financial econometrics: causality granger, geweke

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    Econometrics II

    Seppo Pynnonen

    Department of Mathematics and Statistics, University of Vaasa, Finland

    January 12 February 25, 2016

    Seppo Pynnonen Econometrics II

    http://find/
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    Vector Autoregression (VAR)

    Part V

    Vector Autoregression

    Seppo Pynnonen Econometrics II

    http://find/
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    Vector Autoregression (VAR)

    1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decompositionOn the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysisSeppo Pynnonen Econometrics II

    http://find/
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    Vector Autoregression (VAR)

    Background1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decompositionOn the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysisSeppo Pynnonen Econometrics II

    V A i (VAR)

    http://find/
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    Vector Autoregression (VAR)

    Background

    Example 1

    Consider the following monthly values of FTAS (Financial Times AllShare) and FTA dividend index values from Jan 1965 to Dec 2005

    (source: http://www.lboro.ac.uk/departments/ec/cup/).

    0

    1000

    2000

    3000

    fta

    20

    40

    60

    80

    100

    ftadiv

    40

    20

    0

    20

    40

    ret

    40

    0

    20

    40

    1970 1980 1990 2000

    div

    Time

    FT All Share Monthly Srock and Dividend Indexes

    Seppo Pynnonen Econometrics II

    V t A t i (VAR)

    http://find/http://goback/
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    Vector Autoregression (VAR)

    Background

    Potentially interesting questions:

    1 Does one of the series (dividend) have a tendency to lead theother?

    2 Are there feedbacks between the series?

    3 How about contemporaneous movements?4 How do impulses (shocks, innovations) transfer from one

    series to another?

    5 How about common factors (disturbances, trend, yieldcomponent, risk)?

    Most of these questions can be empirically investigated using tools

    developed in multivariate time series analysis.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Vector Autoregression (VAR)

    Background

    Some of these can be readily investigated by simple tools.For example lead-lag relations can be investigated bycross-autocorrelation function

    xy(k) = xy(k)

    x(0)y(0)

    , (1)

    where

    xy(k) =Tk

    t=1(xt x)(yt+k y) (2)

    is the cross-autocovariance function andx(0)andy(0)are thesample variances ofxt and yt. Note that xy(k) = xy(k).

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Vector Autoregression (VAR)

    Background

    The following autocorrelation functions indicate that dividends tend tolead returns (negative cross-autocorrelation)

    0 5 10 15 20

    0.5

    0.0

    0.5

    1.0

    Lag

    ACF

    Return

    0 5 10 15 20

    0.5

    0.0

    0.5

    1.0

    Lag

    Return & Dividend

    20 15 10 5 0

    0.5

    0.0

    0.5

    1.0

    Lag

    ACF

    Dividend & Return

    0 5 10 15 20

    0.5

    0.0

    0.5

    1.0

    Lag

    Dividend

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Vector Autoregression (VAR)

    Background

    The following scatter diagram of(rett1, divt)demonstrates the firstorder cross-correlation.

    40 20 0 20 40

    40

    20

    0

    20

    40

    Returns and Dividends

    Rett1(%)

    D

    ivt

    (%)

    Vector autoregression (VAR) provides more sophisticated tools for

    deeper analyses.Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Vector Autoregression (VAR)

    The Model1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysisSeppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    g ( )

    The Model

    VAR(1) (i.e., first order vector autoregression)

    The univariate time series generalize straightforwardly tomultivariate time series.

    Consider two time series xt and yt

    xt=1+(1)11xt1+

    (1)12yt1+1t (3)

    yt=2+(1)21xt1+

    (1)22yt1+2t (4)

    it and 2tare white noise error terms that can becontemporaneously correlated

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    g ( )

    The Model

    Generally: Suppose we have m time series yit, i= 1, . . . , m, andt= 1, . . . , T(common length of the time series). Then a

    vector autoregression model is defined as

    y1ty2t

    ...ymt

    = 12

    ...m

    +

    (1)11

    (1)12

    (1)1m

    (1)21

    (1)22

    (1)2m

    ... ... . . . ...

    (1)m1

    (1)m2

    (1)mm

    y1,t1y2,t1

    ...ym,t1

    +

    +

    (p)11

    (p)12

    (p)1m

    (p)21 (p)22 (p)2m...

    ... . . .

    ...

    (p)m1

    (1)m2

    (p)mm

    y1,tp

    y2,tp...ym,tp

    +

    1t

    2t...mt

    .(5)

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    g ( )

    The Model

    In matrix notations

    yt=+ 1yt1+ + pytp+t, (6)

    which can be further simplified by adopting the matric form of alag polynomial

    (L) =I 1L . . . pLp. (7)

    Thus finally we get(L)yt=t. (8)

    Note that in the above model each yitdepends not only its ownhistory but also on other series history (cross dependencies). Thisgives us several additional tools for analyzing causal as well asfeedback effects as we shall see after a while.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    The Model

    A basic assumption in the above model is that the residual vector

    follow a multivariate white noise, i.e.

