? Chris Brooks 2002,陈磊 2004
6-1
Chapter 6
Multivariate models
? Chris Brooks 2002,陈磊 2004
6-2
1 Motivations
? All the models we have looked at thus far have been single
equations models of the form y = X? + u
? All of the variables contained in the X matrix are assumed to
be EXOGENOUS.由系统外因素 决定的变量
? y is an ENDOGENOUS variable,既影响系统 同时又被该系统
及其外部因素所影响的变量,
An example - the demand and supply of a good:
(1)
(2)
(3)
,= quantity of the good demanded / supplied
Pt = price,St = price of a substitute good
Tt = some variable embodying the state of technology
Q P S udt t t t? ? ? ?? ? ?
Q P T vst t t t? ? ? ?? ? ?
Q Qdt st?
Qdt Qst
? Chris Brooks 2002,陈磊 2004
6-3
? Assuming that the market always clears,and dropping the
time subscripts for simplicity
(4)
(5)
This is a simultaneous STRUCTURAL FORM of the model.
? The point is that price and quantity are determined
simultaneously (price affects quantity and quantity affects
price).
? P and Q are endogenous variables,while S and T are
exogenous.
? We can obtain REDUCED FORM equations corresponding to
(4) and (5) by solving equations (4) and (5) for P and for Q.
Simultaneous Equations Models,
The Structural Form
Q P S u? ? ? ?? ? ?
Q P T v? ? ? ?? ? ?
? Chris Brooks 2002,陈磊 2004
6-4
? Solving for Q,
(6)
? Solving for P,
(7)
? Rearranging (6),
(8)
Obtaining the Reduced Form
? ? ?? ? ?P S u? ? ? ?? ? ?P T v
Q S u Q T v
?
?
?
?
? ? ?
?
?
?
? ?? ? ? ? ? ? ?
( ) ( ) ( )? ? ? ? ? ?? ? ? ? ? ? ?P T S v u
P T S v u? ?? ? ? ? ? ? ??? ?? ? ?? ? ?? ? ? ?
? Chris Brooks 2002,陈磊 2004
6-5
? Multiplying (7) through by ??,
(9)
? (8) and (9) are the reduced form equations for P and Q.
Obtaining the Reduced Form
? ?? ?? ? ? ?? ?? ?Q S u Q T v? ? ? ? ? ? ?
( ) ( ) ( )? ? ?? ?? ?? ?? ? ?? ? ? ? ? ? ?Q T S u v
Q T S u v? ?? ? ? ? ? ? ???? ??? ? ??? ? ??? ? ? ?? ?
? Chris Brooks 2002,陈磊 2004
6-6
? But what would happen if we had estimated equations (4) and
(5),i.e,the structural form equations,separately using OLS?
? Both equations depend on P,One of the CLRM assumptions
was that E(X?u) = 0,where X is a matrix containing all the
variables on the RHS of the equation.
? It is clear from (8) that P is related to the errors in (4) and (5)
- i.e,it is stochastic.
? What would be the consequences for the OLS estimator,,if
we ignore the simultaneity?
2 Simultaneous Equations Bias
??
? Chris Brooks 2002,陈磊 2004
6-7
? Recall that and
? So that
? Taking expectations,
? If the X’s are non-stochastic,E(X?u) = 0,which would be
the case in a single equation system,so that,
which is the condition for unbiasedness.
Simultaneous Equations Bias
? ( ' ) '? ? ?X X X y1 y X u? ??
E E E X X X u
X X E X u
( ? ) ( ) (( ' ) ' )
( ' ) ( ' )
? ?
?
? ?
? ?
?
?
1
1
E ( ?)? ??
uXXX
uXXXXXXX
uXXXX
')'(
')'(')'(
)(')'(?
1
11
1
?
??
?
?
?
?
?
?
?
?
?
??
? Chris Brooks 2002,陈磊 2004
6-8
? But,..,if the equation is part of a system,then E(X?u) ? 0,in
general.
? Conclusion,Application of OLS to structural equations
which are part of a simultaneous system will lead to biased
coefficient estimates.
? Is the OLS estimator still consistent,even though it is biased?
? No - In fact the estimator is inconsistent as well.
? Hence it would not be possible to estimate equations (4) and
(5) validly using OLS.
Simultaneous Equations Bias
? Chris Brooks 2002,陈磊 2004
6-9
So What Can We Do?
? Taking equations (8) and (9),we can rewrite them as
(10)
(11)
? We CAN estimate equations (10) & (11) using OLS since all
the RHS variables are exogenous.
? But,.,we probably don’t care what the values of the ?
coefficients are; what we wanted were the original
parameters in the structural equations - ?,?,?,?,?,?.
3 Avoiding Simultaneous Equations Bias
P T S? ? ? ?? ? ? ?10 11 12 1
Q T S? ? ? ?? ? ? ?20 21 22 2
? Chris Brooks 2002,陈磊 2004
6-10
Can We Retrieve the Original Coefficients from the ?’s?
Short answer,sometimes.
? we sometimes encounter another problem,identification.*
? Consider the following demand and supply equations
Supply equation (12)
Demand equation (13)
We cannot tell which is which!
? Both equations are UNIDENTIFIED or UNDERIDENTIFIED.
? The problem is that we do not have enough information from
the equations to estimate 4 parameters,Notice that we would
not have had this problem with equations (4) and (5) since they
have different exogenous variables.
4 Identification of Simultaneous Equations
Q P? ?? ?
Q P? ?? ?
? Chris Brooks 2002,陈磊 2004
6-11
We could have three possible situations:
1,An equation is unidentified
· like (12) or (13)
· we cannot get the structural coefficients from the reduced
form estimates
2,An equation is exactly identified
· e.g,(4) or (5)
· can get unique structural form coefficient estimates
3,An equation is over-identified
· Example given later
· More than one set of structural coefficients could be
obtained from the reduced form.
What Determines whether an
Equation is Identified or not?
? Chris Brooks 2002,陈磊 2004
6-12
? How do we tell if an equation is identified or not?
? There are two conditions we could look at:
- The order 阶 condition - is a necessary but not sufficient
condition for an equation to be identified.
- The rank 秩 condition - is a necessary and sufficient condition
for identification,
在 G个内生变量,G个方程的联立方程组模型中，某一方程是
可识别的，当且仅当该方程没有包含的变量在其他方程中对
应系数组成的矩阵的秩为 G -1。
? 对于相对简单的方程系统，这两个规则将得到同样的结论。
? 事实上，大多数经济和金融方程系统都是过度识别的。
What Determines whether an
Equation is Identified or not?
? Chris Brooks 2002,陈磊 2004
6-13
Statement of the Order Condition
? Let G denote the number of structural equations.
? An equation is just identified if the number of variables
excluded from an equation is G-1.
? If more than G-1 are absent,it is over-identified.
? If less than G-1 are absent,it is not identified.
Example
? the Y’s are endogenous,while the X’s are exogenous.
Determine whether each equation is over-,under-,or just-
identified.
(14)-(16)
Statement of the order condition
Y Y Y X X u
Y Y X u
Y Y u
1 0 1 2 3 3 4 1 5 2 1
2 0 1 3 2 1 2
3 0 1 2 3
? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ?
? ? ?
? ?
? Chris Brooks 2002,陈磊 2004
6-14
Solution
G = 3;
If # excluded variables = 2,the eqn is just identified
If # excluded variables > 2,the eqn is over-identified
If # excluded variables < 2,the eqn is not identified
Equation 14,Not identified
Equation 15,Just identified
Equation 16,Over-identified
如果 模型中每个结构方程都是可识别的，则称结构型联立方程
组模型是可识别的。
Example of the order condition
? Chris Brooks 2002,陈磊 2004
6-15
5 外生性的定义
Leamer（ 1985）,p310
变量 X对变量 Y是外生的，如果变量 Y关于 X的条件分布不随产
生 X的过程的变化而改变。
外生性的两种形式：
? 前定变量：与方程中的当前和未来误差项独立。
? 严格外生变量：与方程中任何时期的误差项独立。
前定变量的通常定义：包括外生变量和滞后的内生变量
? Chris Brooks 2002,陈磊 2004
6-16
? How do we tell whether variables really need to be treated as
endogenous or not?
? Consider again equations (14)-(16),Equation (14) contains Y2
and Y3 - but do we really need equations for them?
