STAT Financial Time Series
|
|
- Aubrey Strickland
- 6 years ago
- Views:
Transcription
1 STAT Financial Time Series Chapter 1 - Introduction Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 42
2 Agenda 1 Basic Description 2 Simple Descriptive Techniques 3 Examples 4 Summary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 2 / 42
3 Basic Description What is Time Series? A collection of values X t : t = 1, 2,..., n t : time (e.g. days, months, years) X t : observed data at different time (e.g. temperature,stock price) Example of Time Series : Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 3 / 42
4 Distinctive feature of time series In classical setting (e.g. t-test, regression), we assume X t s are independent In time series, we assume X t s are dependent!! The dependence in time series is quantified by auto-covariance For small k, Cov(X t, X t+k ) 0 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 4 / 42
5 Some nonstandard features of X t 1) Unequally spaced data (t=1,2,3,4,5,8,9,10,11,12,15,...) Security Price are not available during non-trading days 2) Continuous time series (t can be any real number) Physical phenomenon (e.g. weight) We have {X(t), t R} instead of {X(t)} t=1,2,... 3) Aggregation Data observed : an ACCUMULATION of some quantities over a period of time Examples: Daily return = aggregation of tick by tick return Daily rainfall = aggregation of hourly rainfall Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 5 / 42
6 Some nonstandard features of X t (continue) 4) Replicated time series Data observed : repeated measurement of same quantity across different subjects e.g.: X (j) t total weekly spending for the j-th person at week t where t = 1, 2,.., n are repeated measurement of subject j X (j) t 1 and X (k) t 2 are independent for all t 1, t 2 if j k 5) Multiple time series X t is a vector rather than a scalar e.g. Returns of a portfolio at time t : X t = (X 1,t, X 2,t,..., X p,t ) X i,t : return of equity i at time t Need to consider the cross correlation between different series e.g. Cov(X 1,t, X 2,h ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 6 / 42
7 Importance of Time Series Example 1 - Using correlation to find a better C.I of µ Model : X t = µ + a t θa t 1 where a t N (0, 1) E(X t ) = µ + E(a t ) θe(a t 1 ) = µ There exists time dependence because θ k = 1 Cov(X t, X t k ) = 1 + θ 2 k = 0 0 otherwise Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 7 / 42
8 Importance of Time Series 95% Confidence interval for the mean µ X ± 2 Var(X) Proof:By C.L.T., X µ N (0, 1) Var(X) ( ) P 2 X µ ( Var(X) ) P X 2 Var(X) µ X + 2 Var(X) 0.95 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 8 / 42
9 Importance of Time Series It can be shown that Var(X) = 1 n [(1 θ)2 + 2θ n ] 95% C.I. is X ± 2 [ (1 θ) 2 + 2θ ] 1/2 n n Without dependence (θ = 0), 95% C.I. is X ± 2 n Ratio of length of C.I.s is: { [ 2 L(θ) = (1 θ) 2 + 2θ ] } 1/2 { } [ 2 / n = (1 θ) 2 + 2θ ] 1/2 n n n Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 9 / 42
10 Importance of Time Series Implication: θ L (θ), n = If θ = 1, but we wrongly assumed independence (θ = 0), then our C.I. is too big If θ = 1, but we wrongly assumed independence (θ = 0), then our C.I. is too small Need to incorporate correlation structure to produce a correct C.I. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 10 / 42
11 Agenda 1 Basic Description 2 Simple Descriptive Techniques 3 Examples 4 Summary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 11 / 42
12 Simple Descriptive Techniques Decomposition of a time series X t = T t + S t + N t (Trend) (Seasonality) (Noise) T t + S t Macroscopic Component N t Microscopic Component Series Trend Seasonality Noise Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 12 / 42
13 Trends Examples Linear Trend : T t = α + βt Quadratic Trend : T t = α + βt + γt 2 Linear Trend Quadratic Trend Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 13 / 42
14 Estimation of trends without Seasonality 1) Least Square Method min. (X t T t ) 2 2) Filtering Y t = S m (X t ) 3) Differencing X t = X t X t 1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 14 / 42
15 1. Least Squares Method Least Squares Method: Idea : Find α, β in T t = α + βt such that n RSS = (X t T t ) 2 is minimized t=1 Drawbacks Only simple form of T t is allowed, otherwise the minimization is difficult Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 15 / 42
