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Time Series Analysis Section 1
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Contents

• Time Series Analysis
• Section 1
• Sun 28 Dec. 03 900 1200
• Section 2
• Sun 28 Dec. 03 1300 1600

Time Series Analysis
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Time Series Analysis

• Section 1
• Introduction to Time Series Analysis
• Descriptive Techniques
• Smoothing Methods
• Seasonality
• Introduction to Autocorrelation

Time Series Analysis
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Time Series

• What is Time Series?
• A time series is a set of observations of a
  variable measured at successive points in time or
  over successive periods of time
• The objective of time-series analysis is to
  provide good forecasts or predictions of future
  values of the time series

Introduction
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Objectives of Time Series Analysis

• Description
• Find regular pattern
• Explanation
• Deeper understanding of the mechanism which
  generated a given time series
• Prediction
• Prediction and Forecast
• Control
• Quality control

Introduction
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Simple Descriptive Techniques

• Types of variation
• Stationary Time Series
• The Time Plot
• Transformations

Descriptive Techniques
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Types of Variation

• Traditional methods of time series analysis are
  mainly concerned with decomposing the variation
  in a series into
• Seasonal Component
• Cyclical Component
• Trend Component
• Irregular Component

Descriptive Techniques
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Seasonal Component

• Many time series have a yearly variation
• The seasonal effect is easy to understand
• Examples??
• Easy to measure or remove from the data to give
  deseasonalized data

Descriptive Techniques
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Cyclical Component

• Apart from seasonal effects, some time series
  exhibit variation at a fixed period due to some
  other reasons
• Examples
• Daily variation
• 5 years business cycle
• To some extent, this cycle is predictable

Descriptive Techniques
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Trend Component

• Long-term change in the mean level
• But what is meant by long-term

t
t
Descriptive Techniques
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Irregular Component

• After trend and seasonal (cyclic) variations have
  been removed from a set of data, we are left with
  a series of residuals, which may or may not be
  random

Descriptive Techniques
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Stationary Time Series

• Broadly speaking a time series is said to be
  stationary
• If there is no systematic change in mean (no
  trend)
• If there is no systematic change in variance
• And if strictly periodic variations have been
  removed

Descriptive Techniques
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Transformations

• Like regression Plotting the time series data
  may suggest that it is sensible to consider
  transforming them, e.g. taking logarithms or
  square roots
• To stabilise the variance
• Variance increase as the mean increase take log
• To make the seasonal effect additive
• See later in analysing the seasonal variation
• To make the data normally distributed
• Take log or use other transformation (e.g.
  Box-Cox)

Descriptive Techniques
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Analysing Series which Contain a Trend

• The simplest trend is the familiar linear trend
  noise , for which the observation at time t is
  a random variable Xt given by
• Xt a ßt et
• where a and ß are constants and et denotes a
  random error term with zero mean
• This is a deterministic function of time and is
  sometimes called a global linear trend
• There are also other forms such as
• Quadratic growth Xt a ß1t ß2t2

Descriptive Techniques
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Curve Fitting

• A traditional method of dealing with non-seasonal
  data which contain a trend, particularly yearly
  data, is to fit a simple function such as a
  polynomial curve (linear, quadratic etc.), or
• A Gompertz curve
• log xt a brt
• a, b, r are parameters with 0ltrlt1
• Logistic curve
• xt a / (1 b e-ct)
• Both Gompertz and Logistics curves are S-Shaped
  and approach an asymptotic values as t ? 8
• Fitting these curves to data may lead to
  non-linear simultaneous equations

Descriptive Techniques
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Curve Fitting

• S - Shaped

Descriptive Techniques
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Using Smoothing Methods

• Moving Averages
• Weighted Moving Averages
• Exponential Smoothing
• The objective of these methods is to smooth out
  the random fluctuations caused by the irregular
  component of the time series
• Thus, to use these methods, the time series
  should have no significant trend, seasonal, or
  cyclical effects

Smoothing Methods
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Moving Averages

• The moving averages method uses the average of
  the most recent n data values in the time series
  as the forecast for the next period

Smoothing Methods
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Moving Averages Example
(172119)/3
Smoothing Methods
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Moving Averages Example
Smoothing Methods
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Weighted Moving Averages

