Econometrics II - Univaasalipas.uwasa.fi/~sjp/Teaching/ecmii/lectures/ecmiic6.pdf · 2018. 2....

Econometrics II

Seppo Pynnönen

Department of Mathematics and Statistics, University of Vaasa, Finland

Spring 2018

Seppo Pynnönen Econometrics II

Vector Autoregression (VAR)

Part VI

Vector Autoregression

As of Feb 21, 2018Seppo Pynnönen Econometrics II


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

Geweke’s 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



Background1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)















Background

Example 1

Consider the following monthly values of FTAS (Financial Times All

Share) and FTA dividend index values from Jan 1965 to Dec 2005

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

010

0020

0030

00

fta

2040

6080

100

ftadiv

−40

−20

020

40

ret

−40

020

40

1970 1980 1990 2000

div

Time

FT All Share Monthly Srock and Dividend Indexes



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 oneseries 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.



Background

Some of these can be readily investigated by simple tools.

For example lead-lag relations can be investigated bycross-autocorrelation function

ρ̂xy (k) =γ̂xy (k)√γ̂x(0)γ̂y (0)

, (1)

where

γ̂xy (k) =T−k∑t=1

(xt − x̄)(yt+k − ȳ) (2)

is the cross-autocovariance function and γ̂x(0) and γ̂y (0) are thesample variances of xt and yt . Note that ρ̂xy (k) 6= ρ̂xy (−k).



Background

The following autocorrelation functions indicate that returns tend to leaddividends (negative cross-autocorrelation)

0 5 10 15 20

−0.5

0.00.5

1.0

Lag

ACF

Return

0 5 10 15 20

−0.5

0.00.5

1.0

Lag

Return & Dividend

−20 −15 −10 −5 0

−0.5

0.00.5

1.0

Lag

ACF

Dividend & Return

0 5 10 15 20

−0.5

0.00.5

1.0

Lag

Dividend



Background

The following scatter diagram of (rett−1, divt) demonstrates the firstorder cross-correlation.

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

● ●●

●●

●●

●

●

●

● ●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●●

●

●●

●●●

●●●

●●●●● ●●

●

●

●

●

●

●

●

●●●●

●

● ●●

●

●●

●●

●

●●●

●●

●

●

●●●●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●●●

●

●●

●

●

●

●●●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●●●

●

●

●

●

●

●

●● ●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

● ●●

●● ●● ●●

● ●

●

●●

●●●

●

●

●●● ●

●●

−40 −20 0 20 40

−40

−20

020

40Returns and Dividends

Rett−1 (%)

∆ Div t

(%)

Vector autoregression (VAR) provides more sophisticated tools fordeeper analyses.



The Model1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)















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 = φ1 + φ(1)11 xt−1 + φ

(1)12 yt−1 + �1t (3)

yt = φ2 + φ(1)21 xt−1 + φ

(1)22 yt−1 + �2t (4)

�it and �2t are white noise error terms that can becontemporaneously correlated



The Model

Generally: Suppose we have m time series yit , i = 1, . . . ,m, andt = 1, . . . ,T (common length of the time series). Then avector autoregression model is defined as

y1ty2t...ymt

=

µ1µ2...µm

+

φ(1)11 φ

(1)12 · · · φ

(1)1m

φ(1)21 φ

(1)22 · · · φ

(1)2m

......

. . ....

φ(1)m1 φ

(1)m2 · · · φ

(1)mm

y1,t−1y2,t−1...ym,t−1

+

· · ·+

φ

(p)11 φ

(p)12 · · · φ

(p)1m

φ(p)21 φ

(p)22 · · · φ

(p)2m

......

. . ....

φ(p)m1 φ

(1)m2 · · · φ

(p)mm

y1,t−py2,t−p...ym,t−p

+

�1t�2t...�mt

.(5)



The Model

In matrix notations

yt = µ+ Φ1yt−1 + · · ·+ Φpyt−p + �t , (6)

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

Φ(L) = I− Φ1L− . . .− ΦpLp. (7)

Using this in equation (6), it can be finally written shortly as

Φ(L)yt = µ+ �t . (8)

Note that in the above model each yit depends 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.



The Model

A basic assumption in the above model is that the residual vectorfollow a multivariate white noise, i.e.