16 1. Least Squares Method Least Squares Method: Idea : Find α, β in T t = α + βt such that n RSS = (X t T t ) 2 is minimized t=1 Example: Given the data {1.2, 2.1, 2.9, 3.8}, estimate the trend in the form T t = α + βt. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 16 / 42
17 2. Filtering Filtering Idea : Smooth the series using local data to estimate the trend Y t = S m (X t ) }{{} = s a r X t+r r= q }{{} smoothed series Weighted average of {X t q, X t q 1, X t+s } The weight {a r } are usually assumed to be symmetric (a r = a r ) and normalized ( a r = 1) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 17 / 42
18 2. Filtering Examples 1) Moving Average Filter Y t = 1 2q + 1 q X t+r r= q What is {Y t } if {X t } = {1.1, 2.2, 2.7, 4.1, 5.2, 5.8} and q = 1? If X t = α + βt, Y t = S m (X t ) = 1 2q + 1 α + βt q {[α + β (t + r)] + N t+r } r= q 2) Spencer 15-point filter (a 0, a 1,..., a 7 ) = (74, 67, 46, 21, 3, 5, 6, 3) a r = a r Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 18 / 42
19 2. Filtering Properties of Spencer 15-point filter Does not distort a cubic trend: If X t = T t + N t, where T t = at 3 + bt 2 + ct + d, then S m (X t ) = 7 a rt t+r + 7 a rn t+r r= 7 r= 7 = at 3 + bt 2 + ct + d + 7 a rn t+r r= 7 Since 7 r= 7 a rn t+r 0 (smoothing), we have S m (X t ) at 3 + bt 2 + ct + d = T t We say: the cubic trend passes through the filter Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 19 / 42
20 2. Filtering General Theorem: A k th order polynomial passes through a filter T t (i.e., T t = s r= s a r T t+r = S m (T t ) for T t = c 0 + c 1 t + + c k t k ) if and only if s 1 r= s a r = 1 s 2 r= s r j a r = 0 for j = 1, 2,..., k Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 20 / 42
21 2. Filtering Low-pass filter: Filter out high-frequencies (volatile) signals Retains low-frequencies (smooth) signals Low-Pass e.g. S m (X t ) = Xt+r 2q+1 before Low-pass after Low-pass Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 21 / 42
22 2. Filtering High-pass filter: Filter out low-frequencies (smooth) signals Retains high-frequencies (volatile) signals High-Pass e.g. X t S m (X t ) before High-pass after High-pass Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 22 / 42
23 3. Differencing Differencing X t = X t X t 1 2 X t = ( X t ) Definition: Backshift operator (B): BX t = X t 1 B k X t = X t k, k = 1, 2,... Alternatively, X t = (1 B)X t k X t = (1 B) k X t, k = 1, 2,... Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 23 / 42
24 3. Differencing removes trend If X t = α + βt, then X t = X t X t 1 = α + βt [α + β(t 1)] = β (no Trend!) In general, If X t = T t + N t and T t = p j=0 a jt j, then p X t = p!a p + p N t (The trend T t is eliminated) (Note: If the trend is a j th degree polynomial, then the trend can be eliminated in differencing j times.) Example: What is { X t } if {X t } = {1.1, 2.2, 2.7, 4.1, 5.2, 5.8} and q = 1? Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 24 / 42
25 Seasonal Cycles Additive Seasonal Component X t = T t + S t + N t Multiplicative Seasonal Component X t = T t S t N t Seasonal part: period=d e.g. season: d=4, month:d=12 Requirements 1 S t+d = S t d 2 j=1 Sj = 0 (common effect is explained by Tt) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 25 / 42
26 Estimating/Removing Seasonal effect 1) Moving average method 2) Seasonal Differencing Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 26 / 42
27 1. Moving Average Method STEP 1: Estimate the trend T t by a moving average filter the moving average filter must be of length d the estimated trend is free from seasonal effect because d j=1 S j = 0 { ( 1 1 T t = d 2 X t q + X t q t+q) X, d = 2q 1 qr= q d X t+r, d = 2q + 1 STEP 2: Estimate the seasonal component S j = t=j,d+j,...,kd+j (X t T t ) µ s = 1 dj=1 S d j S j = S j µ s n d n d : number of complete cycle STEP 3: Use any filter for the series X t S t, get an improved T t (May set T t = T t if satisfied with the filter in Step 1.) RESULT: X t = T t + S t + N t, (N t = X t T t S t ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 27 / 42
28 2. Seasonal Differencing Seasonal Differencing d X t = ( 1 B d) X t = X t X t d Seasonal differencing removes seasonal effects: If X t = S t + N t and period= d, then d X t = S t S t d + N t N t d = N t N t d. (Recall S t = S t d ) Drawback - Lose d data points data = (X 1, X 2,..., X n ) differenced data = (X d+1,..., X n ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 28 / 42
