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 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 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 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.
1) This document describes an optimal monetary policy model with 7 endogenous variables and 5 equilibrium conditions, leaving two degrees of freedom.
2) The model maximizes social welfare as the sum of period utilities from consumption and labor, subject to the equilibrium conditions.
3) The first order optimality conditions result in a system of 7 equations that can be solved using log-linearization methods around the non-stochastic steady state, similarly to previous examples.
This document outlines the agenda for the second part of a lecture on Approximate Bayesian Computation (ABC). It begins with a discussion of simulation-based methods in econometrics like simulated method of moments. Next, it discusses the genetic origins and applications of ABC in population genetics, including coalescent theory. The document then covers using indirect inference to provide summary statistics for ABC and estimating demographic parameters from genetic data when the likelihood is intractable.
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.
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.
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 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 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.
1) This document describes an optimal monetary policy model with 7 endogenous variables and 5 equilibrium conditions, leaving two degrees of freedom.
2) The model maximizes social welfare as the sum of period utilities from consumption and labor, subject to the equilibrium conditions.
3) The first order optimality conditions result in a system of 7 equations that can be solved using log-linearization methods around the non-stochastic steady state, similarly to previous examples.
This document outlines the agenda for the second part of a lecture on Approximate Bayesian Computation (ABC). It begins with a discussion of simulation-based methods in econometrics like simulated method of moments. Next, it discusses the genetic origins and applications of ABC in population genetics, including coalescent theory. The document then covers using indirect inference to provide summary statistics for ABC and estimating demographic parameters from genetic data when the likelihood is intractable.
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.
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.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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
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 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
The document discusses three examples of nonlinear and non-Gaussian DSGE models. The first example features Epstein-Zin preferences to allow for a separation between risk aversion and the intertemporal elasticity of substitution. The second example models volatility shocks using time-varying variances. The third example aims to distinguish between the effects of stochastic volatility ("fortune") versus parameter drifting ("virtue") in explaining time-varying volatility in macroeconomic variables. The document outlines the motivation, structure, and solution methods for these three nonlinear DSGE models.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
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.
This document discusses heterogeneous agent models without aggregate uncertainty. It introduces a model with a continuum of agents who face idiosyncratic income fluctuations but no aggregate shocks. There is a unique stationary equilibrium with constant interest rates and wages. The document discusses the recursive competitive equilibrium, existence and uniqueness of the stationary equilibrium, transition functions, computation methods, and some qualitative results from calibrating the model.
The document describes the error correction model (ECM) version of Granger causality testing for determining the causal relationship between two non-stationary time series variables. It involves first testing for cointegration between the variables using the Johansen test or Engle-Granger approach. If cointegrated, the ECM version estimates an error correction model and performs Granger causality tests to examine short-run, long-run, and strong causality. The procedure and hypotheses for each test are provided along with the method for calculating the relevant F-statistics.
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 introduces perturbation methods as a way to solve functional equations that describe economic problems. It presents a basic real business cycle model as an example problem that can be solved using perturbation methods. Specifically, it:
1) Defines the real business cycle model as a functional equation system that is difficult to solve directly.
2) Proposes using perturbation methods by introducing a small perturbation parameter (the standard deviation of technology shocks) and solving the problem when this parameter equals zero.
3) Expands the decision rules as Taylor series in terms of the state variables and perturbation parameter to build a local approximation around the deterministic steady state. This leads to a system of equations that can be solved order-by-order for
Seminar Talk: Multilevel Hybrid Split Step Implicit Tau-Leap for Stochastic R...Chiheb Ben Hammouda
The document describes a multilevel hybrid split-step implicit tau-leap method for simulating stochastic reaction networks. It begins with background on modeling biochemical reaction networks stochastically. It then discusses challenges with existing simulation methods like the chemical master equation and stochastic simulation algorithm. The document introduces the split-step implicit tau-leap method as an improvement over explicit tau-leap for stiff systems. It proposes a multilevel Monte Carlo estimator using this method to efficiently estimate expectations of observables with near-optimal computational work.
