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# Class Note for STAT 635 at OSU 10

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Date Created: 02/06/15

STAT635 LECTURE OUTLINE 1 The world makes the way for a man who knows where he is going Ralph Waldo Emerson OUTLINE I De nition and areas of application I Examples of time series I Summarizing the distribution using moments I Objectives of time series analysis I Stationarity and autocorrelation I Examples of time series models D iid noise D White noise D The random walk process D First order moving average process MAG D First order autoregressive process AR1 I Estimating the mean of the time series I Estimating the ACVE and ACE Source Brockwell and Davis7 sections 11 147 and appendices A17 A2 STATISTICAL ANALYSIS OF TIME SERIES 1 WHAT IS A TIME SERIES D A time series is a set of observations 33 each one being recorded at a specific time t D A time series is a a collection of observations made sequentially through time B quotOne damn thing after another R A Fisher D The statistical analysis of time series refers to all the statistical tech niques used to solve problems in which Observations are collected at regular time intervals There are correlations among successive observations NOTATION D The time series is denoted by Xt7 t E T where T is the index D If T is continuous we have a continuous time series B If T is discrete we have a discrete time series and T Z the set of all integers The time series is sometimes written as 7X727X717X07X17X2739 D This course deals predominantly with discrete time series B For simplicity we will drop the index set and write Xt D A realization of Xt will be denoted by Xt or X17 X2 X3 to indicate that they are observations D In practice the time interval for collection of time series could be seconds minutes hours days weeks months years or function of these or any reasonable regular time intervals STATISTICAL ANALYSIS OF TIME SERIES 2 AREAS OF APPLICATIONS Time series data arise quite naturally in a variety of sectors namely finance economics environmental sciences medicine just to name a few D Financial investment 7 Monthly closings of the Dow Jones industrial index 7 Annual bond yield in the USA 7 Daily SampP 500 index of stocks 7 IBM daily stock closing prices D Macro Economics 7 Monthly percent changes in US wages and salaries US Annual industrial production 7 Quarterly US GNP Billions D Micro Economics 7 Annual Copper prices 7 Daily morning gold prices D Health and Medicine 7 Online Monitoring of Patient condition in Intensive Care 7 Daily morning temperature of adult female 7 Annual US suicide rates D Meteorology Average monthly temperature in Columbus 7 Annual precipitation in inches in the midwest Daily maximum temperatures in the US D Sales 7 Monthly car sales 7 Quarterly sales of toys REMINDER Now is a good time to start gathering your list of potential applications from which you will select the data for your final project STATISTICAL ANALYSIS OF TIME SERIES 3 EXAMPLES OF TIME SERIES Example 1 Monthly Airline passengers Jan 1949 to Dec 1960 700 600 number of passengers housands a a o o o o o o o 200 100 0 20 40 60 80 100 120 140 month WHAT DO YOU SEE D D D CLASSICAL DECOMPOSITION XtTtStCtEt D Trend Long term movement in the mean D Seasonal variation 3 Cyclical uctuations due to calendar D Cycles Ct Cyclical uctuations of larger period eg Business cycles D Residuals Et random and all other unexplained variations WHY IS TRANSFORMATION NECESSARY D D STATISTICAL ANALYSIS OF TIME SERIES GENERAL APPROACH To TIME SERIES MODELING I Plot the series and examine the main features D ls there a trend ls there a seasonality effect Are there cycles D ls variation apparently time dependent D Are any apparent sharp changes in behavior D Are there any outliers I Perform a transformation of the data if necessary D Logarithmic transformation replace X17 X27 Xn with log Xhlog X2 log Xn if uctuations appear to grow linearly with the level of the series I Remove the trend seasonal components and cycles to get stationary residuals D Differencing the data Replacing the original series Xi by Yt Xi Xt1 I Choose a model to fit the residuals D Models will be discussed at length later I Do the forecasting D Be sure to invert the transformations in order to obtain forecasts of the original Xi FOURIER ANALYSIS ALTERNATIVE o Expressing