29 Agenda 1 Basic Description 2 Simple Descriptive Techniques 3 Examples 4 Summary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 29 / 42
30 Example in R Data : Quarterly operating revenues of Washington power company Step 0: Time Series Plot Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 30 / 42
31 Step 0: Time Series Plot Observations 1) Slightly increasing trend 2) Annual Cycle Lower in summer Higher in winter (heating?) R-Codes for time series plot x<-read.delim( C://washpower.dat,head=FALSE) x<-ts(x, frequency = 4, start = c(1980, 1)) ts.plot(x,main= Wasington Water Power Co ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 31 / 42
32 Decomposition X t = T t + S t + N t Step 1: Estimate the trend T t by a filter running over one complete season (d = 4) T 3 = 1 2 X 1 + X 2 + X 3 + X X 5 4 T 4 = 1 2 X 2 + X 3 + X 4 + X X 6 4 T 26 = 1 2 X 24 + X 25 + X 26 + X X 28 4 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 32 / 42
33 Decomposition: R-code for Step 1 x<-read.delim( C://washpower.dat,head=FALSE) x<-ts(x, frequency = 4, start = c(1980, 1)) n<-length(x) t<-rep(0,n-4) for ( k in 1:(n-4)) { t[k]=1/8*x[k]+1/4*x[k+1]+1/4*x[k+2]+1/4*x[k+3]+1/8*x[k+4] } t<-c(na,na,t) t<-ts(t,frequency=4,start=c(1980,1)) ts.plot(t,ylab= Trend ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 33 / 42
34 Decomposition X t = T t + S t + N t Step 2: Estimate seasonal effect from the trend-removed series D t D t = X t T t, D = i=3 D i Estimate the seasonal part S i S 1 = [(D 5 D) + (D 9 D) + + (D 25 D)]/6 S 2 = [(D 6 D) + (D 10 D) + + (D 26 D)]/6 S 3 = [(D 3 D) + (D 7 D) + + (D 23 D)]/6 S 4 = [(D 4 D) + (D 8 D) + + (D 24 D)]/6 S i+4j = S i, all i, j Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 34 / 42
35 Decomposition:R-code for Step 2.2 x<-read.delim( C://washpower.dat,head=FALSE) (refer to R-code for Step 1) t<-ts(t,frequency=4,start=c(1980,1)) d<-rep(0,n-4) for (k in 1:(n-4)){d[k]= x[k+2]-t[k+2]} m<-mean(d) d<-c(na,na,d) s<-rep(0,28) for (k in 1:6){ s[1]= s[1]+1/6*(d[1+4*k]-m)} for (k in 1:6){ s[2]= s[2]+1/6*(d[2+4*k]-m)} for (k in 1:6){ s[3]= s[3]+1/6*(d[4*k-1]-m)} for (k in 1:6){ s[4]= s[4]+1/6*(d[4*k]-m)} for (k in 5:n){ s[k]= s[k-4]} s<-ts(s,frequency=4,start=c(1980,1)) ts.plot(s,ylab= Seasonality ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 35 / 42
36 Seasonal Graph Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 36 / 42
37 Decomposition X t = T t + S t + N t Step 3: Re-estimate the trend from deseasonalized data Q t = X t S t T 3 = 1 2 Q 1 + Q 2 + Q 3 + Q Q 5 4 T 4 = 1 2 Q 2 + Q 3 + Q 4 + Q Q 6 4 T 26 = Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 37 / 42
38 Decomposition: R-code for Step 3 x<-read.delim( C://washpower.dat,head=FALSE) (refer to R-code for Step 2.1) s<-ts(s,frequency=4,start=c(1980,1)) q<-rep(0,n) for (k in 1:n){ q[k]=x[k]-s[k]} t1<-c(na,na,rep(0,n-4)) for (k in 3: (n-2)){ t1[k]=1/8*q[k-2]+1/4*q[k-1]+1/4*q[k]+1/4*q[k+1]+1/8*q[k+2]} t1<-ts(t1,frequency=4,start=c(1980,1)) ts.plot(t1,ylab= Re-estimated Trend ) # par(mfrow=c(2,2)) ts.plot(x) ts.plot(t,ylab= Trend ) ts.plot(s,ylab= Seasonality ) ts.plot(t1,ylab= Re-estimated Trend ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 38 / 42
39 Putting Four Graphs Together Chun Note Yip Yau that (CUHK) the trend and STAT the 6104:Financial re-estimated Time Seriestrend are the same, since39 / 42
40 Decomposition X t = T t + S t + N t What to do after decomposition? Check the residual N t = X t T t S t to detect further structure Sophisticated models for the microscopic structure in N t will be discussed in later chapters Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 40 / 42
41 Agenda 1 Basic Description 2 Simple Descriptive Techniques 3 Examples 4 Summary Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 41 / 42
42 Summary of Chapter 1 1) Time Series Observations with time dependent structure Cov(X t, X t+k ) 0 2) Decomposition X t = T t + S t + N t (Trend) (Seasonality) (Noise) i) Removing Trend only Least Squares Method: Minimizes n (Xt Tt)2 t=1 Filters: S m(x t) = s r= q arxt+r Differencing X t = (1 B)X t = X t X t 1 ii) Removing Seasonality Seasonal Filtering: Step 1: Moving average filter with length d to obtain Tt Step 2: Estimate S t from X t Tt Step 3: Re-estimate T t from X t S t Seasonal Differencing d X t = (1 B d )X t = X t X t d Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 42 / 42
STAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 2 - Probability Models Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 34 Agenda 1 Introduction 2 Stochastic Process Definition 1 Stochastic Definition
