financial econometrics: causality granger, geweke
TRANSCRIPT
-
7/25/2019 Financial Econometrics: Causality Granger, Geweke
1/104
Econometrics II
Seppo Pynnonen
Department of Mathematics and Statistics, University of Vaasa, Finland
January 12 February 25, 2016
Seppo Pynnonen Econometrics II
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
2/104
Vector Autoregression (VAR)
Part V
Vector Autoregression
Seppo Pynnonen Econometrics II
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
3/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
4/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
5/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
6/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
7/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
8/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
9/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
10/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
11/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
12/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
13/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
14/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
15/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
16/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
17/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
18/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
19/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
20/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
21/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
22/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
23/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
24/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
25/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
26/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
27/104
Vector Autoregression (VAR)
D fi i h d f VAR d l
-
7/25/2019 Financial Econometrics: Causality Granger, Geweke
28/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
29/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
30/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
31/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
32/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
33/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
34/104
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
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
35/104
Remark: k=
k.
The diagonal elements, j(k), j= 1, . . . , m ofare the usualautocorrelations.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Autocorrelation and Autocovariance Matrices
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
36/104
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 ).
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Autocorrelation and Autocovariance Matrices
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
37/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Autocorrelation and Autocovariance Matrices
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
38/104
Source: Campbell, Lo, and MackKinlay (1997). Econometrics ofFinancial Markets.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Autocorrelation and Autocovariance Matrices
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
39/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Autocorrelation and Autocovariance Matrices
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
40/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
1 Vector Autoregression (VAR)
http://goforward/http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
41/104
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
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
42/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
43/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
44/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
1 Vector Autoregression (VAR)
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
45/104
( )
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
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
46/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
47/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
48/104
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)
Seppo Pynnonen Econometrics II
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
49/104
Vector Autoregression (VAR)
Exogeneity and Causality
-
7/25/2019 Financial Econometrics: Causality Granger, Geweke
50/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
51/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
52/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
53/104
Granger-causality and measures of feedback
The LR test is then
LR = (T p)
ln |1| ln |11| 2mkp, (52)
ifH0 is true.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
E l 6
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
54/104
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==============================================================
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
G lit d f f db k
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
55/104
Granger-causality and measures of feedback
The p-values indicate that FTA index returns Granger cause 20 year
Gilts, and Gilts lead Treasury bill.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Granger causality and measures of feedback
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
56/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
1 Vector Autoregression (VAR)
B k d
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
57/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
58/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
59/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
60/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
61/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
62/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
63/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Exogeneity and Causality
Gewekes measures of Linear Dependence
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
64/104
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)].
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
65/104
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
Seppo Pynnonen Econometrics II
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
66/104
Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
-
7/25/2019 Financial Econometrics: Causality Granger, Geweke
67/104
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
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
68/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
69/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
70/104
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).
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
71/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
72/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
73/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
74/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
Orthogonalized impulse response function
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
75/104
Noting that E(tt) = , we observe that the components oft
are contemporaneously correlated, meaning that they have
overlapping information to some extend.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
76/104
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
=================================
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
77/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
78/104
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.
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Impulse response analysis
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
79/104
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)
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
Th M d l
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
80/104
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
Seppo Pynnonen Econometrics II
Vector Autoregression (VAR)
Innovation accounting
Variance decomposition
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
81/104
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.,
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
Variance decomposition
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
82/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
The Model
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
83/104
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)
Innovation accounting
On the ordering of variables
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
84/104
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!
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
85/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
Selection of the S matrix
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
86/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
O ll d th d i h i S i t Ch l k
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
87/104
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
.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
88/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
Impulse responses:
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
89/104
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)
Innovation accounting
On the ordering of variables
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
90/104
-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}
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
On the ordering of variables
Variance decomposition graphs of the equity-bond data
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
91/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
The Model
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
92/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
93/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
General impulse response function
Generalized impulse responses:
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
94/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
The Model
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
95/104
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
Accumulated Responses
Accumulated responses for speriods ahead of a unit shock in
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
96/104
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.
Seppo Pynnonen Econometrics II Vector Autoregression (VAR)
Innovation accounting
Accumulated Responses
For the stock returns the impulse responses indicate the return effects
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
97/104
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)
Seppo Pynnonen E t i s II Vector Autoregression (VAR)
Innovation accounting
1 Vector Autoregression (VAR)
Background
The Model
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
98/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
99/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
100/104
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
http://find/http://goback/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
101/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
102/104
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
http://find/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
103/104
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/ -
7/25/2019 Financial Econometrics: Causality Granger, Geweke
104/104
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
http://find/http://goback/