# Econometria dei mercati finanziari c.a. A.A AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna

Save this PDF as:

Size: px
Start display at page:

Download "Econometria dei mercati finanziari c.a. A.A. 2011-2012. 4. AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna"

## Transcription

1 Econometria dei mercati finanziari c.a. A.A AR, MA and ARMA Time Series Models Luca Fanelli University of Bologna

2 For each class of models considered in these slides (AR, MA, ARMA) we focus on: - representation - estimation. We also consider forecasting issues for the AR model alone.

3 AR model: representation We start from the simplest AR model, the autoregressive model of order one, AR(1). We say that the process generating the log-return {r t } belongs to the class of covariance stationary AR(1) models if where r t = β 0 + β 1 r t 1 + u t, u t WN(0,σ 2 u) 1 <β 1 < 1 r 0 is fixed.

4 Two important remarks. 1. The AR(1) can be interpreted in the conventional way: F t := {r t,r t 1,...,r 1 } r t = E(r t F t 1 )+u t u t is a MDS with respect to F t E(r t F t 1 ):=β 0 + β 1 r t 1 hence it is aspecialcaseof the dynamic linear regression model discussed in Slides The fact that u t is a White Noise such that Var(u t ):=σ 2 u t, does not conflict with the possibility that Var(u t F t 1 ):=σ 2 t F t 1. Recall that σ 2 u:=e(u 2 t ):=E(E(u2 t F t 1))!

5 3. The stationarity restriction 1 <β 1 < 1canbe derived by following the definition (Slides 3) and imposing the following conditions: E(r t ):=μ const t Var(r t ):=σ 2 r const t Cov(r t,r t τ ):=depends on τ, not on t. Exercise: prove the statement above!

6 Define the LAG OPERATOR L: L j x t :=x t j, j integer x t :=(1 L)x t. The AR(1) can be re-written as (1 β 1 L)r t = β 0 + u t, u t WN(0,σ 2 u) where β(l):=(1 β 1 L) is known as theautoregressivepolynomial. With this definition, the AR(1) can conventionally be written as β(l)r t = β 0 + u t, u t WN(0,σ 2 u).

7 The covariance stationarity of the AR(1) model is equivalent to the condition (also known as asymptotic stationarity): every solution z C of the characteristic equation is such that z > 1. (1 β 1 z)=0

8 Recall Every real number in R can be uniquely associated with a point that lies on the real line. Every complex number in C can be uniquely associated with a point in the R R plane. Formally, if z C, then z (a, b) a + bi where (a, b) R R, i is such that i 2 =-1, a is its the real part and b is its immaginary part. If z C, z := a + bi :=(a 2 + b 2 ) 1/2 :=( (a, b) 0 ) 1/2 is the absolute value of the complex number.

9 Now consider any (scalar) polynomial of degree n: p n (x):=a 0 + a 1 x + a 2 x a n x n where x R and a i R, i =0, 1,...,n are real numbers. The equation p n (x) =0 may have solutions x that do not belong to R but to C. Consider as an example the quadratic case (n:=2) p 2 (x):=a 0 + a 1 x + a 2 x 2 ; here we know that if (a 1 ) 2 4a 2 a 0 < 0 the equation p 2 (x) =0has no real solution. A fundamental theory of algebra (due to Gauss) says that has always n solutions in C. p n (x) =0 Accordingly, we can represent any of these solutions in the R R plane. In the R R plane we can also represent the unit circle, which is the circle that crosses the points (1,0), (0,-1), (-1,0), (0,1).

10 Therefore, (covariance) stationarity requires that the solutions to the characteristic equation are such that z > 1. β(z)=(1 β 1 z)=0 If it happens that there exist solutions such that z :=1, we say that the AR(1) has a unit root and is nonstationary. (Note that all points (1,0), (0,-1), (-1,0), (0,1) are such that z :=1; it can be shown that the roots (0,- 1), (-1,0), (0,1) are related to the seasonal behavior of r t ). If it happens that there exist solutions such that z <1, we say that the AR(1) is non-stationary and explosive.