More informationTime Series Analysis -- An Introduction -- AMS 586
Time Series Analysis -- An Introduction -- AMS 586 1 Objectives of time series analysis Data description Data interpretation Modeling Control Prediction & Forecasting 2 Time-Series Data Numerical data
More informationStatistics of stochastic processes
Introduction Statistics of stochastic processes Generally statistics is performed on observations y 1,..., y n assumed to be realizations of independent random variables Y 1,..., Y n. 14 settembre 2014
More informationStatistics 153 (Time Series) : Lecture Three
Statistics 153 (Time Series) : Lecture Three Aditya Guntuboyina 24 January 2012 1 Plan 1. In the last class, we looked at two ways of dealing with trend in time series data sets - Fitting parametric curves
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 9 - Heteroskedasticity Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 43 Agenda 1 Introduction 2 AutoRegressive Conditional Heteroskedastic Model (ARCH)
More informationSimple Descriptive Techniques
Simple Descriptive Techniques Outline 1 Types of variation 2 Stationary Time Series 3 The Time Plot 4 Transformations 5 Analysing Series that Contain a Trend 6 Analysing Series that Contain Seasonal Variation
More informationTime Series Outlier Detection
Time Series Outlier Detection Tingyi Zhu July 28, 2016 Tingyi Zhu Time Series Outlier Detection July 28, 2016 1 / 42 Outline Time Series Basics Outliers Detection in Single Time Series Outlier Series Detection
More informationProblem Set 2 Solution Sketches Time Series Analysis Spring 2010
Problem Set 2 Solution Sketches Time Series Analysis Spring 2010 Forecasting 1. Let X and Y be two random variables such that E(X 2 ) < and E(Y 2 )
More information3 Time Series Regression
3 Time Series Regression 3.1 Modelling Trend Using Regression Random Walk 2 0 2 4 6 8 Random Walk 0 2 4 6 8 0 10 20 30 40 50 60 (a) Time 0 10 20 30 40 50 60 (b) Time Random Walk 8 6 4 2 0 Random Walk 0
More informationTime Series 4. Robert Almgren. Oct. 5, 2009
Time Series 4 Robert Almgren Oct. 5, 2009 1 Nonstationarity How should you model a process that has drift? ARMA models are intrinsically stationary, that is, they are mean-reverting: when the value of
More informationSTAT 520: Forecasting and Time Series. David B. Hitchcock University of South Carolina Department of Statistics
David B. University of South Carolina Department of Statistics What are Time Series Data? Time series data are collected sequentially over time. Some common examples include: 1. Meteorological data (temperatures,
More information1.4 Properties of the autocovariance for stationary time-series
1.4 Properties of the autocovariance for stationary time-series In general, for a stationary time-series, (i) The variance is given by (0) = E((X t µ) 2 ) 0. (ii) (h) apple (0) for all h 2 Z. ThisfollowsbyCauchy-Schwarzas
More informationForecasting: The First Step in Demand Planning
Forecasting: The First Step in Demand Planning Jayant Rajgopal, Ph.D., P.E. University of Pittsburgh Pittsburgh, PA 15261 In a supply chain context, forecasting is the estimation of future demand General
More informationIntroduction to Functional Data Analysis A CSCU Workshop. Giles Hooker Biological Statistics and Computational Biology
Introduction to Functional Data Analysis A CSCU Workshop Giles Hooker Biological Statistics and Computational Biology gjh27@cornell.edu www.bscb.cornell.edu/ hooker/fdaworkshop 1 / 26 Agenda What is Functional
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR
More informationSTAT 443 Final Exam Review. 1 Basic Definitions. 2 Statistical Tests. L A TEXer: W. Kong
STAT 443 Final Exam Review L A TEXer: W Kong 1 Basic Definitions Definition 11 The time series {X t } with E[X 2 t ] < is said to be weakly stationary if: 1 µ X (t) = E[X t ] is independent of t 2 γ X
More informationLECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity.
LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series {X t } is a series of observations taken sequentially over time: x t is an observation
More informationFunctional responses, functional covariates and the concurrent model
Functional responses, functional covariates and the concurrent model Page 1 of 14 1. Predicting precipitation profiles from temperature curves Precipitation is much harder to predict than temperature.
