Introduction to Time Series Analysis. Lecture 1.

Transcription

1 Introduction to Time Series Analysis. Lecture 1. Peter Bartlett 1. Organizational issues. 2. Objectives of time series analysis. Examples. 3. Overview of the course. 4. Time series models. 5. Time series modelling: Chasing stationarity. 1

2 Organizational Issues Peter Bartlett. Office hours: Tue 11-12, Thu (Evans 399). Joe Neeman. Office hours: Wed 1:30 2:30, Fri 2-3 (Evans???). bartlett/courses/153-fall2010/ Check it for announcements, assignments, slides,... Text: Time Series Analysis and its Applications. With R Examples, Shumway and Stoffer. 2nd Edition

3 Organizational Issues Classroom and Computer Lab Section: Friday 9 11, in 344 Evans. Starting tomorrow, August 27: Sign up for computer accounts. Introduction to R. Assessment: Lab/Homework Assignments (25%): posted on the website. These involve a mix of pen-and-paper and computer exercises. You may use any programming language you choose (R, Splus, Matlab, python). Midterm Exams (30%): scheduled for October 7 and November 9, at the lecture. Project (10%): Analysis of a data set that you choose. Final Exam (35%): scheduled for Friday, December 17. 3

4 A Time Series

5 A Time Series year 5

6 A Time Series $ year 6

7 A Time Series 400 SP500: $ year 7

8 A Time Series 340 SP500: Jan Jun $ year 8

9 A Time Series 30 SP500 Jan Jun Histogram $ 9

10 A Time Series 340 SP500: Jan Jun Permuted $

11 Objectives of Time Series Analysis 1. Compact description of data. 2. Interpretation. 3. Forecasting. 4. Control. 5. Hypothesis testing. 6. Simulation. 11

12 Classical decomposition: An example Monthly sales for a souvenir shop at a beach resort town in Queensland. (Makridakis, Wheelwright and Hyndman, 1998) 12 x

13 Transformed data

14 Trend

15 Residuals

16 Trend and seasonal variation

17 Objectives of Time Series Analysis 1. Compact description of data. Example: Classical decomposition: X t = T t + S t + Y t. 2. Interpretation. Example: Seasonal adjustment. 3. Forecasting. Example: Predict sales. 4. Control. 5. Hypothesis testing. 6. Simulation. 17

18 Unemployment data Monthly number of unemployed people in Australia. (Hipel and McLeod, 1994) 8 x

19 Trend 8 x

20 Trend plus seasonal variation 8 x

21 Residuals 8 x

22 Predictions based on a (simulated) variable 8 x

23 Objectives of Time Series Analysis 1. Compact description of data: X t = T t + S t + f(y t ) + W t. 2. Interpretation. Example: Seasonal adjustment. 3. Forecasting. Example: Predict unemployment. 4. Control. Example: Impact of monetary policy on unemployment. 5. Hypothesis testing. Example: Global warming. 6. Simulation. Example: Estimate probability of catastrophic events. 23

24 Overview of the Course 1. Time series models 2. Time domain methods 3. Spectral analysis 4. State space models(?) 24

25 Overview of the Course 1. Time series models (a) Stationarity. (b) Autocorrelation function. (c) Transforming to stationarity. 2. Time domain methods 3. Spectral analysis 4. State space models(?) 25

26 Overview of the Course 1. Time series models 2. Time domain methods (a) AR/MA/ARMA models. (b) ACF and partial autocorrelation function. (c) Forecasting (d) Parameter estimation (e) ARIMA models/seasonal ARIMA models 3. Spectral analysis 4. State space models(?) 26

27 Overview of the Course 1. Time series models 2. Time domain methods 3. Spectral analysis (a) Spectral density (b) Periodogram (c) Spectral estimation 4. State space models(?) 27

28 Overview of the Course 1. Time series models 2. Time domain methods 3. Spectral analysis 4. State space models(?) (a) ARMAX models. (b) Forecasting, Kalman filter. (c) Parameter estimation. 28