? We can formally test this using a Hausman test as follows:
1,Obtain the reduced form equations corresponding to (14)-
(16),The reduced forms turn out to be:
(17)-(19)
Estimate the reduced form equations (17)-(19) using OLS,and
obtain the fitted values,
5 Tests for Exogeneity
Y X X v
Y X v
Y X v
1 10 11 1 12 2 1
2 20 21 1 2
3 30 31 1 3
? ? ? ?
? ? ?
? ? ?
? ? ?
? ?
? ?
?,?,?Y Y Y1 2 3
? Chris Brooks 2002,陈磊 2004
6-17
2,Run the regression corresponding to equation (14).
3,Run the regression (14) again,but now also including
the fitted values as additional regressors:
(20)
4,Use an F-test to test the joint restriction that ?2 = 0,and
?3 = 0,If the null hypothesis is rejected,Y2 and Y3 should
be treated as endogenous.
Tests for Exogeneity
?,?Y Y2 3
11331222514332101 ?? uYYXXYYY ???????? ???????
? Chris Brooks 2002,陈磊 2004
6-18
? Consider the following system of equations:
(21-23)
? Assume that the error terms are not correlated with each other.
Can we estimate the equations individually using OLS?
? Equation 21,Contains no endogenous variables,so X1 and X2
are not correlated with u1,So we can use OLS on (21).
? Equation 22,Contains endogenous Y1 together with exogenous
X1 and X2,We can use OLS on (22) if all the RHS variables in
(22) are uncorrelated with that equation’s error term,In fact,Y1
is not correlated with u2 because there is no Y2 term in equation
(21),So we can use OLS on (22).
6 Recursive Systems
Y X X u
Y Y X X u
Y Y Y X X u
1 10 11 1 12 2 1
2 20 21 1 21 1 22 2 2
3 30 31 1 32 2 31 1 32 2 3
? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? Chris Brooks 2002,陈磊 2004
6-19
? Equation 23,Contains both Y1 and Y2; we require these to be
uncorrelated with u3,By similar arguments to the above,
equations (21) and (22) do not contain Y3,so we can use OLS
on (23).
? This is known as a RECURSIVE or TRIANGULAR system.
We do not have a simultaneity problem here.
? But in practice not many systems of equations will be
recursive...
Recursive Systems
? Chris Brooks 2002,陈磊 2004
6-20
Indirect Least Squares (ILS)
? Cannot use OLS on structural equations,but we can validly
apply it to the reduced form equations.
? If the system is just identified,ILS involves estimating the
reduced form equations using OLS,and then using them to
substitute back to obtain the structural parameters.
? However,ILS is not used much because
1,Solving back to get the structural parameters can be
tedious.
2,Most simultaneous equations systems are over-identified.
7 Estimation procedures for Systems
? Chris Brooks 2002,陈磊 2004
6-21
? In fact,we can use this technique for just-identified and
over-identified systems.
? Two stage least squares (2SLS or TSLS) is done in two
stages:
Stage 1:
? Obtain and estimate the reduced form equations using OLS.
Save the fitted values for the dependent variables.
Stage 2:
? Estimate the structural equations,but replace any RHS
endogenous variables with their stage 1 fitted values.
Estimation of Systems
Using Two-Stage Least Squares
? Chris Brooks 2002,陈磊 2004
6-22
Example,Say equations (14)-(16) are required.
Stage 1:
? Estimate the reduced form equations (17)-(19) individually by
OLS and obtain the fitted values,.
Stage 2:
? Replace the RHS endogenous variables with their stage 1
estimated values:
(24)-(26)
? Now and will not be correlated with u1,will not be
correlated with u2,and will not be correlated with u3,
Estimation of Systems
Using Two-Stage Least Squares
?,?,?Y Y Y1 2 3
Y Y Y X X u
Y Y X u
Y Y u
1 0 1 2 3 3 4 1 5 2 1
2 0 1 3 2 1 2
3 0 1 2 3
? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ?
? ? ?
? ?
? ?
?
?
?Y2
?Y3?Y3?Y2
? Chris Brooks 2002,陈磊 2004
6-23
? TSLS是比较经济, 易用的方法 。
? 如果在第一阶段估计时所得到的 R2非常高, 那么古典 OLS估计量与
TSLS估计量将非常接近；如果在第一阶段估计时所得到的 R2非常低,
TSLS估计量将没有太大的实际意义 。
? TSLS估计量是有偏估计量, 但却是一致估计量 。
? It is still of concern in the context of simultaneous systems
whether the CLRM assumptions are supported by the data.
? If the disturbances in the structural equations are
autocorrelated,the 2SLS estimator is not even consistent.
? The standard error estimates also need to be modified
compared with their OLS counterparts,but once this has
been done,we can use the usual t- and F-tests to test
hypotheses about the structural form coefficients.
Estimation of Systems
Using Two-Stage Least Squares
? Chris Brooks 2002,陈磊 2004
6-24
? Recall that the reason we cannot use OLS directly on the
structural equations is that the endogenous variables are
correlated with the errors.
? One solution to this would be not to use Y2 or Y3,but rather
to use some other variables instead.
? We want these other variables to be (highly) correlated with
Y2 and Y3,but not correlated with the errors - they are called
INSTRUMENTS.
? Say we found suitable instruments for Y2 and Y3,z2 and z3
respectively,We do not use the instruments directly,but run
regressions of the form
(27) & (28)
Instrumental Variables
Y z
Y z
2 1 2 2 1
3 3 4 3 2
? ? ?
? ? ?
? ? ?
? ? ?
? Chris Brooks 2002,陈磊 2004
6-25
? Obtain the fitted values from (27) & (28),and,and
replace Y2 and Y3 with these in the structural equation.
? It is typical to use more than one instrument per endogenous
variable.
? If the instruments are the variables in the reduced form
equations,then IV is equivalent to 2SLS.
Instrumental Variables
?Y2 ?Y3
? Chris Brooks 2002,陈磊 2004
6-26
What Happens if We Use IV / 2SLS Unnecessarily?
? The coefficient estimates will still be consistent,but will be
inefficient compared to those that just used OLS directly.
The Problem With IV
? What are the instruments?
Solution,2SLS is easier.
Other Estimation Techniques
1,3SLS - allows for non-zero covariances between the error terms.
2,LIML - estimating reduced form equations by maximum likelihood
3,FIML - estimating all the equations simultaneously using maximum
likelihood
Instrumental Variables
? Chris Brooks 2002,陈磊 2004
6-27
? A natural generalisation of autoregressive models popularised
by Sims.
? A VAR is in a sense a systems regression model i.e,there is
more than one dependent variable.
? Simplest case is a bivariate VAR
where uit is an iid disturbance term with E(uit)=0,i=1,2; E(u1t u2t)=0.
? The analysis could be extended to a VAR(g) model,or so that
there are g variables and g equations.
8 Vector Autoregressive Models
y y y y y u
y y y y y u
t t k t k t k t k t
t t k t k t k t k t
1 10 11 1 1 1 1 11 2 1 1 2 1
2 20 21 2 1 2 2 21 1 1 2 1 2
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
.,,,,,
.,,,,,
? Chris Brooks 2002,陈磊 2004
6-28
? One important feature of VARs is the compactness with
which we can write the notation,For example,consider the
case from above where k=1.
? We can write this as
or
yt = ?0 + ?1 yt-1 + ut
g?1 g?1 g?g g?1 g?1
Vector Autoregressive Models,
Notation and Concepts
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
y
y
y
y
u
u
t
t
t
t
t
t
1
2
10
20
11 11
21 21
1 1
2 1
1
2
?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
?
?
?
? ?
? ?
? Chris Brooks 2002,陈磊 2004
6-29
? VAR模型还具有灵活性和易于一般化的重要特点,
? 例如, 模型可以扩展到包含移动平均误差项, 即 VARMA。
? This model can be extended to the case where there are k
lags of each variable in each equation:
yt = ?0 + ?1 yt-1 + ?2 yt-2 +...+ ?k yt-k + ut
g?1 g?1 g?g g?1 g?g g?1 g?g g?1 g?1
? We can also extend this to the case where the model includes
first difference terms and cointegrating relationships (a
VECM).
