STAT Financial Time Series
|
|
- Gwenda Fitzgerald
- 6 years ago
- Views:
Transcription
1 STAT Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46
2 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 2 / 46
3 Introduction ARIMA(p,d,q) Model φ(b)(1 B) d (Y t µ) = θ(b)z t where Z t WN (0, σ 2 ) Unknown order: (p, d, q) Unknown parameters: ( µ, φ 1, φ 2,..., φ p, θ 1, θ 2,..., θ q, σ 2) Question: Given the order (p, d, q), how to estimate the unknown parameters?? Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 3 / 46
4 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 4 / 46
5 Moment Estimators Sample Moments can be computed from the data Theoretical Moments depend on unknown parameters Match the sample and theoretical moments to solve for the unknown parameters Example: i.i.d. Normal: N (µ, σ 2 ) Sample Theoretical 1 st moment X µ 2 nd x 2 moment i n µ 2 + σ 2 Moment estimators: (match 1st moment): µ = X (match 2nd moment): σ 2 x 2 i = n X 2 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 5 / 46
6 Moment Estimator for Time Series Theoretical Moment Sample Moment µ Y = 1 n nt=1 Y t γ(0) C 0 = 1 n (Yt Y ) 2 γ(k) C k = 1 n n k t=1 (Y t Y )(Y t+k Y ) ρ(0) 1 ρ(k) r k = C k C 0 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 6 / 46
7 Moment Estimator for Time Series Q: How to check if ρ(k) is significantly different from 0? ( whether a time series is serially correlated at lag k ) A: Theorem If Y t WN (0, σ 2 ) then r k = C k C 0 AN (0, 1 n ) }{{} Asymptotically normal as n If r k has a 95% C.I. ± 2 n, there is a significant correlation at lag k Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 7 / 46
8 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 8 / 46
9 Parameter Estimators in AR Model AR(p): Y t = φ 1 Y t φ p Y t p + Z t Express as a regression problem ) Y t = (φ 1 φ p Y t 1. Y t p + Z t = Y t 1φ + Z t (1) where φ = (φ 1 φ p ) and Y t 1 = (Y t 1 Y t p ) Recall in linear regression model, we have Y = Xβ + ε Least square estimate of β is β) = (X X) 1 X Y Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 9 / 46
10 Parameter Estimators in AR Model Least Square Estimate (LSE) Since (1) can be considered as a regression model form, we have the least squares estimate of φ φ = ( n t=p+1 Y t 1Y t 1 Example - AR(1): Y t = φ 1 Y t 1 + Z t ) 1 ( n t=p+1 Y t 1Y t ) φ 1 = nt=2 Y t 1 Y t nt=2 Y 2 t 1 and σ 2 = 1 n n t=2 (Y t φ 1 Y t 1 ) 2 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 10 / 46
11 Parameter Estimators in AR Model Performance of the estimator : n( φ φ) N (0, σ 2 Γ 1 p ) where Γ p = γ(0) γ(1) γ(p 1) γ(1) γ(0) γ(p 2) γ(p 1) γ(p 2) γ(0) Confidence Intervals or hypothesis tests can be applied to make inference on the parameter φ Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 11 / 46
12 Parameter Estimators in AR Model Asymptotic distribution of the estimator: γ(0) γ(1) γ(p 1) n( φ φ) N(0, σ 2 Γ 1 γ(1) γ(0) γ(p 2) p ) Γ p = γ(p 1) γ(p 2) γ(0) Confidence Intervals or hypothesis tests to make inference on φ: Example: Estimated AR(2) model: n = 200, Y t = 0.3Y t Y t 2 + Z t ; ˆγ(0) = 1.11, ˆγ(1) = 0.347, ˆσ 2 = 1. 1 ˆσ 2 Γ p = ( ) 1 = ( C.I. for φ 2 : [0.04 ± /n] = [ 0.101, 0.181]. (Not significantly different from 0) ˆφ Testing for φ 1 = 0: z = 1 var( ˆ ˆφ = 0.3/ 0.998/200 = 4.24 > 2. 1 ) (significantly different from 0) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 12 / 46 )
13 Parameter Estimators in AR Model Yule Walker Equation(Y.W) An alternative method for parameter estimation is via the Yule-Walker Equation (Y.W) Yule-Walker Equation (Y.W): Multiply Y t k to Y t = φ 1 Y t φ p Y t p + Z t where k = 1, 2,..., p Then, take expectation γ(k) = φ 1 γ(k 1) + + φ p γ(k p) ρ(k) = φ 1 ρ(k 1) + + φ p ρ(k p) where k = 1, 2,..., p Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 13 / 46
14 Parameter Estimators in AR Model Matrix Form ρ(1). ρ(p) = Yule-Walker Estimator ρ(0) ρ(1) ρ(p 1) φ ρ(1) ρ(0) ρ(p 2) φ ρ(p 1) ρ(p 2) ρ(0) p 1 r φ 1 r p 1 1 φ =. = r 1 1 r p φ p r p 1 r p r. r p Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 14 / 46
15 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 15 / 46