The document describes the standard Granger causality test procedure for determining whether changes in one time series (Xt) can help forecast changes in another (Yt).
[1] The test involves estimating a VAR model with the two time series and their lags, then comparing the restricted and unrestricted models to calculate an F-statistic.
[2] If the F-statistic exceeds the critical value, the null hypothesis that Xt does not help forecast Yt (or vice versa) is rejected, indicating Granger causality between the two time series.
[3] The test results help explain the relationship between the changes in the two time series by determining the direction of causality (if any
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
The document summarizes the Toda-Yamamoto augmented Granger causality test.
[1] The test allows checking for causality between integrated variables of different orders without needing to determine cointegration. It involves estimating a VAR model with maximal order of integration lags added.
[2] The test procedure involves determining the order of integration (d), selecting the optimal lag length (k), setting the null and alternative hypotheses of no causality and causality, and calculating an F-statistic to test for causality.
[3] If the F-statistic exceeds the critical value, the null of no causality is rejected, indicating causality between the variables.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
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 contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
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
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 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
The document discusses three examples of nonlinear and non-Gaussian DSGE models. The first example features Epstein-Zin preferences to allow for a separation between risk aversion and the intertemporal elasticity of substitution. The second example models volatility shocks using time-varying variances. The third example aims to distinguish between the effects of stochastic volatility ("fortune") versus parameter drifting ("virtue") in explaining time-varying volatility in macroeconomic variables. The document outlines the motivation, structure, and solution methods for these three nonlinear DSGE models.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
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.
This document discusses heterogeneous agent models without aggregate uncertainty. It introduces a model with a continuum of agents who face idiosyncratic income fluctuations but no aggregate shocks. There is a unique stationary equilibrium with constant interest rates and wages. The document discusses the recursive competitive equilibrium, existence and uniqueness of the stationary equilibrium, transition functions, computation methods, and some qualitative results from calibrating the model.
The document describes the error correction model (ECM) version of Granger causality testing for determining the causal relationship between two non-stationary time series variables. It involves first testing for cointegration between the variables using the Johansen test or Engle-Granger approach. If cointegrated, the ECM version estimates an error correction model and performs Granger causality tests to examine short-run, long-run, and strong causality. The procedure and hypotheses for each test are provided along with the method for calculating the relevant F-statistics.
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 introduces perturbation methods as a way to solve functional equations that describe economic problems. It presents a basic real business cycle model as an example problem that can be solved using perturbation methods. Specifically, it:
1) Defines the real business cycle model as a functional equation system that is difficult to solve directly.
2) Proposes using perturbation methods by introducing a small perturbation parameter (the standard deviation of technology shocks) and solving the problem when this parameter equals zero.
3) Expands the decision rules as Taylor series in terms of the state variables and perturbation parameter to build a local approximation around the deterministic steady state. This leads to a system of equations that can be solved order-by-order for
Seminar Talk: Multilevel Hybrid Split Step Implicit Tau-Leap for Stochastic R...Chiheb Ben Hammouda
The document describes a multilevel hybrid split-step implicit tau-leap method for simulating stochastic reaction networks. It begins with background on modeling biochemical reaction networks stochastically. It then discusses challenges with existing simulation methods like the chemical master equation and stochastic simulation algorithm. The document introduces the split-step implicit tau-leap method as an improvement over explicit tau-leap for stiff systems. It proposes a multilevel Monte Carlo estimator using this method to efficiently estimate expectations of observables with near-optimal computational work.
The document describes the standard Granger causality test procedure for determining whether changes in one time series (Xt) can help forecast changes in another (Yt).
[1] The test involves estimating a VAR model with the two time series and their lags, then comparing the restricted and unrestricted models to calculate an F-statistic.