More informationMultivariate Time Series: VAR(p) Processes and Models
Multivariate Time Series: VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t 1 + + Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with
More informationReliability and Risk Analysis. Time Series, Types of Trend Functions and Estimates of Trends
Reliability and Risk Analysis Stochastic process The sequence of random variables {Y t, t = 0, ±1, ±2 } is called the stochastic process The mean function of a stochastic process {Y t} is the function
More informationEmpirical properties of large covariance matrices in finance
Empirical properties of large covariance matrices in finance Ex: RiskMetrics Group, Geneva Since 2010: Swissquote, Gland December 2009 Covariance and large random matrices Many problems in finance require
More informationTime Series Analysis - Part 1
Time Series Analysis - Part 1 Dr. Esam Mahdi Islamic University of Gaza - Department of Mathematics April 19, 2017 1 of 189 What is a Time Series? Fundamental concepts Time Series Decomposition Estimating
More informationSeasonal Autoregressive Integrated Moving Average Model for Precipitation Time Series
Journal of Mathematics and Statistics 8 (4): 500-505, 2012 ISSN 1549-3644 2012 doi:10.3844/jmssp.2012.500.505 Published Online 8 (4) 2012 (http://www.thescipub.com/jmss.toc) Seasonal Autoregressive Integrated
More informationChapter 7 Forecasting Demand
Chapter 7 Forecasting Demand Aims of the Chapter After reading this chapter you should be able to do the following: discuss the role of forecasting in inventory management; review different approaches
More informationLesson 2: Analysis of time series
Lesson 2: Analysis of time series Time series Main aims of time series analysis choosing right model statistical testing forecast driving and optimalisation Problems in analysis of time series time problems
More informationProbabilities & Statistics Revision
Probabilities & Statistics Revision Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 January 6, 2017 Christopher Ting QF
More informationCyclical Effect, and Measuring Irregular Effect
Paper:15, Quantitative Techniques for Management Decisions Module- 37 Forecasting & Time series Analysis: Measuring- Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Principal Investigator
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -33 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Regression on Principal components
More informationLecture Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University
Lecture 15 20 Prepared By: Mohammad Kamrul Arefin Lecturer, School of Business, North South University Modeling for Time Series Forecasting Forecasting is a necessary input to planning, whether in business,
More informationTime Series and Forecasting
Time Series and Forecasting Introduction to Forecasting n What is forecasting? n Primary Function is to Predict the Future using (time series related or other) data we have in hand n Why are we interested?
More informationSimple Linear Regression
Simple Linear Regression Christopher Ting Christopher Ting : christophert@smu.edu.sg : 688 0364 : LKCSB 5036 January 7, 017 Web Site: http://www.mysmu.edu/faculty/christophert/ Christopher Ting QF 30 Week
More informationINTRODUCTION TO FORECASTING (PART 2) AMAT 167
INTRODUCTION TO FORECASTING (PART 2) AMAT 167 Techniques for Trend EXAMPLE OF TRENDS In our discussion, we will focus on linear trend but here are examples of nonlinear trends: EXAMPLE OF TRENDS If you
More information6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series. MA6622, Ernesto Mordecki, CityU, HK, 2006.
6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series MA6622, Ernesto Mordecki, CityU, HK, 2006. References for Lecture 5: Quantitative Risk Management. A. McNeil, R. Frey,
More informationX t = a t + r t, (7.1)
Chapter 7 State Space Models 71 Introduction State Space models, developed over the past 10 20 years, are alternative models for time series They include both the ARIMA models of Chapters 3 6 and the Classical
More informationDecision 411: Class 3
Decision 411: Class 3 Discussion of HW#1 Introduction to seasonal models Seasonal decomposition Seasonal adjustment on a spreadsheet Forecasting with seasonal adjustment Forecasting inflation Poor man
More informationIntroduction to Regression
Introduction to Regression Using Mult Lin Regression Derived variables Many alternative models Which model to choose? Model Criticism Modelling Objective Model Details Data and Residuals Assumptions 1
More informationPart II. Time Series
Part II Time Series 12 Introduction This Part is mainly a summary of the book of Brockwell and Davis (2002). Additionally the textbook Shumway and Stoffer (2010) can be recommended. 1 Our purpose is to
More informationcovariance function, 174 probability structure of; Yule-Walker equations, 174 Moving average process, fluctuations, 5-6, 175 probability structure of
Index* The Statistical Analysis of Time Series by T. W. Anderson Copyright 1971 John Wiley & Sons, Inc. Aliasing, 387-388 Autoregressive {continued) Amplitude, 4, 94 case of first-order, 174 Associated
More informationDecision 411: Class 3
Decision 411: Class 3 Discussion of HW#1 Introduction to seasonal models Seasonal decomposition Seasonal adjustment on a spreadsheet Forecasting with seasonal adjustment Forecasting inflation Poor man
More informationErrata for Campbell, Financial Decisions and Markets, 01/02/2019.
