Chapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29
|
|
- Jordan Ellis
- 5 years ago
- Views:
Transcription
1 Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29
2 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting the period in time in which x occurs. We shall treat x t as a random variable; hence, a time-series is a sequence of random variables ordered in time. Such a sequence is known as a stochastic process. The probability structure of a sequence of random variables is determined by the joint distribution of a stochastic process. The simplest possible probability model for such a joint distribution is: x t = α + ɛ t, ɛ t n.i.d. 0, σ 2 ɛ i.e., x t is normally independently distributed over time with constant variance and mean equal to α. In other words, x t is the sum of a constant and a white-noise process. If a white-noise process were a proper model for financial time-series, forecasting would not be very interesting as the best forecast for the moments of the relevant time series would be their unconditional moments. () Chapter 5 Univariate time-series analysis 2 / 29,
3 Better models The model: x t = α + ɛ t, ɛ t n.i.d. 0, σ 2 ɛ, ˆ α = 1 T ˆ 2 T 1 T x t, σɛ = x t ˆα T t=1 t=1 2 Reflect the traditional approach to portfolio allocation, but it does not reflect the data. At high frequency the variance is not constant and predictable, at low frequency returns are persistent and predictable. () Chapter 5 Univariate time-series analysis 3 / 29
4 Better models US Stock Returns 1m sim wn US Stock Returns 1m sim wn While the CER gives a plausible representation for the 1-month returns, the behaviour over time of the YTM of the 10-Year bond does not resemble at all that of the simulated data. () Chapter 5 Univariate time-series analysis 4 / 29
5 ARMA modelling A more general and more flexible class of models emerges when combinations of ɛ t are used to model x t. We concentrate on a class of models created by taking linear combinations of the white noise, the autoregressive moving average (ARMA) models: AR(1) : x t = ρx t 1 + ɛ t, MA(1) : x t = ɛ t + θɛ t 1, AR(p) : x t = ρ 1 x t 1 + ρ 2 x t ρ p x t p + ɛ t, MA(q) : x t = ɛ t + θ 1 ɛ t θ q ɛ t q, ARMA(p, q) : x t = ρ 1 x t ρ p x t p + θ 1 ɛ t θ q ɛ t q. () Chapter 5 Univariate time-series analysis 5 / 29
6 Analysing time-series models To illustrate empirically all fundamentals we consider a specific member of the ARMA family, the AR model with drift, x t = ρ 0 + ρ 1 x t 1 + ɛ t, (1) ɛ t n.i.d. 0, σ 2 ɛ. Given that each realization of our stochastic process is a random variable, the first relevant fundamental is the density of each observation. In particular, we distinguish between conditional and unconditional densities. () Chapter 5 Univariate time-series analysis 6 / 29
7 Conditional and Unconditional Densities The unconditional density is obtained under the hypothesis that no observation on the time-series is available, while conditional densities are based on the observation of some realization of random variables. In the case of time-series, we derive unconditional density by putting ourselves at the moment preceding the observation of any realization of the time-series. At that moment the information set contains only the knowledge of the process generating the observations. As observations become available, we can compute conditional densities. () Chapter 5 Univariate time-series analysis 7 / 29
8 Conditional Densities Consider the AR(1) model. The moments of the density of x t conditional upon x t 1 are immediately obtained from the relevant process: E (x t j x t 1 ) = ρ 0 + ρ 1 x t 1, Var (x t j x t 1 ) = σ 2 ɛ, Cov (x t j x t 1 ), x t j j x t j 1 = 0 for each j. To derive the moments of the density of x t conditional upon x t need to substitute x t 2 from (1) for x t 1 : 2, we E (x t j x t 2 ) = ρ 0 + ρ 0 ρ 1 + ρ 2 1 x t 2, Var (x t j x t 2 ) = σ 2 ɛ 1 + ρ 2 1, Cov (x t j x t 2 ), x t j j x t j 2 = ρ 1 σ 2 ɛ, for j = 1, Cov (x t j x t 2 ), x t j j x t j 2 = 0, for j > 1. () Chapter 5 Univariate time-series analysis 8 / 29
9 Unconditional Densities Unconditional moments are derived by substituting recursively from to express x t as a function of information available at time t 0, the moment before we start observing realizations of our process. E (x t ) = ρ ρ 1 + ρ 2 1 Var (x t ) = σ 2 ɛ 1 + ρ ρ ρt ρ2t 2 1 γ (j) = Cov x t, x t j = ρ j 1 Var (x t), ρ (j) = + ρ t 1 x 0,, Cov x t, x t p Var (xt ) Var (x t 1 ) = ρ j 1 Var (x t) p Var (xt ) Var (x t 1 ). j Note that γ (j) and ρ (j) are functions of j, known respectively as the autocovariance function and the autocorrelation function. () Chapter 5 Univariate time-series analysis 9 / 29