E[�t ] = 0

E[�t�′s ] ={

Σ� if t = s0 if t 6= 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− Φ1z − Φ2z2 − · · · − Φpzp| = 0 (10)

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



The Model

Example 2

VAR(1) for the return/dividend series:

VAR Estimation Results:

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

Endogenous variables: ret, div

Deterministic variables: const

Sample size: 490

Log 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 0.7995

div.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 div

ret 1.0000 0.8738

div 0.8738 1.0000



Defining the order of a VAR-model1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)
















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




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

� ∼ Nm(0,Σ�). (11)

Akaike’s criterion function (AIC) and Schwarz’s criterion (BIC)have gained popularity in determination of the number of lags inVAR.

In the original forms AIC and BIC are defined as

AIC = −2 log L + 2s (12)

BIC = −2 log L + s logT (13)

where L stands for the Likelihood function, and s denotes thenumber of estimated parameters.




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

For example in a VAR(j) model with m equations there ares = m(1 + jm) + m(m + 1)/2 estimated parameters.




Under the normality assumption BIC can be simplified to

BIC(j) = log∣∣∣Σ̂�,j ∣∣∣+ jm2 logT

T, (14)

and AIC to

AIC(j) = log∣∣∣Σ̂�,j ∣∣∣+ 2jm2

T, (15)

j = 0, . . . , p, where

Σ̂�,j =1

T

T∑t=j+1

�̂t,j �̂′t,j =

1

TÊj Ê

′j (16)

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

Êj = [�̂j+1,j , �̂j+2,j , · · · , �̂T ,j ] (17)

an m × (T − j) matrix.Seppo Pynnönen Econometrics II



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 |Σ̂k | − log |Σ̂p|), (18)

where Σ̂k denotes the maximum likelihood estimate of the residualcovariance matrix of VAR(k) and Σ̂p the 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 : Φk+1 = · · · = Φp = 0 (19)

is true, thenLR ∼ χ2df , (20)

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




In an m variate VAR(k)-model each series has p − k lags less thanthose in VAR(p). Thus the difference in each equation ism(p − k), so that in total df = m2(p − k).

Note that often, when T is small, a modified LR

LR∗ = (T −mg)(log |Σ̂k | − log |Σ̂p|)

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

Yet another method is to use sequential LR

LR(k) = (T −m)(log |Σ̂k | − log |Σ̂k+1|)

which tests the null hypothesis

H0 : Φk+1 = 0

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

EViews uses this method in View -> Lag Structure -> LagLength Criteria. . ..




Example 3

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

year bond yield)0

1000

2000

3000

fta

2040

6080

100

ftadiv

46

810

14

rs91

46

810

14

1970 1980 1990 2000

r20

Time

Bond and Stock Market Series




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

following results. Below are EViews results.

VAR Lag Order Selection Criteria

Endogenous variables: DFTA DDIV DR20 DTBILL

Exogenous 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 21.606

4 -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 21.947

7 -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 22.357

11 -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 criterion

HQ: Hannan-Quinn information criterionSeppo Pynnönen Econometrics II



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.




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

H0 : Υ1 = · · · = Υh = 0 (21)

where Υk = (ρij(k)) is the autocorrelation matrix (see later in thenotes) of the residual series with ρij(k) the cross autocorrelation oforder k of the residuals series i and j . A general purpose(portmanteau) test is the Q-statistic8

Qh = Th∑

k=1

tr(Υ̂′kΥ̂−10 Υ̂kΥ̂

−10 ), (22)

where Υ̂k = (ρ̂ij(k)) are the estimated (residual) autocorrelations,

and Υ̂0 the contemporaneous correlations of the residuals.8See e.g. Lütkepohl, Helmut (1993). Introduction to Multiple Time Series,

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



Alternatively (especially in small samples) a modified statistic isused

Q∗h = T2

h∑k=1

(T − k)−1tr(Υ̂′kΥ̂−10 Υ̂kΥ̂−10 ). (23)




The asymptotic distribution is χ2 with df = p2(h − k). Note that incomputer printouts h is running from 1, 2, . . . h∗ with h∗ specified by theuser.