Vector Autoregressive Models,
Notation and Concepts
? Chris Brooks 2002,陈磊 2004
6-30
? Advantages of VAR Modelling
- Do not need to specify which variables are
endogenous or exogenous - all are endogenous
- Allows the value of a variable to depend on more
than just its own lags or combinations of white noise
terms,so more general than ARMA modelling
- Provided that there are no contemporaneous terms
on the right hand side of the equations,can simply
use OLS separately on each equation,因为方程右边的
变量都是前定变量 。
- Forecasts are often better than,traditional
structural” models.
VAR Models Compared with
Structural Equations Models
? Chris Brooks 2002,陈磊 2004
6-31
? Problems with VAR’s
- VAR’s are a-theoretical (as are ARMA models)。
VAR模型较少用于理论分析和政策建议 。
- How do you decide the appropriate lag length?
- So many parameters! If we have g equations for g
variables and we have k lags of each of the variables
in each equation,we have to estimate (g+kg2)
parameters,e.g,g=3,k=3,parameters = 30
- Do we need to ensure all components of the VAR
are stationary?
- How do we interpret the coefficients?
VAR Models Compared with
Structural Equations Models
? Chris Brooks 2002,陈磊 2004
6-32
Choosing the Optimal Lag Length
Cross-Equation Restrictions
? In the spirit of (unrestricted) VAR modelling,each equation
should have the same lag length
? Suppose that a bivariate VAR(8) estimated using quarterly
data has 8 lags of the two variables in each equation,and we
want to examine a restriction that the coefficients on lags 5
through 8 are jointly zero,This can be done using a
likelihood ratio test
? Denote the variance-covariance matrix of residuals (given by
E ),as, The likelihood ratio test for this joint hypothesis
is given by
??uu???
? ?urTLR ???? ?l o g?l o g
? Chris Brooks 2002,陈磊 2004
6-33
Choosing the Optimal Lag Length
where is the variance-covariance matrix of the residuals for
the restricted model (with 4 lags),is the variance-covariance
matrix of residuals for the unrestricted VAR (with 8 lags),and T
is the sample size.
? The test statistic is asymptotically distributed as a ?2 with
degrees of freedom equal to the total number of restrictions,In
the VAR case above,we are restricting 4 lags of two variables in
each of the two equations = a total of 4 *2 * 2 = 16 restrictions,
? In the general case where we have a VAR with g equations,
and we want to impose the restriction that the last q lags have
zero coefficients,there would be g2q restrictions altogether
? Disadvantages,Conducting the LR test is cumbersome and requires a
normality assumption for the disturbances.
r??
u??
? Chris Brooks 2002,陈磊 2004
6-34Information Criteria for VAR Lag
Length Selection
? Multivariate versions of the information criteria are required.
These can be defined as:
where all notation is as above and k? is the total number of
regressors in all equations,which will be equal to g2k + g for g
equations,each with k lags of the g variables,plus a constant
term in each equation,The values of the information criteria
are constructed for 0,1,… lags (up to some pre-specified
maximum ).k
l n ( l n ( T ))
2?
ln
l n ( T )?ln
/2?ln
T
k
M H QI C
T
k
M S BI C
TkM AI C
?
???
?
???
???? ??? ???
?
? Chris Brooks 2002,陈磊 2004
6-35Does the VAR Include
Contemporaneous Terms?
? So far,we have assumed the VAR is of the form
? But what if the equations had a contemporaneous feedback term?
? We can write this as
? This VAR is in primitive / structural form.
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
y y y y u
y y y y u
t t t t t
t t t t t
1 10 11 1 1 11 2 1 12 2 1
2 20 21 2 1 21 1 1 22 1 2
? ? ? ? ?
? ? ? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
y
y
y
y
y
y
u
u
t
t
t
t
t
t
t
t
1
2
10
20
11 11
21 21
1 1
2 1
12
22
2
1
1
20
0?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
?
?
?
? ?
? ?
?
?
? Chris Brooks 2002,陈磊 2004
6-36
Primitive versus Standard Form VARs
? We can take the contemporaneous terms over to the LHS
and write
or
B yt = ?0 + ?1 yt-1 + ut
? We can then pre-multiply both sides by B-1 to give
yt = B-1?0 + B-1?1 yt-1 + B-1ut
or
yt = A0 + A1 yt-1 + et
? This is known as a standard form VAR,which we can
estimate using OLS.
1
122
12 1
2
10
20
11 11
21 21
1 1
2 1
1
2?
??
??
?
??
?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
??
? ?
?
? ?
? ?
y
y
y
y
u
u
t
t
t
t
t
t
? Chris Brooks 2002,陈磊 2004
6-37
VAR模型的识别
? 粗略地讲，结构型 VAR模型的识别问题是指，能否从一个
简约模型的估计值反导出原来的结构模型的系数。
? 结构型 VAR模型是不可识别的，因为两个方程的等号右边
具有相同的前定变量
? 为了解决这个问题，需要加入一定的约束条件。即同期项的
一个系数 α12或 α22须设为 0，使距阵 B为三角形。
? 最好是依据经济理论加入约束条件
? Chris Brooks 2002,陈磊 2004
6-38
Block Significance and Causality Tests
? It is likely that,when a VAR includes many lags of variables,
it will be difficult to see which sets of variables have significant
effects on each dependent variable and which do not,For
illustration,consider the following bivariate VAR(3):
?
? We might be interested in testing the following hypotheses,
and their implied restrictions on the parameters:
tttttttt
tttttttt
uyyyyyyy
uyyyyyyy
2322231212222212112221121202
1321231112212211112121111101
????????
????????
??????
??????
???????
???????
H y p o t h e s i s I m p l i e d R e s t r i c t i o n
1, L a g s o f y 1 t d o n o t e x p l a i n c u r r e n t y 2 t ? 21 = 0 a n d ? 21 = 0 a n d ? 21 = 0
2, L a g s o f y 1 t d o n o t e x p l a i n c u r r e n t y 1 t ? 11 = 0 a n d ? 11 = 0 a n d ? 11 = 0
3, L a g s o f y 2 t d o n o t e x p l a i n c u r r e n t y 1 t ? 12 = 0 a n d ? 12 = 0 a n d ? 12 = 0
4, L a g s o f y 2 t d o n o t e x p l a i n c u r r e n t y 2 t ? 22 = 0 a n d ? 22 = 0 a n d ? 22 = 0
? Chris Brooks 2002,陈磊 2004
6-39
Block Significance and Causality Tests
? Each of these four joint hypotheses can be tested within the F-test
framework.
? These tests could also be referred to as Granger causality tests.
? Granger causality tests seek to answer questions such as,Do
changes in y1 cause changes in y2?” If y1 causes y2,lags of y1 should
be significant in the equation for y2,If this is the case,we say that
y1,Granger-causes” y2,
? If y2 causes y1,lags of y2 should be significant in the equation for
y1,If both sets of lags are significant,there is,bi-directional
causality”,
? If y2 causes y1,but y1 does not causes y2,then y2 is strong
exogenous (in the equation for y1).
? If neither y2 causes y1,nor y1 causes y2,then y1 y2 are independent
? 此处因果性并不意味着一个变量的变动引起另一个变量的变动
? Chris Brooks 2002,陈磊 2004
6-40
Impulse Responses
? VAR models are often difficult to interpret,one solution is to
construct the impulse responses and variance decompositions.
? Impulse responses trace out the responsiveness of the
dependent variables in the VAR to shocks to the error
term,A unit shock is applied to each variable and its effects
are noted.
? Consider for example a simple bivariate VAR(1):
? A change in u1t will immediately change y1,It will change y2
and also y1 during the next period.
? We can examine how long and to what degree a shock to a
given equation has on all of the variables in the system.
? eg,p341
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
? Chris Brooks 2002,陈磊 2004
6-41
多变量 VAR模型
也可改写为
这里 yt 是一个 k维内生变量向量, ?t 是协方差矩阵为 ? 的扰动向量 。
tptptt yAyAy ????? ?? ?11
ttpp yLALAI ????? )( 1 ?
一般 VAR模型的脉冲响应函数
假如 VAR(p)可逆, 我们可以得到 VMA(∞) 的表达式,
t
t
p
pt
LLI
LALAIy
???
?
)(
)(
2
21
1
1
?
?
????
???? ?
? Chris Brooks 2002,陈磊 2004
6-42
VMA表达式的系数可按下面的方式给出,VAR的系数 A和 VMA的系数
必须满足下面关系：
其中, 。 关于 的条件递归定义了 VMA系数：
?