    E[t] = 0

    E[ts] =

    ift=s0 ift=s

    (9)

    The coefficient matrices must satisfy certain constraints in orderthat the VAR-model is stationary. They are just analogies with theunivariate case, but in matrix terms. It is required that roots of

    |I 1z 2z2

    pzp

    | = 0 (10)

    lie outside the unit circle. Estimation can be carried out by singleequation least squares.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    The Model

    Example 2

    VAR(1) for the return/dividend series:

    VAR Estimation Results:

    =========================Endogenous variables: ret, div

    Deterministic variables: const

    Sample size: 490Log Likelihood: -2785.435

    Roots of the characteristic polynomial:

    0.2756 0.2756

    ret

    Estimate Std. Error t value Pr(>|t|)

    ret.l1 0.01518 0.05974 0.254 0.7995div.l1 0.08980 0.04612 1.947 0.0521 .

    const 0.61045 0.26341 2.318 0.0209 *

    div

    ret.l1 -0.82976 0.06345 -13.078 < 2e-16 ***

    div.l1 0.09620 0.04898 1.964 0.050106 .const 1.05107 0.27974 3.757 0.000193 ***

    ---

    Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1

    Correlation matrix of residuals:

    ret divret 1.0000 0.8738

    div 0.8738 1.0000

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysisSeppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    The number of estimated parameters grows rapidly very large.

    In the first step it is assumed that all the series in the VAR modelhave equal lag lengths. To determine the number of lags that

    should be included, criterion functions can be utilized the samemanner as in the univariate case.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    The underlying assumption is that the residuals follow amultivariate normal distribution, i.e.

    Nm(0, ). (11)

    Akaikes criterion function (AIC) and Schwarzs criterion (BIC)

    have gained popularity in determination of the number of lags inVAR.

    In the original forms AIC and BIC are defined as

    AIC= 2log L+ 2s (12)

    BIC= 2log L+slog T (13)

    where L stands for the Likelihood function, and sdenotes the

    number of estimated parameters.Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    The best fitting model is the one that minimizes the criterionfunction.

    For example in a VAR(j) model with m equations there are

    s=m(1 +jm) +m(m+ 1)/2estimated parameters.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    Under the normality assumption BIC can be simplified to

    BIC(j) = log,j

    +

    jm2 log T

    T , (14)

    and AIC to

    AIC(j) = log,j

    +2jm2

    T , (15)

    j= 0, . . . , p, where

    ,j= 1

    T

    Tt=j+1

    t,jt,j=

    1

    TEjE

    j (16)

    with t,jthe OLS residual vector of the VAR(j) model (i.e. VARmodel estimated with j lags), and

    Ej= [j+1,j, j+2,j, , T,j] (17)

    an m (Tj)matrix.Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/http://goback/
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    Defining the order of a VAR-model

    The likelihood ratio (LR) test can be also used in determining theorder of a VAR. The test is generally of the form

    LR= T(log |k| log |p|), (18)

    where kdenotes the maximum likelihood estimate of the residualcovariance matrix of VAR(k) and pthe estimate of VAR(p)

    (p>k) residual covariance matrix. If VAR(k) (the shorter model)is the true one, i.e., the null hypothesis

    H0 : k+1 = = p=0 (19)

    is true, thenLR 2df, (20)

    where the degrees of freedom, df, equals the difference of in thenumber of estimated parameters between the two models.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    In an m variate VAR(k)-model each series has p k lags less thanthose in VAR(p). Thus the difference in each equation is

    m(p k), so that in total df =m2

    (p k).Note that often, when T is small, a modified LR

    LR = (T mg)(log |k| log |p|)

    is used to correct possible small sample bias, where g=p k.

    Yet another method is to use sequential LR

    LR(k) = (Tm)(log |k| log |k+1|)

    which tests the null hypothesis

    H0 : k+1 =0

    given thatj=0 for j=k+ 2, k+ 3, . . . , p.

    EViews uses this method in View -> Lag Structure -> Lag

    Length Criteria. . ..Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    Example 3

    Consider series: FTAS, FTADIV, R91 (three months bond yield), R20 (20

    year bond yield)

    0

    1000

    2000

    3000

    fta

    20

    40

    60

    80

    100

    ftadiv

    4

    6

    8

    10

    14

    rs91

    4

    6

    8

    10

    14

    1970 1980 1990 2000

    r20

    Time

    Bond and Stock Market Series

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    Let p= 12then in the equity-bond data different VAR models yield the

    following results. Below are EViews results.

    VAR Lag Order Selection CriteriaEndogenous variables: DFTA DDIV DR20 DTBILLExogenous variables: C

    Sample: 1965:01 1995:12

    Included observations: 359=========================================================

    Lag LogL LR FPE AIC SC HQ

    ---------------------------------------------------------0 -3860.59 NA 26324.1 21.530 21.573 21.547

    1 -3810.15 99.473 21728.6* 21.338* 21.554* 21.424*2 -3796.62 26.385 22030.2 21.352 21.741 21.506

    3 -3786.22 20.052 22729.9 21.383 21.945 21.6064 -3783.57 5.0395 24489.4 21.467 22.193 21.750

    5 -3775.66 14.887 25625.4 21.502 22.411 21.864

    6 -3762.32 24.831 26016.8 21.517 22.598 21.9477 -3753.94 15.400 27159.4 21.560 22.814 22.059

    8 -3739.07 27.018* 27348.2 21.566 22.994 22.134

    9 -3731.30 13.933 28656.4 21.612 23.213 22.248

    10 -3722.40 15.774 29843.7 21.651 23.425 22.35711 -3715.54 12.004 31443.1 21.702 23.649 22.476

    12 -3707.28 14.257 32880.6 21.745 23.865 22.588=========================================================

    * indicates lag order selected by the criterion

    LR: sequential modified LR test statistic(each test at 5% level)

    FPE: Final prediction error

    AIC: Akaike information criterion

    SC: Schwarz information criterionHQ: Hannan-Quinn information criterionSeppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    Criterion function minima are all at VAR(1) (SC or BIC just borderline).