10 Stationarity A stochastic process is strictly stationary if its joint density function does not depend on time. More formally, a stochastic process is stationary if, for each j 1, j 2,..., j n, the joint distribution, f x t, x t+j1, x t+j2, x t+jn, does not depend on t. A stochastic process is covariance stationary if its two first unconditional moments do not depend on time, i.e. if the following relations are satisfied for each h, i, j: E (x t ) = E (x t+h ) = µ, E x 2 t = E x 2 t+h = µ 2, E x t+i x t+j = µ ij. () Chapter 5 Univariate time-series analysis 10 / 29
11 Stationarity In the case of our AR(1) process, the condition for stationarity is jρ 1 j < 1. When such a condition is satisfied, we have: Cov x t, x t E (x t ) = E (x t+h ) = ρ 0 1 ρ 1, Var (x t ) = Var (x t+h ) = σ2 ɛ 1 ρ 2, 1 j = ρ j 1 Var (x t). On the other hand, when jρ 1 j = 1, the process is obviously non-stationary: Cov x t, x t E (x t ) = ρ 0 t + x 0, Var (x t ) = σ 2 ɛt, j = σ 2 ɛ (t j). () Chapter 5 Univariate time-series analysis 11 / 29
12 General ARMA processes The Wold decomposition theorem warrants that any stationary stochastic process can be expressed as the sum of a deterministic and a stochastic moving-average component: x t = ɛ t + b 1 ɛ t 1 + b 2 ɛ t b n ɛ t n = 1 + b 1 L + b 2 L b n L n ɛ t = b(l)ɛ t, Represent the polynomial b(l) as the ratio of two polynomials of lower order: x t = b (L) ɛ t = a (L) c (L) ɛ t, c (L) x t = a (L) ɛ t. (2) This is an ARMA process. Stationary requires that the roots of c (L) lie outside the unit circle. Invertibility of the MA component require that the roots of a (L) lie outside the unit circle. () Chapter 5 Univariate time-series analysis 12 / 29
13 General ARMA processes Consider the simplest case, the ARMA(1,1) process: x t = c 1 x t 1 + ɛ t + a 1 ɛ t 1, (1 c 1 L) x t = (1 + a 1 L) ɛ t. The above equation is equivalent to: x t = 1 + a 1L 1 c 1 L ɛ t = (1 + a 1 L) = 1 + c 1 L + (c 1 L) ɛ t h 1 + (a 1 + c 1 ) L + c 1 (a 1 + c 1 ) L 2 + c 2 1 (a 1 + c 1 ) L i ɛ t. Which shows that the ratio of two finite lag polynomials allows us to model an infinite lag polynomial. () Chapter 5 Univariate time-series analysis 13 / 29
14 General ARMA processes We then have, Var (x t ) = = Cov (x t, x t 1 ) = = Hence, h i 1 + (a 1 + c 1 ) 2 + c 2 1 (a 1 + c 1 ) σ 2 ɛ " # 1 + (a 1 + c 1 ) 2 1 c 2 σ 2 ɛ, 1 h i (a 1 + c 1 ) + c 1 (a 1 + c 1 ) + c 2 1 (a 1 + c 1 ) +... " # (a 1 + c 1 ) + c 1 (a 1 + c 1 ) 2 1 c 2 σ 2 ɛ. 1 ρ (1) = Cov (x t,x t 1 ) Var (x t ) = (1 + a 1c 1 ) (a 1 + c 1 ) 1 + c a 1c 1. σ 2 ɛ () Chapter 5 Univariate time-series analysis 14 / 29
15 General ARMA processes For example, suppose c (L) x t = a (L) ɛ t and you want to find x t = d (L) ɛ t. Parameters in d (L) are most easily found by writing c (L) d (L) = a (L) and by matching terms in L j. For an illustration suppose a (L) = 1 + a 1 L, c (L) = 1 + c 1 L. Multiplying out d (L) we have (1 + c 1 L) 1 + d 1 L + d 2 L d n L n = 1 + a 1 L Matching powers of L, d 1 = a 1 c 1 c 1 d 1 + d 2 = 0 c 1 d 2 + d 3 = 0 c 1 d n 1 + d n = 0 x t = ɛ t + (a 1 c 1 ) ɛ t 1 c 1 (a 1 c 1 ) ɛ t ( c 1 ) n 1 (a 1 c 1 ) ɛ t n () Chapter 5 Univariate time-series analysis 15 / 29
16 Insert Clicker 3 here () Chapter 5 Univariate time-series analysis 16 / 29
17 Persistence and the linear model Persistence of time-series destroys one of the crucial properties for implementing valid estimation and inference in the linear model. In the context of the linear model y = Xβ + ɛ. The following property is required to implement valid estimation and inference E (ɛ j X) = 0. (3) Hypothesis (3) implies that E (ɛ i j x 1,...x i,..., x n ) = 0, (i = 1,..., n). Think of the simplest time-series model for a generic variable y: y t = a 0 + a 1 y t 1 + ɛ t. Clearly, if a 1 6= 0, then, although it is still true that E (ɛ t j y t 1 ) = 0, E (ɛ t 1 j y t 1 ) 6= 0 and (3) breaks down. () Chapter 5 Univariate time-series analysis 17 / 29
18 How serious is the problem? To assess intuitively the consequences of persistence, we construct a small Monte-Carlo simulation on the short sample properties of the OLS estimator of the parameters in an AR(1) process. A Monte-Carlo simulation is based on the generation of a sample from a known data generating process (DGP). First we generate a set of random numbers from a given distribution (here a normally independent white-noise disturbance) for a sample size of interest (say 200 observations) and then construct the process of interest (in our case, an AR(1) process). When a sample of observations on the process of interest is available, then we can estimate the relevant parameters and compare their fitted values with the known true value. the Monte-Carlo simulation is a sort of controlled experiment. To overcome the potential dependence of the set of random numbers drawn on the sequence of simulated white-noise residuals, the DGP is replicated many times. () Chapter 5 Univariate time-series analysis 18 / 29