11 If the AR(1) is asymptotically stationary, i.e. if very solution z C of the characteristic equation (1 β 1 z)=0is such that z > 1, then it exists (1 β 1 L) 1 :=1 + β 1 L +(β 1 ) 2 L 2 +(β 1 ) 3 L where := j=0 (β 1 ) j L j := j=0 θ j L j :=θ(l) θ j := (β 1 ) j 0asj and θ(1):= j=0 θ j := j=0 (β 1 ) j (1 β 1 ) 1 <.

12 By multiplying both sides of equation β(l)r t = β 0 + u t by θ(l):=β(l) 1,onegets 1 (1 β 1 L) (1 β 1L)r t = θ(l)β 0 + r t = θ(1)β 0 + θ(l)u t r t = β 0 1 β 1 + r t = β 0 1 β 1 + j=0 j=0 (β 1 ) j u t j θ j u t j. 1 (1 β 1 L) u t This is the infinite moving average MA( ) representation associated with the AR(1).

13 To sum up, the stationary AR(1) model r t = β 0 + β 1 r t 1 + u t, u t WN(0,σ 2 u) admits an equivalent MA( ) representation r t = β 0 1 β 1 + j=0 θ j u t j, θ j := (β 1 ) j. The MA representation is useful to derive the moments of the process: 1. it is immediate to see that μ r :=E(r t ):= β 0 1 β 1 :=θ(1)β 0 ; 2. it is immediate to see that σ 2 r:=var(r t ):=Var j=0 θ j u t j := j=0 ³ θj 2 Var(ut j ) := j=0 ³ β 2 1 j σ 2 u := σ2 u (1 β 2 1 );

14 3. it is immediate to see that e.g. for τ:=1 Cov(r t,r t 1 ):=Cov θ j u t j, j=0 h=0 θ h u t h 1 := j=1 θ j Var(u t j ):= j=1 θ j σ 2 u:= := β 1 1 β 2 σ 2 u. 1 β j 1 σ2 u j=1 More generally, hence σ2 u Cov(r t,r t τ ):= 1 β 2 1 (β 1 ) τ, τ=1,2,... ρ(τ):=corr(r t,r t τ ):= (β 1 ) τ, τ=1,2,...

15 Exercise: show that the AR(1) with β 1 :=1 has a uit root and is not non-stationary. Note that the AR(1) with β 1 :=1 is known as the Random Walk process. Thesequenceofcoefficients θ 1,θ 2,...θ j,... obtained from the MA( ) representation r t = β 0 + u t + θ 1 u t 1 + θ 2 u t β 1 is known as impulse response function. Ifoneinter- prets u t as a shock (an event that can not be predicted on the basis of existing information), then the parameter θ 1 measures the impact of the shock on the variable r t after one period, θ 2 measures the impact of the shock on the variable r t after two periods, and so on.

16 We can generalize the AR(1) to the AR(p), p 2. We say that the process generating the log-return {r t } belongs to the class of covariance stationary AR(p) models if r t = β 0 + β 1 r t β p r t p + u t, u t WN(0,σ 2 u) where r 0, r 1,..., r 1 p are fixed and the autoregressive polynomial β(l):=(1 β 1 L β 2 L 2... β p L p ) is such that every solution z C of the characteristic equation is such that z > 1. β(z) =0

17 Also in this case if the AR(p) is stationary it exists the inverse polynomial such that θ(l):=β(l) 1 := j=0 θ j L j θ j 0asj and θ(1):= j=0 θ j :=β(1) 1 :=(1 β 1 β 2... β p ) 1 <.

18 The AR(p) model β(l)r t = β 0 + u t, u t WN(0,σ 2 u) can also be represented as the MA( ): r t = β(1) 1 β 0 + θ(l)u t. E(r t ):=θ(1)β 0 := β 0 (1 β 1 β 2... β p ) ; ρ(τ):=corr(r t,r t τ ) slightly more involving.

