries
Similar restrictions apply as in four-variate analyses.
We conclude that
• strong correlations among GDPs for short-term cycles but less
evidence for long-term cycles, especially for Germany
• low correlations among HP series.
• for short-term cycle, these correlations for HP Fra-Ger is 0.65
and for HP Spa-Ger is -0.95.
• only a few positive correlations for the long-term cycle in HP
have been found: Fra-Spa (0.58) and Ger-Ita (0.57)
• correlations HP-GDP are only found for long-term cycle,
especially for France and Spain.
60 / 91
61. Eight-variate results: short cycle correlations
France Germany Italy Spain
GDP HP GDP HP GDP HP GDP HP
F-G 1 -0.33 0.67 0.10 0.81 -0.59 0.77 0.13
F-H 1 0.075 0.65 -0.35 -0.13 -0.12 -0.64
G-G 1 0.17 0.80 -0.27 0.88 -0.011
G-H 1 0.055 -0.26 -0.10 -0.95
I-G 1 -0.037 0.66 0.034
I-H 1 -0.55 -0.040
S-G 1 0.34
S-H 1
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62. Eight-variate results: long cycle correlations
France Germany Italy Spain
GDP HP GDP HP GDP HP GDP HP
F-G 1 0.95 0.19 0.043 0.72 0.41 0.54 0.50
F-H 1 0.44 0.24 0.63 0.43 0.57 0.58
G-G 1 0.41 -0.31 0.26 0.44 0.21
G-H 1 -0.005 0.57 0.036 0.29
I-G 1 0.045 0.12 0.37
I-H 1 0.13 0.099
S-G 1 0.61
S-H 1
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64. Time Series of Four Maturities
Yield (in %) Time to maturity: 3 month
10 Yield (in %) Time to maturity: 1 year
8
8
6
6
4
4
Date Date
1985 1990 1995 2000 1985 1990 1995 2000
Yield (in %) Yield (in %)
Time to maturity: 3 year Time to maturity: 10 year
10
10.0
8
7.5
6
5.0
Date Date
1985 1990 1995 2000 1985 1990 1995 2000
64 / 91
65. Term Structure of Interest Rates over Time
10.0
Yield (Percent)
7.5
5.0
125
100
75 2000.0
Mat 1997.5
urity 50 1995.0
(Mo
nths 1992.5
) 25 1990.0 Time
1987.5
65 / 91
66. Literature Review
Earlier analyses of this data:
• Affine Term Structure Models (ATSM):
Vasicek (1977), Cox, Ingersoll, and Ross (1985), Duffie and
Kan (1996), Dai and Singleton (2000), and De Jong (2000)
• Nelson-Siegel Model (NS):
Nelson and Siegel (1987), Diebold and Li (2006), Diebold,
Rudebusch and Aruoba (2006), De Pooter (2007), and
Koopman, Mallee, and Van der Wel (2009)
• Arbitrage-Free Nelson-Siegel Model (AFNS):
Christensen, Diebold, and Rudebusch (2007)
• Functional Signal plus Noise (FSN):
Bowsher and Meeks (2008)
In all cases: a dynamic factor model set-up !
66 / 91
67. Still Some Outstanding Issues...
• Which of these models provides an accurate description of the
data?
• Duffee (2002) and Bams and Schotman (2003) provide
evidence against affine term structure models
• What are the dynamics of the underlying factors:
Stationary or Nonstationary?
• Stationary: Affine Term Structure Models, Nelson-Siegel,
Arbitrage-Free Nelson-Siegel
• Nonstationary: Campbell and Shiller (1987), Hall, Anderson
and Granger (1992), Engsted and Tanggaard (1994) and
Bowsher and Meeks (2008)
• What are the dynamics of the underlying factors: #lags?
• Almost all models take a VAR(1) for the factor dynamics
• Exception: Bowsher and Meeks (2008)
67 / 91
68. Further Outline
• (General) Dynamic Factor Model (DFM)
• General Set-Up
• Stationary and Nonstationary
• Smooth Dynamic Factor Model (SDFM)
• Specification
• Knot Selection
• Other Restrictions of the DFM
• Results
• Conclusion
68 / 91
69. The Dynamic Factor Model (DFM)
• Time series panel of N monthly yield observations
yt = (yt (τ1 ), . . . , yt (τN ))′ with yt (τi ) the yield at time t with
maturity τi
• The general dynamic factor model is given by:
yt = µy + Λft + εt , εt ∼ N(0, H),
ft = Uαt
αt+1 = µα + T αt + R ηt , ηt ∼ N(0, Q),
for t = 1, . . . , n
• In here ft is an r -dimensional stochastic process that is
generated by the p-dimensional state vector αt and ηt is a
q × 1 vector
• We take H diagonal and have for the p × 1 initial state
α1 ∼ N(a1 , P1 ), and assume N >> r , r ≤ p and p ≥ q
69 / 91
70. The Dynamic Factor Model (DFM) – Cont’d
• Vectors µy and µα , and matrices Λ, H, U, T and Q are
system matrices, R is selection matrix of ones and zeros
• Special case of state space model.
