Chapter 1 - Introduction

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1 Admin. 2 Basic description. 3 Some examples. 4 Trend and seasonal cycles. 5 Course overview. 6 On statistical modelling. Y. Chen, S. Tavakoli. Chapter 1 - ...
Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Chapter 1 - Introduction

Y. Chen S. Tavakoli A Cambridge Part III Course, Lent 2015

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Outline 1 2 3 4 5 6

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Administration and examination

24 lectures in total, 12 on Time Series and 12 on Monte Carlo Inference 4 example sheets and 4 example classes 6 exam questions in total, 3 on Time Series and 3 on Monte Carlo Inference Need to pick 4 out of 6 exam questions

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Course website

My email address is [email protected] Course website: http://www.statslab.cam.ac.uk/~yc319/ts.html Feedback form on the course website

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Books - I

(by P. J. Brockwell and R. A. Davis) serves as a good introduction, especially for those completely new to time series analysis. Introduction to Time Series and Forecasting

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Books - II Time Series: Applications to Finance with R and S-Plus

(by N. H. Chan) covers large parts of this course, presented in a less mathematical and very concise style.

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Books - III Time Series Analysis and its Applications: with R Examples

(by R. H. Shummway and D. S. Stoer) is another great book on time series analysis aimed at roughly the same level as our course.

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Books - more advanced (by P. J. Brockwell and R. A. Davis) covers the rst part of our course (linear time series models) in much greater depth than we do. It also contains rigorous proofs of many theoretical results that we state but do not prove in the lectures.

Time Series: Theory and Methods

GARCH Models: Structure, Statistical Inference and Financial

Applications (by C. Francq and J-M. Zakoian) is an excellent read for those who are keen to know more about non-linear time series models. It provides a comprehensive and systematic approach from a mathematical (theoretical) perspective.

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Outline 1 2 3 4 5 6

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

What is a time series

A time series is a set of data collected at successive (discrete) time-points In this course we will conne ourselves to observations made at regularly spaced time-points, without loss of generality taking the interval between such points to be 1 We shall write the series as {Xt , t ∈ Z} Other time series: continuous time series, discrete observed values, multivariate time series etc. Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Applications

economics  e.g., monthly data for unemployment, hospital admissions nance  e.g., daily exchange rate, a share price environmental  e.g., daily rainfall, air quality readings medicine  e.g., brain wave activity

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Outline 1 2 3 4 5 6

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

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Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

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Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

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Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Classical decomposition I

One simple method of describing a series is that of classical decomposition. The notion is that the series can be decomposed into three elements: Trend (Tt ): long term movements in the mean Seasonal eects (St ): cyclical uctuations related to the calendar or business cycles Microscopic part (Mt ): other random or systematic uctuations Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Classical decomposition II

The idea is to create separate models for these three elements and then combine them either additively Xt = Tt + St + Mt or multiplicatively Xt = Tt St Mt

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Classical decomposition - example

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Estimation of trends and seasonal cycles I

Least squares method for linear trend: suppose that the seasonal part is absent a simple linear time trend structure Tt = a + bt estimate a and b by minimising ∑(Xt − Tt ) Major drawbacks: cannot deal with trend changing over time linear form might be too restrictive 2

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Estimation of trends and seasonal cycles II

Smoothing or ltering: s

Tt = ∑ ar Xt+r r =−q

where the weights {ar } of the lter are usually assumed to be symmetric and normalized, so ar = a r and ∑ ar = 1, This lter is also known as the moving average lter. −

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Estimation of trends and seasonal cycles III

Dierencing: First order dierencing: Yt = Xt − Xt Higher order dierencing: e.g. second order, Yt = (Xt − Xt ) − (Xt − Xt ) Seasonal dierencing: Yt = Xt − Xt d , if the cycle is of length −1

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Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Estimation of trends and seasonal cycles IV

Reasons behind dierencing: if Xt = Tt + Mt with Tt = a + bt , then Tt is eliminated in the new seriesYt = Xt − Xt same argument works for order-k dierencing, where Tt is a k -degree polynomial of t −1

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Outline 1 2 3 4 5 6

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Dierent approaches

In this course, we mainly focus on modelling the microscopic part Mt : Time domain

linear models: autoregressive (AR) models, moving average (MA) models, ARMA models, ARIMA models non-linear models: generalised autogressive heteroskedasticity (GARCH) models, threshold models

Frequency domain

Spectral analysis

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Statistical software

Most of the common time series data analysis can be done in R Some useful packages include: ARIMA, fGarch, stochvol Most frequently used functions include: lter, acf, pacf, arima, garch, spectrum, etc More details can be found here: cran.r-project.org/web/views/TimeSeries.html Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Outline 1 2 3 4 5 6

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

General framework

Data generating mechanism Probability structure Estimation procedure Model selection Model diagnostics

Y. Chen, S. Tavakoli

Chapter 1 - Introduction

Admin Basic description Some examples Trend and seasonal cycles Course overview On statistical modelling

Some quotes

All models are wrong, but some are useful. (George Box) There are no routine statistical questions, only questionable statistical routines. (David Cox) If you torture the data enough, nature will always confess. (Ronald Coase) In theory there is no dierence between theory and practice. But, in practice, there is. (Attributed to multiple people) Y. Chen, S. Tavakoli

Chapter 1 - Introduction