VAR(1) Residual Portmanteau Tests for Autocorrelations

H0: no residual autocorrelations up to lag h

Sample: 1966:02 1995:12


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

Lags Q-Stat Prob. Adj Q-Stat Prob. 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 32

4 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 96

8 137.0931 0.0538 139.0485 0.0424 112

9 152.9130 0.0659 155.2751 0.0507 128

10 168.4887 0.0797 171.2972 0.0599 144

11 179.3347 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 Pynnönen Econometrics II



VAR Residual Portmanteau Tests for Autocorrelations

H0: no residual autocorrelations up to lag h

Sample: 1965:01 1995:12


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

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 48

6 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 96

9 121.8202 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 144

12 157.4354 0.5425 160.6003 0.4718 160

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

*The test is valid only for lags larger than

the VAR lag order.

df is degrees of freedom for (approximate)

chi-square distribution




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 there

ordinary differences (opposed to log-differences) are analyzed. Here,

however, log transformations are preferred.



Vector ARMA (VARMA)1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)















Vector ARMA (VARMA)

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

yt = µ+

p∑i=1

Φiyt−i + �t −q∑

j=1

Θj�t−j (24)

orΦ(L)yt = µ+ Θ(L)�t , (25)

where yt , µ, and �t are m × 1 vectors, and Φi ’s and Θj ’s arem ×m matrices, and

Φ(L) = I− Φ1L− . . .− ΦpLp

Θ(L) = I−Θ1L− . . .−ΘqLq.(26)

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



Autocorrelation and Autocovariance Matrices1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)
















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

γij(k) = E (yit−k − µi )(yjt − µj). (27)

Although for the usual autocovariance γk = γ−k , the same is nottrue for the cross autocovariance, but γij(k) 6= γij(−k). Thecorresponding cross autocorrelations are

ρi ,j(k) =γij(k)√γi (0)γj(0)

. (28)

The kth autocorrelation matrix is then

Υk =

ρ1(k) ρ1,2(k) . . . ρ1,m(k)ρ2,1(k) ρ2(k) . . . ρ2,m(k)...

.... . .

...ρm,1(k) ρm,2(k) . . . ρm(k)

. (29)Seppo Pynnönen Econometrics II



Remark: Υk = Υ′−k .

The diagonal elements, ρj(k), j = 1, . . . ,m of Υ are the usualautocorrelations.




Example 4

Cross autocorrelations of the equity-bond data.

Υ̂1 =

Divt Ftat R20t TbltDivt−1Ftat−1R20t−1Tblt−1

0.0483 −0.0099 0.0566 0.0779−0.0160 0.1225∗ −0.2968∗ −0.1620∗

0.0056 −0.1403∗ 0.2889∗ 0.3113∗0.0536 −0.0266 0.1056∗ 0.3275∗

Note: Correlations with absolute value exceeding 2× std err = 2× /

√371 ≈ 0.1038

are statistically significant at the 5% level (indicated by ∗).




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 to

be positively correlated!

iFrench, K. and R. Roll (1986). Stock return variances: The arrival ofinformation and reaction of traders. Journal of Financial Economics, 17, 5–26.




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




One explanation could be that the cross autocorrelations arepositive, which can be partially proved as follows: Let

rpt =1

m

m∑i=1

rit =1

mι′rt (30)

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

is a vector of ones, and rt = (r1t , . . . , rmt)′ is the vector of returnsof the securities. Then

cov(rpt−1, rpt) = cov

[ι′rt−1m

,ι′rtm

]=ι′Γ1ι

m2, (31)

where Γ1 is the first order autocovariance matrix.




Therefore

m2cov(rpt−1, rpt) = ι′Γ1ι =

(ι′Γ1ι− tr(Γ1)

)+ tr(Γ1), (32)

where tr(·) is the trace operator which sums the diagonal elementsof a square matrix.

Consequently, because the right hand side tends to be positive andtr(Γ1) tends to be negative, ι

′Γ1ι− tr(Γ1), which contains onlycross autocovariances, must be positive, and larger than theabsolute value of tr(Γ1), the autocovariances of individual stockreturns.