• All vector autoregressive moving average processes can be
formulated in this framework (see, e.g., Box, Jenkins and
Reinsel (1994))
• In this paper: VAR and Cointegrated VAR (CVAR) for ft .
Obtained by suitable specification of U, T and R
• Elements of system matrices µy , Λ, H, µα , T and Q generally
contain parameters that need to be estimated
70 / 91
71. The Dynamic Factor Model (DFM) – Cont’d
• Need to impose restrictions on loading and variance matrices
to ensure identification, see Jungbacker and Koopman (2008):
• Vectors µy and µα not both estimated without restrictions
=¿ Restrict µα = 0 to focus on loading matrix Λ
• Impose restrictions on Λ, T & Q that govern covariances
=¿ Restrict r rows in Λ to be r × r identity matrix:
1 0 0
0 1 0
0 0 1
Λ = λ4,1 λ4,2 λ4,3
. . .
. .
.
. . .
λN,1 λN,2 λN,3
71 / 91
72. Dynamic Factor Model (DFM) – Stationary Case
• Take a VAR(k) model for the r × 1 vector ft :
k−1
ft+1 = Γt−j ft−j + ζt , ζt ∼ NID(0, Qζ )
j=0
• Stationarity imposed by restriction that |Γ(z)| = 0 has all
roots outside the unit circle
• Can write this ft in DFM. For example, for k = 1 have
αt = ft ,
U = Ir ,
R = Ir ,
T = Γ0 ,
Q = Qζ
72 / 91
73. Dynamic Factor Model (DFM) – Nonstationary Case
• Now the ft are generated by a cointegrated vector
autoregressive process:
k−1
∆ft+1 = βγ ′ ft + Γj ∆ft−j + ζt , ζt ∼ N(0, Qζ ),
j=0
• The r × s matrices β and γ have full column rank; in the
nonstationary case s < r and all ft nonstationary
• We propose an alternative but observationally equivalent
specification for ft via factor rotation:
f¯N ′
f¯ = t
= β⊥ γ ft ,
t
f¯S
t
¯
also need to construct new loading matrix Λ = ¯
ΛN ¯
ΛS
73 / 91
74. Dynamic Factor Model (DFM) – Estimation
• As noted earlier, the Dynamic Factor Model (DFM) can be
seen as a special case of state space model
• Generally we can use likelihood-based methods: direct ML
and/or EM methods
• However. . .
• . . . the dimension of the observations vector is much larger
than the state vector
• . . . there is a large number of parameters (DFM-VAR(1),
N = 17, r = 3: 91 parameters)
• We therefore estimate the models using the methodology of
Jungbacker and Koopman (2008) and estimate parameters by
direct ML using analytical score expressions
74 / 91
75. Smooth Dynamic Factor Model (SDFM)
• For cross-sectional observation i we can write the DFM as:
r
yt (τi ) = µy ,i + λij fjt +εit , t = 1, . . . T , i = 1, . . . , N,
j=1
where λij is the loading of factor j on maturity i
• We propose to let the loading parameter be an unknown
function gj (·) for each factor j, where the argument of the
function relates to i
• Assume these functions g1 (·), . . . , gr (·) smooth functions of
time to maturity:
λij = gj (τi )
• In practice: cubic spline for each gj (·)
• Call this Smooth Dynamic Factor Model (SDFM)
75 / 91
76. Smooth Dynamic Factor Model (SDFM) – Spline
• In a spline the location of the knots determines how the factor
loadings behave for varying maturities
j
• Knot k for column j: sk
• Order the knots by time to maturity:
j j
τ 1 = s 1 < · · · < s Kj = τ N , j = 1, . . . , r
• Get following loading function:
j
gj (s1 )
.
.
gj (τi ) = wij λj , λj = . , j = 1, . . . , r ,
j
gj (sKj )
with wij a 1 × Kj vector (only depends on the knot locations)
and λj a Kj × 1 parameter vector
76 / 91
77. Smooth Dynamic Factor Model (SDFM) – Select Knots
• But how many knots Kj to select in the spline W λ?