19 From the figure we note that the estimate of a 1 is heavily biased in small samples, but the bias decreases as the sample gets larger, and disappears eventually. One can show analytically that the average of 2 the OLS estimate of a is a 1 () Chapter 5 Univariate. time-series analysis 19 / 29 How serious is the problem? We report the averages across replications in the following figure A1MEAN TRUEA1 Figure: Small sample bias
20 The Maximum Likelihood Method The likelihood function is the joint probability distribution of the data, treated as a function of the unknown coefficients The maximum likelihood estimator (MLE) consists of value of the coefficients that maximize the likelihood function The MLE selects the value of parameters to maximize the probability of drawing the data that have been effectively observed () Chapter 5 Univariate time-series analysis 20 / 29
21 MLE of an MA process Consider an MA process for a return r t+1 : r t+1 = θ 0 + ε t+1 + θ 1 ε t The time series of the residuals can be computed as ε t+1 = r t+1 θ 0 θ 1 ε t ε 0 = 0 If ε t+1 is normally distributed, than we have! 1 f (ε t+1 ) = (2πσ 2 ε ) 1/2 exp ε2 t+1 2σ 2 ε () Chapter 5 Univariate time-series analysis 21 / 29
22 MLE of an MA process If the ε t+1 are independent over time the likelihood function can be written as follows f (ε 1, ε 2,...ε t+1 ) = T Π i=1 f (ε i ) = Π T 1 exp ε2 i=1 (2πσ 2 1/2 ε ) i 2σ 2 ε! The MLE chooses θ 0, θ 1, σ 2 ε to maximize the probability that the estimated model has generated the observed data-set. The optimum is not always found analically, iterative search is the standard method. () Chapter 5 Univariate time-series analysis 22 / 29
23 MLE of an AR process Consider a vector x t containing observations on time-series variables at time t. A sample of T time-series observations on all the variables is represented as: 2 3 x 1 X 1. T = x T In general, estimation is performed by considering the joint sample density function, known also as the likelihood function, which can be expressed as D X 1 T j X 0, θ. The likelihood function is defined on the parameter space ˆ, given the observation of the observed sample X 1 T and of a set of initial conditions X 0. One can interpret such initial conditions as the pre-sample observations on the relevant variables (which are usually unavailable). () Chapter 5 Univariate time-series analysis 23 / 29
24 MLE of an AR process In case of independent observations the likelihood function can be written as the product of the density functions for each observation. However, this is not the relevant case for time-series, as time-series observations are in general sequentially correlated. In the case of time-series, the sample density is constructed using the concept of sequential conditioning. The likelihood function, conditioned with respect to initial conditions, can always be written as the product of a marginal density and a conditional density: Obviously, D X 1 T j X 0, θ = D (x 1 j X 0, θ) D X 2 T j X 1, θ. D X 2 T j X 0, θ = D (x 2 j X 1, θ) D X 3 T j X 2, θ, and, by recursive substitution: D X 1 T j X 0, θ = T D (x t j X t 1, θ). t=1 () Chapter 5 Univariate time-series analysis 24 / 29
25 MLE of an AR process Having obtained D X 1 T j X 0, θ, we can in theory derive D X 1 T, θ by integrating with respect to X 0 the density conditional on pre-sample observations. In practice this could be intractable analytically, as D (X 0 ) is not known. The hypothesis of stationarity becomes crucial at this stage, as stationarity restricts the memory of time-series and limits the effects of pre-sample observations to the first observations in the sample. This is why, in the case of stationary processes, one can simply ignore initial conditions. Clearly, the larger the sample, the better, as the weight of lost information becomes smaller. Moreover, note that even by omitting initial conditions, we have: D X 1 T T j X 0, θ = D (x 1 j X 0, θ) D (x t j X t 1, θ). Therefore, the likelihood function is separated in the product on T 1 conditional distributions and one unconditional distribution. In the case of non-stationarity, the unconditional distribution is undefined. On the other hand, in the case of stationarity, the DGP is completely () Chapter 5 Univariate time-series analysis 25 / 29 t=2