    LR-tests suggest VAR(8). Let us look next at the residual

    autocorrelations.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    http://find/
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    Defining the order of a VAR-model

    In order to investigate whether the VAR residuals are White Noise,the hypothesis to be tested is

    H0 : 1 = = h = 0 (21)

    wherek= (ij(k))is the autocorrelation matrix (see later in thenotes) of the residual series with ij(k)the cross autocorrelation of

    order kof the residuals series i and j. A general purpose(portmanteau) test is the Q-statistic8

    Qh =Th

    k=1

    tr(k10

    k10 ), (22)

    where k= (ij(k))are the estimated (residual) autocorrelations,

    and 0 the contemporaneous correlations of the residuals.8See e.g. Lutkepohl, Helmut (1993). Introduction to Multiple Time Series,

    2nd Ed., Ch. 4.4Seppo Pynnonen Econometrics II

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    Vector Autoregression (VAR)

    D fi i h d f VAR d l

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    Defining the order of a VAR-model

    The asymptotic distribution is 2 with df =p2(h k). Note that incomputer printouts h is running from1, 2, . . . h with h specified by theuser.

    VAR(1) Residual Portmanteau Tests for AutocorrelationsH0: no residual autocorrelations up to lag h

    Sample: 1966:02 1995:12

    Included observations: 359================================================

    Lags Q-Stat Prob. Adj Q-Stat P rob. df

    ------------------------------------------------1 1.847020 NA* 1.852179 NA* NA*

    2 27.66930 0.0346 27.81912 0.0332 16

    3 44.05285 0.0761 44.34073 0.0721 324 53.46222 0.2725 53.85613 0.2603 48

    5 72.35623 0.2215 73.01700 0.2059 64

    6 96.87555 0.0964 97.95308 0.0843 80

    7 110.2442 0.1518 111.5876 0.1320 968 137.0931 0.0538 139.0485 0.0424 1 12

    9 152.9130 0.0659 155.2751 0.0507 1 28

    10 168.4887 0.0797 171.2972 0.0599 14411 179.3347 0.1407 182.4860 0.1076 160

    12 189.0256 0.2379 192.5120 0.1869 176

    ================================================*The test is valid only for lags larger than the

    VAR lag order. df is degrees of freedom for

    (approximate) chi-square distribution

    There is still left some autocorrelation into the VAR(1) residuals. Let us

    next check the residuals of the VAR(2) model.Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    D fi i th d f VAR d l

    http://find/
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    Defining the order of a VAR-model

    VAR Residual Portmanteau Tests for Autocorrelations

    H0: no residual autocorrelations up to lag hSample: 1965:01 1995:12Included observations: 369

    ==============================================

    Lags Q-Stat Prob. Adj Q-Stat Prob. df----------------------------------------------

    1 0.438464 NA* 0.439655 NA* NA*

    2 1.623778 NA* 1.631428 NA* NA*3 17.13353 0.3770 17.26832 0.3684 16

    4 27.07272 0.7143 27.31642 0.7027 32

    5 44.01332 0.6369 44.48973 0.6175 486 66.24485 0.3994 67.08872 0.3717 64

    7 80.51861 0.4627 81.63849 0.4281 80

    8 104.3903 0.2622 106.0392 0.2271 969 121.8202 0.2476 123.9049 0.2081 112

    10 136.8909 0.2794 139.3953 0.2316 128

    11 147.3028 0.4081 150.1271 0.3463 14412 157.4354 0.5425 160.6003 0.4718 160

    ==============================================

    *The test is valid only for lags larger thanthe VAR lag order.df is degrees of freedom for (approximate)

    chi-square distribution

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Defining the order of a VAR model

    http://find/
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    Defining the order of a VAR-model

    Now the residuals pass the white noise test. On the basis of these residualanalyses we can select VAR(2) as the specification for further analysis.

    Mills (1999) finds VAR(6) as the most appropriate one. Note that thereordinary differences (opposed to log-differences) are analyzed. Here,

    however, log transformations are preferred.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Vector ARMA (VARMA)

    http://find/http://goback/
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    Vector ARMA (VARMA)1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Vector ARMA (VARMA)

    http://find/
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    Vector ARMA (VARMA)

    Similarly as is done in the univariate case one can extend the VARmodel to the vector ARMA model

    yt=+

    pi=1

    iyti+tq

    j=1

    jtj (24)

    or(L)yt=+ (L)t, (25)

    where yt, , and t arem 1vectors, andis andjs arem m matrices, and

    (L) = I 1L . . . pLp

    (L) = I 1L . . . qLq

    .

    (26)

    Provided that(L) is invertible, we always can write theVARMA(p, q)-model as a VAR() model with(L) = 1(L)(L). The presence of a vector MA component,however, complicates the analysis to some extend.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Autocorrelation and Autocovariance Matrices

    ( )

    http://find/http://goback/
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    Autocorrelation and Autocovariance Matrices1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Autocorrelation and Autocovariance Matrices

    http://find/
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    The kth cross autocorrelation of the ith and jth time series, yitand yjt is defined as

    ij(k) =E(yitk i)(yjt j). (27)

    Although for the usual autocovariance k=k, the same is nottrue for the cross autocovariance, but ij(k) =ij(k). Thecorresponding cross autocorrelations are

    i,j(k) = ij(k)

    i(0)j(0). (28)

    The kth autocorrelation matrix is then

    k=

    1(k) 1,2(k) . . . 1,m(k)2,1(k) 2(k) . . . 2,m(k)...