ILCLCI ???? ?221
021 ??? ?CC
11 A??
2112 AA ?? ??
?
ILLILALAI pP ??????? ))(( 2211 ?? ??
从而可知 VMA的系数可以由 VAR的系数递归得到 。
qC
pqPqqq AAA ??? ???? ????,..2211
一般 VAR模型的脉冲响应函数
? Chris Brooks 2002,陈磊 2004
6-43
考虑 VMA(∞) 的表达式
设, y 的第 i个变量 可以写成：
其中 k 是变量个数 。
仅考虑 2个变量 (k = 2)的情形：
ity
)( 3,32,21,1,0
1
???????? ???
? jtijjtijjtijjtij
k
jit
y ????????
现在假定在基期给 一个单位的脉冲, 即：
–2 –1 0 1 2 3 4 5 ……… t
1y
??
? ??
e lse
t
t,0
0,1
1?
tt LLIy ???? )( 2210 ?????
)(,ijqq ?? ?
????
?
?
???
?
???
?
???
??
???
?
???
?
???
?
???
??
???
?
???
?
???
?
???
??
???
?
???
?
?
?
?
?
2,2
2,1
22,221,2
12,211,2
1,2
1,1
22,121,1
12,111,1
,2
,1
22,021,0
12,011,0
2
1
t
t
t
t
t
t
t
t
y
y
?
?
??
??
?
?
??
??
?
?
??
??
tt ??,02?
110??
? Chris Brooks 2002,陈磊 2004
6-44
由 的脉冲引起的 的响应函数：
1y ?21,221,121,0,,???
21,44,2
21,33,2
21,22,2
21,11,2
21,00,2
,4
,3
,2
,1
,0
?
?
?
?
?
??
??
??
??
??
yt
yt
yt
yt
yt
? ?
由上述推导可知由 的脉冲引起的 的响应函数序列是由 VMA(∞) 中
系数矩阵 第 2行, 第 1列的元素组成,q = 1,2,… 。 因此, 一般
地, 由 的脉冲引起的 的响应函数可以求出如下：
其中, 代表着对第 j个变量的单位冲击引起第 i个变量的第 q期滞后反映 。
jy
iy
??,,,,,,4,3,2,1,0 ijijijijij ?????
)(,ijqq ?? ?
1y
2y
2y
ijq,?
? Chris Brooks 2002,陈磊 2004
6-45
Variance Decompositions
? Variance decompositions offer a slightly different method
of examining VAR dynamics,They give the proportion of
the movements in the dependent variables that are due to
their,own” shocks,versus shocks to the other variables.
?
? This is done by determining how much of the s-step ahead
forecast error variance for each variable is explained by
innovations to each explanatory variable (s = 1,2,…).
? The variance decomposition gives information about the
relative importance of each shock to the variables in the
VAR.
? 脉冲响应函数和方差分解常常提供一定程度上的相似信息,
? Chris Brooks 2002,陈磊 2004
6-46
The Ordering of the Variables
? But for calculating impulse responses and variance
decompositions,the ordering of the variables is important.
? The main reason for this is that above,we assumed that the
VAR error terms were statistically independent of one another,
? This is generally not true,however,The error terms will
typically be correlated to some degree,Therefore,the notion of
examining the effect of the innovations separately has little
meaning,since they have a common component,
? What is done is to,orthogonalise” the innovations.
? In the bivariate VAR,this problem would be approached by
attributing all of the effect of the common component to the first
of the two variables in the VAR,
? In the general case where there are more variables,the situation
is more complex but the interpretation is the same,
? Chris Brooks 2002,陈磊 2004
6-47
由以上讨论可知, 对第 i个变量的冲击不仅直接影响第 i个变量, 并且通
过 VAR模型的动态 （ 滞后 ） 结构传导给所有的其它内生变量 。 脉冲响应函
数描绘了在一个扰动项上加上一次性的一个冲击 (one-time shock),对于内
生变量的当前值和未来值所带来的影响 。
假如扰动项 同期不相关, 那么脉冲响应的解释就很直接 。 第 i个扰动
项 就只对第 i个内生变量有一个冲击 。 然而, 扰动之间大都是相关的, 可
以描述为它们有一个与被指定变量不相关的公共成分 。 为了解释脉冲, 最
常用的方法是引进一个转换矩阵, 使扰动项变成不相关：
这里 D是对角协方差矩阵 。 如下面所说明的, EViews提供多种关于 P的选择
方法 。
t?
ti,?
),0(~ DNPv tt ??
? Chris Brooks 2002,陈磊 2004
6-48
乔利斯基 （ Cholesky） 分解：
对于任意实对称正定矩阵 ?,存在惟一一个主对角线元素为 1的下三角形
矩阵 A和惟一一个主对角线元素为正的对角矩阵 D使得
（ 20.12）
利用这一矩阵 A可以构造一个 向量, 构造方法为,
则
AAD ???
)1( ?T
)( ttE ?? ???
tv tt Av ?1??
DAAADAAAAEAvvE tttt ??????????? ?????? 111111 ][][][][])[(][)( ??
? Chris Brooks 2002,陈磊 2004
6-49An Example,The Interaction between
Property Returns and the Macroeconomy.
? Brooks and Tsolacos (1999) employ a VAR methodology for
investigating the interaction between the UK property market and
various macroeconomic variables,
? Monthly data are used for 1986 to 1997,
? It is assumed that stock returns are related to macroeconomic and
business conditions.
? The variables included in the VAR are
– FTSE Property Total Return Index (with general stock market
effects removed)
– The rate of unemployment
– Nominal interest rates
– The spread between long and short term interest rates
– Unanticipated inflation
– The dividend yield,
The property index and unemployment are I(1) and hence are
differenced,
? Chris Brooks 2002,陈磊 2004
6-50Marginal Significance Levels associated with Joint F-
tests that all 14 Lags have not Explanatory Power
for that particular Equation in the VAR
? Multivariate AIC selected 14 lags of each variable in the VAR
L a g s o f V a r i a b l e
D e p e n d e n t v a r i a b l e SI R D I V Y SP R E A D U N E M U N I N F L PR O PR E S
SI R 0, 0 0 0 0 0, 0 0 9 1 0, 0 2 4 2 0, 0 3 2 7 0, 2 1 2 6 0, 0 0 0 0
D I V Y 0, 5 0 2 5 0, 0 0 0 0 0, 6 2 1 2 0, 4 2 1 7 0, 5 6 5 4 0, 4 0 3 3
SP R E A D 0, 2 7 7 9 0, 1 3 2 8 0, 0 0 0 0 0, 4 3 7 2 0, 6 5 6 3 0, 0 0 0 7
U N E M 0, 3 4 1 0 0, 3 0 2 6 0, 1 1 5 1 0, 0 0 0 0 0, 0 7 5 8 0, 2 7 6 5
U N I N F L 0, 3 0 5 7 0, 5 1 4 6 0, 3 4 2 0 0, 4 7 9 3 0, 0 0 0 4 0, 3 8 8 5
PR O PR E S 0, 5 5 3 7 0, 1 6 1 4 0, 5 5 3 7 0, 8 9 2 2 0, 7 2 2 2 0, 0 0 0 0
? Chris Brooks 2002,陈磊 2004
6-51Variance Decompositions for the
Property Sector Index Residuals
? Ordering for Variance Decompositions and Impulse Responses:
– Order I,PROPRES,DIVY,UNINFL,UNEM,SPREAD,SIR
– Order II,SIR,SPREAD,UNEM,UNINFL,DIVY,PROPRES.