Errata for Campbell, Financial Decisions and Markets, 01/02/2019. Page xi, section title for Section 11.4.3 should be Endogenous Margin Requirements. Page 20, equation 1.49), expectations operator E should
More informationLesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions
(A) Lesson Context BIG PICTURE of this UNIT: CONTEXT of this LESSON: How do we analyze and then work with a data set that shows both increase and decrease What is a parabola and what key features do they
More informationEvaluation of Some Techniques for Forecasting of Electricity Demand in Sri Lanka
Appeared in Sri Lankan Journal of Applied Statistics (Volume 3) 00 Evaluation of Some echniques for Forecasting of Electricity Demand in Sri Lanka.M.J. A. Cooray and M.Indralingam Department of Mathematics
More informationAssociation studies and regression
Association studies and regression CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar Association studies and regression 1 / 104 Administration
More informationFinancial Time Series Analysis: Part II
Department of Mathematics and Statistics, University of Vaasa, Finland Spring 2017 1 Unit root Deterministic trend Stochastic trend Testing for unit root ADF-test (Augmented Dickey-Fuller test) Testing
More informationSTATISTICAL LOAD MODELING
STATISTICAL LOAD MODELING Eugene A. Feinberg, Dora Genethliou Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, NY 11794-3600, USA Janos T. Hajagos
More informationBasics: Definitions and Notation. Stationarity. A More Formal Definition
Basics: Definitions and Notation A Univariate is a sequence of measurements of the same variable collected over (usually regular intervals of) time. Usual assumption in many time series techniques is that
More informationLinear-Quadratic-Gaussian (LQG) Controllers and Kalman Filters
Linear-Quadratic-Gaussian (LQG) Controllers and Kalman Filters Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 204 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationForecasting. Operations Analysis and Improvement Spring
Forecasting Operations Analysis and Improvement 2015 Spring Dr. Tai-Yue Wang Industrial and Information Management Department National Cheng Kung University 1-2 Outline Introduction to Forecasting Subjective
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49
State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics - 2013 1 / 49 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing
More informationStat 565. (S)Arima & Forecasting. Charlotte Wickham. stat565.cwick.co.nz. Feb
Stat 565 (S)Arima & Forecasting Feb 2 2016 Charlotte Wickham stat565.cwick.co.nz Today A note from HW #3 Pick up with ARIMA processes Introduction to forecasting HW #3 The sample autocorrelation coefficients
More informationYourCabs Forecasting Analytics Project. Team A6 Pratyush Kumar Shridhar Iyer Nirman Sarkar Devarshi Das Ananya Guha
YourCabs Forecasting Analytics Project Team A6 Pratyush Kumar Shridhar Iyer Nirman Sarkar Devarshi Das Ananya Guha Business Assumptions YourCabs acts as an aggregator of radio-cabs from several operators
More informationThe Australian Operational Daily Rain Gauge Analysis
The Australian Operational Daily Rain Gauge Analysis Beth Ebert and Gary Weymouth Bureau of Meteorology Research Centre, Melbourne, Australia e.ebert@bom.gov.au Daily rainfall data and analysis procedure
More informationMAT 3379 (Winter 2016) FINAL EXAM (PRACTICE)
MAT 3379 (Winter 2016) FINAL EXAM (PRACTICE) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes. Only
More informationDCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 4 DIFFERENTIATION
DCDM BUSINESS SCHOOL FACULTY OF MANAGEMENT ECONOMIC TECHNIUES 1 LECTURE 4 DIFFERENTIATION 1 Differentiation Managers are often concerned with the way that a variable changes over time Prices, for example,
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Fin. Econometrics / 53
State-space Model Eduardo Rossi University of Pavia November 2014 Rossi State-space Model Fin. Econometrics - 2014 1 / 53 Outline 1 Motivation 2 Introduction 3 The Kalman filter 4 Forecast errors 5 State
More informationApplied Time Series Analysis FISH 507. Eric Ward Mark Scheuerell Eli Holmes
Applied Time Series Analysis FISH 507 Eric Ward Mark Scheuerell Eli Holmes Introduc;ons Who are we? Who & why you re here? What are you looking to get from this class? Days and Times Lectures When: Tues
More informationRegression of Time Series
Mahlerʼs Guide to Regression of Time Series CAS Exam S prepared by Howard C. Mahler, FCAS Copyright 2016 by Howard C. Mahler. Study Aid 2016F-S-9Supplement Howard Mahler hmahler@mac.com www.howardmahler.com/teaching
More informationSTOR 356: Summary Course Notes
STOR 356: Summary Course Notes Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 7599-360 rls@email.unc.edu February 19, 008 Course text: Introduction
More informationARMA models with time-varying coefficients. Periodic case.