26 To give more empirical content to our case, let us consider again the case of the univariate first-order autoregressive process, X t j X t 1 N λx t 1, σ 2, (4) D X 1 T j λ, σ 2 = D X 1 j λ, σ 2 T t=2 D X t j X t 1, λ, σ 2. (5) From (5), the likelihood function clearly involves T 1 conditional densities and one unconditional density. The conditional densities are given by (4), the unconditional density can be derived only in the case of stationarity: x t = λx t 1 + u t, u t N.I.D 0, σ 2. () Chapter 5 Univariate time-series analysis 26 / 29
27 We can obtain by recursive substitution: x t = u t + λu t λ n 1 u 1 + λ n x 0. Only if jλj < 1, the effect of the initial condition disappears and we can write the unconditional density of x t as: D x t j λ, σ 2 σ = 2 N 0, 1 λ 2. Under stationarity we can derive the exact likelihood function: D X 1 T j λ, σ 2 = (2π) T 2 σ T 1 λ exp " 1 2σ 2 1 λ 2 x T t=2 (x t λx t 1 ) 2!# and estimates of the parameters of interest are derived by maximizing this function. Note that bλ cannot be derived analytically, using the exact likelihood function; but it requires conditioning the likelihood and operating a grid search. () Chapter 5 Univariate time-series analysis 27 / 29,
28 Putting ARMA models at work There are four main steps in the Box-Jenkins approach: PRE WHITENING: make sure that the time series is stationary.. MODEL SELECTION: Information criteria are a useful tool to this end. The Akaike s information criteria (AIC) and the.schwarz Bayesian Criterion (SBC) are the most commonly used criteria: AIC = 2 log(l) + 2(p + q) SBC = 2 log(l) + log(n)(p + q) MODEL CHECKING: residual tests. Make sure that residuals are not autocorrelated and check whether their distribution is normal, also ex-post evaluation technique based on RMSE and MAE are implemented (Diebold-Mariano, Giacomini-White). FORECASTING, the selected model is typically simulated forward after estimation of the estimation of parameters to produce forecasts for the variable of interests at the relevant horizon. () Chapter 5 Univariate time-series analysis 28 / 29
29 An Illustration To illustrate how ARMA model can be put at work consider the case of forecasting the returns on portfolio 15 of the 25 FF portfolios. () Chapter 5 Univariate time-series analysis 29 / 29
Chapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 59
Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 59 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting
More informationCourse information FN3142 Quantitative finance
Course information 015 16 FN314 Quantitative finance This course is aimed at students interested in obtaining a thorough grounding in market finance and related empirical methods. Prerequisite If taken
More informationEconometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland
Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2018 Part IV Financial Time Series As of Feb 5, 2018 1 Financial Time Series Asset Returns Simple returns Log-returns Portfolio
More informationEmpirical Analysis of the US Swap Curve Gough, O., Juneja, J.A., Nowman, K.B. and Van Dellen, S.
WestminsterResearch http://www.westminster.ac.uk/westminsterresearch Empirical Analysis of the US Swap Curve Gough, O., Juneja, J.A., Nowman, K.B. and Van Dellen, S. This is a copy of the final version
More informationAmath 546/Econ 589 Univariate GARCH Models
Amath 546/Econ 589 Univariate GARCH Models Eric Zivot April 24, 2013 Lecture Outline Conditional vs. Unconditional Risk Measures Empirical regularities of asset returns Engle s ARCH model Testing for ARCH
More informationFinancial Econometrics
Financial Econometrics Volatility Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) Volatility 01/13 1 / 37 Squared log returns for CRSP daily GPD (TCD) Volatility 01/13 2 / 37 Absolute value
More informationConditional Heteroscedasticity
1 Conditional Heteroscedasticity May 30, 2010 Junhui Qian 1 Introduction ARMA(p,q) models dictate that the conditional mean of a time series depends on past observations of the time series and the past
More informationARCH and GARCH models
ARCH and GARCH models Fulvio Corsi SNS Pisa 5 Dic 2011 Fulvio Corsi ARCH and () GARCH models SNS Pisa 5 Dic 2011 1 / 21 Asset prices S&P 500 index from 1982 to 2009 1600 1400 1200 1000 800 600 400 200
More informationModelling Returns: the CER and the CAPM
Modelling Returns: the CER and the CAPM Carlo Favero Favero () Modelling Returns: the CER and the CAPM 1 / 20 Econometric Modelling of Financial Returns Financial data are mostly observational data: they
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 14th February 2006 Part VII Session 7: Volatility Modelling Session 7: Volatility Modelling
More information12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006.
