STAT 497 LECTURE NOTE 11

1
STAT 497LECTURE NOTE 11

• VAR MODELS AND GRANGER CAUSALITY

2
VECTOR TIME SERIES

• A vector series consists of multiple single
  series.
• Why we need multiple series?
• To be able to understand the relationship between
  several components
• To be able to get better forecasts

3
VECTOR TIME SERIES

• Price movements in one market can spread easily
  and instantly to another market. For this reason,
  financial markets are more dependent on each
  other than ever before. So, we have to consider
  them jointly to better understand the dynamic
  structure of global market. Knowing how markets
  are interrelated is of great importance in
  finance.
• For an investor or a financial institution
  holding multiple assets play an important role in
  decision making.

4
VECTOR TIME SERIES

5
VECTOR TIME SERIES

6
VECTOR TIME SERIES

7

8

9
VECTOR TIME SERIES

• Consider an m-dimensional time series
  Yt(Y1,Y2,,Ym). The series Yt is weakly
  stationary if its first two moments are time
  invariant and the cross covariance between Yit
  and Yjs for all i and j are functions of the time
  difference (s?t) only.

10
VECTOR TIME SERIES

• The mean vector
• The covariance matrix function

11
VECTOR TIME SERIES

• The correlation matrix function
• where D is a diagonal matrix in which the i-th
  diagonal element is the variance of the i-th
  process, i.e.
• The covariance and correlation matrix functions
  are positive semi-definite.

12
VECTOR WHITE NOISE PROCESS

• atWN(0,?) iff at is stationary with mean 0
  vector and

13
VECTOR TIME SERIES

• Yt is a linear process if it can be expressed
  as
• where ?j is a sequence of mxn matrix whose
  entries are absolutely summable, i.e.

14
VECTOR TIME SERIES

• For a linear process, E(Yt)0 and

15
MA (WOLD) REPRESENTATION

• For the process to be stationary, ?s should be
  square summable in the sense that each of the mxm
  sequence ?ij.s is square summable.

16
AR REPRESENTATION

• For the process to be invertible, ?s should be
  absolute summable.

17
THE VECTOR AUTOREGRESSIVE MOVING AVERAGE (VARMA)
PROCESSES

• VARMA(p,q) process

18
VARMA PROCESS

• VARMA process is stationary, if the zeros of
  ?p(B) are outside the unit circle.
• VARMA process is invertible, if the zeros of
  ?q(B) are outside the unit circle.

19
IDENTIFIBILITY PROBLEM

• Multiplying matrices by some arbitrary matrix
  polynomial may give us an identical covariance
  matrix. So, the VARMA(p,q) model is not
  identifiable. We cannot uniquely determine p and
  q.

20
IDENTIFIBILITY PROBLEM

• Example VARMA(1,1) process

MA(?)VMA(1)
21

22
IDENTIFIBILITY

• To eliminate this problem, there are three
  methods suggested by Hannan (1969, 1970, 1976,
  1979).
• From each of the equivalent models, choose the
  minimum MA order q and AR order p. The resulting
  representation will be unique if Rank(?p(B))m.
• Represent ?p(B) in lower triangular form. If the
  order of ?ij(B) for i,j1,2,,m, then the model
  is identifiable.
• Represent ?p(B) in a form ?p(B) ?p(B)I where
  ?p(B) is a univariate AR(p). The model is
  identifiable if ?p?0.

23
VAR(1) PROCESS

• Yi,t depends not only the lagged values of Yit
  but also the lagged values of the other
  variables.
• Always invertible.
• Stationary if outside the
  unit circle. Let ?B?1.

The zeros of I??B is related to the eigenvalues
of ?.
24
VAR(1) PROCESS

• Hence, VAR(1) process is stationary if the
  eigenvalues of ? ?i, i1,2,,m are all inside
  the unit circle.
• The autocovariance matrix

25
VAR(1) PROCESS

• k1,

26
VAR(1) PROCESS

• Then,

27
VAR(1) PROCESS

• Example

The process is stationary.
28
VMA(1) PROCESS

• Always stationary.
• The autocovariance function
• The autocovariance matrix function cuts of after
  lag 1.

