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Transcript of Chapter 1 Basics - University of empirics/Courses/Econ916/Chap_1.pdfآ  Chapter 1 Basics 1.1...

  • Chapter 1

    Basics

    1.1 Definition

    A time series (or stochastic process) is a function Xpt, ωq such that for

    each fixed t, Xpt, ωq is a random variable [denoted by Xtpωq]. For a fixed ω,

    Xpt, ωq is simply a function of t and is called a realization of the stochastic

    process.

    $''&''% Time Domain Approach

    Frequency Domain Approach

    $''&''% Continuous Time

    Discrete Time

    11

  • 12 CHAPTER 1. BASICS

    1.2 Characterization of a time series

    A time series is characterized by the joint distribution function of any

    subset Xt1 , � � � , Xtn that is FXt1 , ��� , Xtn pxt1 , � � � , xtnq.

    Using the joint distribution function, define

    µt � EpXtq

    γt,s � CovpXt, Xsq, provided that they exist.

    The first two monents, {µt}, {γt,s} completely characterize a Gaussian process.

    FXt1 , ��� , Xtn is in general, very difficult to analyze. In particular, esti-

    mation of {µt}, {γt,s} appears to be impossible unless a number of different

    realizations, i.e. repeated observations are available.

    1.3 Stationarity

    Assume FXt1 ,��� ,Xtn is invariant under transformation of the time indices,

    that is

    FXt1�h, ��� , Xtn�hpxt1 , � � � , xtnq � FXt1 , ��� ,Xtn pxt1 , � � � , xtnq p1q

    for all sets of indices (t1, � � � , tn). This is called the strict stationarity.

  • 1.3. STATIONARITY 13

    Under this assumption, the joint distribution function depends only on

    the distance between the elements in the index set.

    If tXtu is strictly stationary and E|Xt|   8, then

    EpXtq � µ, @t p2q

    CovpXt, Xsq � γ|t�s|, @t, s p3q

    tXtu is said to be weakly stationary (or ”covariance stationary” or sim-

    ply, ”stationary”) if (2) and (3) holds.

    REMARKS

    1) (Weak) stationarity does not imply strict stationarity. Nor does

    strict stationarity imply (weak) stationarity. e.g. A strict

    stationary process may not possess finite moments (e.g. Cauchy).

    2) For a Gaussian process, weak and strict stationarity are equivalent.

  • 14 CHAPTER 1. BASICS

    µ and γ|t�s| can be estimated by

    pµ � sX � 1 T

    t�1 Xt

    pγh � ch � 1 T

    t�h�1 pXt � sXqpXt�h � sXq, h � 0, 1, � � � .

    If the process is also ergodic (average asymptotic independence), sX and ch are consistent estimators of µ and γh.

    1.4 Autocovariance and Autocorrelation Func-

    tions

    The sequence tγhu viewed as a function of h is called the autocovariance function.

    The autocorrelation function is defined by

    ρh � γh γ0 ,

    note ρ0 � 1

    Example 1. (White noise process)

    tXtu, Xt � iid p0, σ2q, 0   σ2   8.

    µ � 0, γh �

    $''&''% σ2 if h � 0,

    0 otherwise.

  • 1.4. AUTOCOVARIANCE AND AUTOCORRELATION FUNCTIONS 15

    Example 2. (MA(1) process)

    Let tεtu be a white noise process with finite variance σ2. Let

    Xt � εt � θεt�1.

    Then

    µ � EpXtq � 0

    γ0 � Epεt � θεt�1q2 � Epε2t q � θ2Epε2t�1q

    � p1� θ2qσ2

    γh � Erpεt � θεt�1qpεt�h � θεt�h�1qs

    $''&''% θσ2 if |h| � 1,

    0 if |h| ¥ 2.

    Therefore,

    ρh �

    $''''''&''''''% 1 if h � 0, θ

    1� θ2 if |h| � 1,

    0 otherwise.

  • 16 CHAPTER 1. BASICS

    Suppose θ � 0.6. Then ρ1 � 0.61� 0.62 � 0.44.

    1 2 3 4 5

    1

    0.5

    0 h

    ρh

    ρh � ρphq : the autocorrelation function.

    Sample mean of tXtuTt�1 � sXT Sample variance = c0 pγ0 Sample autocovariance = ch, h � 1, 2, � � � pγh Sample autocorrelation = rhpor pρhq � ch

    c0 , h � 1, 2, � � �

    Note r0 � 1.

  • 1.5. LAG OPERATOR 17

    A plot of rh against h � 0, 1, � � � is called correlogram.

    1 2 3 4 5

    1

    0.5

    0 h

    rh

    The sample autocorrelations are estimates of the corresponding theo-

    retical autocorrelations and are therefore subject to sampling errors.

    1.5 Lag Operator

    The operator L (sometimes denoted by B) is defined by

    LXt � Xt�1.

  • 18 CHAPTER 1. BASICS

    Formally, L operates on the whole sequence.

    L :

    ����������������

    ...

    Xt�1

    Xt

    Xt�1

    ...

    ���������������� ÝÑ

    ����������������

    ...

    Xt�2

    Xt�1

    Xt

    ...