19 AR model: estimation The estimation of stationary AR models does not need a separate treatment. Everything is the Slides 3. To see this, whatever the lag order p, thearmodel can be compacted in the expression r t = x 0 tβ + u t, t =1,...T where β:= β 0 β 1. β p, x t:= 1 r t 1. r t p. OLS and ML estimation of θ:=(β 0,σ 2 u) 0 is done. Testing hypotheses on β is done.

20 Forecasting issues: the AR(p) model Given the AR(p): r t = β 0 + β 1 r t β p r t p + u t u t WN(0,σ 2 u) t =1, 2,...,T we consider the problem of computing forecasts of actual returns. We start from the one step-ahead forecast: ˆr T +1 :=E(r T +1 F T ). From the AR(p) it follows that the quantity is given by ˆr T +1 :=E(r T +1 F T ) :=β 0 + β 1 r T + β 2 r T β p r T p.

21 In practice, the actual one step-ahed forecast is ˆr a T +1 :=x0 T +1ˆβ := ˆβ 0 + ˆβ 1 r T + ˆβ 2 r T ˆβ p r T p that means that we need to estimate the parameters from the data in order to obtain the forecast! We also need forecast intervals. Define the forecast error: e T +1 :=r T +1 ˆr T +1 :=u T +1 If u T +1 F T N(0,σ 2 u), then e T +1 F T N(0,σ 2 u) which implies r T +1 ˆr T +1 F T N(0,σ 2 u).

22 From r T +1 ˆr T +1 F T N(0,σ 2 u) we have Pr q 1 α r T +1 ˆr T +1 q 1 α :=1 α σ u where q 1 α is a quantile of the N(0, 1) distribution, also known as (1 α)100 percentile. Hence Pr ˆr T +1 σ u q 1 α r T +1 ˆr T +1 + σ u q 1 α :=1 α is the probability that the unknown future return r T +1 falls within the interval ˆr T +1 ± σ u q 1 α. In practice, in large samples (T ), we have that Pr hˆr a T +1 ˆσ uq 1 α r T +1 ˆr a T +1 +ˆσ uq 1 α i :=1 α

23 Two step-ahead forecast: ˆr T +2 :=E(r T +2 F T ) :=E(β 0 + β 1 r T +1 + β 2 r T β p r T p+1 F T ) :=β 0 + β 1 E(r T +1 F T )+β 2 r T β p r T p+1 :=β 0 + β 1ˆr T +1 + β 2 r T β p r T p+1. The actual forecast will be: ˆr T a +2 :=ˆβ 0 + ˆβ 1ˆr T a +1 + ˆβ 2 r T ˆβ p r T p+1.

24 Forecast error: e T +2 :=r T +2 ˆr T +2 :=β 1 (r T +1 ˆr T +1 )+u T +2 therefore Note that :=β 1 e T +1 + u T +2 e T +2 F T N(0, (1 + β 2 1 )σ2 u). Var(e T +2 ) Var(e T +1 ):=σ 2 u because as the forecast horizon increases, the uncertainty of the forecast increases as well. The (1 α)100 forecast interval is therefore given by ˆr a T +2 ± (1 + ˆβ 2 1) 1/2ˆσ u q 1 α. In general, the step-ahed forecast will be function of the 1 step-ahed forecast and the variance of the corresponding forecast error will tend to increase.

25 It is possible to show that ˆr T + :=E(r T + F T ) E(r t ) and this is due to the mean-reversion property of the stationary AR(p). Exercise: model. prove the property above for the AR(1)

26 MA model: representation We start from the simplest MA model, the moving average model of order one, MA(1). We say that the process generating the log-return {r t } belongs to the class of MA(1) models if r t = β 0 + u t + θ 1 u t 1, u t WN(0,σ 2 u) where θ 1 is any real number 6= 1andu 0 is fixed. Important remark: the MA(1) is covariance stationary for any value of θ 1 (including θ 1 :=1). Exercise: prove this statement! Every MA(1) (MA(q))processisastationaryprocess.