    . . . ...

    m,1(k) m,2(k) . . . m(k)

    . (29)

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Autocorrelation and Autocovariance Matrices

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    Remark: k=

    k.

    The diagonal elements, j(k), j= 1, . . . , m ofare the usualautocorrelations.

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    Example 4

    Cross autocorrelations of the equity-bond data.

    1 =

    Divt Ftat R20t TbltDivt1

    Ftat1R20t1Tblt1

    0.0483

    0.0099 0.0566 0.0779

    0.0160 0.1225 0.2968

    0.1620

    0.0056 0.1403 0.2889 0.31130.0536 0.0266 0.1056 0.3275

    Note: Correlations with absolute value exceeding2 std err = 2 /

    371 0.1038are statistically significant at the5%level (indicated by ).

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    Example 5

    Consider individual security returns and portfolio returns.

    For example French and Rolli have found that daily returns of individualsecurities are slightly negatively correlated.

    The tables below, however suggest that daily returns of portfolios tend tobe positively correlated!

    iFrench, K. and R. Ross (1986). Stock return variances: The arrival ofinformation and reaction of traders. Journal of Financial Economics, 17, 526.

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    Source: Campbell, Lo, and MackKinlay (1997). Econometrics ofFinancial Markets.

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    One explanation could be that the cross autocorrelations are

    positive, which can be partially proved as follows: Let

    rpt= 1

    m

    mi=1

    rit= 1

    mrt (30)

    denote the return of an equal weighted index, where = (1, . . . , 1)

    is a vector of ones, and rt= (r1t, . . . , rmt) is the vector of returns

    of the securities. Then

    cov(rpt1, rpt) = cov rt1

    m ,rt

    m

    = 1

    m2 , (31)

    where1 is the first order autocovariance matrix.

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    Therefore

    m2cov(rpt1, rpt) =1=

    1 tr(1)

    + tr(1), (32)

    wheretr()is the trace operator which sums the diagonal elementsof a square matrix.

    Consequently, because the right hand side tends to be positive andtr(1)tends to be negative,

    1 tr(1), which contains onlycross autocovariances, must be positive, and larger than theabsolute value oftr(1), the autocovariances of individual stockreturns.

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    1 Vector Autoregression (VAR)

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedbackGewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

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    Consider the following extension of the VAR-model (multivariatedynamic regression model)

    yt=c+

    pi=1

    Aiyti+p

    i=0

    Bixti+t, (33)

    where p+ 1 t T, yt= (y1t, . . . , ymt), c is an m 1vector ofconstants, A1, . . . , Ap arem m matrices of lag coefficients,

    xt= (x1t, . . . , xkt)is a k 1vector of regressors, B0, B1, . . . , Bparek m coefficient matrices, and t is an m 1vector of errorshaving the properties

    E(t) =E{E(t|Yt

    1, xt)} = 0 (34)

    and

    E(ts) =E{E(t

    s|Yt1, xt)} =

    t=s0 t=s,

    (35)

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    whereYt1 = (yt1, yt2, . . . , y1). (36)

    We can compile this into a matrix form

    Y=XB+U, (37)

    where

    Y = (yp+1, . . . , yT)X = (zp+1. . . . , zT)

    zt = (1, yt1, . . . , y

    tp, x

    t, . . . , x

    tp)

    U = (p+1, . . . , T),

    (38)

    and B= (c, A1, . . . , Ap, B

    0, . . . , B

    p). (39)

    The estimation theory for this model is basically the same as forthe univariate linear regression.

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    For example, the LS and (approximate) ML estimator ofB is

    B= (

    XX

    )1X

    Y

    , (40)

    and the ML estimator of is

    = 1

    TUU, U= Y XB. (41)

    We say that xt is weakly exogenous1 if the stochastic structure of

    x contains no information that is relevant for estimation of theparameters of interest, B, and.

    If the conditional distribution ofxtgiven the past is independent ofthe history ofyt then xt is said to be strongly exogenous.

    1For an in depth discussion ofExogeneity, see Engle, R.F., D.F. Hendry andJ.F. Richard (1983). Exogeneity. Econometrica, 51:2,277304.

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    ( )

    Background

    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedbackGewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

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    Granger-causality and measures of feedback

    One of the key questions that can be addressed to VAR-models ishow useful some variables are for forecasting others.

    If the history ofxdoes not help to predict the future values ofy,we say that xdoes not Granger-cause y.2

    2Granger, C.W. (1969). Econometrica 37, 424438. Sims, C.A. (1972).American Economic Review, 62, 540552.

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    Granger-causality and measures of feedback

    Usually the prediction ability is measured in terms of the MSE(Mean Square Error).

    Thus, x fails to Granger-cause y, if for all s>0

    MSE(yt+s|yt, yt1, . . .)= MSE(yt+s|yt, yt1, . . . , xt, xt1, . . .), (42)

    where (e.g.)

    MSE(yt+

    s|yt, yt1

    , . . .)=E

    (yt+s yt+s)

    2|yt, yt1, . . .