16 Moving Average Models Moment Estimator MA(q): Y t = Z t θ 1 Z t 1 Z t q Express ρ(k) in terms of unknown parameters θ 1, θ 2,..., θ q Solve r k = ρ(k) for unknown parameters Example - MA(1): Y t = Z t θz t 1 ρ(1) = θ (Yt Y )(Y r 1+θ 2 1 = t+1 Y ) (Yt Y ) 2 The estimate satisfies r 1 = θ r 1 θ 2 + θ + 1 = 0 θ = 1± 1 4r θ2 2r 1 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 16 / 46
17 Moving Average Models Conditional Least Square(CLS) Method Idea - Find the noise sequence {Z t } from the observation {Y t } Minimize the sum of squares n S (θ) = Zt 2 t=1 Example - MA(1): Y t = Z t θz t 1 Conditional on Z 0 = 0, then Z 1 = Y 1, Z 2 = Y 1 + θz 1,..., Z k = Y k + θy k 1 Minimize S (θ) = n t=1 Z 2 t by numerical optimization Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 17 / 46
18 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 18 / 46
19 ARMA Model Method of moment and Conditional Least Square can be applied similarly as in pure MA model Example (CLS) - ARMA(1,1): Y t φy t 1 = Z t θz t 1 Conditional on Z 0 = 0, Y 0 = 0 Z 1 (φ, θ) = Y 1, Z 2 (φ, θ) = Y 2 φy 1 + θz 1,..., Z k (φ, θ) = Y k φy k 1 + θz k 1 Minimizes S (θ) = n t=1 Z 2 t (φ, θ) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 19 / 46
20 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 20 / 46
21 Maximum Likelihood Estimator Maximum Likelihood Estimator (MLE) MLE = Most probable parameter value that corresponds closely with the observed data Let X 1, X 2,..., X n be i.i.d. random variables with probability density function f (x, θ) f (x i, θ) is the probability of observing x i The likelihood function L(x, θ) = n i=1 f (x i, θ) is the probability of observing the data set The MLE θ maximizes L(x, θ) The most probable parameter value such that the current data set is obtained Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 21 / 46
22 Maximum Likelihood Estimator MLE for Time Series Idea: Write down the Likelihood function for Y 1, Y 2,..., Y n AR(1): Y t = φy t 1 + Z t Given Y 1, (Y 2,..., Y n ) and (Z 1,..., Z n ) are of one to one correspondence i.e. f (Y 2,..., Y n Y 1 ) = f (Z 2,..., Z n Y 1 ) = f (Z 2,..., Z n ) ( ) n 1 ( = 1 2 exp 1 ) nt=2 Z 2πσ 2 2σ 2 2 t Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 22 / 46
23 Maximum Likelihood Estimator Likelihood L(Y 1, Y 2,..., Y n ) = f (Y 2,..., Y n Y 1 )f (Y 1 ) = = ( ) n exp 1 n 2πσ 2 2σ 2 t=2 Z t 2 ( ) n φ 2πσ 2 exp 2 1 φ 2 2πσ 2 exp 1 2σ 2 (1 φ2 )Y 2 1 { 1 [ nt=2 (Y 2σ 2 t φy t 1 ) 2 + (1 φ 2 )Y1 2 ] } Maximize L(Y 1,..., Y n ) or equivalently log L(Y 1,..., Y n ) by numerical method Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 23 / 46
24 Maximum Likelihood Estimator Likelihood vs Least Square for AR(1) Log likelihood l = log L log(1 φ 2 ) n log σ 2 (1 φ2 )Y 2 1 σ 2 1 σ 2 nt=2 (Y t φy t 1 ) 2 Conditional Least Square S (θ) = n Z 2 t=2 t = n (Y t φy t 1 ) 2 t=2 Unconditional Least Square S(θ) = (1 φ 2 )Y n t=2 (Y t φy t 1 ) 2 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 24 / 46
25 Maximum Likelihood Estimator If n is large, the term n t=2 (Y t φy t 1 ) 2 dominates Minimizing S(θ), S (θ) and maximizing loglikelihood are similar Three methods will give similar results Except for some simple models such as AR(1), we have to employ numerical optimization for the estimation. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 25 / 46
26 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 26 / 46
27 Partial ACF (PACF) PACF measures the correlation between Y t and Y t k that is not explained by Y t 1,..., Y t k+1 PACF - ) φ kk = Corr (Y k+1 P sp{y2,...,y k }Y k+1, Y 1 P sp{y2,...,y k }Y 1 where P sp{x,z} Y is the projection of random variable Y onto the linear subspace spanned by the random variables {X, Z} In particular, P sp{x,z} Y = αx + βz where E[(Y αx + βz) 2 ] is minimized over α and β Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 27 / 46
28 Example on PACF Example Assume that {Y 1, Y 2, Y 3 } is a zero mean stationary sequence with ACF of first lag equal to ρ 1 ) φ 22 = Corr (Y 3 P sp{y2 }Y 3, Y 1 P sp{y2 }Y 1 P sp{y2 }Y 3 = βy 2 where E[(Y 3 βy 2 ) 2 ] is minimized E[(Y 3 βy 2 ) 2 ] = γ(0) 2βγ(1) + β 2 γ(0) [ = γ(0) β γ(1) ] 2 γ(0) + γ(0) γ(1) 2 γ(0) β = γ(1) γ(0) = ρ 1 i.e.p sp{y2 }Y 3 = ρ 1 Y 2 Similarly, P sp{y2 }Y 1 = ρ 1 Y 2 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 28 / 46
29 Example on PACF (continue) Therefore, φ 22 = Corr(Y 3 ρ 1 Y 2, Y 1 ρ 1 Y 2 ) = Cov(Y 3 ρ 1 Y 2,Y 1 ρ 1 Y 2 ) Var(Y3 ρ 1 Y 2 )Var(Y 1 ρ 1 Y 2 ) = ρ 2 ρ ρ 2 1 In particular, if Y t AR(1), then ρ k = φ k, φ 22 = 0 Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 29 / 46