[2] If the F-statistic exceeds the critical value, the null hypothesis that Xt does not help forecast Yt (or vice versa) is rejected, indicating Granger causality between the two time series.
[3] The test results help explain the relationship between the changes in the two time series by determining the direction of causality (if any
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
The document summarizes the Toda-Yamamoto augmented Granger causality test.
[1] The test allows checking for causality between integrated variables of different orders without needing to determine cointegration. It involves estimating a VAR model with maximal order of integration lags added.
[2] The test procedure involves determining the order of integration (d), selecting the optimal lag length (k), setting the null and alternative hypotheses of no causality and causality, and calculating an F-statistic to test for causality.
[3] If the F-statistic exceeds the critical value, the null of no causality is rejected, indicating causality between the variables.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
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.
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.
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.
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.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
The document discusses the design of IIR digital filters using different methods. It begins by describing the difference equation and transfer function of IIR filters. It then covers the Impulse Invariant Method and Bilinear Z-Transform (BZT) Method for designing IIR filters by transforming analog prototypes. Key steps include prewarping frequencies, designing analog filters, and applying the bilinear transform. Examples demonstrate applying these methods to design Butterworth filters.
The document summarizes two models:
1. The Lo-Zivot Threshold Cointegration Model, which uses a threshold vector error correction model (TVECM) to analyze the dynamic adjustment of cointegrated time series variables to their long-run equilibrium. It allows for nonlinear and asymmetric adjustment speeds.
2. A bivariate vector error correction model (VECM) and band-threshold vector error correction model (BAND-TVECM) that extend the VECM to allow for nonlinear and discontinuous adjustments to long-run equilibrium across multiple regimes defined by thresholds on a variable. This captures asymmetric adjustment speeds and dynamic behavior.
The BAND-TVECM allows modeling of
1. Position angle (θ) is measured in revolutions, degrees, or radians. Common units include 1 revolution = 360° = 2π radians.
2. Angular displacement (Δθ) is the change in position angle between an initial angle (θ1) and final angle (θ2).
3. Angular velocity (ω) is the rate of change of the angular displacement with respect to time and is measured in radians/second. Average and instantaneous angular velocity can be calculated.
The document outlines auto-regressive (AR) processes of order p. It begins by introducing AR(p) processes formally and discussing white noise. It then derives the first and second moments of an AR(p) process. Specific details are provided about AR(1) and AR(2) processes, including equations for their variance as a function of the noise variance and AR coefficients. Examples of simulated AR(1) processes are shown for different coefficient values.
The document analyzes the motion of an inverted pendulum with an oscillating support. It uses Lagrangian mechanics to derive an equation of motion for the pendulum's angle θ. For small oscillations of θ, the equation is approximated as θ̈ + θaω2cos(ωt) - ω02 = 0. This shows the pendulum's motion depends on the frequency and amplitude of the support oscillations. If the support frequency ω is high enough, the pendulum undergoes stable oscillations instead of falling over. An analytical approximation of θ as a small oscillation correlated with the support motion is derived, explaining this stabilizing effect.
The document analyzes the motion of an inverted pendulum with an oscillating support. It uses Lagrangian mechanics to derive an equation of motion for the pendulum's angle θ. For small oscillations of θ, the equation is approximated as θ̈ + θaω2cos(ωt) - ω02 = 0, where a and ω are constants related to the support oscillation. If the support oscillation frequency ω is high enough, θ undergoes small oscillations correlated with cos(ωt), preventing the pendulum from falling over on average. The full motion of θ consists of these small oscillations modulated by an overall oscillation with frequency Ω related to ω and a.
This document provides an introduction to Hidden Markov Models (HMMs). It begins by explaining the key differences between Markov Models and HMMs, noting that in HMMs the states are hidden and can only be indirectly observed through observations. It then outlines the main elements of an HMM - the number of states, observations, state transition probabilities, observation probabilities, and initial state distribution. An example HMM is provided. Finally, it briefly introduces three common problems in HMMs - determining the most likely model given observations, determining the most likely state sequence, and determining the model parameters that are most likely to have generated the observations.