27 We can re-write the MA(1) model as r t = β 0 + θ(l)u t where θ(l):=1 + θ 1 L is the MA polynomial. As we know, if every solution z C of the characteristic equation (1 + θ 1 z)=0 is such that z > 1, then it exists such that θ(l) 1 :=β(l):=1+ j=1 β j L j β j := (θ 1 ) j 0asj and β(1):=1+ j=1 β j <.

28 In this case we say that the MA(1) process is invertible, i.e. it can be represented as AR( ): θ(l) 1 r t = θ(1) 1 β 0 + u t β(l)r t = β(1)β 0 + u t The invertibility condition allow us to express r t as linear function of infinite past returns plus the White Noise term: this means that one can forecast r t by using past values r t 1,r t 2,... If the MA(1) is not invertible, it can be shown that the poynomial θ(l) 1 also involves powers of the type L j (not L j ): we have the paradox that in order to forecast r t one needs also future values of r t!! MA model: estimation, see ARMA

29 ARMA model: representation We start from the simplest ARMA process, which is the ARMA(1,1) process. We say that the process generating the log-return {r t } belongs to the class of covariance stationary and invertible ARMA(1, 1) models if r t = β 0 + β 1 r t 1 + u t + θ 1 u t 1, u t WN(0,σ 2 u) where r 0 and r 1,arefixed, u 0 := 0=:u 1,where 1 <β 1 < 1, 1 <θ 1 < 1. In this model, the condition on 1 <β 1 < 1 ensures the (covariance) stationarity of the process because β(z):=(1 β 1 z)willhavesolutions z > 1, while the condition 1 <θ 1 < 1 ensures the invertibility of the MA part, beacuse θ(z):=(1 + θ 1 z) will have solutions z > 1.

30 Moments of stationary ARMA(1,1) process. Given r t = β 0 + β 1 r t 1 + u t + θ 1 u t, u t WN(0,σ 2 u) 1 <β 1 < 1 we have μ r :=E(r t ):= β 0 1 β 1 ; σ 2 r:=var(r t ):= (1 + θ2 1 +2β 1θ 1 )σ 2 u 1 β 2 1 Exercise: prove the formulas above. ρ(τ):=corr(r t,r t τ ):=β 1 + β 1σ 2 u σ 2. r

31 Generalization to the ARMA(p, q) case,p 2, q 2. We say that the process generating the log-return {r t } belongs to the class of covariance stationary and invertible ARMA(p, q) models if β(l)r t = β 0 + θ(l)u t, u t WN(0,σ 2 u) where r 0, r 1,..., r 1 p are fixed, u 0 := 0=:u 1,...,=:u 1 q are fixed, β(l):=(1 β 1 L β 2 L 2... β p L p ) θ(l):=(1 + θ 1 L + θ 2 L θ q L q ) and the polynomials β(l) and θ(l) are such that the solutions, z 1,z 2 C of the characteristic equations β(z 1 )=0, θ(z 2 )=0 are such that z 1 > 1and z 2 > 1.

32 ARMA model: estimation We consider the estimation problem of the ARMA(p,q) model: (1 β 1 L... β p L p )r t = β 0 +(1+θ 1 L+...+θ q L q )u t u t WN(0,σ 2 u) r 0,r 1,...,r 1 p are fixed u 0 :=u 1 :=...:=u 1 q :=0 ϑ:=(β 0,β 1,..., β p,θ 1,...,θ q ) unknown parameters. Hp: u t F t 1 is Gaussian.

33 With this additional hypothesis, we are able to write the log-likelihood function of the ARMA(p,q). First, given the time series r 1,...,r T where r 0,r 1,..., r 1 p are treated as given (nonstochastic), the likelihood function is L(ϑ):=f(r 1,...,r T ; r 0,r 1,..., r 1 p,ϑ). We know that for dependent observations the likelihood function admits the conditional factorization: L(ϑ):=Π T t=1 f(r t r t 1,...,r 1 ; r 0,r 1,..., r 1 p,ϑ).