30 Another Interpretation of PACF In the regression Y = β 1 X β k X k + β k+1 X k+1 + ε, the parameter β k+1 can be interpreted as the relationship between Y and X k+1, after accounting for the effect of X 1,..., X k 1, X k It can be shown that, the definition ) φ kk = Corr (Y k+1 P sp{y2,...,y k }Y k+1, Y 1 P sp{y2,...,y k }Y 1 is equivalent to the coefficients φ kk in the representation Y k+1 = φ k1 Y k + φ k2 Y k φ kk Y 1 i.e. φ kk measures the relationship between Y k+1 and Y 1 after accounting for the effect of Y 2,..., Y k Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 30 / 46
31 Computational formula for PACF φ kk is the coefficient in the representation Y k+1 = φ k1 Y k + φ k2 Y k φ kk Y 1 This representation can be found by searching for the minimum of E[(Y k+1 β 1 Y k β k Y 1 ) 2 ] It means that projecting Y k+1 on the space spanned by Y 1,..., Y k Differentiating β 1, β 2,..., β k in terms and set the equations to zero, φ k1,..., φ kk can be obtained by solving ρ(0) ρ(k 1) φ k ρ(k 1) ρ(0) φ kk = ρ(1). ρ(k) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 31 / 46
32 PACF for AR Models In general, AR(p): Y t = φ 1 Y t φ p Y t p + Z t φ 11, φ 22,..., φ pp are non-zero if Y t is linearly related to Y t k for k p Y p+1 P sp{y1,...,y p}y p+1 = Y p+1 φ 1 Y p φ p Y 1 = Z p+1 which is uncorrelated with all Y k, k p, thus, φ p+1,p+1 ) = Corr (Y p+2, P sp{y2,...,y p+1 }Y p+2, Y 1 P sp{y2,...,y p+1 }Y 1 ) = Corr (Z p+2, Y 1 P sp{y2,...,y p+1 }Y 1 = 0 Similarly, φ kk = 0 for k > p Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 32 / 46
33 Theorem of PACF for AR Models For an AR(p) Model, nφkk AN (0, 1) for k > p To test whether φ kk is significant, Significant if φ kk > 2 n Can conclude that the time series follows AR(p), p k NOT Significant otherwise Can conclude that the time series follows AR(p), p < k Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 33 / 46
34 Using ACF and PACF to determine order of AR and MA model AR(p) Model - ACF plot: No clear pattern except AR(1) shows exponential decay pattern PACF plot: Cut-off at lag p MA(q) Model - ACF plot: Cut-off at lag q PACF plot: No clear pattern except MA(1) shows exponential decay pattern Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 34 / 46
35 Examples on ACF and PACF ACF PACF AR(1) AR(3) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 35 / 46
36 Examples on ACF and PACF ACF PACF MA(1) MA(4) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 36 / 46
37 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 37 / 46
38 Order Selection ACF and PACF are graphical methods to determine order of AR and MA model It is more desirable to have a systematic order selection criterion for a general ARMA model Some common model selection criterion: FPE(Final Prediction Error) AIC (Akaike s Information Criterion) BIC (Bayesian Information Criterion) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 38 / 46
39 Order Selection - Final Prediction Error Final Prediction Error (FPE) FPE = σ ( ) 2 n+p n p where σ 2 is the MLE of σ 2 Idea for an AR(p) Model : 1. Mean Square Error of parameter estimation is ( ) n + p MSE σ 2 n 2. The best unbiased estimator for σ 2 is σ 2 n n p 3. Substituting σ 2 by σ 2 n n p to the MSE gives the formula of FPE 4. Note the tradeoff between goodness of fit and the model complexity Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 39 / 46
40 Order Selection - Akaike s Information Criterion Akaike s Information Criterion (AIC) AIC - 2 log L ( β, S x( β) ) + 2(p + q + 1) n AICC (AIC corrected) - 2 log L ( β, S x( β) ) 2(p + q + 1)n + n n p q 2 where S x ( β) = n t=1 (X t φ 1 X t 1 φ p X t p θ 1 Z t 1 θ q Z t q ) 2 β = ( φ 1,..., φ p, θ 1,..., θ q ) L( β, σ 2 ) = ( ) n 1 2 exp 2π σ 2 [ 1 2π σ 2 S x( β) ] Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 40 / 46
41 Order Selection - Akaike s Information Criterion Idea - Similar to FPE, but AIC and AICC estimates the expected likelihood function E[L Y ( β, σ 2 )] where β, σ 2 are MLE computed From the observation X = {X 1,..., X n } and Y is a new observation independent of X X and Y are different to avoid the problem of overfitting Note the tradeoff between model complexity and goodness of fit Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 41 / 46