1) The document discusses several models for pricing derivatives in energy markets, including a 2-factor model, an Exode model, and models based on Levy processes.
2) It proposes an asset price model that incorporates mean-reversion, seasonal variations in forward curves, and the leptokurtic behavior of log-spot prices using a Levy process.
3) The model discretizes the Levy process using a Euler scheme and estimates the model parameters using maximum likelihood methods.
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.
The document summarizes key concepts about the Laplace transform. It defines the Laplace transform, discusses properties like linearity and time shifting. It provides examples of taking the Laplace transform of unit step functions. It also covers computing the inverse Laplace transform using partial fraction expansion and handling cases with repeated or complex poles.
Introduction to modern time series analysisSpringer
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1. State space methods for signal extraction,
parameter estimation and forecasting
Siem Jan Koopman
http://personal.vu.nl/s.j.koopman
Department of Econometrics
VU University Amsterdam
Tinbergen Institute
2012
2. Regression Model in time series
For a time series of a dependent variable yt and a set of regressors
Xt , we can consider the regression model
yt = µ + Xt β + εt , εt ∼ NID(0, σ 2 ), t = 1, . . . , n.
For a large n, is it reasonable to assume constant coefficients µ, β
and σ 2 ?
Assume we have strong reasons to believe that µ and β are
time-varying, the model may be written as
yt = µt + Xt βt + εt , µt+1 = µt + ηt , βt+1 = βt + ξt .
This is an example of a linear state space model.
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3. State Space Model
Linear Gaussian state space model is defined in three parts:
→ State equation:
αt+1 = Tt αt + Rt ζt , ζt ∼ NID(0, Qt ),
→ Observation equation:
yt = Zt αt + εt , εt ∼ NID(0, Ht ),
→ Initial state distribution α1 ∼ N (a1 , P1 ).
Notice that
• ζt and εs independent for all t, s, and independent from α1 ;
• observation yt can be multivariate;
• state vector αt is unobserved;
• matrices Tt , Zt , Rt , Qt , Ht determine structure of model.
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4. State Space Model
• state space model is linear and Gaussian: therefore properties
and results of multivariate normal distribution apply;
• state vector αt evolves as a VAR(1) process;
• system matrices usually contain unknown parameters;
• estimation has therefore two aspects:
• measuring the unobservable state (prediction, filtering and
smoothing);
• estimation of unknown parameters (maximum likelihood
estimation);
• state space methods offer a unified approach to a wide range
of models and techniques: dynamic regression, ARIMA, UC
models, latent variable models, spline-fitting and many ad-hoc
filters;
• next, some well-known model specifications in state space
form ...
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5. Regression with time-varying coefficients
General state space model:
αt+1 = Tt αt + Rt ζt , ζt ∼ NID(0, Qt ),
yt = Zt αt + εt , εt ∼ NID(0, Ht ).
Put regressors in Zt (that is Zt = Xt ),
Tt = I , Rt = I ,
Result is regression model yt = Xt βt + εt with time-varying
coefficient βt = αt following a random walk.
In case Qt = 0, the time-varying regression model reduces to the
standard linear regression model yt = Xt β + εt with β = α1 .
Many dynamic specifications for the regression coefficient can be
considered, see below.
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6. AR(1) process in State Space Form
Consider AR(1) model yt+1 = φyt + ζt , in state space form:
αt+1 = Tt αt + Rt ζt , ζt ∼ NID(0, Qt ),
yt = Zt αt + εt , εt ∼ NID(0, Ht ).
with state vector αt and system matrices
Zt = 1, Ht = 0,
Tt = φ, Rt = 1, Qt = σ 2
we have
• Zt and Ht = 0 imply that α1t = yt ;
• State equation implies yt+1 = φ1 yt + ζt with ζt ∼ NID(0, σ 2 );
• This is the AR(1) model !