34 Under HP: where f(u t F t 1 ):=(2πσ 2 u) 1/2 exp =(2πσ 2 u) 1/2 exp ( ( u2 t 2σ 2 u ) [r t E(r t F t 1 )] 2 2σ 2 u u t :=r t E(r t F t 1 ) ) :=r t β 0 β 1 r t 1... β p r t p θ 1 u t 1... θ q u t q :=r t μ t so that if u t F t 1 N(0,σ 2 u), t =1, 2,...,T then r t F t 1 N(μ t,σ 2 u), t =1, 2,...,T.

35 This means that L(ϑ):=Π T t=1 f(r t r t 1,..., r 1 ; r 0,r 1,...,r 1 p,ϑ) and = Π T t=1 (2πσ2 u) 1/2 exp ( u2 t 2σ 2 u ) log L(ϑ):=- T 2 log(2π) T 2 log(σ2 u) TX t=1 u 2 t 2σ 2 u. There is not analytic solution to ˆϑ:= max ϑ log L(ϑ) hence log L(ϑ) must be maximized numerically. ˆϑ will be the ML estimator of the parameters of the ARMA(p, q) model.

### Univariate Time Series Analysis; ARIMA Models

Econometrics 2 Spring 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing

### Univariate Time Series Analysis; ARIMA Models

Econometrics 2 Fall 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Univariate Time Series Analysis We consider a single time series, y,y 2,..., y T. We want to construct simple

### Sales forecasting # 2

Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

### Estimating an ARMA Process

Statistics 910, #12 1 Overview Estimating an ARMA Process 1. Main ideas 2. Fitting autoregressions 3. Fitting with moving average components 4. Standard errors 5. Examples 6. Appendix: Simple estimators

### Non-Stationary Time Series andunitroottests

Econometrics 2 Fall 2005 Non-Stationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases:

### Time Series Analysis

Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and García-Martos

### Lecture 2: ARMA(p,q) models (part 3)

Lecture 2: ARMA(p,q) models (part 3) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.

### Some useful concepts in univariate time series analysis

Some useful concepts in univariate time series analysis Autoregressive moving average models Autocorrelation functions Model Estimation Diagnostic measure Model selection Forecasting Assumptions: 1. Non-seasonal

### Time Series Analysis

Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos (UC3M-UPM)

### 1 Short Introduction to Time Series

ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

### 2. What are the theoretical and practical consequences of autocorrelation?

Lecture 10 Serial Correlation In this lecture, you will learn the following: 1. What is the nature of autocorrelation? 2. What are the theoretical and practical consequences of autocorrelation? 3. Since

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### Note on the EM Algorithm in Linear Regression Model

International Mathematical Forum 4 2009 no. 38 1883-1889 Note on the M Algorithm in Linear Regression Model Ji-Xia Wang and Yu Miao College of Mathematics and Information Science Henan Normal University

### INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its

### Time Series Analysis

Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:

### A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices

A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria Email: ettetuk@yahoo.com

### Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation

Parametric Models Part I: Maximum Likelihood and Bayesian Density Estimation Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Fall 2015 CS 551, Fall 2015

### An Introduction to Time Series Regression

An Introduction to Time Series Regression Henry Thompson Auburn University An economic model suggests examining the effect of exogenous x t on endogenous y t with an exogenous control variable z t. In

### Analysis of Financial Time Series with EViews

Analysis of Financial Time Series with EViews Enrico Foscolo Contents 1 Asset Returns 2 1.1 Empirical Properties of Returns................. 2 2 Heteroskedasticity and Autocorrelation 4 2.1 Testing for

### Time Series Analysis 1. Lecture 8: Time Series Analysis. Time Series Analysis MIT 18.S096. Dr. Kempthorne. Fall 2013 MIT 18.S096

Lecture 8: Time Series Analysis MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis 1 Outline Time Series Analysis 1 Time Series Analysis MIT 18.S096 Time Series Analysis 2 A stochastic

### Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.

Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be

### ARMA, GARCH and Related Option Pricing Method

ARMA, GARCH and Related Option Pricing Method Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook September

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

### Forecasting in supply chains

1 Forecasting in supply chains Role of demand forecasting Effective transportation system or supply chain design is predicated on the availability of accurate inputs to the modeling process. One of the

### Time Series Analysis

Time Series Analysis Time series and stochastic processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

### Exam Solutions. X t = µ + βt + A t,

Exam Solutions Please put your answers on these pages. Write very carefully and legibly. HIT Shenzhen Graduate School James E. Gentle, 2015 1. 3 points. There was a transcription error on the registrar

### Impulse Response Functions

Impulse Response Functions Wouter J. Den Haan University of Amsterdam April 28, 2011 General definition IRFs The IRF gives the j th -period response when the system is shocked by a one-standard-deviation

### Univariate and Multivariate Methods PEARSON. Addison Wesley

Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

### Rob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1

Rob J Hyndman Forecasting using 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Example: Japanese cars 3 Using Fourier terms for seasonality 4

### 1 Prior Probability and Posterior Probability

Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

### ITSM-R Reference Manual

ITSM-R Reference Manual George Weigt June 5, 2015 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................

### 17.0 Linear Regression

17.0 Linear Regression 1 Answer Questions Lines Correlation Regression 17.1 Lines The algebraic equation for a line is Y = β 0 + β 1 X 2 The use of coordinate axes to show functional relationships was

### Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

### Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

### Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi

Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi Module No. # 02 Simple Solutions of the 1 Dimensional Schrodinger Equation Lecture No. # 7. The Free

### Analysis and Computation for Finance Time Series - An Introduction

ECMM703 Analysis and Computation for Finance Time Series - An Introduction Alejandra González Harrison 161 Email: mag208@exeter.ac.uk Time Series - An Introduction A time series is a sequence of observations

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

### Introduction to ARMA Models

Statistics 910, #8 1 Overview Introduction to ARMA Models 1. Modeling paradigm 2. Review stationary linear processes 3. ARMA processes 4. Stationarity of ARMA processes 5. Identifiability of ARMA processes

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

Chapter 7 Chapter Table of Contents OVERVIEW...193 GETTING STARTED...194 TheThreeStagesofARIMAModeling...194 IdentificationStage...194 Estimation and Diagnostic Checking Stage...... 200 Forecasting Stage...205

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### Topic 5: Stochastic Growth and Real Business Cycles

Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar

### Forecasting methods applied to engineering management

Forecasting methods applied to engineering management Áron Szász-Gábor Abstract. This paper presents arguments for the usefulness of a simple forecasting application package for sustaining operational

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### Linear Threshold Units

Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

### 3.1 Stationary Processes and Mean Reversion

3. Univariate Time Series Models 3.1 Stationary Processes and Mean Reversion Definition 3.1: A time series y t, t = 1,..., T is called (covariance) stationary if (1) E[y t ] = µ, for all t Cov[y t, y t

### Time Series - ARIMA Models. Instructor: G. William Schwert

APS 425 Fall 25 Time Series : ARIMA Models Instructor: G. William Schwert 585-275-247 schwert@schwert.ssb.rochester.edu Topics Typical time series plot Pattern recognition in auto and partial autocorrelations

### C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}

C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression

### Approximate Likelihoods for Spatial Processes

Approximate Likelihoods for Spatial Processes Petruţa C. Caragea, Richard L. Smith Department of Statistics, University of North Carolina at Chapel Hill KEY WORDS: Maximum likelihood for spatial processes,

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Testing on proportions

Testing on proportions Textbook Section 5.4 April 7, 2011 Example 1. X 1,, X n Bernolli(p). Wish to test H 0 : p p 0 H 1 : p > p 0 (1) Consider a related problem The likelihood ratio test is where c is

### Java Modules for Time Series Analysis

Java Modules for Time Series Analysis Agenda Clustering Non-normal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### In this paper we study how the time-series structure of the demand process affects the value of information

MANAGEMENT SCIENCE Vol. 51, No. 6, June 25, pp. 961 969 issn 25-199 eissn 1526-551 5 516 961 informs doi 1.1287/mnsc.15.385 25 INFORMS Information Sharing in a Supply Chain Under ARMA Demand Vishal Gaur

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### Introduction to Time Series Analysis. Lecture 6.