E xpl a i n e d by i nn o v a t i o n s i n
S IR D IV Y S P R E A D U N E M U N IN F L P R O P R E S
M o n t h s a h e a d I II I II I II I II I II I II
1 0,0 0,8 0,0 38,2 0,0 9,1 0,0 0,7 0,0 0,2 100, 0 51,0
2 0,2 0,8 0,2 35,1 0,2 12,3 0,4 1,4 1,6 2,9 97,5 47,5
3 3,8 2,5 0,4 29,4 0,2 17,8 1,0 1,5 2,3 3,0 92,3 45,8
4 3,7 2,1 5,3 22,3 1,4 18,5 1,6 1,1 4,8 4,4 83,3 51,5
12 2,8 3,1 15,5 8,7 15,3 19,5 3,3 5,1 17,0 13,5 46,1 50,0
24 8,2 6,3 6,8 3,9 38,0 36,2 5,5 14,7 18,1 16,9 23,4 22,0
? Chris Brooks 2002,陈磊 2004
6-52
Impulse Responses and Standard Error Bands
for Innovations in Dividend Yield
Innov at ions in D ividend Y ie lds
-0.06
-0.04
-0.02
0
0, 0 2
0, 0 4
0, 0 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Ste ps Ahea d
? Chris Brooks 2002,陈磊 2004
6-53
? P350 图 6-1
Impulse Responses and Standard Error Bands
for Innovations in unexpected inflation
? Chris Brooks 2002,陈磊 2004
6-54
Innov at ions in t he T- Bil l Y ie ld
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
S t eps Ah ea d
Impulse Responses and Standard Error Bands
for Innovations in the Treasury Bill Yield
6-1
Chapter 6
Multivariate models
? Chris Brooks 2002,陈磊 2004
6-2
1 Motivations
? All the models we have looked at thus far have been single
equations models of the form y = X? + u
? All of the variables contained in the X matrix are assumed to
be EXOGENOUS.由系统外因素 决定的变量
? y is an ENDOGENOUS variable,既影响系统 同时又被该系统
及其外部因素所影响的变量,
An example - the demand and supply of a good:
(1)
(2)
(3)
,= quantity of the good demanded / supplied
Pt = price,St = price of a substitute good
Tt = some variable embodying the state of technology
Q P S udt t t t? ? ? ?? ? ?
Q P T vst t t t? ? ? ?? ? ?
Q Qdt st?
Qdt Qst
? Chris Brooks 2002,陈磊 2004
6-3
? Assuming that the market always clears,and dropping the
time subscripts for simplicity
(4)
(5)
This is a simultaneous STRUCTURAL FORM of the model.
? The point is that price and quantity are determined
simultaneously (price affects quantity and quantity affects
price).
? P and Q are endogenous variables,while S and T are
exogenous.
? We can obtain REDUCED FORM equations corresponding to
(4) and (5) by solving equations (4) and (5) for P and for Q.
Simultaneous Equations Models,
The Structural Form
Q P S u? ? ? ?? ? ?
Q P T v? ? ? ?? ? ?
? Chris Brooks 2002,陈磊 2004
6-4
? Solving for Q,
(6)
? Solving for P,
(7)
? Rearranging (6),
(8)
Obtaining the Reduced Form
? ? ?? ? ?P S u? ? ? ?? ? ?P T v
Q S u Q T v
?
?
?
?
? ? ?
?
?
?
? ?? ? ? ? ? ? ?
( ) ( ) ( )? ? ? ? ? ?? ? ? ? ? ? ?P T S v u
P T S v u? ?? ? ? ? ? ? ??? ?? ? ?? ? ?? ? ? ?
? Chris Brooks 2002,陈磊 2004
6-5
? Multiplying (7) through by ??,
(9)
? (8) and (9) are the reduced form equations for P and Q.
Obtaining the Reduced Form
? ?? ?? ? ? ?? ?? ?Q S u Q T v? ? ? ? ? ? ?
( ) ( ) ( )? ? ?? ?? ?? ?? ? ?? ? ? ? ? ? ?Q T S u v
Q T S u v? ?? ? ? ? ? ? ???? ??? ? ??? ? ??? ? ? ?? ?
? Chris Brooks 2002,陈磊 2004
6-6
? But what would happen if we had estimated equations (4) and
(5),i.e,the structural form equations,separately using OLS?
? Both equations depend on P,One of the CLRM assumptions
was that E(X?u) = 0,where X is a matrix containing all the
variables on the RHS of the equation.
? It is clear from (8) that P is related to the errors in (4) and (5)
- i.e,it is stochastic.
? What would be the consequences for the OLS estimator,,if
we ignore the simultaneity?
2 Simultaneous Equations Bias
??
? Chris Brooks 2002,陈磊 2004
6-7
? Recall that and
? So that
? Taking expectations,
? If the X’s are non-stochastic,E(X?u) = 0,which would be
the case in a single equation system,so that,
which is the condition for unbiasedness.
Simultaneous Equations Bias
? ( ' ) '? ? ?X X X y1 y X u? ??
E E E X X X u
X X E X u
( ? ) ( ) (( ' ) ' )
( ' ) ( ' )
? ?
?
? ?
? ?
?
?
1
1
E ( ?)? ??
uXXX
uXXXXXXX
uXXXX
')'(
')'(')'(
)(')'(?
1
11
1
?
??
?
?
?
?
?
?
?
?
?
??
? Chris Brooks 2002,陈磊 2004
6-8
? But,..,if the equation is part of a system,then E(X?u) ? 0,in
general.
? Conclusion,Application of OLS to structural equations
which are part of a simultaneous system will lead to biased
coefficient estimates.
? Is the OLS estimator still consistent,even though it is biased?
? No - In fact the estimator is inconsistent as well.
? Hence it would not be possible to estimate equations (4) and
(5) validly using OLS.
Simultaneous Equations Bias
? Chris Brooks 2002,陈磊 2004
6-9
So What Can We Do?
? Taking equations (8) and (9),we can rewrite them as
(10)
(11)
? We CAN estimate equations (10) & (11) using OLS since all
the RHS variables are exogenous.
? But,.,we probably don’t care what the values of the ?
coefficients are; what we wanted were the original
parameters in the structural equations - ?,?,?,?,?,?.
3 Avoiding Simultaneous Equations Bias
P T S? ? ? ?? ? ? ?10 11 12 1
Q T S? ? ? ?? ? ? ?20 21 22 2
? Chris Brooks 2002,陈磊 2004
6-10
Can We Retrieve the Original Coefficients from the ?’s?
Short answer,sometimes.
? we sometimes encounter another problem,identification.*
? Consider the following demand and supply equations
Supply equation (12)
Demand equation (13)
We cannot tell which is which!
? Both equations are UNIDENTIFIED or UNDERIDENTIFIED.
? The problem is that we do not have enough information from
the equations to estimate 4 parameters,Notice that we would
not have had this problem with equations (4) and (5) since they
have different exogenous variables.
4 Identification of Simultaneous Equations
Q P? ?? ?
Q P? ?? ?
? Chris Brooks 2002,陈磊 2004
6-11
We could have three possible situations:
1,An equation is unidentified
· like (12) or (13)
· we cannot get the structural coefficients from the reduced
form estimates
2,An equation is exactly identified
· e.g,(4) or (5)
· can get unique structural form coefficient estimates
3,An equation is over-identified
· Example given later
· More than one set of structural coefficients could be
obtained from the reduced form.
What Determines whether an
Equation is Identified or not?
? Chris Brooks 2002,陈磊 2004
6-12
? How do we tell if an equation is identified or not?
? There are two conditions we could look at:
- The order 阶 condition - is a necessary but not sufficient
condition for an equation to be identified.
- The rank 秩 condition - is a necessary and sufficient condition
for identification,
在 G个内生变量,G个方程的联立方程组模型中，某一方程是
可识别的，当且仅当该方程没有包含的变量在其他方程中对
应系数组成的矩阵的秩为 G -1。
? 对于相对简单的方程系统，这两个规则将得到同样的结论。
? 事实上，大多数经济和金融方程系统都是过度识别的。
What Determines whether an
Equation is Identified or not?
? Chris Brooks 2002,陈磊 2004
6-13
Statement of the Order Condition
? Let G denote the number of structural equations.
? An equation is just identified if the number of variables
excluded from an equation is G-1.
? If more than G-1 are absent,it is over-identified.
? If less than G-1 are absent,it is not identified.
Example
? the Y’s are endogenous,while the X’s are exogenous.
Determine whether each equation is over-,under-,or just-
identified.
(14)-(16)
Statement of the order condition
Y Y Y X X u
Y Y X u
Y Y u
1 0 1 2 3 3 4 1 5 2 1
2 0 1 3 2 1 2
3 0 1 2 3
? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ?
? ? ?
? ?