29
VMA(1) PROCESS

• Hence, VMA(1) process is invertible if the
  eigenvalues of ? ?i, i1,2,,m are all inside
  the unit circle.

30
IDENTIFICATION OF VARMA PROCESSES

• Same as univariate case.
• SAMPLE CORRELATION MATRIC FUNCTION Given a
  vector series of n observations, the sample
  correlation matrix function is
• where s are the crosscorrelation for
  the i-th and j-th component series.
• It is very useful to identify VMA(q).

31
SAMPLE CORRELATION MATRIC FUNCTION

• Tiao and Box (1981) They have proposed to use
  ,? and . signs to show the significance of the
  cross correlations.
• sign the value is greater than 2 times the
  estimated standard error
• sign the value is less than 2 times the
  estimated standard error
• . sign the value is within the 2 times estimated
  standard error

32
PARTIAL AUTOREGRESSION OR PARTIAL LAG CORRELATION
MATRIX FUNCTION

• They are useful to identify VAR order. The
  partial autoregression matrix function is
  proposed by Tiao and Box (1981) but it is not a
  proper correlation coefficient. Then, Heyse and
  Wei (1985) have proposed the partial lag
  correlation matrix function which is a proper
  correlation coefficient. Both of them can be used
  to identify the VARMA(p,q).

33
GRANGER CAUSALITY

• In time series analysis, sometimes, we would like
  to know whether changes in a variable will have
  an impact on changes other variables.
• To find out this phenomena more accurately, we
  need to learn more about Granger Causality Test.

34
GRANGER CAUSALITY

• In principle, the concept is as follows
• If X causes Y, then, changes of X happened first
  then followed by changes of Y.

35
GRANGER CAUSALITY

• If X causes Y, there are two conditions to be
  satisfied
• 1. X can help in predicting Y. Regression of X on
  Y has a big R2
• 2. Y can not help in predicting X.

36
GRANGER CAUSALITY

• In most regressions, it is very hard to discuss
  causality. For instance, the significance of the
  coefficient ? in the regression
• only tells the co-occurrence of x and y, not
  that x causes y.
• In other words, usually the regression only tells
  us there is some relationship between x and y,
  and does not tell the nature of the relationship,
  such as whether x causes y or y causes x.

37
GRANGER CAUSALITY

• One good thing of time series vector
  autoregression is that we could test causality
  in some sense. This test is first proposed by
  Granger (1969), and therefore we refer it Granger
  causality.
• We will restrict our discussion to a system of
  two variables, x and y. y is said to
  Granger-cause x if current or lagged values of y
  helps to predict future values of x. On the other
  hand, y fails to Granger-cause x if for all s gt
  0, the mean squared error of a forecast of xts
  based on (xt, xt-1, . . .) is the same as that is
  based on (yt, yt-1, . . .) and (xt, xt-1, . . .).

38
GRANGER CAUSALITY

• If we restrict ourselves to linear functions, x
  fails to Granger-cause x if
• Equivalently, we can say that x is exogenous in
  the time series sense with respect to y, or y is
  not linearly informative about future x.

39
GRANGER CAUSALITY

• A variable X is said to Granger cause another
  variable Y, if Y can be better predicted from the
  past of X and Y together than the past of Y
  alone, other relevant information being used in
  the prediction (Pierce, 1977).

40
GRANGER CAUSALITY

• In the VAR equation, the example we proposed
  above implies a lower triangular coefficient
  matrix
• Or if we use MA representations,

41
GRANGER CAUSALITY

• Consider a linear projection of yt on past,
  present and future xs,
• where E(etx? ) 0 for all t and ?. Then y fails
  to Granger-cause x iff dj 0 for j 1, 2, . . ..

