arma autocorrelation functions - nc state autocorrelation functions ... the partial...

Download ARMA Autocorrelation Functions - Nc State   Autocorrelation Functions ... The Partial Autocorrelation Function An MA(q) can be identi ed from its ACF: non-zero to lag q, and zero afterwards

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  • ARMA Autocorrelation Functions

    For a moving average process, MA(q):

    xt = wt + 1wt1 + 2wt2 + + qwtq.

    So (with 0 = 1)

    (h) = cov(xt+h, xt

    )= E

    qj=0

    jwt+hj

    qk=0

    kwtk

    =

    2w

    qhj=0

    jj+h, 0 h q

    0 h > q.

    1

  • So the ACF is

    (h) =

    qhj=0

    jj+h

    qj=0

    2j

    , 0 h q

    0 h > q.

    Notes:

    In these expressions, 0 = 1 for convenience.

    (q) 6= 0 but (h) = 0 for h > q. This characterizesMA(q).

    2

  • For an autoregressive process, AR(p):

    xt = 1xt1 + 2xt2 + + pxtp + wt.

    So

    (h) = cov(xt+h, xt

    )= E

    pj=1

    jxt+hj + wt+h

    xt

    =p

    j=1

    j(h j) + cov(wt+h, xt

    ).

    3

  • Because xt is causal, xt is wt+ a linear combination of wt1, wt2, . . . .

    So

    cov(wt+h, xt

    )=

    2w h = 00 h > 0. Hence

    (h) =p

    j=1

    j(h j), h > 0

    and

    (0) =p

    j=1

    j(j) + 2w.

    4

  • If we know the parameters 1, 2, . . . , p and 2w, these equa-tions for h = 0 and h = 1,2, . . . , p form p+ 1 linear equations

    in the p+ 1 unknowns (0), (1), . . . , (p).

    The other autocovariances can then be found recursivelyfrom the equation for h > p.

    Alternatively, if we know (or have estimated) (0), (1), . . . , (p),they form p + 1 linear equations in the p + 1 parameters

    1, 2, . . . , p and 2w.

    These are the Yule-Walker equations.

    5

  • For the ARMA(p, q) model with p > 0 and q > 0:

    xt =1xt1 + 2xt2 + + pxtp+ wt + 1wt1 + 2wt2 + + qwtq,

    a generalized set of Yule-Walker equations must be used.

    The moving average models ARMA(0, q) = MA(q) are theonly ones with a closed form expression for (h).

    For AR(p) and ARMA(p, q) with p > 0, the recursive equationmeans that for h > max(p, q + 1), (h) is a sum of geomet-

    rically decaying terms, possibly damped oscillations.

    6

  • The recursive equation is

    (h) =p

    j=1

    j(h j), h > q.

    What kinds of sequences satisfy an equation like this?

    Try (h) = zh for some constant z.

    The equation becomes

    0 = zh p

    j=1

    jz(hj) = zh

    1 pj=1

    jzj

    = zh(z).

    7

  • So if (z) = 0, then (h) = zh satisfies the equation.

    Since (z) is a polynomial of degree p, there are p solutions,say z1, z2, . . . , zp.

    So a more general solution is

    (h) =p

    l=1

    clzhl ,

    for any constants c1, c2, . . . , cp.

    If z1, z2, . . . , zp are distinct, this is the most general solution;if some roots are repeated, the general form is a little morecomplicated.

    8

  • If all z1, z2, . . . , zp are real, this is a sum of geometricallydecaying terms.

    If any root is complex, its complex conjugate must also be aroot, and these two terms may be combined into geometri-cally decaying sine-cosine terms.

    The constants c1, c2, . . . , cp are determined by initial condi-tions; in the ARMA case, these are the Yule-Walker equa-tions.

    Note that the various rates of decay are the zeros of (z),the autoregressive operator, and do not depend on (z), themoving average operator.