    . (43)

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    Granger-causality and measures of feedback

    Remark: This is essentially the same that y is strongly exogenous to x.In terms of VAR models this can be expressed as follows:

    Consider the g=m+k dimensional vector zt= (yt, x

    t), which is

    assumed to follow a VAR(p) model

    zt=

    pi=1

    izti+t, (44)

    where

    E(t) = 0

    E(ts) =

    , t=s0, t=s.

    (45)

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    Vector Autoregression (VAR)

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    Granger-causality and measures of feedback

    Nowx does not Granger-cause y if and only if, C2i 0, orequivalently, if and only if, |11| = |1|, where1 =E(1t

    1t)

    with 1t from the regression

    yt=p

    i=1

    C1iyti+1t. (48)

    Changing the roles of the variables we get the necessary andsufficient condition ofy notGranger-causing x.

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    Granger-causality and measures of feedback

    It is also said that x is block-exogenouswith respect to y.

    Testing for the Granger-causality ofx on y reduces to testing forthe hypothesisH0 : C2i=0. (49)

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    This can be done with the likelihood ratio test by estimating withOLS the restricted3 and non-restricted4 regressions, and

    calculating the respective residual covariance matrices:

    Unrestricted:

    11= 1

    T p

    T

    t=p+11t

    1t. (50)

    Restricted:

    1 = 1

    T p

    Tt=p+1

    1t1t. (51)

    3Perform OLS regressions of each of the elements in y on a constant, p lagsof the elements ofx and plags of the elements ofy.

    4Perform OLS regressions of each of the elements in y on a constant and plags of the elements ofy.

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    Granger-causality and measures of feedback

    The LR test is then

    LR = (T p)

    ln |1| ln |11| 2mkp, (52)

    ifH0 is true.

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    Example 6

    Granger causality between pairwise equity-bond market series

    Pairwise Granger Causality TestsSample: 1965:01 1995:12

    Lags: 12

    ================================================================Null Hypothesis: Obs F-Statistic Probability

    ================================================================

    DFTA does not Granger Cause DDIV 365 0.71820 0.63517DDIV does not Granger Cause DFTA 1.43909 0.19870

    DR20 does not Granger Cause DDIV 365 0.60655 0.72511DDIV does not Granger Cause DR20 0.55961 0.76240

    DTBILL does not Granger Cause DDIV 365 0.83829 0.54094

    DDIV does not Granger Cause DTBILL 0.74939 0.61025

    DR20 does not Granger Cause DFTA 365 1.79163 0.09986

    DFTA does not Granger Cause DR20 3.85932 0.00096

    DTBILL does not Granger Cause DFTA 365 0.20955 0.97370DFTA does not Granger Cause DTBILL 1.25578 0.27728

    DTBILL does not Granger Cause DR20 365 0.33469 0.91843

    DR20 does not Granger Cause DTBILL 2.46704 0.02377==============================================================

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    Granger-causality and measures of feedback

    The p-values indicate that FTA index returns Granger cause 20 year

    Gilts, and Gilts lead Treasury bill.

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    Granger-causality and measures of feedback

    Let us next examine the block exogeneity between the bond and equitymarkets (two lags). Test results are in the table below.

    ===================================================================

    Direction LoglU LoglR 2(LU-LR) df p-value

    --------------------------------------------------------------------(Tbill, R20) --> (FTA, Div) -1837.01 -1840.22 6.412 8 0.601

    (FTA,Div) --> (Tbill, R20) -2085.96 -2096.01 20.108 8 0.010

    ===================================================================

    The test results indicate that the equity markets are Granger-causing

    bond markets. That is, to some extend previous changes in stock markets

    can be used to predict bond markets.

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    Background

    The Model

    Defining the order of a VAR-modelVector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Exogeneity and Causality

    Gewekes measures of Linear Dependence

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    Geweke s measures of Linear Dependence

    Above we tested Granger-causality, but there are several other

    interesting relations that are worth investigating.

    Geweke4has suggested a measure for linear feedback from x toybased on the matrices1 and11 as

    Fxy = ln(|1|/|11|), (53)so that the statement that x does not (Granger) cause y isequivalent to Fxy = 0.

    Similarly the measure of linear feedback from y tox is defined by

    Fyx = ln(|2|/|22|). (54)

    4Geweke (1982) Journal of the American Statistical Association, 79,304324.

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    Geweke s measures of Linear Dependence

    It may also be interesting to investigate the instantaneous causality

    between the variables.

    For the purpose, premultiplying the earlier VAR system ofy and x by Im 12

    122

    12111 Ik

    (55)

    gives a new system of equations, where the first m equations become

    yt=

    pi=0

    C3ixti+

    pi=1

    D3iyti+1t, (56)

    with the error 1t=1t 12122 2tthat is uncorrelated with v2t

    5 andconsequently with xt (important!).

    5cov[1t,2t] = cov

    1t12

    122 2t,2t

    = cov[1t,2t]12

    122 cov[2t,2t] = 1212 = 0.

    Note further thatcov[1t] = cov1t12

    122 2t

    = 11 12

    122 21 =: 11.2.

    Similarlycov[2t] = 22.1.

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    u p

    Similarly, the last kequations can be written as

    xt=p

    i=1

    E3ixti+p

    i=0

    F3iyti+2t. (57)

    Denotingi=E(it

    it), i= 1, 2, there is instantaneous causalitybetween y and x if and only ifC30 =0 and F30 =0 or, equivalently,

    |11| > |1| and |22| > |2|.