Exogeneity and Causality1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)
















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

yt = c +

p∑i=1

A′iyt−i +

p∑i=0

B′ixt−i + �t , (33)

where p + 1 ≤ t ≤ T , y′t = (y1t , . . . , ymt), c is an m × 1 vector ofconstants, A1, . . . ,Ap are m ×m matrices of lag coefficients,x′t = (x1t , . . . , xkt) is a k × 1 vector of regressors, B0,B1, . . . ,Bpare k ×m coefficient matrices, and �t is an m × 1 vector of errorshaving the properties

E[�t ] = E[E[�t |Yt−1, xt ]] = 0 (34)

and

E[�t�′s

]= E

[E[�t�′s |Yt−1, xt

]]=

{Σ� t = s0 t 6= s, (35)




whereYt−1 = (yt−1, yt−2, . . . , y1). (36)

We can compile this into a matrix form

Y = XB + U, (37)

whereY = (yp+1, . . . , yT )

′

X = (zp+1. . . . , zT )′

zt = (1, y′t−1, . . . , y′t−p, x

′t , . . . , x

′t−p)

′

U = (�p+1, . . . , �T )′,

(38)

andB = (c,A′1, . . . ,A

′p,B

′0, . . . ,B

′p)′. (39)

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




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

B̂ = (X′X)−1X′Y, (40)

and the ML estimator of Σ� is

Σ̂� =1

TÛ′Û, Û = Y − XB̂. (41)

We say that xt is weakly exogenous2 if the stochastic structure ofx contains no information that is relevant for estimation of theparameters of interest, B, and Σ�.

If the conditional distribution of xt given the past is independent ofthe history of yt then xt is said to be strongly exogenous.

2For an in depth discussion of Exogeneity, see Engle, R.F., D.F. Hendry andJ.F. Richard (1983). Exogeneity. Econometrica, 51:2, 277–304.




Background

The Model


Vector ARMA (VARMA)

















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

If the history of x does not help to predict the future values of y ,we say that x does not Granger-cause y .3

3Granger, C.W. (1969). Econometrica 37, 424–438. Sims, C.A. (1972).American Economic Review, 62, 540–552.





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(ŷt+s |yt , yt−1, . . .)= MSE(ŷt+s |yt , yt−1, . . . , xt , xt−1, . . .),

(42)

where (e.g.)

MSE(ŷt+s |yt , yt−1, . . .)= E

[(yt+s − ŷt+s)2|yt , yt−1, . . .

].

(43)





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 z′t = (y′t , x′t), which is

assumed to follow a VAR(p) model

zt =

p∑i=1

Πizt−i + νt , (44)

whereE[νt ] = 0

E[νtν ′s ] ={

Σν , t = s0, t 6= s.

(45)





Partition the VAR of z as

yt =∑p

i=1 Cixt−i +∑p

i=1 Diyt−i + ν1t

xt =∑p

i=1 Eixt−i +∑p

i=1 Fiyt−i + ν2t(46)

where ν ′t = (ν′1t ,ν

′2t) and Πi and Σν are partitioned respectively,

Πi =

(Ci DiEi Fi

), i = 1, . . . , p (47)

and

Σν =

(Σ11 Σ21Σ21 Σ22

)(48)

with E[νktν ′lt ] = Σkl , k, l = 1, 2.





Now x does not Granger-cause y if and only if, Ci = 0 for alli = 1, . . . , p, or equivalently, if and only if, |Σ11| = |Σ1|, whereΣ1 = E[η1tη′1t ] with η1t from the regression

yt =

p∑i=1

Diyt−i + η1t . (49)

Changing the roles of the variables we get the necessary andsufficient condition of y not Granger-causing x.





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

Testing for the Granger-causality of x on y reduces to testing forthe hypothesis

H0 : C1 = · · · = Cp = 0. (50)




This can be done with the likelihood ratio test by estimating withOLS the restricted4 and non-restricted4 regressions, andcalculating the respective residual covariance matrices:

Unrestricted:

Σ̂11 =1

T − p

T∑t=p+1

ν̂1t ν̂′1t . (51)

Restricted:

Σ̂1 =1

T − p

T∑t=p+1

η̂1t η̂′1t . (52)

4Perform OLS regressions of each of the elements in y on a constant, p lagsof the elements of x and p lags of the elements of y.

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





The LR test is then

LR = (T − p)(

log |Σ̂1| − log |Σ̂11|)∼ χ2mkp, (53)

if H0 is true.