42 Order Selection - Bayesian Information Criterion Bayesian Information Criterion (BIC) [ n σ 2 ] BIC = (n p q) log + n(1 + log 2π) n p 1 [ ni=1 Xi 2 n σ 2 ] +(p + q) log p + q Motivated by Bayesian argument: BIC the posterior probability of a particular model given data Consistent order selection procedure As n, the order selected by BIC will equal to the true order that will equal to the true order that the observation follow with probability going to 1 AIC is not consistent (i.e. get a wrong model) Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 42 / 46
43 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 43 / 46
44 Residual Analysis Check the goodness of fit of the model by studying the residual Ẑ t = Y t Ŷt STEPS - 1. Time series plot of Ẑt (should look like white noises) 2. ACF of Ẑt (should not have serial correlation) ( ) 3. Confidence interval for r j is 2 2 n, n 4. Portmanteau Statistics Q = n(n + 2) h j=1 r 2 Z (j) n j A common choice of h is between 15 to 30 Q χ 2 (r p q), thus, if Q is bigger than the 95% percentile of a χ 2 (n p q) distribution, we reject the null hypothesis that {Ẑt} is WN. Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 44 / 46
45 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR model) 4 Moving Average Models (MA model) 5 ARMA Models 6 Maximum Likelihood Estimator 7 partial ACF 8 Order Selection 9 Residual Analysis 10 Model Building Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 45 / 46
46 Model Building Three Stages - 1. Model Specification Trend, seasonal effect, choosing ARIMA model by FPE/AIC/BIC 2. Model Identification (estimating coefficient) CLS,MLE,LSE 3. Model Checking(diagnostic) Residual analysis (plot of {Ẑt}) Portmanteau test Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 46 / 46
Parameter estimation: ACVF of AR processes
Parameter estimation: ACVF of AR processes Yule-Walker s for AR processes: a method of moments, i.e. µ = x and choose parameters so that γ(h) = ˆγ(h) (for h small ). 12 novembre 2013 1 / 8 Parameter estimation:
More informationUniversity of Oxford. Statistical Methods Autocorrelation. Identification and Estimation
University of Oxford Statistical Methods Autocorrelation Identification and Estimation Dr. Órlaith Burke Michaelmas Term, 2011 Department of Statistics, 1 South Parks Road, Oxford OX1 3TG Contents 1 Model
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 informationTime Series Analysis
Time Series Analysis Christopher Ting http://mysmu.edu.sg/faculty/christophert/ christopherting@smu.edu.sg Quantitative Finance Singapore Management University March 3, 2017 Christopher Ting Week 9 March
More informationUnivariate Time Series Analysis; ARIMA Models
Econometrics 2 Fall 24 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing
More informationAutoregressive Moving Average (ARMA) Models and their Practical Applications
Autoregressive Moving Average (ARMA) Models and their Practical Applications Massimo Guidolin February 2018 1 Essential Concepts in Time Series Analysis 1.1 Time Series and Their Properties Time series:
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 informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA AR MA Model
More informationStationary Stochastic Time Series Models
Stationary Stochastic Time Series Models When modeling time series it is useful to regard an observed time series, (x 1,x,..., x n ), as the realisation of a stochastic process. In general a stochastic
More informationReview Session: Econometrics - CLEFIN (20192)
Review Session: Econometrics - CLEFIN (20192) Part II: Univariate time series analysis Daniele Bianchi March 20, 2013 Fundamentals Stationarity A time series is a sequence of random variables x t, t =
More informationFORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL
FORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL B. N. MANDAL Abstract: Yearly sugarcane production data for the period of - to - of India were analyzed by time-series methods. Autocorrelation
More informationUnivariate ARIMA Models
Univariate ARIMA Models ARIMA Model Building Steps: Identification: Using graphs, statistics, ACFs and PACFs, transformations, etc. to achieve stationary and tentatively identify patterns and model components.