29 Time Series Models A time series model specifies the joint distribution of the sequence {X t } of random variables. For example: P[X 1 x 1,...,X t x t ] for all t and x 1,...,x t. Notation: X 1, X 2,... is a stochastic process. x 1, x 2,... is a single realization. We ll mostly restrict our attention to second-order properties only: EX t, E(X t1, X t2 ). 29

30 Time Series Models Example: White noise: X t WN(0, σ 2 ). i.e., {X t } uncorrelated, EX t = 0, VarX t = σ 2. Example: i.i.d. noise: {X t } independent and identically distributed. P[X 1 x 1,...,X t x t ] = P[X 1 x 1 ] P[X t x t ]. Not interesting for forecasting: P[X t x t X 1,...,X t 1 ] = P[X t x t ]. 30

31 Gaussian white noise P[X t x t ] = Φ(x t ) = 1 2π xt e x2 /2 dx

32 Gaussian white noise

33 Time Series Models Example: Binary i.i.d. P[X t = 1] = P[X t = 1] = 1/

34 Random walk S t = t i=1 X i. Differences: S t = S t S t 1 = X t

35 Random walk ES t? VarS t?

36 Random Walk Recall S&P500 data. (Notice that it s smooth) 340 SP500: Jan Jun $ year 36

37 Random Walk Differences: S t = S t S t 1 = X t. 10 SP500, Jan Jun first differences $ year 37

38 Trend and Seasonal Models X t = T t + S t + E t = β 0 + β 1 t + i (β i cos(λ i t) + γ i sin(λ i t)) + E t

39 Trend and Seasonal Models X t = T t + E t = β 0 + β 1 t + E t

40 Trend and Seasonal Models X t = T t + S t + E t = β 0 + β 1 t + i (β i cos(λ i t) + γ i sin(λ i t)) + E t

41 Trend and Seasonal Models: Residuals

42 Time Series Modelling 1. Plot the time series. Look for trends, seasonal components, step changes, outliers. 2. Transform data so that residuals are stationary. (a) Estimate and subtract T t, S t. (b) Differencing. (c) Nonlinear transformations (log, ). 3. Fit model to residuals. 42

43 Nonlinear transformations Recall: Monthly sales. (Makridakis, Wheelwright and Hyndman, 1998) 12 x

44 Time Series Modelling 1. Plot the time series. Look for trends, seasonal components, step changes, outliers. 2. Transform data so that residuals are stationary. (a) Estimate and subtract T t, S t. (b) Differencing. (c) Nonlinear transformations (log, ). 3. Fit model to residuals. 44

45 Differencing Recall: S&P 500 data. 340 SP500: Jan Jun SP500, Jan Jun first differences $ 280 $ year year 45

46 Differencing and Trend Define the lag-1 difference operator, (think first derivative ) X t = X t X t 1 = (1 B)X t, where B is the backshift operator, BX t = X t 1. If X t = β 0 + β 1 t + Y t, then If X t = k i=0 β it i + Y t, then X t = β 1 + Y t. k X t = k!β k + k Y t, where k X t = ( k 1 X t ) and 1 X t = X t. 46

47 Differencing and Seasonal Variation Define the lag-s difference operator, s X t = X t X t s = (1 B s )X t, where B s is the backshift operator applied s times, B s X t = B(B s 1 X t ) and B 1 X t = BX t. If X t = T t + S t + Y t, and S t has period s (that is, S t = S t s for all t), then s X t = T t T t s + s Y t. 47

48 Time Series Modelling 1. Plot the time series. Look for trends, seasonal components, step changes, outliers. 2. Transform data so that residuals are stationary. (a) Estimate and subtract T t, S t. (b) Differencing. (c) Nonlinear transformations (log, ). 3. Fit model to residuals. 48

49 Outline 1. Objectives of time series analysis. Examples. 2. Overview of the course. 3. Time series models. 4. Time series modelling: Chasing stationarity. 49