ARMA models with time-varying coefficients. Periodic case. Agnieszka Wy lomańska Hugo Steinhaus Center Wroc law University of Technology ARMA models with time-varying coefficients. Periodic case. 1 Some
More informationTime Series. Karanveer Mohan Keegan Go Stephen Boyd. EE103 Stanford University. November 15, 2016
Time Series Karanveer Mohan Keegan Go Stephen Boyd EE103 Stanford University November 15, 2016 Outline Introduction Linear operations Least-squares Prediction Introduction 2 Time series data represent
More informationA nonparametric test for seasonal unit roots
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna To be presented in Innsbruck November 7, 2007 Abstract We consider a nonparametric test for the
More informationpeak half-hourly South Australia
Forecasting long-term peak half-hourly electricity demand for South Australia Dr Shu Fan B.S., M.S., Ph.D. Professor Rob J Hyndman B.Sc. (Hons), Ph.D., A.Stat. Business & Economic Forecasting Unit Report
More informationIntroduction to Regression Analysis. Dr. Devlina Chatterjee 11 th August, 2017
Introduction to Regression Analysis Dr. Devlina Chatterjee 11 th August, 2017 What is regression analysis? Regression analysis is a statistical technique for studying linear relationships. One dependent
More informationRegression and correlation. Correlation & Regression, I. Regression & correlation. Regression vs. correlation. Involve bivariate, paired data, X & Y
Regression and correlation Correlation & Regression, I 9.07 4/1/004 Involve bivariate, paired data, X & Y Height & weight measured for the same individual IQ & exam scores for each individual Height of
More informationForecasting. Al Nosedal University of Toronto. March 8, Al Nosedal University of Toronto Forecasting March 8, / 80
Forecasting Al Nosedal University of Toronto March 8, 2016 Al Nosedal University of Toronto Forecasting March 8, 2016 1 / 80 Forecasting Methods: An Overview There are many forecasting methods available,
More informationarxiv: v1 [stat.co] 11 Dec 2012
Simulating the Continuation of a Time Series in R December 12, 2012 arxiv:1212.2393v1 [stat.co] 11 Dec 2012 Halis Sak 1 Department of Industrial and Systems Engineering, Yeditepe University, Kayışdağı,
More informationX random; interested in impact of X on Y. Time series analogue of regression.
Multiple time series Given: two series Y and X. Relationship between series? Possible approaches: X deterministic: regress Y on X via generalized least squares: arima.mle in SPlus or arima in R. We have
More information1. Fundamental concepts
. Fundamental concepts A time series is a sequence of data points, measured typically at successive times spaced at uniform intervals. Time series are used in such fields as statistics, signal processing
More informationCh 6. Model Specification. Time Series Analysis
We start to build ARIMA(p,d,q) models. The subjects include: 1 how to determine p, d, q for a given series (Chapter 6); 2 how to estimate the parameters (φ s and θ s) of a specific ARIMA(p,d,q) model (Chapter
More informationb) (1) Using the results of part (a), let Q be the matrix with column vectors b j and A be the matrix with column vectors v j :
Exercise assignment 2 Each exercise assignment has two parts. The first part consists of 3 5 elementary problems for a maximum of 10 points from each assignment. For the second part consisting of problems
More informationHigher Portfolio Quadratics and Polynomials
Higher Portfolio Quadratics and Polynomials Higher 5. Quadratics and Polynomials Section A - Revision Section This section will help you revise previous learning which is required in this topic R1 I have
More informationStatistics 910, #15 1. Kalman Filter
Statistics 910, #15 1 Overview 1. Summary of Kalman filter 2. Derivations 3. ARMA likelihoods 4. Recursions for the variance Kalman Filter Summary of Kalman filter Simplifications To make the derivations
More informationCHANGE DETECTION IN TIME SERIES
CHANGE DETECTION IN TIME SERIES Edit Gombay TIES - 2008 University of British Columbia, Kelowna June 8-13, 2008 Outline Introduction Results Examples References Introduction sunspot.year 0 50 100 150 1700
More informationTime Series and Forecasting
Time Series and Forecasting Introduction to Forecasting n What is forecasting? n Primary Function is to Predict the Future using (time series related or other) data we have in hand n Why are we interested?
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 II. Basic definitions A time series is a set of observations X t, each
More informationModule 9: Stationary Processes
Module 9: Stationary Processes Lecture 1 Stationary Processes 1 Introduction A stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space.
More informationMultivariate Analysis and Likelihood Inference
Multivariate Analysis and Likelihood Inference Outline 1 Joint Distribution of Random Variables 2 Principal Component Analysis (PCA) 3 Multivariate Normal Distribution 4 Likelihood Inference Joint density
More informationSample Exam 1 KEY NAME: 1. CS 557 Sample Exam 1 KEY. These are some sample problems taken from exams in previous years. roughly ten questions.
Sample Exam 1 KEY NAME: 1 CS 557 Sample Exam 1 KEY These are some sample problems taken from exams in previous years. roughly ten questions. Your exam will have 1. (0 points) Circle T or T T Any curve
More informationFitting Linear Statistical Models to Data by Least Squares: Introduction
Fitting Linear Statistical Models to Data by Least Squares: Introduction Radu Balan, Brian R. Hunt and C. David Levermore University of Maryland, College Park University of Maryland, College Park, MD Math
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 12: Gaussian Belief Propagation, State Space Models and Kalman Filters Guest Kalman Filter Lecture by
More information(6, 4) Is there arbitrage in this market? If so, find all arbitrages. If not, find all pricing kernels.
Advanced Financial Models Example sheet - Michaelmas 208 Michael Tehranchi Problem. Consider a two-asset model with prices given by (P, P 2 ) (3, 9) /4 (4, 6) (6, 8) /4 /2 (6, 4) Is there arbitrage in
More informationIn terms of measures: Exercise 1. Existence of a Gaussian process: Theorem 2. Remark 3.
1. GAUSSIAN PROCESSES A Gaussian process on a set T is a collection of random variables X =(X t ) t T on a common probability space such that for any n 1 and any t 1,...,t n T, the vector (X(t 1 ),...,X(t
More informationR = µ + Bf Arbitrage Pricing Model, APM
4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required.
More informationModeling conditional distributions with mixture models: Applications in finance and financial decision-making
Modeling conditional distributions with mixture models: Applications in finance and financial decision-making John Geweke University of Iowa, USA Journal of Applied Econometrics Invited Lecture Università
More information9 Electric Load Modeling for
9 Electric Load Modeling for Long-Term Forecasting 9.1 Introduction Long-term electric peak-load forecasting is an important issue in effective and efficient planning. Over- or underestimation can greatly
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationChapter 12: An introduction to Time Series Analysis. Chapter 12: An introduction to Time Series Analysis
Chapter 12: An introduction to Time Series Analysis Introduction In this chapter, we will discuss forecasting with single-series (univariate) Box-Jenkins models. The common name of the models is Auto-Regressive
More informationExact and high order discretization schemes. Wishart processes and their affine extensions
for Wishart processes and their affine extensions CERMICS, Ecole des Ponts Paris Tech - PRES Université Paris Est Modeling and Managing Financial Risks -2011 Plan 1 Motivation and notations 2 Splitting
More information1 Linear Difference Equations
ARMA Handout Jialin Yu 1 Linear Difference Equations First order systems Let {ε t } t=1 denote an input sequence and {y t} t=1 sequence generated by denote an output y t = φy t 1 + ε t t = 1, 2,... with
More informationThe Slow Convergence of OLS Estimators of α, β and Portfolio. β and Portfolio Weights under Long Memory Stochastic Volatility
The Slow Convergence of OLS Estimators of α, β and Portfolio Weights under Long Memory Stochastic Volatility New York University Stern School of Business June 21, 2018 Introduction Bivariate long memory
More informationDecision 411: Class 3
Decision 411: Class 3 Discussion of HW#1 Introduction to seasonal models Seasonal decomposition Seasonal adjustment on a spreadsheet Forecasting with seasonal adjustment Forecasting inflation Log transformation
More informationTime series and Forecasting
Chapter 2 Time series and Forecasting 2.1 Introduction Data are frequently recorded at regular time intervals, for instance, daily stock market indices, the monthly rate of inflation or annual profit figures.
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationMultivariate GARCH models.
Multivariate GARCH models. Financial market volatility moves together over time across assets and markets. Recognizing this commonality through a multivariate modeling framework leads to obvious gains
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationForecasting. Simon Shaw 2005/06 Semester II
Forecasting Simon Shaw s.c.shaw@maths.bath.ac.uk 2005/06 Semester II 1 Introduction A critical aspect of managing any business is planning for the future. events is called forecasting. Predicting future
More informationTime Series I Time Domain Methods
Astrostatistics Summer School Penn State University University Park, PA 16802 May 21, 2007 Overview Filtering and the Likelihood Function Time series is the study of data consisting of a sequence of DEPENDENT
More informationInteractions. Interactions. Lectures 1 & 2. Linear Relationships. y = a + bx. Slope. Intercept
Interactions Lectures 1 & Regression Sometimes two variables appear related: > smoking and lung cancers > height and weight > years of education and income > engine size and gas mileage > GMAT scores and
More information