Introduction to Time Series Analysis. Lecture 6. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153-fall2010 Last lecture: 1. Causality 2. Invertibility 3. AR(p) models 4. ARMA(p,q) models 1 Introduction

### Time Series Analysis and Forecasting

Time Series Analysis and Forecasting Math 667 Al Nosedal Department of Mathematics Indiana University of Pennsylvania Time Series Analysis and Forecasting p. 1/11 Introduction Many decision-making applications

### Linear Models for Continuous Data

Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear

### Lecture 2: Simple Linear Regression

DMBA: Statistics Lecture 2: Simple Linear Regression Least Squares, SLR properties, Inference, and Forecasting Carlos Carvalho The University of Texas McCombs School of Business mccombs.utexas.edu/faculty/carlos.carvalho/teaching

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Testing against a Change from Short to Long Memory

Testing against a Change from Short to Long Memory Uwe Hassler and Jan Scheithauer Goethe-University Frankfurt This version: January 2, 2008 Abstract This paper studies some well-known tests for the null

### Chapter 6: Multivariate Cointegration Analysis

Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### Determining distribution parameters from quantiles

Determining distribution parameters from quantiles John D. Cook Department of Biostatistics The University of Texas M. D. Anderson Cancer Center P. O. Box 301402 Unit 1409 Houston, TX 77230-1402 USA cook@mderson.org

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### Principle of Data Reduction

Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

### Efficiency and the Cramér-Rao Inequality

Chapter Efficiency and the Cramér-Rao Inequality Clearly we would like an unbiased estimator ˆφ (X of φ (θ to produce, in the long run, estimates which are fairly concentrated i.e. have high precision.

### ADVANCED FORECASTING MODELS USING SAS SOFTWARE

ADVANCED FORECASTING MODELS USING SAS SOFTWARE Girish Kumar Jha IARI, Pusa, New Delhi 110 012 gjha_eco@iari.res.in 1. Transfer Function Model Univariate ARIMA models are useful for analysis and forecasting

### Centre for Central Banking Studies

Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics

### AR(p) + MA(q) = ARMA(p, q)

AR(p) + MA(q) = ARMA(p, q) Outline 1 3.4: ARMA(p, q) Model 2 Homework 3a Arthur Berg AR(p) + MA(q) = ARMA(p, q) 2/ 12 ARMA(p, q) Model Definition (ARMA(p, q) Model) A time series is ARMA(p, q) if it is

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 7 Multiple Linear Regression (Contd.) This is my second lecture on Multiple Linear Regression

### Normal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.

Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e -(y -µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:

### The Engle-Granger representation theorem

The Engle-Granger representation theorem Reference note to lecture 10 in ECON 5101/9101, Time Series Econometrics Ragnar Nymoen March 29 2011 1 Introduction The Granger-Engle representation theorem is

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### 3. Regression & Exponential Smoothing

3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Vector autoregressions, VAR

1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS15/16 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,

### THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

### Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

### Random Vectors and the Variance Covariance Matrix

Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write

### Epipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.

Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations).

### P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

### APPLICATION OF THE VARMA MODEL FOR SALES FORECAST: CASE OF URMIA GRAY CEMENT FACTORY

APPLICATION OF THE VARMA MODEL FOR SALES FORECAST: CASE OF URMIA GRAY CEMENT FACTORY DOI: 10.2478/tjeb-2014-0005 Ramin Bashir KHODAPARASTI 1 Samad MOSLEHI 2 To forecast sales as reliably as possible is

### Analysis of algorithms of time series analysis for forecasting sales

SAINT-PETERSBURG STATE UNIVERSITY Mathematics & Mechanics Faculty Chair of Analytical Information Systems Garipov Emil Analysis of algorithms of time series analysis for forecasting sales Course Work Scientific

### Performing Unit Root Tests in EViews. Unit Root Testing

Página 1 de 12 Unit Root Testing The theory behind ARMA estimation is based on stationary time series. A series is said to be (weakly or covariance) stationary if the mean and autocovariances of the series

### 1 Teaching notes on GMM 1.

Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

### Sales forecasting # 1

Sales forecasting # 1 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary