# Matrix Time Series Analysis by Seyed Yaser Samadi (Under the Direction of Professor Lynne Billard)

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• Autocovariance Function Estimation Via Penalized Regression
Autocovariance Function Estimation via Penalized Regression Lina Liao, Cheolwoo Park Department of Statistics, University of Georgia, Athens, GA 30602, USA Jan Hannig Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA Kee-Hoon Kang Department of Statistics, Hankuk University of Foreign Studies, Yongin, 449-791, Korea Abstract The work revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with ﬂexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero mean time series, we apply a penalized regression technique to a diﬀerenced time series, which does not require a separate detrending procedure. In penalized regression, selection of tuning parameters is critical and we propose four diﬀerent data-driven criteria to determine them. A simulation study shows eﬀectiveness of the tuning parameter selection and that the proposed approach is superior to three existing methods. We also brieﬂy discuss the extension of the proposed approach to interval-valued time series. Some key words: Autocovariance function; Diﬀerenced time series; Regularization; Time series; Tuning parameter selection. 1 1 Introduction Let fYt; t 2 T g be a stochastic process such that V ar(Yt) < 1, for all t 2 T . The autoco- variance function of fYtg is given as γ(s; t) = Cov(Ys;Yt) for all s; t 2 T . In this work, we consider a regularly sampled time series f(i; Yi); i = 1; ··· ; ng. Its model can be written as Yi = g(i) + i; i = 1; : : : ; n; (1.1) where g is a smooth deterministic function and the error is assumed to be a weakly stationary 2 process with E(i) = 0, V ar(i) = σ and Cov(i; j) = γ(ji − jj) for all i; j = 1; : : : ; n: The estimation of the autocovariance function γ (or autocorrelation function) is crucial to determine the degree of serial correlation in a time series.
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• LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a speciﬁc time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we deﬁne (i) Def. Distribution function: (ﬁ-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an inﬁnite matrix).
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• Lecture 1: Stationary Time Series∗
Lecture 1: Stationary Time Series∗ 1 Introduction If a random variable X is indexed to time, usually denoted by t, the observations {Xt, t ∈ T} is called a time series, where T is a time index set (for example, T = Z, the integer set). Time series data are very common in empirical economic studies. Figure 1 plots some frequently used variables. The upper left ﬁgure plots the quarterly GDP from 1947 to 2001; the upper right ﬁgure plots the the residuals after linear-detrending the logarithm of GDP; the lower left ﬁgure plots the monthly S&P 500 index data from 1990 to 2001; and the lower right ﬁgure plots the log diﬀerence of the monthly S&P. As you could see, these four series display quite diﬀerent patterns over time. Investigating and modeling these diﬀerent patterns is an important part of this course. In this course, you will ﬁnd that many of the techniques (estimation methods, inference proce- dures, etc) you have learned in your general econometrics course are still applicable in time series analysis. However, there are something special of time series data compared to cross sectional data. For example, when working with cross-sectional data, it usually makes sense to assume that the observations are independent from each other, however, time series data are very likely to display some degree of dependence over time. More importantly, for time series data, we could observe only one history of the realizations of this variable. For example, suppose you obtain a series of US weekly stock index data for the last 50 years.
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• STA 6857---Autocorrelation and Cross-Correlation & Stationary Time Series
Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c STA 6857—Autocorrelation and Cross-Correlation & Outline Stationary Time Series (§1.4, 1.5) 1 Announcements 2 Autocorrelation and Cross-Correlation 3 Stationary Time Series 4 Homework 1c Arthur Berg STA 6857—Autocorrelation and Cross-Correlation & Stationary Time Series (§1.4, 1.5) 2/ 25 Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c Announcements Autocorrelation and Cross-Correlation Stationary Time Series Homework 1c Homework Questions from Last Time We’ve seen the AR(p) and MA(q) models. Which one do we use? In scientiﬁc modeling we wish to choose the model that best Our TA, Aixin Tan, will have ofﬁce hours on Thursdays from 1–2pm describes or explains the data. Later we will develop many in 218 Grifﬁn-Floyd. techniques to help us choose and ﬁt the best model for our data. Homework 1c will be assigned today and the last part of homework Is this white noise process in the models unique? 1, homework 1d, will be assigned on Friday. Homework 1 will be We will see later that any stationary time series can be described as collected on Friday, September 7. Don’t wait till the last minute to do a MA(1) + “deterministic part” by the Wold decomposition, where the homework, because more homework will follow next week. the white noise process in the MA(1) part is unique. So in short, the answer is yes. We will also see later how to estimate the white noise process which will aid in forecasting.
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• Statistics 153 (Time Series) : Lecture Five
Statistics 153 (Time Series) : Lecture Five Aditya Guntuboyina 31 January 2012 1 Contents 1. Deﬁnitions of strong and weak stationary sequences. 2. Examples of stationary sequences. 2 Stationarity The concept of a stationary time series is the most important thing in this course. Stationary time series are typically used for the residuals after trend and seasonality have been removed. Stationarity allows a systematic study of time series forecasting. In order to avoid confusion, we shall, from now on, make a distinction be- tween the observed time series data x1; : : : ; xn and the sequence of random vari- ables X1;:::;Xn. The observed data x1; : : : ; xn is a realization of X1;:::;Xn. It is convenient of think of the random variables fXtg as forming a doubly inﬁnite sequence: :::;X−2;X−1;X0;X1;X2;:::: The notion of stationarity will apply to the doubly inﬁnite sequence of ran- dom variables fXtg. Strictly speaking, stationarity does not apply to the data x1; : : : ; xn. However, one frequently abuses terminology and uses stationary data to mean that the random variables of which the observed data are a real- ization have a stationary distribution. 1 Deﬁnition 2.1 (Strict or Strong Stationarity). A doubly inﬁnite se- quence of random variables fXtg is strictly stationary if the joint dis- tribution of (Xt1 ;Xt2 ;:::;Xtk ) is the same as the joint distribution of (Xt1+h;Xt2+h;:::;Xtk+h) for every choice of times t1; : : : ; tk and h. Roughly speaking, stationarity means that the joint distribution of the ran- dom variables remains constant over time. For example, under stationarity, the joint distribution of today's and tomorrow's random variables is the same as the joint distribution of the variables from any two successive days (past or future).
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• Cos(X+Y) = Cos(X) Cos(Y) - Sin(X) Sin(Y)
On 1.9, you will need to use the facts that, for any x and y, sin(x+y) = sin(x) cos(y) + cos(x) sin(y). cos(x+y) = cos(x) cos(y) - sin(x) sin(y). (sin(x))2 + (cos(x))2 = 1. 28 1 Characteristics of Time Series n/2 1/2 1 1 f(xxx )=(2⇡)− Γ − exp (xxx µµµ )0Γ − (xxx µµµ ) , (1.31) | | −2 − − ⇢ where denotes the determinant. This distribution forms the basis for solving problems|·| involving statistical inference for time series. If a Gaussian time series, xt , is weakly stationary, then µt = µ and γ(ti,tj)=γ( ti tj ), so that{ the} vector µµµ and the matrix Γ are independent of time. These| − facts| imply that all the ﬁnite distributions, (1.31), of the series xt depend only on time lag and not on the actual times, and hence the series{ must} be strictly stationary. 1.6 Estimation of Correlation Although the theoretical autocorrelation and cross-correlation functions are useful for describing the properties of certain hypothesized models, most of the analyses must be performed using sampled data. This limitation means the sampled points x1,x2,...,xn only are available for estimating the mean, autocovariance, and autocorrelation functions. From the point of view of clas- sical statistics, this poses a problem because we will typically not have iid copies of xt that are available for estimating the covariance and correlation functions. In the usual situation with only one realization, however, the as- sumption of stationarity becomes critical. Somehow, we must use averages over this single realization to estimate the population means and covariance functions.
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• 13 Stationary Processes
13 Stationary Processes In time series analysis our goal is to predict a series that typically is not deterministic but contains a random component. If this random component is stationary, then we can develop powerful techniques to forecast its future values. These techniques will be developed and discussed in this chapter. 13.1 Basic Properties In Section 12.4 we introduced the concept of stationarity and deﬁned the autocovariance function (ACVF) of a stationary time series Xt at lag h as f g γX (h) = Cov(Xt+h;Xt); h = 0; 1; 2 ::: ± ± and the autocorrelation function as γX (h) ρX (h) := : γX (0) The autocovariance function and autocorrelation function provide a useful measure of the degree of dependence among the values of a time series at diﬀerent times and for this reason play an important role when we consider the prediction of future values of the series in terms of the past and present values. They can be estimated from observations of X1;:::;Xn by computing the sample autocovariance function and autocorrelation function as described in Deﬁnition 12.4.4. Proposition 13.1.1. Basic properties of the autocovariance function γ( ): · γ(0) 0; ≥ γ(h) γ(0) for all h; j j ≤ γ(h) = γ( h) for all h; i.e., γ( ) is even. − · Deﬁnition 13.1.2. A real-valued function κ deﬁned on the integers is non-negative deﬁnite if n aiκ(i j)aj 0 i;j=1 − ≥ X 0 for all positive integers n and vectors a = (a1; : : : ; an) with real-valued components ai.
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• Cross Correlation Functions 332 D