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 informationMAT3379 (Winter 2016)
MAT3379 (Winter 2016) Assignment 4 - SOLUTIONS The following questions will be marked: 1a), 2, 4, 6, 7a Total number of points for Assignment 4: 20 Q1. (Theoretical Question, 2 points). Yule-Walker estimation
More informationForecasting using R. Rob J Hyndman. 2.4 Non-seasonal ARIMA models. Forecasting using R 1
Forecasting using R Rob J Hyndman 2.4 Non-seasonal ARIMA models Forecasting using R 1 Outline 1 Autoregressive models 2 Moving average models 3 Non-seasonal ARIMA models 4 Partial autocorrelations 5 Estimation
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1
ECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1 Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN
More informationModule 3. Descriptive Time Series Statistics and Introduction to Time Series Models
Module 3 Descriptive Time Series Statistics and Introduction to Time Series Models Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W Q Meeker November 11, 2015
More informationProblem Set 2: Box-Jenkins methodology
Problem Set : Box-Jenkins methodology 1) For an AR1) process we have: γ0) = σ ε 1 φ σ ε γ0) = 1 φ Hence, For a MA1) process, p lim R = φ γ0) = 1 + θ )σ ε σ ε 1 = γ0) 1 + θ Therefore, p lim R = 1 1 1 +
More informationEconometrics II Heij et al. Chapter 7.1
Chapter 7.1 p. 1/2 Econometrics II Heij et al. Chapter 7.1 Linear Time Series Models for Stationary data Marius Ooms Tinbergen Institute Amsterdam Chapter 7.1 p. 2/2 Program Introduction Modelling philosophy
More informationIntroduction to ARMA and GARCH processes
Introduction to ARMA and GARCH processes Fulvio Corsi SNS Pisa 3 March 2010 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March 2010 1 / 24 Stationarity Strict stationarity: (X 1,
More informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Forecasting Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA
More informationEASTERN MEDITERRANEAN UNIVERSITY ECON 604, FALL 2007 DEPARTMENT OF ECONOMICS MEHMET BALCILAR ARIMA MODELS: IDENTIFICATION
ARIMA MODELS: IDENTIFICATION A. Autocorrelations and Partial Autocorrelations 1. Summary of What We Know So Far: a) Series y t is to be modeled by Box-Jenkins methods. The first step was to convert y t
More informationAR(p) + I(d) + MA(q) = ARIMA(p, d, q)
AR(p) + I(d) + MA(q) = ARIMA(p, d, q) Outline 1 4.1: Nonstationarity in the Mean 2 ARIMA Arthur Berg AR(p) + I(d)+ MA(q) = ARIMA(p, d, q) 2/ 19 Deterministic Trend Models Polynomial Trend Consider the
More informationChapter 4: Models for Stationary Time Series
Chapter 4: Models for Stationary Time Series Now we will introduce some useful parametric models for time series that are stationary processes. We begin by defining the General Linear Process. Let {Y t
More informationIntroduction to Time Series Analysis. Lecture 11.
Introduction to Time Series Analysis. Lecture 11. Peter Bartlett 1. Review: Time series modelling and forecasting 2. Parameter estimation 3. Maximum likelihood estimator 4. Yule-Walker estimation 5. Yule-Walker
More informationChapter 6: Model Specification for Time Series
Chapter 6: Model Specification for Time Series The ARIMA(p, d, q) class of models as a broad class can describe many real time series. Model specification for ARIMA(p, d, q) models involves 1. Choosing
More informationTIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA
CHAPTER 6 TIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA 6.1. Introduction A time series is a sequence of observations ordered in time. A basic assumption in the time series analysis
More informationLecture on ARMA model
Lecture on ARMA model Robert M. de Jong Ohio State University Columbus, OH 43210 USA Chien-Ho Wang National Taipei University Taipei City, 104 Taiwan ROC October 19, 2006 (Very Preliminary edition, Comment
More informationProf. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis
Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation
More informationAR, MA and ARMA models
AR, MA and AR by Hedibert Lopes P Based on Tsay s Analysis of Financial Time Series (3rd edition) P 1 Stationarity 2 3 4 5 6 7 P 8 9 10 11 Outline P Linear Time Series Analysis and Its Applications For
More information4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2. Mean: where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore,
61 4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 Mean: y t = µ + θ(l)ɛ t, where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore, E(y t ) = µ + θ(l)e(ɛ t ) = µ 62 Example: MA(q) Model: y t = ɛ t + θ 1 ɛ
More informationStat 5100 Handout #12.e Notes: ARIMA Models (Unit 7) Key here: after stationary, identify dependence structure (and use for forecasting)
Stat 5100 Handout #12.e Notes: ARIMA Models (Unit 7) Key here: after stationary, identify dependence structure (and use for forecasting) (overshort example) White noise H 0 : Let Z t be the stationary
More informationAPPLIED ECONOMETRIC TIME SERIES 4TH EDITION
APPLIED ECONOMETRIC TIME SERIES 4TH EDITION Chapter 2: STATIONARY TIME-SERIES MODELS WALTER ENDERS, UNIVERSITY OF ALABAMA Copyright 2015 John Wiley & Sons, Inc. Section 1 STOCHASTIC DIFFERENCE EQUATION
More informationMidterm Suggested Solutions
CUHK Dept. of Economics Spring 2011 ECON 4120 Sung Y. Park Midterm Suggested Solutions Q1 (a) In time series, autocorrelation measures the correlation between y t and its lag y t τ. It is defined as. ρ(τ)
More informationChapter 8: Model Diagnostics
Chapter 8: Model Diagnostics Model diagnostics involve checking how well the model fits. If the model fits poorly, we consider changing the specification of the model. A major tool of model diagnostics
More informationWe will only present the general ideas on how to obtain. follow closely the AR(1) and AR(2) cases presented before.