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7. AR(2) in State Space Form
Consider AR(2) model yt+1 = φ1 yt + φ2 yt−1 + ζt , in state space
form:
αt+1 = Tt αt + Rt ζt , ζt ∼ NID(0, Qt ),
yt = Zt αt + εt , εt ∼ NID(0, Ht ).
with 2 × 1 state vector αt and system matrices:
Zt = 1 0 , Ht = 0
φ1 1 1
Tt = , Rt = , Qt = σ 2
φ2 0 0
we have
• Zt and Ht = 0 imply that α1t = yt ;
• First state equation implies yt+1 = φ1 yt + α2t + ζt with
ζt ∼ NID(0, σ 2 );
• Second state equation implies α2,t+1 = φ2 yt ;
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8. MA(1) in State Space Form
Consider MA(1) model yt+1 = ζt + θζt−1 , in state space form:
αt+1 = Tt αt + Rt ζt , ζt ∼ NID(0, Qt ),
yt = Zt αt + εt , εt ∼ NID(0, Ht ).
with 2 × 1 state vector αt and system matrices:
Zt = 1 0 , Ht = 0
0 1 1
Tt = , Rt = , Qt = σ 2
0 0 θ
we have
• Zt and Ht = 0 imply that α1t = yt ;
• First state equation implies yt+1 = α2t + ζt with
ζt ∼ NID(0, σ 2 );
• Second state equation implies α2,t+1 = θζt ;
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9. General ARMA in State Space Form
Consider ARMA(2,1) model
yt = φ1 yt−1 + φ2 yt−2 + ζt + θζt−1
in state space form
yt
αt =
φ2 yt−1 + θζt
Zt = 1 0 , Ht = 0,
φ1 1 1
Tt = , Rt = , Qt = σ 2
φ2 0 θ
All ARIMA(p, d, q) models have a (non-unique) state space
representation.
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12. Kalman Filter
• The Kalman filter calculates the mean and variance of the
unobserved state, given the observations.
• The state is Gaussian: the complete distribution is
characterized by the mean and variance.
• The filter is a recursive algorithm; the current best estimate is
updated whenever a new observation is obtained.
• To start the recursion, we need a1 and P1 , which we assumed
given.
• There are various ways to initialize when a1 and P1 are
unknown, which we will not discuss here.
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13. Kalman Filter
The unobserved state αt can be estimated from the observations
with the Kalman filter:
vt = yt − Z t a t ,
Ft = Zt Pt Zt′ + Ht ,
Kt = Tt Pt Zt′ Ft−1 ,
at+1 = Tt at + Kt vt ,
Pt+1 = Tt Pt Tt′ + Rt Qt Rt′ − Kt Ft Kt′ ,
for t = 1, . . . , n and starting with given values for a1 and P1 .
• Writing Yt = {y1 , . . . , yt },
at+1 = E(αt+1 |Yt ), Pt+1 = Var(αt+1 |Yt ).
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14. Lemma I : multivariate Normal regression
Suppose x and y are jointly Normally distributed vectors with
means E(x) and E(y ), respectively, with variance matrices Σxx and
Σyy , respectively, and covariance matrix Σxy . Then
−1
E(x|y ) = µx + Σxy Σyy (y − µy ),
Var(x|y ) = Σxx − Σxy Σyy Σ′ .
−1
xy
Define e = x − E(x|y ). Then
−1
Var(e) = Var([x − µx ] − Σxy Σyy [y − µy ])
= Σxx − Σxy Σyy Σ′
−1
xy
= Var(x|y ),
and
−1
Cov(e, y ) = Cov([x − µx ] − Σxy Σyy [y − µy ], y )
= Σxy − Σxy = 0.