• Small number of knots: Loadings lie on same polynomial for
considerable number of maturities
• Large number of knots: Get closer to unrestricted DFM
• Propose using a Wald test procedure to determine the knots
• This is standard as we are testing linear restrictions
• Amounts to an iterative general-to-specific approach:
1. Start with all knots
2. Calculate test statistic for all knots
3. Remove knot with smallest non-significant statistic
4. Continue with 2 and 3 until all knots are significant
77 / 91
78. Dynamic Factor Models for the Term Structure
• The general dynamic factor model is given by:
yt = µy + Λft + εt , εt ∼ N(0, H),
ft = Uαt
αt+1 = µα + T αt + R ηt , ηt ∼ N(0, Q),
• It nests the term structure models mentioned earlier
• Functional Signal plus Noise – Bowsher and Meeks (2008)
• Rather than a spline for the factor loadings they adopt the
Harvey and Koopman (1993) time-varying spline for the yield
curve:
yt = µy + Wft + εt , εt ∼ NID(0, H),
with W as before and ft time-varying knot values
• Take a CVAR(k) for ft and have restrictions on Λ
78 / 91
79. Dynamic Factor Models for the Term Structure – Cont’d
• Nelson-Siegel – Nelson and Siegel (1987), Diebold and Li
(2006), Diebold, Rudebusch and Aruoba (2006)
• The yield curve is expressed as a linear combination of smooth
factors
1 − e −λτ 1 − e −λτ
gns (τ ) = ξ1 + · ξ2 + − e −λτ · ξ3
λτ λτ
which gives
yt = µy + Λns ft + εt , εt ∼ NID(0, H)
• Interpretation as level, slope and curvature for the factors
• Typically: (restricted) VAR(1) for the state, µy = 0
• Restrictions on Λ
79 / 91
80. Dynamic Factor Models for the Term Structure – Cont’d
• Arbitrage-Free NS – Christensen, Diebold and Rudebusch
(2007)
• The NS model is not arbitrage free
• CDR employ “reverse engineering” and obtain an
Arbitrage-Free NS model
• Dynamics of latent factors now coming from solution of SDE,
plus ‘correction’ term for µy
• Restrictions on Λ, T and µy
• Affine Term Structure Models – Duffie and Kan (1996)
• Term structure can be explained by dynamics of unobserved
short rate
• Short rate depends on unobserved factors
• Focus on Gaussian case
• Restrictions on Λ, T and µy
80 / 91
81. Results
Strategy:
• Following, e.g., Litterman and Scheinkman (1991) we only
look at 3-factor models
• Restrict ourselves to Gaussian models
• Use an existing dataset: unsmoothed Fama-Bliss
• For DFM, SDFM, FSN and NS: VAR and CVAR
• For CVAR case focus on 1 random walk
We show the following results:
• VAR and CVAR for DFM
• Results for SDFM
• Estimation results NS, FSN, AFNS and ATSM
• In-sample fit of all models
• Validity of restrictions
81 / 91
82. DFM Likelihoods and AIC
Below we show the value of the loglikelihood at the ML value
(ℓ(ψ)) and AIC (AIC) for the Dynamic Factor Model (DFM):
Model ℓ(ψ) AIC Model ℓ(ψ) AIC
VAR(1) 3894.5 -7595 CVAR(1) 3899.0 -7606
VAR(2) 3918.5 -7625 CVAR(2) 3923.7 -7637
VAR(3) 3922.6 -7615 CVAR(3) 3927.7 -7627
VAR(4) 3932.2 -7616 CVAR(4) 3937.3 -7628
Note: Similar results hold for the NS and FSN model
82 / 91
88. NS Influence of Factor Dynamics on Loadings
To get a feeling how the choice of factor dynamics affects the
factor loadings we estimate the factor loadings parameter λ in the
Nelson-Siegel model for different choices of factor dynamics:
Model p=1 p=2 p=3 p=4
VAR(p) 0.07303 0.07211 0.07216 0.07193
CVAR(p) 0.07302 0.07210 0.07213 0.07191
Recall that for the Nelson-Siegel model we have
1 − e −λτ 1 − e −λτ
gns (τ ) = ξ1 + · ξ2 + − e −λτ · ξ3
λτ λτ
88 / 91
89. All Models - Overview
Finally, we provide an overview of all models and test the
restrictions imposed on the DFM by the various models
Model ℓ(ψ) nψ AIC
DFM-VAR(2) 3918.5 106 -7625
DFM-CVAR(2) 3923.7 105 -7637
SDFM-VAR(2) 3906.8 85 -7644
SDFM-CVAR(2) 3913.6 85 -7657
FSN-VAR(2) 3479.0 64 -6830
FSN-CVAR(2) 3483.7 63 -6841
NS-VAR(2) 3808.4 65 -7487
NS-CVAR(2) 3813.5 64 -7499
AFNS 3253.3 42 -6423
ATSM 3393.0 30 -6726
89 / 91