Example 6

Granger causality between pairwise equity-bond market series (by EViews)

Pairwise Granger Causality Tests

Sample: 1965:01 1995:12

Lags: 12

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

Null Hypothesis: Obs F-Statistic Probability

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

DFTA does not Granger Cause DDIV 365 0.71820 0.63517

DDIV does not Granger Cause DFTA 1.43909 0.19870

DR20 does not Granger Cause DDIV 365 0.60655 0.72511

DDIV 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 0.97370

DFTA 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

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





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

Gilts, and Gilts lead Treasury bill.





Let us next examine the block exogeneity between the bond and equity

markets (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.




Background

The Model


Vector ARMA (VARMA)

















Above we tested Granger-causality, but there are several otherinteresting relations that are worth investigating.

Geweke5 has suggested a measure for linear feedback from x to ybased on the matrices Σ1 and Σ11 as

Fx→y = log(|Σ1|/|Σ11|), (54)

so that the statement that ”x does not (Granger) cause y” isequivalent to Fx→y = 0.

Similarly the measure of linear feedback from y to x is defined by

Fy→x = log(|Σ2|/|Σ22|). (55)

5Geweke (1982) Journal of the American Statistical Association, 79,304–324.





It may also be interesting to investigate the instantaneous causalitybetween the variables.

For the purpose, define regression

yt =

p∑i=0

Cixt−i +

p∑i=1

Diyt−i + ω1t , (56)

with C0 capturing the contemporaneous effect of xt on yt .

Note: The coefficient matricese Ci and Di are just generic notations, sothey are logically different in different specifications of the models.





Similarly, the last k equations can be written as

xt =

p∑i=1

Eixt−i +

p∑i=0

Fiyt−i + ω2t . (57)

Denoting Σωi = E[ωitω′it ], i = 1, 2, there is instantaneous causalitybetween y and x if and only if C0 6= 0 and F0 6= 0 or, equivalently,|Σ11| > |Σω1 | and |Σ22| > |Σω2 |.

Analogously to the linear feedback we can define instantaneous linearfeedback

Fx·y = log(|Σ11|/|Σω1 |) = log(|Σ22|/|Σω2 |). (58)Linear dependence, defined as

Fx,y = Fx→y + Fy→x + Fx·y. (59)

measures overall (linear) dependence between x and y.

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





Absence of a particular causal ordering is then equivalent to one of thesefeedback measures being zero.

Using the method of least squares we get estimates for the variousmatrices above as

Σ̂i =1

T − p

T∑t=p+1

η̂it η̂′it , (60)

Σ̂ii =1

T − p

T∑t=p+1

ν̂it ν̂′it , (61)

Σ̂ωi =1

T − p

T∑t=p+1

ω̂it ω̂′it , (62)

for i = 1, 2.

For exampleF̂x→y = log(|Σ̂1|/|Σ̂11|). (63)





With these estimates one can test in particular:

No Granger-causality: x→ y, H01 : Fx→y = 0

(T − p)F̂x→y ∼ χ2mkp. (64)

No Granger-causality: y→ x, H02 : Fy→x = 0

(T − p)F̂y→x ∼ χ2mkp. (65)

No instantaneous feedback: H03 : Fx·y = 0

(T − p)F̂x·y ∼ χ2mk . (66)

No linear dependence: H04 : Fx,y = 0

(T − p)F̂x,y ∼ χ2mk(2p+1). (67)

This last is due to the asymptotic independence of the measuresFx→y, Fy→x and Fx·y.





Alternatively one can use asymptotically equivalent Wald andLagrange Multiplier (LM) statistics for testing these hypotheses.

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





Example 7

The LR-statistics of the above measures and the associated χ2 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 0.00011

x,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)].



Innovation accounting1 Vector Autoregression (VAR)

Background

The Model


Vector ARMA (VARMA)
















Consider a bivariate VAR(1)(xtyt

)=

(φ11 φ12φ21 φ22

)(xt−1yt−1

)+

(�1t�2t

). (68)

The vector moving average (VMA) representation of is(xtyt

)=∞∑i=0

(ψ11(i) ψ12(i)ψ21(i) ψ22(i)

)(�1,t−i�2,t−i

)(69)

The coefficients ψjk(i) can be used in one hand to trace the shocks(�) on the future values of x and y or in the other hand on thevariance of the prediction errors of x and y .