    Analogously to the linear feedback we can define instantaneous linearfeedback

    Fxy = ln(|11|/|1|) = ln(|22|/|2|). (58)

    A concept closely related to the idea of linear feedback is that of lineardependence, a measure of which is given by

    Fx,y =Fxy+ Fyx+Fxy. (59)

    Consequently the linear dependence can be decomposed additively intothree forms of feedback.

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    p

    Absence of a particular causal ordering is then equivalent to one of these

    feedback measures being zero.Using the method of least squares we get estimates for the variousmatrices above as

    i= 1

    T p

    T

    t=p+1 it

    it, (60)

    ii= 1

    T p

    Tt=p+1

    it

    it, (61)

    i=

    1

    T p

    T

    t=p+1

    it

    it

    , (62)

    for i= 1, 2.

    For exampleFxy = ln(|1|/|11|). (63)

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    With these estimates one can test in particular:

    No Granger-causality: x y, H01 :Fxy = 0

    (T p)Fxy 2mkp. (64)

    No Granger-causality: y x, H02 :Fyx = 0

    (T p)Fyx 2mkp. (65)

    No instantaneous feedback: H03 :Fxy = 0

    (T p)Fxy 2mk. (66)

    No linear dependence: H04 :Fx,y = 0

    (T p)Fx,y 2mk(2p+1). (67)

    This last is due to the asymptotic independence of the measuresFx

    y, Fy

    x and Fxy.

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    Alternatively one can use asymptotically equivalent Wald and

    Lagrange Multiplier (LM) statistics for testing these hypotheses.

    Note that in each case(T p)F is the LR-statistic.

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    Example 2.8: The LR-statistics of the above measures and the associated2 values for the equity-bond data are reported in the following table

    with p= 2.

    [y = ( log FTAt, log DIVt)and x = ( log Tbillt, log r20t)]

    ==================================

    LR DF P-VALUE----------------------------------x-->y 6.41 8 0.60118

    y-->x 20.11 8 0.00994

    x.y 23.31 4 0.00011x,y 49.83 20 0.00023

    ==================================

    Note. The results lead to the same inference as in Mills (1999), p. 251,

    although numerical values are different [in Mills VAR(6) is analyzed and

    here VAR(2)].

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    g

    The Model

    Defining the order of a VAR-modelVector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

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    Vector Autoregression (VAR)

    Innovation accounting

    1 Vector Autoregression (VAR)

    Background

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    g

    The Model

    Defining the order of a VAR-modelVector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

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    Consider the VAR(p) model

    (L)yt=t, (70)

    where(L) =I 1L 2L

    2 pLp (71)

    is the (matric) lag polynomial.

    Provided that the stationary conditions hold we have analogouslyto the univariate case the vector MA representation ofyt as

    yt= 1(L)t=t+

    i=1

    iti, (72)

    wherei is an m m coefficient matrix.

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    The error terms t represent shocks in the system.

    Suppose we have a unit shock in tthen its effect in y, speriodsahead is

    yt+s

    t= s. (73)

    Accordingly the interpretation of thematrices is that theyrepresent marginal effects, or the models response to a unit shock(or innovation) at time point t in each of the variables.

    Economists call such parameters are as dynamic multipliers.

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    For example, if we were told that the first element in tchanges by1, that the second element changes by 2, . . . , and the mthelement changes by m, then the combined effect of these changeson the value of the vector yt+swould be given by

    yt+s=yt+s

    1t1+ +

    yt+smt

    m= s,

    (74)

    where = (1, . . . , m).

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    The response ofyito a unit shock in yj is given the sequence,known as the impulse response function,

    ij,1, ij,2, ij,3, . . . , (75)

    where ij,k is the ijth element of the matrixk (i,j= 1, . . . , m).

    Generally an impulse response function traces the effect of a

    one-time shock to one of the innovations on current and future

    values of the endogenous variables.

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    What about exogeneity (or Granger-causality)?

    Suppose we have a bivariate VAR system such that xtdoes notGranger cause y. Then we can write

    ytxt

    =

    (1)11 0

    (1)21

    (1)22

    yt1xt1

    +

    + (p)11 0(p)21 (p)22

    ytpxtp

    + 1t2t

    .(76)

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    Then under the stationarity condition

    (I (L))1 =I+

    i=1iL

    i, (77)

    where

    i =

    (i)11 0

    (i)21

    (i)22

    . (78)

    Hence, we see that variable y does not react to a shock ofx.

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    Generally, if a variable, or a block of variables, are strictlyexogenous, then the implied zero restrictions ensure that thesevariables do not react to a shock to any of the endogenousvariables.

    Nevertheless it is advised to be careful when interpreting thepossible (Granger) causalities in the philosophical sense of causalitybetween variables.

    Remark: See also the critique of impulse response analysis in the end of

    this section.

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    Noting that E(tt) = , we observe that the components oft

    are contemporaneously correlated, meaning that they have

    overlapping information to some extend.

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    Example 7

    For example, in the equity-bond data the contemporaneousVAR(2)-residual correlations are

    =================================FTA DIV R20 TBILL

    ---------------------------------

    FTA 1

    DIV 0.123 1

    R20 -0.247 -0.013 1

    TBILL -0.133 0.081 0.456 1

    =================================

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    Many times, however, it is of interest to know how newinformation on yjtmakes us revise our forecasts on yt+s.

    In order to single out the individual effects the residuals must be

    first orthogonalized, such that they become contemporaneouslyuncorrelated (they are already serially uncorrelated).

    Remark: If the error terms (t) are already contemporaneously

    uncorrelated, naturally no orthogonalization is needed.