12. Conditional heteroscedastic models (ARCH) MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. Autoregressive Conditional Heteroscedasticity with Estimates of Variance
More informationInstitute of Actuaries of India Subject CT6 Statistical Methods
Institute of Actuaries of India Subject CT6 Statistical Methods For 2014 Examinations Aim The aim of the Statistical Methods subject is to provide a further grounding in mathematical and statistical techniques
More informationA potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Consider
More informationPredictive Regressions: A Present-Value Approach (van Binsbe. (van Binsbergen and Koijen, 2009)
Predictive Regressions: A Present-Value Approach (van Binsbergen and Koijen, 2009) October 5th, 2009 Overview Key ingredients: Results: Draw inference from the Campbell and Shiller (1988) present value
More informationUniversity of New South Wales Semester 1, Economics 4201 and Homework #2 Due on Tuesday 3/29 (20% penalty per day late)
University of New South Wales Semester 1, 2011 School of Economics James Morley 1. Autoregressive Processes (15 points) Economics 4201 and 6203 Homework #2 Due on Tuesday 3/29 (20 penalty per day late)
More informationAmath 546/Econ 589 Univariate GARCH Models: Advanced Topics
Amath 546/Econ 589 Univariate GARCH Models: Advanced Topics Eric Zivot April 29, 2013 Lecture Outline The Leverage Effect Asymmetric GARCH Models Forecasts from Asymmetric GARCH Models GARCH Models with
More informationIntroduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book.
Simulation Methods Chapter 13 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 26, 2017 Christopher
More informationRisk Management and Time Series
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Risk Management and Time Series Time series models are often employed in risk management applications. They can be used to estimate
More informationModelling financial data with stochastic processes
Modelling financial data with stochastic processes Vlad Ardelean, Fabian Tinkl 01.08.2012 Chair of statistics and econometrics FAU Erlangen-Nuremberg Outline Introduction Stochastic processes Volatility
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationForecasting Financial Markets. Time Series Analysis
Forecasting Financial Markets Time Series Analysis Copyright 1999-2011 Investment Analytics Copyright 1999-2011 Investment Analytics Forecasting Financial Markets Time Series Analysis Slide: 1 Overview
More informationList of tables List of boxes List of screenshots Preface to the third edition Acknowledgements
Table of List of figures List of tables List of boxes List of screenshots Preface to the third edition Acknowledgements page xii xv xvii xix xxi xxv 1 Introduction 1 1.1 What is econometrics? 2 1.2 Is
More informationFinancial Time Series Analysis (FTSA)
Financial Time Series Analysis (FTSA) Lecture 6: Conditional Heteroscedastic Models Few models are capable of generating the type of ARCH one sees in the data.... Most of these studies are best summarized
More informationActuarial Society of India EXAMINATIONS
Actuarial Society of India EXAMINATIONS 7 th June 005 Subject CT6 Statistical Models Time allowed: Three Hours (0.30 am 3.30 pm) INSTRUCTIONS TO THE CANDIDATES. Do not write your name anywhere on the answer
More informationWeek 7 Quantitative Analysis of Financial Markets Simulation Methods
Week 7 Quantitative Analysis of Financial Markets Simulation Methods Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 November
More informationDiscussion Paper No. DP 07/05
SCHOOL OF ACCOUNTING, FINANCE AND MANAGEMENT Essex Finance Centre A Stochastic Variance Factor Model for Large Datasets and an Application to S&P data A. Cipollini University of Essex G. Kapetanios Queen
More informationForecasting: an introduction. There are a variety of ad hoc methods as well as a variety of statistically derived methods.
Forecasting: an introduction Given data X 0,..., X T 1. Goal: guess, or forecast, X T or X T+r. There are a variety of ad hoc methods as well as a variety of statistically derived methods. Illustration
More informationEstimation of dynamic term structure models
Estimation of dynamic term structure models Greg Duffee Haas School of Business, UC-Berkeley Joint with Richard Stanton, Haas School Presentation at IMA Workshop, May 2004 (full paper at http://faculty.haas.berkeley.edu/duffee)
More informationFinancial Econometrics Notes. Kevin Sheppard University of Oxford
Financial Econometrics Notes Kevin Sheppard University of Oxford Monday 15 th January, 2018 2 This version: 22:52, Monday 15 th January, 2018 2018 Kevin Sheppard ii Contents 1 Probability, Random Variables
More informationHigh-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5]
1 High-Frequency Data Analysis and Market Microstructure [Tsay (2005), chapter 5] High-frequency data have some unique characteristics that do not appear in lower frequencies. At this class we have: Nonsynchronous
More information1. You are given the following information about a stationary AR(2) model:
Fall 2003 Society of Actuaries **BEGINNING OF EXAMINATION** 1. You are given the following information about a stationary AR(2) model: (i) ρ 1 = 05. (ii) ρ 2 = 01. Determine φ 2. (A) 0.2 (B) 0.1 (C) 0.4
More informationForecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models
The Financial Review 37 (2002) 93--104 Forecasting Stock Index Futures Price Volatility: Linear vs. Nonlinear Models Mohammad Najand Old Dominion University Abstract The study examines the relative ability
More informationStatistical Models and Methods for Financial Markets
Tze Leung Lai/ Haipeng Xing Statistical Models and Methods for Financial Markets B 374756 4Q Springer Preface \ vii Part I Basic Statistical Methods and Financial Applications 1 Linear Regression Models
More informationدرس هفتم یادگیري ماشین. (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی
یادگیري ماشین توزیع هاي نمونه و تخمین نقطه اي پارامترها Sampling Distributions and Point Estimation of Parameter (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی درس هفتم 1 Outline Introduction