The former is called impulse response analysis and the lattervariance decomposition. Together these are called innovationaccounting.




Background

The Model


Vector ARMA (VARMA)

















Consider the VAR(p) model

Φ(L)yt = �t , (70)

whereΦ(L) = I− Φ1L− Φ2L2 − · · · − ΦpLp (71)

is the (matric) lag polynomial.

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

yt = Φ−1(L)�t = �t +

∞∑i=1

Ψi�t−i , (72)

where Ψi is an m ×m coefficient matrix.





The error terms �t represent shocks in the system.

Suppose we have a unit shock in �t then its effect in y, s periodsahead is

∂yt+s∂�t

= Ψs . (73)

Accordingly the interpretation of the Ψ matrices is that theyrepresent marginal effects, or the model’s response to a unit shock(or innovation) at time point t in each of the variables.

Economists call such parameters are as dynamic multipliers.





For example, if we were told that the first element in �t changes byδ1, that the second element changes by δ2, . . . , and the mthelement changes by δm, then the combined effect of these changeson the value of the vector yt+s would be given by

∆yt+s =∂yt+s∂�1t

δ1 + · · ·+∂yt+s∂�mt

δm = Ψsδ,

(74)

where δ′ = (δ1, . . . , δm).





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

ψij ,1, ψij ,2, ψij ,3, . . ., (75)

where ψij ,k is the ijth element of the matrix Ψk (i , j = 1, . . . ,m).

Generally an impulse response function traces the effect of aone-time shock to one of the innovations on current and futurevalues of the endogenous variables.





What about exogeneity (or Granger-causality)?

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

xt

= ( φ(1)11 0φ

(1)21 φ

(1)22

)(yt−1xt−1

)+ · · ·

+

(φ

(p)11 0

φ(p)21 φ

(p)22

)(yt−pxt−p

)+

(�1t�2t

).

(76)





Then under the stationarity condition

(I− Φ(L))−1 = I +∞∑i=1

ΨiLi , (77)

where

Ψi =

(ψ

(i)11 0

ψ(i)21 ψ

(i)22

). (78)

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





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.





Orthogonalized impulse response function

Noting that E (�t�′t) = Σ�, we observe that the components of �t

are contemporaneously correlated, meaning that they haveoverlapping information to some extend.





Example 8

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

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





Many times, however, it is of interest to know how ”new”information on yjt makes us revise our forecasts on yt+s .

In order to single out the individual effects the residuals must befirst 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.





Unfortunately, orthogonalization is not unique in the sense thatchanging 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 S suchthat SS′ = Σ�, and

νt = S−1�t , (79)

then I = S−1Σ�S′−1, implying

E[νtν ′t ] = S−1E[�t�′t ]S′−1 = S−1Σ�S′

−1 = I.

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





The new vector MA representation becomes

yt =∞∑i=0

Ψ∗i νt−i , (80)

where Ψ∗i = ΨiS (m ×m matrices) so that Ψ∗0 = S. The impulseresponse function of yi to a unit shock in yj is then given by

ψ∗ij ,0, ψ∗ij ,1, ψ

∗ij ,2, . . . (81)




Background

The Model


Vector ARMA (VARMA)

















The uncorrelatedness of νts allow the error variance of the sstep-ahead forecast of yit to 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

ψ∗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 i at different step-aheadforecasts, i.e.,





R2ij ,s = 100

∑sk=0 ψ

∗2ij ,k∑m

h=1

∑sk=0 ψ

∗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 endogenousvariable into the component shocks to the VAR.

Letting s increase to infinity one gets the portion of the totalvariance of yj that is due to the disturbance term �j of yj .




Background

The Model


Vector ARMA (VARMA)

















As was mentioned earlier, when there is contemporaneouscorrelation between the residuals, i.e., cov(�t) = Σ� 6= I theorthogonalized impulse response coefficients are not unique. Thereare no statistical methods to define the ordering. It must be doneby the analyst!





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

The second variable may have an immediate impact on the lastm − 2 components of yt , but not on y1t , the first component, andso on. Of course this is usually a difficult task in practice.