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    Unfortunately, orthogonalization is not unique in the sense that

    changing the order of variables in y chances the results.

    Nevertheless, there usually exist some guidelines (based on theeconomic theory) how the ordering could be done in a specificsituation.

    Whatever the case, if we define a lower triangular matrix Ssuchthat SS = , and

    t=S1t, (79)

    then I=S1S1, implying

    E[tt] =S1E[tt] S1 =S1S1 =I.

    Consequently the new residuals are both uncorrelated over time aswell as across equations. Furthermore, they have unit variances.

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    The new vector MA representation becomes

    yt=

    i=0iti, (80)

    wherei = iS (m m matrices) so that0 =S. The impulse

    response function ofyito a unit shock in yj is then given by

    ij,0, ij,1,

    ij,2, . . . (81)

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    The Model

    Defining the order of a VAR-modelVector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variablesGeneral impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

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    The uncorrelatedness ofts allow the error variance of the s

    step-ahead forecast ofyitto be decomposed into componentsaccounted for by these shocks, or innovations (this is why thistechnique is usually called innovation accounting).

    Because the innovations have unit variances (besides theuncorrelatedness), the components of this error variance accountedfor by innovations to yj is given by

    s

    k=0

    2

    ij,k. (82)

    Comparing this to the sum of innovation responses we get arelative measure how important variable js innovations are inexplaining the variation in variable iat different step-aheadforecasts, i.e.,

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    R2ij,s= 100

    sk=0

    2ij,k

    mh=1

    sk=0

    2ih,k

    . (83)

    Thus, while impulse response functions trace the effects of a shockto one endogenous variable on to the other variables in the VAR,

    variance decomposition separates the variation in an endogenous

    variable into the component shocks to the VAR.

    Letting s increase to infinity one gets the portion of the totalvariance ofyjthat is due to the disturbance term j ofyj.

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    The Model

    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variablesGeneral impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II Vector Autoregression (VAR)

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    On the ordering of variables

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    As was mentioned earlier, when there is contemporaneouscorrelation between the residuals, i.e., cov(t) = =I theorthogonalized impulse response coefficients are not unique. There

    are no statistical methods to define the ordering. It must be doneby the analyst!

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    It is recommended that various orderings should be tried to seewhether the resulting interpretations are consistent.

    The principle is that the first variable should be selected such that

    it is the only one with potential immediate impact on all othervariables.

    The second variable may have an immediate impact on the lastm 2components ofyt, but not on y1t, the first component, and

    so on. Of course this is usually a difficult task in practice.

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    Selection of the S matrix, where SS = , actually defines alsothe ordering of variables.

    Selecting it as a lower triangular matrix implies that the first

    variable is the one affecting (potentially) the all others, the secondto the m 2rest (besides itself) and so on.

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    O ll d th d i h i S i t Ch l k

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    One generally used method in choosing S is to use Cholesky

    decomposition which results to a lower triangular matrix with positivemain diagonal elements.6

    6For example, if the covariance matrix is

    =

    21 1221 22

    then if =SS, where

    S=

    s11 0s21 s22

    We get

    =

    21 1221

    22

    =

    s11 0

    s21 s22

    s11 s210 s22

    =

    s211 s11s21

    s21s11 s221+ s

    222

    .

    Thus s211 =

    21 , s21s11 =21 =12, and s

    221+ s

    222 =

    22 . Solving these yields s11 =1,

    s21 =21/1, and s22 =

    22 221/21 . That is,

    S=

    1 0

    21/1 (22 221/21 )12

    .

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    Example 8Let us choose in our example two orderings. One with stock marketseries first followed by bond market series

    [(I: FTA, DIV, R20, TBILL)],

    and an ordering with interest rate variables first followed by stockmarkets series

    [(II: TBILL, R20, DIV, FTA)].

    In EViews the order is simply defined in the Cholesky ordering option.

    Below are results in graphs with I: FTA, DIV, R20, TBILL; II: R20,TBILL DIV, FTA, and III: General impulse response function.

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    Impulse responses:

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DR20

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DTBILL

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DFTA

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DDIV

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DR20

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DR20

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DR20

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DTBILL

    Response to Cholesky One S.D. Innovations 2 S.E.

    Order {FTA, DIV, R20, TBILL}

    Seppo Pynnonen Econometrics II Vector Autoregression (VAR)

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    On the ordering of variables

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    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DR20

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 1 0

    Response of DFTA to DTBILL

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DFTA

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DDIV

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DR20

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 1 0

    Response of DDIV to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DR20

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 1 0

    Response of DR20 to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DR20

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 1 0

    Response of DTBILL to DTBILL

    Response to Cholesky One S.D. Innovations 2 S.E.

    Order {TBILL, R20, DIV, FTA}

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    On the ordering of variables

    Variance decomposition graphs of the equity-bond data

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    Variance decomposition graphs of the equity bond data

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DFTA variance due to DFTA

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DFTA variance due to DDIV

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DFTA variance due to DR20

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DFTA variance due to DTBILL

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DDIV variance due to DFTA

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DDIV variance due to DDIV

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DDIV variance due to DR20

    -20

    0

    20

    40

    60

    80

    100

    120

    1 2 3 4 5 6 7 8 9 10

    Percent DDIV variance due to DTBILL

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DR20 variance due to DFTA

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DR20 variance due to DDIV

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DR20 variance due to DR20

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DR20 variance due to DTBILL

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DTBILL variance due to DFTA

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DTBILL variance due to DDIV

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DTBILL variance due to DR20

    -20

    0

    20

    40

    60

    80

    100

    1 2 3 4 5 6 7 8 9 10

    Percent DTBILL variance due to DTBILL

    Variance Decomposition 2 S.E.