42
TESTING GRANGER CAUSALITY

• Procedure
• 1) Check that both series are stationary in mean,
  variance and covariance (if necessary transform
  the data via logs, differences to ensure this)
• 2) Estimate AR(p) models for each series, where p
  is large enough to ensure white noise residuals.
  F tests and other criteria (e.g. Schwartz or
  Akaike) can be used to establish the maximum lag
  p that is needed.
• 3) Re-estimate both model, now including all the
  lags of the other variable
• 4) Use F tests to determine whether, after
  controlling for past Y, past values of X can
  improve forecasts Y (and vice versa)

43
TEST OUTCOMES

• 1. X Granger causes Y but Y does not Granger
  cause X
• 2. Y Granger causes X but X does not Granger
  cause Y
• 3. X Granger causes Y and Y Granger causes X
  (i.e., there is a feedback system)
• 4. X does not Granger cause Y and Y does not
  Granger cause X

44
TESTING GRANGER CAUSALITY

• The simplest test is to estimate the regression
  which is based on
• using OLS and then conduct a F-test of the null
  hypothesis
• H0 ?1 ?2 . . . ?p 0.

45
TESTING GRANGER CAUSALITY

• 2.Run the following regression, and calculate RSS
  (full model)
• 3.Run the following limited regression, and
  calculate RSS (Restricted model).

46
TESTING GRANGER CAUSALITY

• 4.Do the following F-test using RSS obtained from
  stages 2 and 3
• F (n-k) /q .(RSSrestricted-RSSfull) /
  RSSfull
• n number of observations
• k number of parameters from full model
• q number of parameters from restricted model

47
TESTING GRANGER CAUSALITY

• 5. If H0 rejected, then X causes Y.
• This technique can be used in investigating
  whether or not Y causes X.

48
Example of the Usage of Granger Test

• World Oil Price and Growth of US Economy
• Does the increase of world oil price influence
  the growth of US economy or does the growth of US
  economy effects the world oil price?
• James Hamilton did this study using the following
  model
• Zt a0 a1 Zt-1...amZt-mb1Xt-1 bmXt-met
• Zt ?Pt changes of world price of oil
• Xt log (GNPt/ GNPt-1)

49
World Oil Price and Growth of US Economy

• There are two causalities that need to be
  observed
• (i) H0 Growth of US Economy does not influence
  world oil price
• Full
• Zt a0 a1 Zt-1...amZt-mb1Xt-1 bmXt-met
• Restricted
• Zt a0 a1 Zt-1...amZt-m et

50
World Oil Price and Growth of US Economy

• (ii) H0 World oil price does not influence
  growth of US Economy
• Full
• Xt a0 a1 Xt-1 amXt-m b1Zt-1bmZt-m et
• Restricted
• Xt a0 a1 Xt-1 amXt-m et

51
World Oil Price and Growth of US Economy

• F Tests Results
• 1. Hypothesis that world oil price does not
  influence US economy is rejected. It means that
  the world oil price does influence US economy .
• 2. Hypothesis that US economy does not affect
  world oil price is not rejected. It means that
  the US economy does not have effect on world oil
  price.

52
World Oil Price and Growth of US Economy

• Summary of James Hamiltons Results

Null Hypothesis (H0) (I)F(4,86) (II)F(8,74)
I. Economic growth ??World Oil Price 0.58 0.71
II. World Oil Price??Economic growth 5.55 3.28
53
World Oil Price and Growth of US Economy

• Remark The first experiment used the data
  1949-1972 (95 observations) and m4 while the
  second experiment used data 1950-1972 (91
  observations) and m8.

54
Chicken vs. Egg

• This causality test is also can be used in
  explaining which comes first chicken or egg.
  More specifically, the test can be used in
  testing whether the existence of egg causes the
  existence of chicken or vise versa.
• Thurman and Fisher did this study using yearly
  data of chicken population and egg productions in
  the US from 1930 to1983
• The results
• 1. Egg causes the chicken.
• 2. There is no evidence that chicken causes egg.
  

55
Chicken vs. Egg

• Remark Hypothesis that egg has no effect on
  chicken population is rejected while the other
  hypothesis that chicken has no effect on egg is
  not rejected. Why?

56
GRANGER CAUSALITY

• We have to be aware of that Granger causality
  does not equal to what we usually mean by
  causality. For instance, even if x1 does not
  cause x2, it may still help to predict x2, and
  thus Granger-causes x2 if changes in x1 precedes
  that of x2 for some reason.
• A naive example is that we observe that a
  dragonfly flies much lower before a rain storm,
  due to the lower air pressure. We know that
  dragonflies do not cause a rain storm, but it
  does help to predict a rain storm, thus
  Granger-causes a rain storm.