Selection of the S matrix

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

Selecting it as a lower triangular matrix implies that the firstvariable is the one affecting (potentially) the all others, the secondto the m − 2 rest (besides itself) and so on.





One generally used method in choosing S is to use Choleskydecomposition which results to a lower triangular matrix with positivemain diagonal elements.6

6For example, if the covariance matrix is

Σ =

(σ21 σ12σ21 σ

22

)then if Σ = SS′, where

S =

(s11 0s21 s22

)We get

Σ =

(σ21 σ12σ21 σ

22

)=

(s11 0s21 s22

)(s11 s210 s22

)=

(s211 s11s21s21s11 s221 + s

222

).

Thus s211 = σ21 , s21s11 = σ21 = σ12, and s

221 + s

222 = σ

22 . Solving these yields s11 = σ1,

s21 = σ21/σ1, and s22 =√σ22 − σ221/σ21 . That is,

S =

(σ1 0

σ21/σ1 (σ22 − σ221/σ21)12

).





Example 9

Let 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.





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}





-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


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

Order {TBILL, R20, DIV, FTA}





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.




Background

The Model


Vector ARMA (VARMA)

















The general impulse response function are defined as9

GI (j , δi ,Ft−1) = E[yt+j |�it = δi ,Ft−1]− E[yt+j |Ft−1]. (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 Σ is diagonal.

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





Generalized impulse responses:

-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10


-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10


-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10


-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-0.4

0.0

0.4

0.8

1.2

1.6

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


-2

-1

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10


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




Background

The Model


Vector ARMA (VARMA)

















Accumulated responses for s periods ahead of a unit shock invariable i on variable j may be determined by summing up thecorresponding response coefficients. That is,

ψ(s)ij =

s∑k=0

ψij ,k . (85)

The total accumulated effect is obtained by

ψ(∞)ij =

∞∑k=0

ψij ,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.





For the stock returns the impulse responses indicate the return effectswhile the accumulated responses indicate the price effects.

All accumulated responses are obtained by summing the MA-matrices

Ψ(s) =s∑

k=0

Ψk , (87)

with Ψ(∞) = Ψ(1), where

Ψ(L) = 1 + Ψ1L + Ψ2L2 + · · ·. (88)

is the lag polynomial of the MA-representation

yt = Ψ(L)�t . (89)




Background

The Model


Vector ARMA (VARMA)

















Consider the VAR(p) model

yt = Φ1yt−1 + · · ·+ Φpyt−p + �t (90)

orΦ(L)yt = �t . (91)





Then under stationarity the vector MA representation is

yt = �t + Ψ1�t−1 + Ψ2�t−2 + · · · (92)

When we have estimates of the AR-matrices Φi denoted by Φ̂i ,i = 1, . . . , p the next problem is to construct estimates for MAmatrices Ψj . It can be shown that

Ψj =

j∑i=1

Ψj−iΦi (93)

with Ψ0 = I, and Φj = 0 when i > p. Consequently, the estimatescan be obtained by replacing Φi ’s by the corresponding estimates.





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

Σ� = SS′, (94)

we can writeyt =

∑∞i=0 Ψi�t−i

=∑∞

i=0 ΨiSS−1�t−i

=∑∞

i=0 Ψ∗i νt−i ,

(95)





whereΨ∗i = ΨiS (96)

and νt = S−1�t . Then

cov[νt ] = S−1Σ�S

′−1 = I. (97)

The estimates for Ψ∗i are obtained by replacing Ψt with its

estimates and using Cholesky decomposition of Σ̂�.




Background

The Model


Vector ARMA (VARMA)

















Critical quaestions:

Ordering of variables is one problem.

Interpretations related to Granger-causality from the orderedimpulse 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 structuralinterpretations of the results.


Vector AutoregressionVector Autoregression (VAR)BackgroundThe ModelDefining the order of a VAR-modelVector ARMA (VARMA)Autocorrelation and Autocovariance MatricesExogeneity and CausalityInnovation accounting

Econometrics II - Univaasalipas.uwasa.fi/~sjp/Teaching/ecmii/lectures/ecmiic6.pdf · 2018. 2....

Documents

Transcript of Econometrics II - Univaasalipas.uwasa.fi/~sjp/Teaching/ecmii/lectures/ecmiic6.pdf · 2018. 2....