? Chris Brooks 2002,陈磊 2004
6-14
Solution
G = 3;
If # excluded variables = 2,the eqn is just identified
If # excluded variables > 2,the eqn is over-identified
If # excluded variables < 2,the eqn is not identified
Equation 14,Not identified
Equation 15,Just identified
Equation 16,Over-identified
如果 模型中每个结构方程都是可识别的，则称结构型联立方程
组模型是可识别的。
Example of the order condition
? Chris Brooks 2002,陈磊 2004
6-15
5 外生性的定义
Leamer（ 1985）,p310
变量 X对变量 Y是外生的，如果变量 Y关于 X的条件分布不随产
生 X的过程的变化而改变。
外生性的两种形式：
? 前定变量：与方程中的当前和未来误差项独立。
? 严格外生变量：与方程中任何时期的误差项独立。
前定变量的通常定义：包括外生变量和滞后的内生变量
? Chris Brooks 2002,陈磊 2004
6-16
? How do we tell whether variables really need to be treated as
endogenous or not?
? Consider again equations (14)-(16),Equation (14) contains Y2
and Y3 - but do we really need equations for them?
? We can formally test this using a Hausman test as follows:
1,Obtain the reduced form equations corresponding to (14)-
(16),The reduced forms turn out to be:
(17)-(19)
Estimate the reduced form equations (17)-(19) using OLS,and
obtain the fitted values,
5 Tests for Exogeneity
Y X X v
Y X v
Y X v
1 10 11 1 12 2 1
2 20 21 1 2
3 30 31 1 3
? ? ? ?
? ? ?
? ? ?
? ? ?
? ?
? ?
?,?,?Y Y Y1 2 3
? Chris Brooks 2002,陈磊 2004
6-17
2,Run the regression corresponding to equation (14).
3,Run the regression (14) again,but now also including
the fitted values as additional regressors:
(20)
4,Use an F-test to test the joint restriction that ?2 = 0,and
?3 = 0,If the null hypothesis is rejected,Y2 and Y3 should
be treated as endogenous.
Tests for Exogeneity
?,?Y Y2 3
11331222514332101 ?? uYYXXYYY ???????? ???????
? Chris Brooks 2002,陈磊 2004
6-18
? Consider the following system of equations:
(21-23)
? Assume that the error terms are not correlated with each other.
Can we estimate the equations individually using OLS?
? Equation 21,Contains no endogenous variables,so X1 and X2
are not correlated with u1,So we can use OLS on (21).
? Equation 22,Contains endogenous Y1 together with exogenous
X1 and X2,We can use OLS on (22) if all the RHS variables in
(22) are uncorrelated with that equation’s error term,In fact,Y1
is not correlated with u2 because there is no Y2 term in equation
(21),So we can use OLS on (22).
6 Recursive Systems
Y X X u
Y Y X X u
Y Y Y X X u
1 10 11 1 12 2 1
2 20 21 1 21 1 22 2 2
3 30 31 1 32 2 31 1 32 2 3
? ? ? ?
? ? ? ? ?
? ? ? ? ? ?
? ? ?
? ? ? ?
? ? ? ? ?
? Chris Brooks 2002,陈磊 2004
6-19
? Equation 23,Contains both Y1 and Y2; we require these to be
uncorrelated with u3,By similar arguments to the above,
equations (21) and (22) do not contain Y3,so we can use OLS
on (23).
? This is known as a RECURSIVE or TRIANGULAR system.
We do not have a simultaneity problem here.
? But in practice not many systems of equations will be
recursive...
Recursive Systems
? Chris Brooks 2002,陈磊 2004
6-20
Indirect Least Squares (ILS)
? Cannot use OLS on structural equations,but we can validly
apply it to the reduced form equations.
? If the system is just identified,ILS involves estimating the
reduced form equations using OLS,and then using them to
substitute back to obtain the structural parameters.
? However,ILS is not used much because
1,Solving back to get the structural parameters can be
tedious.
2,Most simultaneous equations systems are over-identified.
7 Estimation procedures for Systems
? Chris Brooks 2002,陈磊 2004
6-21
? In fact,we can use this technique for just-identified and
over-identified systems.
? Two stage least squares (2SLS or TSLS) is done in two
stages:
Stage 1:
? Obtain and estimate the reduced form equations using OLS.
Save the fitted values for the dependent variables.
Stage 2:
? Estimate the structural equations,but replace any RHS
endogenous variables with their stage 1 fitted values.
Estimation of Systems
Using Two-Stage Least Squares
? Chris Brooks 2002,陈磊 2004
6-22
Example,Say equations (14)-(16) are required.
Stage 1:
? Estimate the reduced form equations (17)-(19) individually by
OLS and obtain the fitted values,.
Stage 2:
? Replace the RHS endogenous variables with their stage 1
estimated values:
(24)-(26)
? Now and will not be correlated with u1,will not be
correlated with u2,and will not be correlated with u3,
Estimation of Systems
Using Two-Stage Least Squares
?,?,?Y Y Y1 2 3
Y Y Y X X u
Y Y X u
Y Y u
1 0 1 2 3 3 4 1 5 2 1
2 0 1 3 2 1 2
3 0 1 2 3
? ? ? ? ? ?
? ? ? ?
? ? ?
? ? ? ? ?
? ? ?
? ?
? ?
?
?
?Y2
?Y3?Y3?Y2
? Chris Brooks 2002,陈磊 2004
6-23
? TSLS是比较经济, 易用的方法 。
? 如果在第一阶段估计时所得到的 R2非常高, 那么古典 OLS估计量与
TSLS估计量将非常接近；如果在第一阶段估计时所得到的 R2非常低,
TSLS估计量将没有太大的实际意义 。
? TSLS估计量是有偏估计量, 但却是一致估计量 。
? It is still of concern in the context of simultaneous systems
whether the CLRM assumptions are supported by the data.
? If the disturbances in the structural equations are
autocorrelated,the 2SLS estimator is not even consistent.
? The standard error estimates also need to be modified
compared with their OLS counterparts,but once this has
been done,we can use the usual t- and F-tests to test
hypotheses about the structural form coefficients.
Estimation of Systems
Using Two-Stage Least Squares
? Chris Brooks 2002,陈磊 2004
6-24
? Recall that the reason we cannot use OLS directly on the
structural equations is that the endogenous variables are
correlated with the errors.
? One solution to this would be not to use Y2 or Y3,but rather
to use some other variables instead.
? We want these other variables to be (highly) correlated with
Y2 and Y3,but not correlated with the errors - they are called
INSTRUMENTS.
? Say we found suitable instruments for Y2 and Y3,z2 and z3
respectively,We do not use the instruments directly,but run
regressions of the form
(27) & (28)
Instrumental Variables
Y z
Y z
2 1 2 2 1
3 3 4 3 2
? ? ?
? ? ?
? ? ?
? ? ?
? Chris Brooks 2002,陈磊 2004
6-25
? Obtain the fitted values from (27) & (28),and,and
replace Y2 and Y3 with these in the structural equation.
? It is typical to use more than one instrument per endogenous
variable.
? If the instruments are the variables in the reduced form
equations,then IV is equivalent to 2SLS.
Instrumental Variables
?Y2 ?Y3
? Chris Brooks 2002,陈磊 2004
6-26
What Happens if We Use IV / 2SLS Unnecessarily?
? The coefficient estimates will still be consistent,but will be
inefficient compared to those that just used OLS directly.
The Problem With IV
? What are the instruments?
Solution,2SLS is easier.
Other Estimation Techniques
1,3SLS - allows for non-zero covariances between the error terms.
2,LIML - estimating reduced form equations by maximum likelihood
3,FIML - estimating all the equations simultaneously using maximum
likelihood
Instrumental Variables
? Chris Brooks 2002,陈磊 2004
6-27
? A natural generalisation of autoregressive models popularised
by Sims.
? A VAR is in a sense a systems regression model i.e,there is
more than one dependent variable.
? Simplest case is a bivariate VAR
where uit is an iid disturbance term with E(uit)=0,i=1,2; E(u1t u2t)=0.
? The analysis could be extended to a VAR(g) model,or so that
there are g variables and g equations.
8 Vector Autoregressive Models
y y y y y u
y y y y y u
t t k t k t k t k t
t t k t k t k t k t
1 10 11 1 1 1 1 11 2 1 1 2 1
2 20 21 2 1 2 2 21 1 1 2 1 2
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
.,,,,,
.,,,,,
? Chris Brooks 2002,陈磊 2004
6-28
? One important feature of VARs is the compactness with
which we can write the notation,For example,consider the
case from above where k=1.
? We can write this as
or
yt = ?0 + ?1 yt-1 + ut
g?1 g?1 g?g g?1 g?1
Vector Autoregressive Models,
Notation and Concepts
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
y
y
y
y
u
u
t
t
t
t
t
t
1
2
10
20
11 11
21 21
1 1
2 1
1
2
?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
?