ACF and PACF of an AR(p) We will only present the general ideas on how to obtain the ACF and PACF of an AR(p) model since the details follow closely the AR(1) and AR(2) cases presented before. Recall that
More informationMEI Exam Review. June 7, 2002
MEI Exam Review June 7, 2002 1 Final Exam Revision Notes 1.1 Random Rules and Formulas Linear transformations of random variables. f y (Y ) = f x (X) dx. dg Inverse Proof. (AB)(AB) 1 = I. (B 1 A 1 )(AB)(AB)
More informationLecture 7: Model Building Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 7: Model Building Bus 41910, Time Series Analysis, Mr R Tsay An effective procedure for building empirical time series models is the Box-Jenkins approach, which consists of three stages: model
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 III. Stationary models 1 Purely random process 2 Random walk (non-stationary)
More informationEconometrics I: Univariate Time Series Econometrics (1)
Econometrics I: Dipartimento di Economia Politica e Metodi Quantitativi University of Pavia Overview of the Lecture 1 st EViews Session VI: Some Theoretical Premises 2 Overview of the Lecture 1 st EViews
More information2. An Introduction to Moving Average Models and ARMA Models
. An Introduction to Moving Average Models and ARMA Models.1 White Noise. The MA(1) model.3 The MA(q) model..4 Estimation and forecasting of MA models..5 ARMA(p,q) models. The Moving Average (MA) models
More informationSTA 6857 Estimation ( 3.6)
STA 6857 Estimation ( 3.6) Outline 1 Yule-Walker 2 Least Squares 3 Maximum Likelihood Arthur Berg STA 6857 Estimation ( 3.6) 2/ 19 Outline 1 Yule-Walker 2 Least Squares 3 Maximum Likelihood Arthur Berg
More informationModelling using ARMA processes
Modelling using ARMA processes Step 1. ARMA model identification; Step 2. ARMA parameter estimation Step 3. ARMA model selection ; Step 4. ARMA model checking; Step 5. forecasting from ARMA models. 33
More informationComment about AR spectral estimation Usually an estimate is produced by computing the AR theoretical spectrum at (ˆφ, ˆσ 2 ). With our Monte Carlo
Comment aout AR spectral estimation Usually an estimate is produced y computing the AR theoretical spectrum at (ˆφ, ˆσ 2 ). With our Monte Carlo simulation approach, for every draw (φ,σ 2 ), we can compute
More information3 Theory of stationary random processes
3 Theory of stationary random processes 3.1 Linear filters and the General linear process A filter is a transformation of one random sequence {U t } into another, {Y t }. A linear filter is a transformation
More informationLab: Box-Jenkins Methodology - US Wholesale Price Indicator
Lab: Box-Jenkins Methodology - US Wholesale Price Indicator In this lab we explore the Box-Jenkins methodology by applying it to a time-series data set comprising quarterly observations of the US Wholesale
More informationCh 8. MODEL DIAGNOSTICS. Time Series Analysis
Model diagnostics is concerned with testing the goodness of fit of a model and, if the fit is poor, suggesting appropriate modifications. We shall present two complementary approaches: analysis of residuals
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 informationAdvanced Econometrics
Advanced Econometrics Marco Sunder Nov 04 2010 Marco Sunder Advanced Econometrics 1/ 25 Contents 1 2 3 Marco Sunder Advanced Econometrics 2/ 25 Music Marco Sunder Advanced Econometrics 3/ 25 Music Marco
More informationCircle the single best answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 6, 2017 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. There are 32 multiple choice
More informationEstimating AR/MA models
September 17, 2009 Goals The likelihood estimation of AR/MA models AR(1) MA(1) Inference Model specification for a given dataset Why MLE? Traditional linear statistics is one methodology of estimating
More informationA time series is called strictly stationary if the joint distribution of every collection (Y t
5 Time series A time series is a set of observations recorded over time. You can think for example at the GDP of a country over the years (or quarters) or the hourly measurements of temperature over a
More information{ } Stochastic processes. Models for time series. Specification of a process. Specification of a process. , X t3. ,...X tn }
Stochastic processes Time series are an example of a stochastic or random process Models for time series A stochastic process is 'a statistical phenomenon that evolves in time according to probabilistic
More informationCircle a single answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 4, 215 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. There are 31 questions. Circle
More informationEconometric Forecasting
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 1, 2014 Outline Introduction Model-free extrapolation Univariate time-series models Trend
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