the series in terms of its Fourier cornponents We will only briefly talk about this approach Little thinking interlude D Provide two examples of time series with both seasonal e ects and cycles D X17X2 7X45 is a set of observed heights for 45 American adults y17y27 y45 is a set of observed daily closing prices of stock epf on Wall Street over a 45 day period Provide your list of what makes the analysis of these two data set qualitative and quantitative di erent STATISTICAL ANALYSIS OF TIME SERIES 5 Example 2 Australian wine Jan 1980 to Dec 1981 3000 2500 7 7 2000 7 7 15007 7 thousands 10007 7 500 7 0 1 1 1 1 1980 1982 1984 1986 1988 1990 1992 Source Brockwell amp Davis 2002 Example 3 The monthly accidental deaths 65 1 1 1 1 1 1 1 0 10 20 30 40 50 60 70 80 Source Brockwell amp Davis 2002 STATISTICAL ANALYSIS OF TIME SERIES Example 4 US Population at 10 year intervals 250 200 150 100 50 0 l l l l l l 1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 Source Brockwell amp Davis 2002 Example 5 Value of the SampP500 from June 6 1960 to June 6 2000 1600 1400 1200 1000 800 value 600 400 200 0 l l l l l l l 1960 1965 1970 1975 1980 1985 1990 1995 2000 Source Chris Chat eld 2000 STATISTICAL ANALYSIS OF TIME SERIES Example 6 The series from cosine 0 20 40 60 80 100 120 140 160 180 200 Source Brockwell amp Davis 2002 Example 7 Strikes in the USA 1951 1980 65 55 7 thousands on 1 P m I 35 7 3 I I I I I 1950 1955 1960 1965 1970 1975 1980 Source Brockwell amp Davis 2002 STATISTICAL ANALYSIS OF TIME SERIES OBJECTIVE OF TIME SERIES ANALYSIS Given an observed data set X75 one may want to I Modeling or Model building 7 Set up a hypothetical model or family of models to represent the data generating mechanism More about this in the next section I Estimation Estimate the parameters of the postulated model I Model checking Check the goodness of t of the model to the data I Understanding and insight Use the tted model to enhance understanding of the data generating mechanism 7 Compact description of the data eg decomposition into a Trend b Seasonal c random terms 7 Seasonal adjustment removal of the seasonal component so that it is not confused with the long term trend I Filtering 7 Separation of noise from signal I Hypothesis testing 7 Do the recorded temperatures provide evidence of global warming I Forecasting Predict the future price of a share of a given stock I Simulation 7 For very complicated systems simulation is sometimes the only way to gain insights into the data generating mechanism Remark Note that the term time series is used to mean both the observed data and the process of which it is a realization STATISTICAL ANALYSIS OF TIME SERIES 9 MODELING AND ESTIMATION WITH TIME SERIES 77All models are wrong but some are useful 7 George E P Box 1979 p 202 D A time series model for the observed data X75 is a speci cation of the joint distributions or possibly only the means and covariances of a sequence of random variables Xi of which Xt is postulated to be a realization U Need to specify all the joint distributions FX1X2 Xn of the random vectors X1X2 Xn for n 12 D Problem Far too many parameters to be estimated In fact FX1 X2 Xn is potentially infinite dimensional D Decision Focus sequences that depend only on second order properties 7 IE Xi Expected values 7 IE XthXt t 12 h 0 l 2 Expected products D Question How realistic is such a restriction D Well There is some loss but further theory show that one should not be too alarmed A NOTEWORTHY AND FUNDAMENTAL CONTRAST Traditional statistics Random sampling procedures enable its to ob tain replicated observations under identical conditions Besides these observations are independent By contrast Time series We have only one single realization at each time point and also dependence over time The observed time series is just one out of the in nite number of possible realizations More precisely it is a sample of size one WHY DOES THE CONTRAST MATTER For any inference to be possible we must recreate some notion of replicability STATISTICAL ANALYSIS OF TIME SERIES 10 ZERO MEAN TIME SERIES MODELS THE HD NOISE MODEL I Simplest time series model B No trend Zero Inean D No seasonal variations D Independence of obs from the same distribution iid I Notation D X17X2 Xn are iid I Distributionally 7 iid ness 7 implies FX1X2Xn FltX1gtFltX2quot39FXn TIME PLOT OF AN IID SERIES 25 o i 725 0 i i i