More informationPairs trading. Gesina Gorter
Pairs trading Gesina Gorter December 12, 2006 Contents 1 Introduction 3 11 IMC 3 12 Pairs trading 4 13 Graduation project 5 14 Outline 6 2 Trading strategy 7 21 Introductory example 8 22 Data 14 23 Properties
More informationStochastic Volatility (SV) Models
1 Motivations Stochastic Volatility (SV) Models Jun Yu Some stylised facts about financial asset return distributions: 1. Distribution is leptokurtic 2. Volatility clustering 3. Volatility responds to
More informationForecasting Exchange Rate between Thai Baht and the US Dollar Using Time Series Analysis
Forecasting Exchange Rate between Thai Baht and the US Dollar Using Time Series Analysis Kunya Bowornchockchai International Science Index, Mathematical and Computational Sciences waset.org/publication/10003789
More informationThailand Statistician January 2016; 14(1): Contributed paper
Thailand Statistician January 016; 141: 1-14 http://statassoc.or.th Contributed paper Stochastic Volatility Model with Burr Distribution Error: Evidence from Australian Stock Returns Gopalan Nair [a] and
More informationIdentifying Long-Run Risks: A Bayesian Mixed-Frequency Approach
Identifying : A Bayesian Mixed-Frequency Approach Frank Schorfheide University of Pennsylvania CEPR and NBER Dongho Song University of Pennsylvania Amir Yaron University of Pennsylvania NBER February 12,
More informationMathematical Annex 5 Models with Rational Expectations
George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Mathematical Annex 5 Models with Rational Expectations In this mathematical annex we examine the properties and alternative solution methods for
More informationValue at Risk and Self Similarity
Value at Risk and Self Similarity by Olaf Menkens School of Mathematical Sciences Dublin City University (DCU) St. Andrews, March 17 th, 2009 Value at Risk and Self Similarity 1 1 Introduction The concept
More informationOnline Appendix to Grouped Coefficients to Reduce Bias in Heterogeneous Dynamic Panel Models with Small T
Online Appendix to Grouped Coefficients to Reduce Bias in Heterogeneous Dynamic Panel Models with Small T Nathan P. Hendricks and Aaron Smith October 2014 A1 Bias Formulas for Large T The heterogeneous
More informationChapter 6 Forecasting Volatility using Stochastic Volatility Model
Chapter 6 Forecasting Volatility using Stochastic Volatility Model Chapter 6 Forecasting Volatility using SV Model In this chapter, the empirical performance of GARCH(1,1), GARCH-KF and SV models from
More informationThis thesis is protected by copyright which belongs to the author.
A University of Sussex PhD thesis Available online via Sussex Research Online: http://sro.sussex.ac.uk/ This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced
More informationINFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE
INFORMATION EFFICIENCY HYPOTHESIS THE FINANCIAL VOLATILITY IN THE CZECH REPUBLIC CASE Abstract Petr Makovský If there is any market which is said to be effective, this is the the FOREX market. Here we
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2009, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (42 pts) Answer briefly the following questions. 1. Questions
More informationLecture Notes of Bus (Spring 2013) Analysis of Financial Time Series Ruey S. Tsay
Lecture Notes of Bus 41202 (Spring 2013) Analysis of Financial Time Series Ruey S. Tsay Simple AR models: (Regression with lagged variables.) Motivating example: The growth rate of U.S. quarterly real
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay. Solutions to Final Exam.
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2011, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (32 pts) Answer briefly the following questions. 1. Suppose
More informationThe Economic and Social BOOTSTRAPPING Review, Vol. 31, No. THE 4, R/S October, STATISTIC 2000, pp
The Economic and Social BOOTSTRAPPING Review, Vol. 31, No. THE 4, R/S October, STATISTIC 2000, pp. 351-359 351 Bootstrapping the Small Sample Critical Values of the Rescaled Range Statistic* MARWAN IZZELDIN
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationMuch of what appears here comes from ideas presented in the book:
Chapter 11 Robust statistical methods Much of what appears here comes from ideas presented in the book: Huber, Peter J. (1981), Robust statistics, John Wiley & Sons (New York; Chichester). There are many
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
More informationFE570 Financial Markets and Trading. Stevens Institute of Technology
FE570 Financial Markets and Trading Lecture 6. Volatility Models and (Ref. Joel Hasbrouck - Empirical Market Microstructure ) Steve Yang Stevens Institute of Technology 10/02/2012 Outline 1 Volatility
More informationINTERTEMPORAL ASSET ALLOCATION: THEORY
INTERTEMPORAL ASSET ALLOCATION: THEORY Multi-Period Model The agent acts as a price-taker in asset markets and then chooses today s consumption and asset shares to maximise lifetime utility. This multi-period
More informationModeling Credit Risk of Loan Portfolios in the Presence of Autocorrelation (Part 2)
Practitioner Seminar in Financial and Insurance Mathematics ETH Zürich Modeling Credit Risk of Loan Portfolios in the Presence of Autocorrelation (Part 2) Christoph Frei UBS and University of Alberta March
More informationPractical example of an Economic Scenario Generator
Practical example of an Economic Scenario Generator Martin Schenk Actuarial & Insurance Solutions SAV 7 March 2014 Agenda Introduction Deterministic vs. stochastic approach Mathematical model Application
More informationIntroductory Econometrics for Finance
Introductory Econometrics for Finance SECOND EDITION Chris Brooks The ICMA Centre, University of Reading CAMBRIDGE UNIVERSITY PRESS List of figures List of tables List of boxes List of screenshots Preface
More informationReturn Decomposition over the Business Cycle
Return Decomposition over the Business Cycle Tolga Cenesizoglu March 1, 2016 Cenesizoglu Return Decomposition & the Business Cycle March 1, 2016 1 / 54 Introduction Stock prices depend on investors expectations
More informationBack to estimators...
Back to estimators... So far, we have: Identified estimators for common parameters Discussed the sampling distributions of estimators Introduced ways to judge the goodness of an estimator (bias, MSE, etc.)
More informationOptimal Window Selection for Forecasting in The Presence of Recent Structural Breaks
Optimal Window Selection for Forecasting in The Presence of Recent Structural Breaks Yongli Wang University of Leicester Econometric Research in Finance Workshop on 15 September 2017 SGH Warsaw School
More informationDefinition 9.1 A point estimate is any function T (X 1,..., X n ) of a random sample. We often write an estimator of the parameter θ as ˆθ.
9 Point estimation 9.1 Rationale behind point estimation When sampling from a population described by a pdf f(x θ) or probability function P [X = x θ] knowledge of θ gives knowledge of the entire population.
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationGRANULARITY ADJUSTMENT FOR DYNAMIC MULTIPLE FACTOR MODELS : SYSTEMATIC VS UNSYSTEMATIC RISKS
GRANULARITY ADJUSTMENT FOR DYNAMIC MULTIPLE FACTOR MODELS : SYSTEMATIC VS UNSYSTEMATIC RISKS Patrick GAGLIARDINI and Christian GOURIÉROUX INTRODUCTION Risk measures such as Value-at-Risk (VaR) Expected
More informationLONG MEMORY IN VOLATILITY
LONG MEMORY IN VOLATILITY How persistent is volatility? In other words, how quickly do financial markets forget large volatility shocks? Figure 1.1, Shephard (attached) shows that daily squared returns
More informationOptimal Hedge Ratio and Hedging Effectiveness of Stock Index Futures Evidence from India
Optimal Hedge Ratio and Hedging Effectiveness of Stock Index Futures Evidence from India Executive Summary In a free capital mobile world with increased volatility, the need for an optimal hedge ratio
More informationThis homework assignment uses the material on pages ( A moving average ).
Module 2: Time series concepts HW Homework assignment: equally weighted moving average This homework assignment uses the material on pages 14-15 ( A moving average ). 2 Let Y t = 1/5 ( t + t-1 + t-2 +
More informationLecture 2 Dynamic Equilibrium Models: Three and More (Finite) Periods
Lecture 2 Dynamic Equilibrium Models: Three and More (Finite) Periods. Introduction In ECON 50, we discussed the structure of two-period dynamic general equilibrium models, some solution methods, and their
More information1 Dynamic programming
1 Dynamic programming A country has just discovered a natural resource which yields an income per period R measured in terms of traded goods. The cost of exploitation is negligible. The government wants
More informationChapter 9 Dynamic Models of Investment
George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chapter 9 Dynamic Models of Investment In this chapter we present the main neoclassical model of investment, under convex adjustment costs. This
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationBrooks, Introductory Econometrics for Finance, 3rd Edition
P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Chris Brooks,
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I January
More informationInternet Appendix for Asymmetry in Stock Comovements: An Entropy Approach
Internet Appendix for Asymmetry in Stock Comovements: An Entropy Approach Lei Jiang Tsinghua University Ke Wu Renmin University of China Guofu Zhou Washington University in St. Louis August 2017 Jiang,
More informationCalibration of Interest Rates
WDS'12 Proceedings of Contributed Papers, Part I, 25 30, 2012. ISBN 978-80-7378-224-5 MATFYZPRESS Calibration of Interest Rates J. Černý Charles University, Faculty of Mathematics and Physics, Prague,
More informationStrategies for Improving the Efficiency of Monte-Carlo Methods
Strategies for Improving the Efficiency of Monte-Carlo Methods Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu Introduction The Monte-Carlo method is a useful
More informationFinancial Econometrics Jeffrey R. Russell. Midterm 2014 Suggested Solutions. TA: B. B. Deng
Financial Econometrics Jeffrey R. Russell Midterm 2014 Suggested Solutions TA: B. B. Deng Unless otherwise stated, e t is iid N(0,s 2 ) 1. (12 points) Consider the three series y1, y2, y3, and y4. Match
More informationRATIONAL BUBBLES AND LEARNING
RATIONAL BUBBLES AND LEARNING Rational bubbles arise because of the indeterminate aspect of solutions to rational expectations models, where the process governing stock prices is encapsulated in the Euler
More informationFinancial Econometrics Lecture 5: Modelling Volatility and Correlation
Financial Econometrics Lecture 5: Modelling Volatility and Correlation Dayong Zhang Research Institute of Economics and Management Autumn, 2011 Learning Outcomes Discuss the special features of financial
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2010, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationThe Constant Expected Return Model
Chapter 1 The Constant Expected Return Model Date: February 5, 2015 The first model of asset returns we consider is the very simple constant expected return (CER) model. This model is motivated by the
More informationMarket Risk Analysis Volume I
Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationFiscal and Monetary Policies: Background
Fiscal and Monetary Policies: Background Behzad Diba University of Bern April 2012 (Institute) Fiscal and Monetary Policies: Background April 2012 1 / 19 Research Areas Research on fiscal policy typically
More informationRisk Management. Risk: the quantifiable likelihood of loss or less-than-expected returns.
ARCH/GARCH Models 1 Risk Management Risk: the quantifiable likelihood of loss or less-than-expected returns. In recent decades the field of financial risk management has undergone explosive development.
More informationThe Econometrics of Financial Returns
The Econometrics of Financial Returns Carlo Favero December 2017 Favero () The Econometrics of Financial Returns December 2017 1 / 55 The Econometrics of Financial Returns Predicting the distribution of
More informationIdiosyncratic risk, insurance, and aggregate consumption dynamics: a likelihood perspective
Idiosyncratic risk, insurance, and aggregate consumption dynamics: a likelihood perspective Alisdair McKay Boston University June 2013 Microeconomic evidence on insurance - Consumption responds to idiosyncratic
More informationBayesian Linear Model: Gory Details
Bayesian Linear Model: Gory Details Pubh7440 Notes By Sudipto Banerjee Let y y i ] n i be an n vector of independent observations on a dependent variable (or response) from n experimental units. Associated
More informationKey Moments in the Rouwenhorst Method
Key Moments in the Rouwenhorst Method Damba Lkhagvasuren Concordia University CIREQ September 14, 2012 Abstract This note characterizes the underlying structure of the autoregressive process generated
More informationMacroeconometrics - handout 5
Macroeconometrics - handout 5 Piotr Wojcik, Katarzyna Rosiak-Lada pwojcik@wne.uw.edu.pl, klada@wne.uw.edu.pl May 10th or 17th, 2007 This classes is based on: Clarida R., Gali J., Gertler M., [1998], Monetary
More informationStrategies for High Frequency FX Trading
Strategies for High Frequency FX Trading - The choice of bucket size Malin Lunsjö and Malin Riddarström Department of Mathematical Statistics Faculty of Engineering at Lund University June 2017 Abstract
More informationBooth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay. Solutions to Midterm
Booth School of Business, University of Chicago Business 41202, Spring Quarter 2014, Mr. Ruey S. Tsay Solutions to Midterm Problem A: (30 pts) Answer briefly the following questions. Each question has
More informationTechnical Appendix: Policy Uncertainty and Aggregate Fluctuations.
Technical Appendix: Policy Uncertainty and Aggregate Fluctuations. Haroon Mumtaz Paolo Surico July 18, 2017 1 The Gibbs sampling algorithm Prior Distributions and starting values Consider the model to
More informationAnalyzing Oil Futures with a Dynamic Nelson-Siegel Model
Analyzing Oil Futures with a Dynamic Nelson-Siegel Model NIELS STRANGE HANSEN & ASGER LUNDE DEPARTMENT OF ECONOMICS AND BUSINESS, BUSINESS AND SOCIAL SCIENCES, AARHUS UNIVERSITY AND CENTER FOR RESEARCH
More informationSolving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function?
DOI 0.007/s064-006-9073-z ORIGINAL PAPER Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function? Jules H. van Binsbergen Michael W. Brandt Received:
More informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
More informationLECTURE NOTES 10 ARIEL M. VIALE
LECTURE NOTES 10 ARIEL M VIALE 1 Behavioral Asset Pricing 11 Prospect theory based asset pricing model Barberis, Huang, and Santos (2001) assume a Lucas pure-exchange economy with three types of assets:
More information4 Reinforcement Learning Basic Algorithms
Learning in Complex Systems Spring 2011 Lecture Notes Nahum Shimkin 4 Reinforcement Learning Basic Algorithms 4.1 Introduction RL methods essentially deal with the solution of (optimal) control problems
More informationThe mean-variance portfolio choice framework and its generalizations
The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014 Outline and objectives The backward, three-step solution
More informationFinancial Time Series and Their Characterictics
Financial Time Series and Their Characterictics Mei-Yuan Chen Department of Finance National Chung Hsing University Feb. 22, 2013 Contents 1 Introduction 1 1.1 Asset Returns..............................
More information