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 informationLecture 4a: ARMA Model
Lecture 4a: ARMA Model 1 2 Big Picture Most often our goal is to find a statistical model to describe real time series (estimation), and then predict the future (forecasting) One particularly popular model
More informationFE570 Financial Markets and Trading. Stevens Institute of Technology
FE570 Financial Markets and Trading Lecture 5. Linear Time Series Analysis and Its Applications (Ref. Joel Hasbrouck - Empirical Market Microstructure ) Steve Yang Stevens Institute of Technology 9/25/2012
More informationITSM-R Reference Manual
ITSM-R Reference Manual George Weigt February 11, 2018 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................
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 informationWhite Noise Processes (Section 6.2)
White Noise Processes (Section 6.) Recall that covariance stationary processes are time series, y t, such. E(y t ) = µ for all t. Var(y t ) = σ for all t, σ < 3. Cov(y t,y t-τ ) = γ(τ) for all t and τ
More informationLecture 2: Univariate Time Series
Lecture 2: Univariate Time Series Analysis: Conditional and Unconditional Densities, Stationarity, ARMA Processes Prof. Massimo Guidolin 20192 Financial Econometrics Spring/Winter 2017 Overview Motivation:
More informationARIMA Models. Jamie Monogan. January 16, University of Georgia. Jamie Monogan (UGA) ARIMA Models January 16, / 27
ARIMA Models Jamie Monogan University of Georgia January 16, 2018 Jamie Monogan (UGA) ARIMA Models January 16, 2018 1 / 27 Objectives By the end of this meeting, participants should be able to: Argue why
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 informationEstimation and application of best ARIMA model for forecasting the uranium price.
Estimation and application of best ARIMA model for forecasting the uranium price. Medeu Amangeldi May 13, 2018 Capstone Project Superviser: Dongming Wei Second reader: Zhenisbek Assylbekov Abstract This
More information9. Model Selection. statistical models. overview of model selection. information criteria. goodness-of-fit measures
FE661 - Statistical Methods for Financial Engineering 9. Model Selection Jitkomut Songsiri statistical models overview of model selection information criteria goodness-of-fit measures 9-1 Statistical models
More informationCh 4. Models For Stationary Time Series. Time Series Analysis
This chapter discusses the basic concept of a broad class of stationary parametric time series models the autoregressive moving average (ARMA) models. Let {Y t } denote the observed time series, and {e
More informationNote: The primary reference for these notes is Enders (2004). An alternative and more technical treatment can be found in Hamilton (1994).
Chapter 4 Analysis of a Single Time Series Note: The primary reference for these notes is Enders (4). An alternative and more technical treatment can be found in Hamilton (994). Most data used in financial
More informationSTOR 356: Summary Course Notes Part III
STOR 356: Summary Course Notes Part III Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu April 23, 2008 1 ESTIMATION
More informationLecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
More informationForecasting. This optimal forecast is referred to as the Minimum Mean Square Error Forecast. This optimal forecast is unbiased because
Forecasting 1. Optimal Forecast Criterion - Minimum Mean Square Error Forecast We have now considered how to determine which ARIMA model we should fit to our data, we have also examined how to estimate
More informationElements of Multivariate Time Series Analysis
Gregory C. Reinsel Elements of Multivariate Time Series Analysis Second Edition With 14 Figures Springer Contents Preface to the Second Edition Preface to the First Edition vii ix 1. Vector Time Series
More informationLecture note 2 considered the statistical analysis of regression models for time
DYNAMIC MODELS FOR STATIONARY TIME SERIES Econometrics 2 LectureNote4 Heino Bohn Nielsen March 2, 2007 Lecture note 2 considered the statistical analysis of regression models for time series data, and
More informationContents. 1 Time Series Analysis Introduction Stationary Processes State Space Modesl Stationary Processes 8
A N D R E W T U L L O C H T I M E S E R I E S A N D M O N T E C A R L O I N F E R E N C E T R I N I T Y C O L L E G E T H E U N I V E R S I T Y O F C A M B R I D G E Contents 1 Time Series Analysis 5
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 informationMarcel Dettling. Applied Time Series Analysis SS 2013 Week 05. ETH Zürich, March 18, Institute for Data Analysis and Process Design
Marcel Dettling Institute for Data Analysis and Process Design Zurich University of Applied Sciences marcel.dettling@zhaw.ch http://stat.ethz.ch/~dettling ETH Zürich, March 18, 2013 1 Basics of Modeling