    9

  • Example: ARMA(1,1)

    xt = xt1 + wt1 + wt.

    The recursion is

    (h) = (h 1), h = 2,3, . . .

    So (h) = ch for h = 1,2, . . . , but c 6= 1.

    Graphically, the ACF decays geometrically, but with a differ-ent value at h = 0.

    10

  • 5 10 15 20 25

    0.2

    0.4

    0.6

    0.8

    1.0

    Index

    AR

    MA

    acf(

    ar =

    0.9

    , ma

    =

    0.5,

    24)

    11

  • The Partial Autocorrelation Function

    An MA(q) can be identified from its ACF: non-zero to lag q,and zero afterwards.

    We need a similar tool for AR(p).

    The partial autocorrelation function (PACF) fills that role.

    12

  • Recall: for multivariate random variables X,Y,Z, the partialcorrelations of X and Y given Z are the correlations of:

    the residuals of X from its regression on Z; and

    the residuals of Y from its regression on Z.

    Here regression means conditional expectation, or best lin-ear prediction, based on population distributions, not a sam-ple calculation.

    In a time series, the partial autocorrelations are defined as

    h,h = partial correlation of xt+h and xtgiven xt+h1, xt+h2, . . . , xt+1.

    13

  • For an autoregressive process, AR(p):

    xt = 1xt1 + 2xt2 + + pxtp + wt,

    If h > p, the regression of xt+h on xt+h1, xt+h2, . . . , xt+1 is

    1xt+h1 + 2xt+h2 + + pxt+hp

    So the residual is just wt+h, which is uncorrelated withxt+h1, xt+h2, . . . , xt+1 and xt.

    14

  • So the partial autocorrelation is zero for h > p:

    h,h = 0, h > p.

    We can also show that p,p = p, which is non-zero by as-sumption.

    So p,p 6= 0 but h,h = 0 for h > p. This characterizes AR(p).

    15

  • The Inverse Autocorrelation Function

    SASs proc arima also shows the Inverse Autocorrelation Func-tion (IACF).

    The IACF of the ARMA(p, q) model

    (B)xt = (B)wt

    is defined to be the ACF of the inverse (or dual) process

    (B)x(inverse)t = (B)wt.

    The IACF has the same property as the PACF: AR(p) ischaracterized by an IACF that is nonzero at lag p but zerofor larger lags.

    16

  • Summary: Identification of ARMA processes

    AR(p) is characterized by a PACF or IACF that is:

    nonzero at lag p;

    zero for lags larger than p.

    MA(q) is characterized by an ACF that is:

    nonzero at lag q;

    zero for lags larger than q.

    For anything else, try ARMA(p, q) with p > 0 and q > 0.

    17

  • For p > 0 and q > 0:

    AR(p) MA(q) ARMA(p, q)

    ACF Tails off Cuts off after lag q Tails off

    PACF Cuts off after lag p Tails off Tails off

    IACF Cuts off after lag p Tails off Tails off

    Note: these characteristics are used to guide the initial choiceof a model; estimation and model-checking will often lead to

    a different model.

    18

  • Other ARMA Identification Techniques

    SASs proc arima offers the MINIC option on the identifystatement, which produces a table of SBC criteria for various

    values of p and q.

    The identify statement has two other options: ESACF andSCAN.

    Both produce tables in which the pattern of zero and non-zero values characterize p and q.

    See Section 3.4.10 in Brocklebank and Dickey.19

  • options linesize = 80;

    ods html file = varve3.html;

    data varve;

    infile ../data/varve.dat;

    input varve;

    lv = log(varve);

    run;

    proc arima data = varve;

    title Use identify options to identify a good model;

    identify var = lv(1) minic esacf scan;

    estimate q = 1 method = ml;

    estimate q = 2 method = ml;

    estimate p = 1 q = 1 method = ml;

  • run;

    proc arima output

    http://www.stat.ncsu.edu/people/bloomfield/courses/st730/varve3.html

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