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    The Model

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    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II Vector Autoregression (VAR)

    Innovation accounting

    General impulse response function

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    The general impulse response function are defined as9

    GI(j, i,Ft1) = E[yt+j|it=i,Ft1] E[yt+j|Ft1]. (84)

    That is difference of conditional expectation given an one timeshock occurs in series j. These coincide with the orthogonalizedimpulse responses if the residual covariance matrix is diagonal.

    9Pesaran, M. Hashem and Yongcheol Shin (1998). Impulse ResponseAnalysis in Linear Multivariate Models, Economics Letters,58,17-29.

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    Generalized impulse responses:

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    p p

    -4

    -2

    0

    2

    4

    6

    8

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DFTA

    -4

    -2

    0

    2

    4

    6

    8

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DDIV

    -4

    -2

    0

    2

    4

    6

    8

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DR20

    -4

    -2

    0

    2

    4

    6

    8

    1 2 3 4 5 6 7 8 9 10

    Response of DFTA to DTBILL

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DFTA

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DDIV

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DR20

    -0.4

    0.0

    0.4

    0.8

    1.2

    1.6

    1 2 3 4 5 6 7 8 9 10

    Response of DDIV to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DR20

    -2

    -1

    0

    1

    2

    3

    4

    1 2 3 4 5 6 7 8 9 10

    Response of DR20 to DTBILL

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DFTA

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DDIV

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DR20

    -2

    -1

    0

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7 8 9 10

    Response of DTBILL to DTBILL

    Response to Generalized One S.D. Innovations 2 S.E.

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    1 Vector Autoregression (VAR)

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    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II Vector Autoregression (VAR)

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    Accumulated Responses

    Accumulated responses for speriods ahead of a unit shock in

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    p p

    variable ion variable jmay be determined by summing up thecorresponding response coefficients. That is,

    (s)ij =

    s

    k=0ij,k. (85)

    The total accumulated effect is obtained by

    ()ij =

    k=0ij,k. (86)

    In economics this is called the total multiplier.

    Particularly these may be of interest if the variables are firstdifferences, like the stock returns.

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    For the stock returns the impulse responses indicate the return effects

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    For the stock returns the impulse responses indicate thereturn effects

    while the accumulated responses indicate the price effects.All accumulated responses are obtained by summing the MA-matrices

    (s) =

    s

    k=0k, (87)

    with() = (1), where

    (L) = 1 + 1L+ 2L2 + . (88)

    is the lag polynomial of the MA-representation

    yt= (L)t. (89)

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    1 Vector Autoregression (VAR)

    Background

    The Model

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    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo P nnonen E t i II Vector Autoregression (VAR)

    Innovation accounting

    On estimation of the impulse response coefficients

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    Consider the VAR(p) model

    yt= 1yt1+ + pytp+t (90)

    or(L)yt=t. (91)

    Seppo PynnonenE t i II Vector Autoregression (VAR)

    Innovation accounting

    On estimation of the impulse response coefficients

    Th d t ti it th t MA t ti i

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    Then under stationarity the vector MA representation is

    yt=t+ 1t1+ 2t2+ (92)

    When we have estimates of the AR-matrices idenoted by i,i= 1, . . . , pthe next problem is to construct estimates for MA

    matricesj. It can be shown that

    j=

    j

    i=1jii (93)

    with0 =I, andj=0 when i>p. Consequently, the estimatescan be obtained by replacingis by the corresponding estimates.

    Seppo PynnonenE t i II Vector Autoregression (VAR)

    Innovation accounting

    On estimation of the impulse response coefficients

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    Next we have to obtain the orthogonalized impulse responsecoefficients. This can be done easily, for letting S be the Choleskydecomposition of such that

    =SS, (94)

    we can writeyt =

    i=0iti

    = i=0iSS

    1ti

    =i=0iti,

    (95)

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Innovation accounting

    On estimation of the impulse response coefficients

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    wherei = iS (96)

    and t=S1t. Then

    cov[t] =S1S1 =I. (97)

    The estimates foriare obtained by replacingtwith itsestimates and using Cholesky decomposition of.

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Innovation accounting

    1 Vector Autoregression (VAR)

    Background

    The Model

    D fi i th d f VAR d l

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    Defining the order of a VAR-model

    Vector ARMA (VARMA)

    Autocorrelation and Autocovariance Matrices

    Exogeneity and Causality

    Granger-causality and measures of feedback

    Gewekes measures of Linear Dependence

    Innovation accounting

    Impulse response analysis

    Variance decomposition

    On the ordering of variables

    General impulse response function

    Accumulated Responses

    On estimation of the impulse response coefficients

    Critique of impulse response analysis

    Seppo Pynnonen Econometrics II

    Vector Autoregression (VAR)

    Innovation accounting

    Critique of impulse response analysis

    http://find/
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    Critical quaestions:

    Ordering of variables is one problem.

    Interpretations related to Granger-causality from the ordered

    impulse response analysis may not be valid.If important variables are omitted from the system, theireffects go to the residuals and hence may lead to majordistortions in the impulse responses and the structural

    interpretations of the results.

    Seppo Pynnonen Econometrics II

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