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15. Lemma II : an extension
The proof of the Kalman filter uses a slight extension of the lemma
for multivariate Normal regression theory.
Suppose x, y and z are jointly Normally distributed vectors with
E(z) = 0 and Σyz = 0. Then
−1
E(x|y , z) = E(x|y ) + Σxz Σzz z,
Var(x|y , z) = Var(x|y ) − Σxz Σzz Σ′ ,
−1
xz
Can you give a proof of Lemma II ?
Hint : apply Lemma I for x and y ∗ = (y ′ , z ′ )′ .
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16. Kalman Filter : derivation
State space model: αt+1 = Tt αt + Rt ζt , yt = Zt αt + εt .
• Writing Yt = {y1 , . . . , yt }, define
at+1 = E(αt+1 |Yt ), Pt+1 = Var(αt+1 |Yt );
• The prediction error is
vt = yt − E(yt |Yt−1 )
= yt − E(Zt αt + εt |Yt−1 )
= yt − Zt E(αt |Yt−1 )
= yt − Z t a t ;
• It follows that vt = Zt (αt − at ) + εt and E(vt ) = 0;
• The prediction error variance is Ft = Var(vt ) = Zt Pt Zt′ + Ht .
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17. Kalman Filter : derivation
State space model: αt+1 = Tt αt + Rt ζt , yt = Zt αt + εt .
• We have Yt = {Yt−1 , yt } = {Yt−1 , vt } and E(vt yt−j ) = 0 for
j = 1, . . . , t − 1;
• Lemma E(x|y , z) = E(x|y ) + Σxz Σ−1 z, and take x = αt+1 ,
zz
y = Yt−1 and z = vt = Zt (αt − at ) + εt ;
• It follows that E(αt+1 |Yt−1 ) = Tt at ;
• Further, E(αt+1 vt′ ) = Tt E(αt vt′ ) + Rt E(ζt vt′ ) = Tt Pt Zt′ ;
• We carry out lemma and obtain the state update
at+1 = E(αt+1 |Yt−1 , yt )
= Tt at + Tt Pt Zt′ Ft−1 vt
= Tt at + Kt vt ;
with Kt = Tt Pt Zt′ Ft−1
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18. Kalman Filter : derivation
• Our best prediction of yt is Zt at . When the actual
observation arrives, calculate the prediction error
vt = yt − Zt at and its variance Ft = Zt Pt Zt′ + Ht . The new
best estimates of the state mean is based on both the old
estimate at and the new information vt :
at+1 = Tt at + Kt vt ,
similarly for the variance:
Pt+1 = Tt Pt Tt′ + Rt Qt Rt′ − Kt Ft Kt′ .
Can you derive the updating equation for Pt+1 ?
• The Kalman gain
Kt = Tt Pt Zt′ Ft−1
is the optimal weighting matrix for the new evidence.
• You should be able to replicate the proof of the Kalman filter
for the Local Level Model (DK, Chapter 2).
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20. Prediction and Filtering
This version of the Kalman filter computes the predictions of the
states directly:
at+1 = E(αt+1 |Yt ), Pt+1 = Var(αt+1 |Yt ).
The filtered estimates are defined by
at|t = E(αt |Yt ), Pt|t = Var(αt |Yt ),
for t = 1, . . . , n.
The Kalman filter can be “re-ordered” to compute both:
′
at|t = at + Mt vt , Pt|t = Pt − Mt Ft Mt ,
at+1 = Tt at|t , Pt+1 = Tt Pt|t Tt′ + Rt Qt Rt′ ,
with Mt = Pt Zt′ Ft−1 for t = 1, . . . , n and starting with a1 and P1 .
Compare Mt with Kt . Verify this version of KF using earlier
derivations.
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21. Smoothing
• The filter calculates the mean and variance conditional on Yt ;
• The Kalman smoother calculates the mean and variance
conditional on the full set of observations Yn ;
• After the filtered estimates are calculated, the smoothing
recursion starts at the last observations and runs until the
first.