• In the Moving Averages method, each observation
  in the calculation receives the same weight
• Weighted Moving Averages involves selecting
  different weight for each data value and then
  computing a weighted average of the most recent n
  data values as the forecast
• In most cases, the most recent observation
  receives the most weight, and the weight
  decreases for older data values

Smoothing Methods
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Weighted Moving Averages Example
(3/6)19(2/6)21(1/6)17
19.33
Smoothing Methods
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Weighted Moving Averages Example
Smoothing Methods
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Exponential Smoothing

• This method uses a weighted average of past time
  series values as the forecast it is a special
  case of the weighted moving averages method in
  which we select only one weight the weight for
  the most recent observation
• Ft1 aYt (1 a)Ft
• Where Ft1 forecast of the time series for
  period t1
• Yt actual value of the time series in period
  t
• Ft forecast of the time series for period t
• a smoothing constant (0 a 1)

Smoothing Methods
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Exponential Smoothing Example
Assume a 0.2
F2 0.2(17) (1-0.2)(17) 17.00
F3 0.2(21) (1-0.2)(17) 17.80
F10 0.2(22) (1-0.2)(18.49) 17.80
Smoothing Methods
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Mean Squared Error (MSE)

• The Mean Squared Error (MSE) is an often-used
  measure of the accuracy of a forecasting method

Smoothing Methods
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Analysing Series which Contain Seasonal Variation

• Three seasonal models in common use are
• A Xt mt St et
• B Xt mt St et
• C Xt mt St et
• Where mt is the deseasonalized mean level at
  time t
• St is the seasonal effect at time t
• et is the random error
• Model A describes the additive case
• Model B and C describe the multiplicative cases

Seasonality
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Seasonal Variation

• The analysis of time series which exhibit
  seasonal variation depends on whether one wants
  to
• Measure the seasonal effect and/or
• Eliminate seasonality

Seasonality
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Seasonal Indices

• We consider two models
• A Xt mt St et (Additive model)
• C Xt mt St et (Multiplicative model)

Seasonality
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Seasonal Indices Example
Seasonality
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Multiplicative Model

• The model is assumed to be
• Xt mtStet
• So we just divide the original data Xt with the
  estimated trend series (from centred moving
  average)
• See example

Seasonality
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Multiplicative Model Example
Seasonality
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Multiplicative Model Example
3.9945

• Multiply each index by 4/3.9945
• Season 1 2 3 4
• Index 1.1462 1.0313 0.8546 0.9679

Seasonality
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Addictive Model

• The model is assumed to be
• Xt mt St et
• So we just subtract the estimated trend series
  (from centred moving average)
• See example

Seasonality
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Additive Model Example
Seasonality
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Additive Model Example
-0.1208

• Adding 0.1208/4 to each index
• Seasonal 1 2 3 4
• Index 3.6052 0.8052 -3.6156 -0.7948

Seasonality
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Time Series Analysis

• What we did is only the descriptive analysis
  without any probabilistic function
• Next we consider various probability models for
  time series, which are collectively called
  stochastic processes
• A Stochastic Process can be described as a
  statistical phenomenon that evolves in time
  according to probabilistic laws

Autocorrelation
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A Simple Time Series Model

• Today is a linear function of yesterday
• Yt f(Yt-1)
• Back to regression analysis
• Y ß0 ß1 X1 ß2 X2 ßn Xn e
• Whats a different??
• Whats a problem??

Autocorrelation
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Time Series vs. Regression

• Regression
• Y ß0 ß1 X1
• ? Y1 ß0 ß1 X1
• Y2 ß0 ß1 X2
• Yn ß0 ß1 Xn
• Regression assumes Y is a random variable
• In Time Series Analysis, however, this assumption
  is invalid since Yt f(Yt-1) which means there
  is a dependence across observations.
• This calls Autocorrelation or Serial Correlation

Autocorrelation
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Autocorrelation

• Let X Yt-1
• Yt f(X) f(Yt-1)
• If the correlation is
• The correlation for time series models is

Autocorrelation
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Cant we use Regression?

• Yes we can still use Least-Squares (LS)
  Regression
• BUT
• Cautions are
• Results from regression are coefficients (ß) and
  standard error (? t-stat)
• Coefficients are still the same
• BUT variance (? standard error) is not the same
• So What?

Autocorrelation
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Questions?