More informationLesson 13: Box-Jenkins Modeling Strategy for building ARMA models
Lesson 13: Box-Jenkins Modeling Strategy for building ARMA models Facoltà di Economia Università dell Aquila umberto.triacca@gmail.com Introduction In this lesson we present a method to construct an ARMA(p,
More informationCh 9. FORECASTING. Time Series Analysis
In this chapter, we assume the model is known exactly, and consider the calculation of forecasts and their properties for both deterministic trend models and ARIMA models. 9.1 Minimum Mean Square Error
More informationDynamic Time Series Regression: A Panacea for Spurious Correlations
International Journal of Scientific and Research Publications, Volume 6, Issue 10, October 2016 337 Dynamic Time Series Regression: A Panacea for Spurious Correlations Emmanuel Alphonsus Akpan *, Imoh
More information5 Autoregressive-Moving-Average Modeling
5 Autoregressive-Moving-Average Modeling 5. Purpose. Autoregressive-moving-average (ARMA models are mathematical models of the persistence, or autocorrelation, in a time series. ARMA models are widely
More informationMultivariate Time Series
Multivariate Time Series Notation: I do not use boldface (or anything else) to distinguish vectors from scalars. Tsay (and many other writers) do. I denote a multivariate stochastic process in the form
More informationSTAT 443 (Winter ) Forecasting
Winter 2014 TABLE OF CONTENTS STAT 443 (Winter 2014-1141) Forecasting Prof R Ramezan University of Waterloo L A TEXer: W KONG http://wwkonggithubio Last Revision: September 3, 2014 Table of Contents 1
More informationIntroduction to Time Series Analysis. Lecture 12.
Last lecture: Introduction to Time Series Analysis. Lecture 12. Peter Bartlett 1. Parameter estimation 2. Maximum likelihood estimator 3. Yule-Walker estimation 1 Introduction to Time Series Analysis.
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
More informationApplied time-series analysis
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 18, 2011 Outline Introduction and overview Econometric Time-Series Analysis In principle,
More informationARIMA Modelling and Forecasting
ARIMA Modelling and Forecasting Economic time series often appear nonstationary, because of trends, seasonal patterns, cycles, etc. However, the differences may appear stationary. Δx t x t x t 1 (first
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN
More informationCovariances of ARMA Processes
Statistics 910, #10 1 Overview Covariances of ARMA Processes 1. Review ARMA models: causality and invertibility 2. AR covariance functions 3. MA and ARMA covariance functions 4. Partial autocorrelation
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationMAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS)
MAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS) 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.
More informationVector autoregressions, VAR
1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS17/18 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,
More informationA Data-Driven Model for Software Reliability Prediction
A Data-Driven Model for Software Reliability Prediction Author: Jung-Hua Lo IEEE International Conference on Granular Computing (2012) Young Taek Kim KAIST SE Lab. 9/4/2013 Contents Introduction Background
More informationRoss Bettinger, Analytical Consultant, Seattle, WA
ABSTRACT DYNAMIC REGRESSION IN ARIMA MODELING Ross Bettinger, Analytical Consultant, Seattle, WA Box-Jenkins time series models that contain exogenous predictor variables are called dynamic regression
More information3. ARMA Modeling. Now: Important class of stationary processes
3. ARMA Modeling Now: Important class of stationary processes Definition 3.1: (ARMA(p, q) process) Let {ɛ t } t Z WN(0, σ 2 ) be a white noise process. The process {X t } t Z is called AutoRegressive-Moving-Average
More informationUnit root problem, solution of difference equations Simple deterministic model, question of unit root
Unit root problem, solution of difference equations Simple deterministic model, question of unit root (1 φ 1 L)X t = µ, Solution X t φ 1 X t 1 = µ X t = A + Bz t with unknown z and unknown A (clearly X
More informationThe Role of "Leads" in the Dynamic Title of Cointegrating Regression Models. Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji
he Role of "Leads" in the Dynamic itle of Cointegrating Regression Models Author(s) Hayakawa, Kazuhiko; Kurozumi, Eiji Citation Issue 2006-12 Date ype echnical Report ext Version publisher URL http://hdl.handle.net/10086/13599
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
More informationBIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation
BIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation Yujin Chung November 29th, 2016 Fall 2016 Yujin Chung Lec13: MLE Fall 2016 1/24 Previous Parametric tests Mean comparisons (normality assumption)
More information