αt = E(αt |Yn ),
ˆ Vt = Var(αt |Yn ),
rt = weighted sum of future innovations, Nt = Var(rt ),
Lt = Tt − Kt Zt .
Starting with rn = 0, Nn = 0, the smoothing recursions are given
by
rt−1 = Ft−1 vt + Lt rt , Nt−1 = Ft−1 + L′ Nt Lt ,
t
αt = at + Pt rt−1 ,
ˆ Vt = Pt − Pt Nt−1 Pt .
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24. Missing Observations
Missing observations are very easy to handle in Kalman filtering:
• suppose yj is missing
• put vj = 0, Kj = 0 and Fj = ∞ in the algorithm
• proceed further calculations as normal
The filter algorithm extrapolates according to the state equation
until a new observation arrives. The smoother interpolates
between observations.
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27. Forecasting
Forecasting requires no extra theory: just treat future observations
as missing:
• put vj = 0, Kj = 0 and Fj = ∞ for j = n + 1, . . . , n + k
• proceed further calculations as normal
• forecast for yj is Zj aj
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29. Parameter Estimation
The system matrices in a state space model typically depends on a
parameter vector ψ. The model is completely Gaussian; we
estimate by Maximum Likelihood.
The loglikelihood af a time series is
n
log L = log p(yt |Yt−1 ).
t=1
In the state space model, p(yt |Yt−1 ) is a Gaussian density with
mean at and variance Ft :
n
nN 1
log L = − log 2π − log |Ft | + vt′ Ft−1 vt ,
2 2
t=1
with vt and Ft from the Kalman filter. This is called the prediction
error decomposition of the likelihood. Estimation proceeds by
numerically maximising log L.
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30. ML Estimate of Nile Data
1400
observation level (q = 0)
level (q = 1000) level (q = 0.0973 = ML estimate)
1300
1200
1100
1000
900
800
700
600
500
1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970
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31. Diagnostics
• Null hypothesis: standardised residuals
√
vt / F t ∼ NID(0, 1)
• Apply standard test for Normality, heteroskedasticity, serial
correlation;
• A recursive algorithm is available to calculate smoothed
disturbances (auxilliary residuals), which can be used to detect
breaks and outliers;
• Model comparison and parameter restrictions: use likelihood
based procedures (LR test, AIC, BIC).
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33. Assignment 1 - Exercises 1 and 2
1. Formulate the following models in state space form:
• ARMA(3,1) process;
• ARMA(1,3) process;
• ARIMA(3,1,1) process;
• AR(3) plus noise model;
• Random Walk plus AR(3) process.
Please discuss the initial conditions for αt in all cases.
2. Derive a Kalman filter for the local level model
2 2
yt = µ t + ξ t , ξt ∼ N(0, σξ ), ∆µt+1 = ηt ∼ N(0, ση ),
with E (ξt ηt ) = σξη = 0 and E (ξt ηs ) = 0 for all t, s and t = s.
Also discuss the problem of missing obervations in this case.
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34. Assignment 1 - Exercise 3
3.
Consider a time series for yt that is modelled by a state space
model and consider a time series vector of k exogenous variables
Xt . The dependent time series yt depend on the explanatory
variables Xt in a linear way.
• How would you introduce Xt as regression effects in the state
space model ?
• Can you allow the transition matrix Tt to depend on Xt too ?
• Can you allow the transition matrix Tt to depend on yt ?
For the last two questions, please also consider maximum
likelihood estimation of coefficients within the system matrices.
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35. Assignment 1 - Computing work
• Consider Chapter 2 of Durbin and Koopman book.
• There are 8 figures in Chapter 2.
• Write computer code that can reproduce all these figures in
Chapter 2, except Fig. 2.4.
• Write a short documentation for the Ox program.
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