    ���������������� The lead operator L�1 (sometimes denoted by F ) is defined by

    L�1Xt � Xt�1.

    Successive application of L yields LhXt � Xt�h, h � 1, 2, � � �

    We define L0Xt � Xt.

    The lag operator is a linear operator :

    LpcXtq � cLXt

    LpXt � Ytq � LXt � LYt

    and can be manipulated like a usual algebraic quantity.

    For example, suppose

    yt � φyt�1 � εt, and |φ|   1.

  • 1.6. GENERAL LINEAR PROCESS 19

    Then, p1� φLqyt � εt

    So, yt � εt1� φL � 8̧

    j�0 pφLqjεt

    ∆ � 1� L is called the difference operator.

    ∆2yt � p1� Lq2yt � p1� 2L� L2qyt

    = yt � 2yt�1 � yt�2

    ∆2yt � ∆p∆ytq � ∆pyt � yt�1q

    = pyt � yt�1q � pyt�1 � yt�2q � yt � 2yt�1 � yt�2

    1.6 General Linear Process

    yt � εt � ψ1εt�1 � ψ2εt�2 � � � �

    � ψpLqεt ÐÝ linear process

    where εt � iid p0, σ2q and

    ψpLq � 1� ψ1L� ψ2L2 � � � �

    polynomial in lag operators ψpLq is sometimes called transfer function

  • 20 CHAPTER 1. BASICS

    A time series tytu can be viewed as the result of applying a backward (linear)

    filter to a white noise process.

    εt inputÝÑ

    linear filter

    ψpLq outputÝÑ Yt � ψpLqεt

    The sequence tψj : j � 0, 1, � � � u can be finite or infinite. If it is finite

    of order q, we obtain MApqq process. This is clearly a stationary process. If

    tψju is infinite, we usually assume it is absolutely summable. i.e. 8̧

    j�0 |ψj|   8.

    Then the resulting process is stationary.

    To see this,

    µ � 0 � 8̧

    j�0 ψj � 0

    γ0 � σ2 8̧

    j�0 ψ2j   σ2

    � 8̧ j�0

    |ψj|

    2

      8

    γh � σ2 8̧

    j�0 ψjψj�h   σ2

    j�0 |ψj||ψj�h|   σ2

    � 8̧ j�0

    |ψj|

    2

      8

  • 1.7. AUTOREGRESSIVE PROCESS 21

    The stationary condition is embodied in the condition that ψpzq must

    converge for |z| ¤ 1, i.e. for z on or within the unit circle.

    Note that absolute summability of tψju is sufficient but not necessary

    for stationarity.

    7

    j�0 ψjz

    j   8̧

    j�0 |ψj|   8

    1.7 Autoregressive Process

    The process yt defined by

    yt � φ1yt�1 � � � � � φpyt�p � εt p�q

    is called a p-th order autoregressive process and is denoted by

    yt � ARppq

    The equation p�q is sometimes called a stochastic difference equation.

  • 22 CHAPTER 1. BASICS

    1.7.1 First - Order Autoregressive Process

    ARp1q is given by

    yt � φyt�1 � εt.

    By successive substitution,

    yt � εt � φyt�1

    � εt � φεt�1 � φ2yt�2

    � εt � φεt�1 � φ2εt�2 � φ3yt�3

    � J�1̧

    j�0 φjεt�j � φJyt�J ,

    implying Epyt|yt�Jq � φJyt�J

    If |φ| ¥ 1, the value of yt�J can affect the prediction of future yt, no

    matter how far ahead.

    If |φ|   1,

    yt � lim JÑ8

    J�1̧

    j�0 φjεt�j � lim

    JÑ8 φJyt�J

    � 8̧

    j�0 φjεt�j.

    Note that 8̧

    j�0 |φj| �

    j�0 |φ|j � 11� |φ|   8, if |φ|   1,

  • 1.7. AUTOREGRESSIVE PROCESS 23

    so, tφju is absolutely summable and tytu is a linear process.

    ψpLq � p1� φLq�1 � 8̧

    j�0 φjLj, or ψj � φj

    Now,

    Epy2t q � Epφ2y2t�1 � ε2t � 2φyt�1εtq

    γ0 � φ2γ0 � σ2

    because Epyt�1εtq

    � ErEpyt�1εt|yt�1qs

    � Eryt�1Epεt|yt�1qs

    � Epyt�1 � 0q � 0

    Therefore, γ0 � σ 2

    1� φ2   8, if |φ|   1.

    1.7.2 Second - Order Autoregressive Process

    ARp2q is given by

    yt � φ1yt�1 � φ2yt�2 � εt

    or

    φpLqyt � εt,

    where, φpLq � 1� φ1L� φ2L2.

  • 24 CHAPTER 1. BASICS

    Now, suppose φpzq can be written as

    φpzq � p1� λ1zqp1� λ2zq.

    Then

    ψpzq � φ�1pzq � 1p1� λ1zqp1� λ2zq � K1

    1� λ1z � K2

    1� λ2z ,

    where K1 � λ1 λ1 � λ2 and K2 �

    �λ2 λ1 � λ2 .

    Therefore, ψpzq converges for |z| ¤ 1, iff |λ1|   1 and |λ2|   1.

    In other words