i i i i i i 20 40 60 80 100 120 140 160 180 200 Great limitation CANNOT BE USED To FORECAST ANYTHING Note Though apparently uninteresting the iid model serve a building block for more sophisticated time series models STATISTICAL ANALYSIS OF TIME SERIES 11 A RANDOM WALK SERIES A BINARY PROCESS First de ned the iid series Xi such that PrXt 1 and PrXt 11 The process such de ned is a binary process It is a valid iid noise process A RANDOM WALK PROCESS Let Sh t012With 300and StX1X2 Xt7 t 1727 Where Xi is our binary process de ned earlier St is called a simple symmetric random walk TIME PLOT OF A SIMPLE SYMMETRIC RANDOM WALK 15 i i i i i i i i i 0 20 40 60 80 100 120 140 160 180 200 E Find an expression for St SP1 and explain STATISTICAL ANALYSIS OF TIME SERIES 12 STATIONARY MODELS AND AUTOCORRELATION D A time series Xt t 0 l1 l2 is said to be stationary if it has statis tical properties similar to those of the ltime shifted 7 series Xth t 0 l1 l2 for each integer n STRONG STATIONARITY De nition 1 A time series Xi is said to be stroneg or strictly stationary if the joint density functions depend only on the relation location of the observations so that fltxt1h7 xt2h7 39 39 39 7mtkh fltxt17xt27 39 39 39 7xtk7 meaning that Xt1hXt2h thh and Xt1Xt2 th have the samejoint distributions for all h and for all choices of the time points Note Strong stationarity is well too strong Besides specifying the densities fxt1xt2 xtk is usually very complicated SECOND ORDER MOMENTS FOR TIME SERIES ANALYSIS D The mean function of Xi is mt EX D The variance function of Xi is 0amp0 WXt E 09 MXOWZI D The covariance function of Xi is 7X5t COVX5Xt E XS MX5Xt uxtl Using second order moments we can provide a much more tractable and usable de nition of stationarity known as weak stationarity QUESTION WHY IS STATIONARITY So IMPORTANT EVEN CRUCIAL STATISTICAL ANALYSIS OF TIME SERIES 13 De nition 2 A time series Xi is weakly stationary if i aXt is independent oft ie aXt aX for all t ii 7X05 h t is independent oft for each h h is called the lag Weak stationarlty is also referred to as second order stationarity covariance sta tlonarlty or even wide sense stationarity From now on we shall say stationary to mean weakly stationary QUESTION WHY IS STATIONARITY so IMPORTANT EVEN CRUCIAL ANSWER STATIONARITY ALLOWS THE RE CREATION OF THE NOTION OF REPLICABILITY THAT IS CRUCIAL TO STATISTICAL INFERENCE INTUITIVELY Let Xt be the value of the series at time t Holding t xed imagine an infinite series of repetitions of essentially the same generating process giving rise to a population of values of Xt similar to What happens in traditional regression models For a stationary time series the popula tion probability distribution of Xt is independent of t Independence on t captures the replicability crucial to statistical inference Note The above intuition captures both strong and weak stationarity IN A WEAK SENSE STATIONARITY MEANS D Constant mean D Constant variance D Covariance function independent of h STATISTICAL ANALYSIS OF TIME SERIES 14 STATIONARY MODELS AND AUTOCOVARIANCE AUTOCOVARIANCES AND AUTOCORRELATIONS De nition 3 Let Xi be a stationary time series with mean function uX The autocovariance function ACVF of Xi at lag h is yXUL covXthXt E 09 MXXt MXI The autocorrelation function ACF of Xi at lag h is pXh corrXthXt Note They measure the amount of dependence between Xi and XHh SOME PROPERTIES D 7X00 VXt D AyXh 7X h for each h B When Xi is Gaussian the distribution is completely speci ed by uX and 7X EXAMPLES OF STATIONARY PROCESSES THE IID NOISE PROCESS If Xi is an iid noise process we write Xi N D002 We also have 02 ifh0 tht w gt 0 ifhy O STATISTICAL ANALYSIS OF TIME SERIES 15 THE WHITE NOISE PROCESS Let Xi be a sequence of o Uncorrelated random variables ie AyXh 0 for h y 0 0 Each variable having zero mean ie lEXt 0 0 Each variable having nite variance ie VXt 02 lt 00 Such a sequence is referred to as white noise with mean 0 and variance 02 and indicated by Xi N WN002 NOTEWORTHY REMARKS Every D0 02 is WN002 but not conversely Xi N WN0 02 is clearly a stationary process The covariance function of Xi N WN0 02 is the same as that of D0 02 namely 02 ifh0 7 tht M l 0 ifhy O For Xt N WN0 02 the distributions of Xi and Xth may be different LET7S DO EXERCISE 18 PART 1 STATISTICAL ANALYSIS OF TIME SERIES 16

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