?
?
? ?
? ?
? Chris Brooks 2002,陈磊 2004
6-29
? VAR模型还具有灵活性和易于一般化的重要特点,
? 例如, 模型可以扩展到包含移动平均误差项, 即 VARMA。
? This model can be extended to the case where there are k
lags of each variable in each equation:
yt = ?0 + ?1 yt-1 + ?2 yt-2 +...+ ?k yt-k + ut
g?1 g?1 g?g g?1 g?g g?1 g?g g?1 g?1
? We can also extend this to the case where the model includes
first difference terms and cointegrating relationships (a
VECM).
Vector Autoregressive Models,
Notation and Concepts
? Chris Brooks 2002,陈磊 2004
6-30
? Advantages of VAR Modelling
- Do not need to specify which variables are
endogenous or exogenous - all are endogenous
- Allows the value of a variable to depend on more
than just its own lags or combinations of white noise
terms,so more general than ARMA modelling
- Provided that there are no contemporaneous terms
on the right hand side of the equations,can simply
use OLS separately on each equation,因为方程右边的
变量都是前定变量 。
- Forecasts are often better than,traditional
structural” models.
VAR Models Compared with
Structural Equations Models
? Chris Brooks 2002,陈磊 2004
6-31
? Problems with VAR’s
- VAR’s are a-theoretical (as are ARMA models)。
VAR模型较少用于理论分析和政策建议 。
- How do you decide the appropriate lag length?
- So many parameters! If we have g equations for g
variables and we have k lags of each of the variables
in each equation,we have to estimate (g+kg2)
parameters,e.g,g=3,k=3,parameters = 30
- Do we need to ensure all components of the VAR
are stationary?
- How do we interpret the coefficients?
VAR Models Compared with
Structural Equations Models
? Chris Brooks 2002,陈磊 2004
6-32
Choosing the Optimal Lag Length
Cross-Equation Restrictions
? In the spirit of (unrestricted) VAR modelling,each equation
should have the same lag length
? Suppose that a bivariate VAR(8) estimated using quarterly
data has 8 lags of the two variables in each equation,and we
want to examine a restriction that the coefficients on lags 5
through 8 are jointly zero,This can be done using a
likelihood ratio test
? Denote the variance-covariance matrix of residuals (given by
E ),as, The likelihood ratio test for this joint hypothesis
is given by
??uu???
? ?urTLR ???? ?l o g?l o g
? Chris Brooks 2002,陈磊 2004
6-33
Choosing the Optimal Lag Length
where is the variance-covariance matrix of the residuals for
the restricted model (with 4 lags),is the variance-covariance
matrix of residuals for the unrestricted VAR (with 8 lags),and T
is the sample size.
? The test statistic is asymptotically distributed as a ?2 with
degrees of freedom equal to the total number of restrictions,In
the VAR case above,we are restricting 4 lags of two variables in
each of the two equations = a total of 4 *2 * 2 = 16 restrictions,
? In the general case where we have a VAR with g equations,
and we want to impose the restriction that the last q lags have
zero coefficients,there would be g2q restrictions altogether
? Disadvantages,Conducting the LR test is cumbersome and requires a
normality assumption for the disturbances.
r??
u??
? Chris Brooks 2002,陈磊 2004
6-34Information Criteria for VAR Lag
Length Selection
? Multivariate versions of the information criteria are required.
These can be defined as:
where all notation is as above and k? is the total number of
regressors in all equations,which will be equal to g2k + g for g
equations,each with k lags of the g variables,plus a constant
term in each equation,The values of the information criteria
are constructed for 0,1,… lags (up to some pre-specified
maximum ).k
l n ( l n ( T ))
2?
ln
l n ( T )?ln
/2?ln
T
k
M H QI C
T
k
M S BI C
TkM AI C
?
???
?
???
???? ??? ???
?
? Chris Brooks 2002,陈磊 2004
6-35Does the VAR Include
Contemporaneous Terms?
? So far,we have assumed the VAR is of the form
? But what if the equations had a contemporaneous feedback term?
? We can write this as
? This VAR is in primitive / structural form.
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
y y y y u
y y y y u
t t t t t
t t t t t
1 10 11 1 1 11 2 1 12 2 1
2 20 21 2 1 21 1 1 22 1 2
? ? ? ? ?
? ? ? ? ?
? ?
? ?
? ? ? ?
? ? ? ?
y
y
y
y
y
y
u
u
t
t
t
t
t
t
t
t
1
2
10
20
11 11
21 21
1 1
2 1
12
22
2
1
1
20
0?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
?
?
?
? ?
? ?
?
?
? Chris Brooks 2002,陈磊 2004
6-36
Primitive versus Standard Form VARs
? We can take the contemporaneous terms over to the LHS
and write
or
B yt = ?0 + ?1 yt-1 + ut
? We can then pre-multiply both sides by B-1 to give
yt = B-1?0 + B-1?1 yt-1 + B-1ut
or
yt = A0 + A1 yt-1 + et
? This is known as a standard form VAR,which we can
estimate using OLS.
1
122
12 1
2
10
20
11 11
21 21
1 1
2 1
1
2?
??
??
?
??
?
??
?
?? ?
?
??
?
?? ?
?
??
?
??
?
??
?
?? ?
?
??
?
??
?
??
? ?
?
? ?
? ?
y
y
y
y
u
u
t
t
t
t
t
t
? Chris Brooks 2002,陈磊 2004
6-37
VAR模型的识别
? 粗略地讲，结构型 VAR模型的识别问题是指，能否从一个
简约模型的估计值反导出原来的结构模型的系数。
? 结构型 VAR模型是不可识别的，因为两个方程的等号右边
具有相同的前定变量
? 为了解决这个问题，需要加入一定的约束条件。即同期项的
一个系数 α12或 α22须设为 0，使距阵 B为三角形。
? 最好是依据经济理论加入约束条件
? Chris Brooks 2002,陈磊 2004
6-38
Block Significance and Causality Tests
? It is likely that,when a VAR includes many lags of variables,
it will be difficult to see which sets of variables have significant
effects on each dependent variable and which do not,For
illustration,consider the following bivariate VAR(3):
?
? We might be interested in testing the following hypotheses,
and their implied restrictions on the parameters:
tttttttt
tttttttt
uyyyyyyy
uyyyyyyy
2322231212222212112221121202
1321231112212211112121111101
????????
????????
??????
??????
???????
???????
H y p o t h e s i s I m p l i e d R e s t r i c t i o n
1, L a g s o f y 1 t d o n o t e x p l a i n c u r r e n t y 2 t ? 21 = 0 a n d ? 21 = 0 a n d ? 21 = 0
2, L a g s o f y 1 t d o n o t e x p l a i n c u r r e n t y 1 t ? 11 = 0 a n d ? 11 = 0 a n d ? 11 = 0
3, L a g s o f y 2 t d o n o t e x p l a i n c u r r e n t y 1 t ? 12 = 0 a n d ? 12 = 0 a n d ? 12 = 0
4, L a g s o f y 2 t d o n o t e x p l a i n c u r r e n t y 2 t ? 22 = 0 a n d ? 22 = 0 a n d ? 22 = 0
? Chris Brooks 2002,陈磊 2004
6-39
Block Significance and Causality Tests
? Each of these four joint hypotheses can be tested within the F-test
framework.
? These tests could also be referred to as Granger causality tests.
? Granger causality tests seek to answer questions such as,Do
changes in y1 cause changes in y2?” If y1 causes y2,lags of y1 should
be significant in the equation for y2,If this is the case,we say that
y1,Granger-causes” y2,
? If y2 causes y1,lags of y2 should be significant in the equation for
y1,If both sets of lags are significant,there is,bi-directional
causality”,
? If y2 causes y1,but y1 does not causes y2,then y2 is strong
exogenous (in the equation for y1).
? If neither y2 causes y1,nor y1 causes y2,then y1 y2 are independent
? 此处因果性并不意味着一个变量的变动引起另一个变量的变动
? Chris Brooks 2002,陈磊 2004
6-40
Impulse Responses
? VAR models are often difficult to interpret,one solution is to
construct the impulse responses and variance decompositions.
? Impulse responses trace out the responsiveness of the
dependent variables in the VAR to shocks to the error
term,A unit shock is applied to each variable and its effects
are noted.
? Consider for example a simple bivariate VAR(1):
? A change in u1t will immediately change y1,It will change y2
and also y1 during the next period.
? We can examine how long and to what degree a shock to a
given equation has on all of the variables in the system.
? eg,p341
y y y u
y y y u
t t t t
t t t t
1 10 11 1 1 11 2 1 1
2 20 21 2 1 21 1 1 2
? ? ? ?
? ? ? ?
? ?
? ?
? ? ?
? ? ?
? Chris Brooks 2002,陈磊 2004
6-41
多变量 VAR模型
也可改写为
这里 yt 是一个 k维内生变量向量, ?t 是协方差矩阵为 ? 的扰动向量 。
tptptt yAyAy ????? ?? ?11
ttpp yLALAI ????? )( 1 ?
一般 VAR模型的脉冲响应函数
假如 VAR(p)可逆, 我们可以得到 VMA(∞) 的表达式,
t
t
p
pt
LLI
LALAIy
???
?
)(
)(
2
21
1
1
?
?
????
???? ?
? Chris Brooks 2002,陈磊 2004
6-42
VMA表达式的系数可按下面的方式给出,VAR的系数 A和 VMA的系数
必须满足下面关系：
其中, 。 关于 的条件递归定义了 VMA系数：
?
ILCLCI ???? ?221
021 ??? ?CC
11 A??
2112 AA ?? ??
?
ILLILALAI pP ??????? ))(( 2211 ?? ??
从而可知 VMA的系数可以由 VAR的系数递归得到 。
qC
pqPqqq AAA ??? ???? ????,..2211
一般 VAR模型的脉冲响应函数
? Chris Brooks 2002,陈磊 2004
6-43
考虑 VMA(∞) 的表达式
设, y 的第 i个变量 可以写成：
其中 k 是变量个数 。
仅考虑 2个变量 (k = 2)的情形：
ity
)( 3,32,21,1,0
1
???????? ???
? jtijjtijjtijjtij
k
jit
y ????????
现在假定在基期给 一个单位的脉冲, 即：
–2 –1 0 1 2 3 4 5 ……… t
1y
??
? ??
e lse
t
t,0
0,1
1?
tt LLIy ???? )( 2210 ?????
)(,ijqq ?? ?
????
?
?
???
?
???
?
???
??
???
?
???
?
???
?
???
??
???
?
???
?
???
?
???
??
???
?
???
?
?
?
?
?
2,2
2,1
22,221,2
12,211,2
1,2
1,1
22,121,1
12,111,1
,2
,1
22,021,0
12,011,0
2
1
t
t
t
t
t
t
t
t
y
y
?
?
??
??
?
?
??
??
?
?
??
??
tt ??,02?
110??
? Chris Brooks 2002,陈磊 2004
6-44
由 的脉冲引起的 的响应函数：
1y ?21,221,121,0,,???
21,44,2
21,33,2
21,22,2
21,11,2
21,00,2
,4
,3
,2
,1
,0
?
?
?
?
?
??
??
??
??
??
yt
yt
yt
yt
yt
? ?
由上述推导可知由 的脉冲引起的 的响应函数序列是由 VMA(∞) 中
系数矩阵 第 2行, 第 1列的元素组成,q = 1,2,… 。 因此, 一般
地, 由 的脉冲引起的 的响应函数可以求出如下：
其中, 代表着对第 j个变量的单位冲击引起第 i个变量的第 q期滞后反映 。
jy
iy
??,,,,,,4,3,2,1,0 ijijijijij ?????
)(,ijqq ?? ?
1y
2y
2y
ijq,?
? Chris Brooks 2002,陈磊 2004
6-45
Variance Decompositions
? Variance decompositions offer a slightly different method
of examining VAR dynamics,They give the proportion of
the movements in the dependent variables that are due to
their,own” shocks,versus shocks to the other variables.
?
? This is done by determining how much of the s-step ahead
forecast error variance for each variable is explained by
innovations to each explanatory variable (s = 1,2,…).
? The variance decomposition gives information about the
relative importance of each shock to the variables in the
VAR.
? 脉冲响应函数和方差分解常常提供一定程度上的相似信息,
? Chris Brooks 2002,陈磊 2004
6-46
The Ordering of the Variables
? But for calculating impulse responses and variance
decompositions,the ordering of the variables is important.
? The main reason for this is that above,we assumed that the
VAR error terms were statistically independent of one another,
? This is generally not true,however,The error terms will
typically be correlated to some degree,Therefore,the notion of
examining the effect of the innovations separately has little
meaning,since they have a common component,
? What is done is to,orthogonalise” the innovations.
? In the bivariate VAR,this problem would be approached by
attributing all of the effect of the common component to the first
of the two variables in the VAR,
? In the general case where there are more variables,the situation
is more complex but the interpretation is the same,
? Chris Brooks 2002,陈磊 2004
6-47
由以上讨论可知, 对第 i个变量的冲击不仅直接影响第 i个变量, 并且通
过 VAR模型的动态 （ 滞后 ） 结构传导给所有的其它内生变量 。 脉冲响应函
数描绘了在一个扰动项上加上一次性的一个冲击 (one-time shock),对于内
生变量的当前值和未来值所带来的影响 。
假如扰动项 同期不相关, 那么脉冲响应的解释就很直接 。 第 i个扰动
项 就只对第 i个内生变量有一个冲击 。 然而, 扰动之间大都是相关的, 可
以描述为它们有一个与被指定变量不相关的公共成分 。 为了解释脉冲, 最
常用的方法是引进一个转换矩阵, 使扰动项变成不相关：
这里 D是对角协方差矩阵 。 如下面所说明的, EViews提供多种关于 P的选择
方法 。
t?
ti,?
),0(~ DNPv tt ??
? Chris Brooks 2002,陈磊 2004
6-48
乔利斯基 （ Cholesky） 分解：
对于任意实对称正定矩阵 ?,存在惟一一个主对角线元素为 1的下三角形
矩阵 A和惟一一个主对角线元素为正的对角矩阵 D使得
（ 20.12）
利用这一矩阵 A可以构造一个 向量, 构造方法为,
则
AAD ???
)1( ?T
)( ttE ?? ???
tv tt Av ?1??
DAAADAAAAEAvvE tttt ??????????? ?????? 111111 ][][][][])[(][)( ??
? Chris Brooks 2002,陈磊 2004
6-49An Example,The Interaction between
Property Returns and the Macroeconomy.
? Brooks and Tsolacos (1999) employ a VAR methodology for
investigating the interaction between the UK property market and
various macroeconomic variables,
? Monthly data are used for 1986 to 1997,
? It is assumed that stock returns are related to macroeconomic and
business conditions.
? The variables included in the VAR are
– FTSE Property Total Return Index (with general stock market
effects removed)
– The rate of unemployment
– Nominal interest rates
– The spread between long and short term interest rates
– Unanticipated inflation
– The dividend yield,
The property index and unemployment are I(1) and hence are
differenced,
? Chris Brooks 2002,陈磊 2004
6-50Marginal Significance Levels associated with Joint F-
tests that all 14 Lags have not Explanatory Power
for that particular Equation in the VAR
? Multivariate AIC selected 14 lags of each variable in the VAR
L a g s o f V a r i a b l e
D e p e n d e n t v a r i a b l e SI R D I V Y SP R E A D U N E M U N I N F L PR O PR E S
SI R 0, 0 0 0 0 0, 0 0 9 1 0, 0 2 4 2 0, 0 3 2 7 0, 2 1 2 6 0, 0 0 0 0
D I V Y 0, 5 0 2 5 0, 0 0 0 0 0, 6 2 1 2 0, 4 2 1 7 0, 5 6 5 4 0, 4 0 3 3
SP R E A D 0, 2 7 7 9 0, 1 3 2 8 0, 0 0 0 0 0, 4 3 7 2 0, 6 5 6 3 0, 0 0 0 7
U N E M 0, 3 4 1 0 0, 3 0 2 6 0, 1 1 5 1 0, 0 0 0 0 0, 0 7 5 8 0, 2 7 6 5
U N I N F L 0, 3 0 5 7 0, 5 1 4 6 0, 3 4 2 0 0, 4 7 9 3 0, 0 0 0 4 0, 3 8 8 5
PR O PR E S 0, 5 5 3 7 0, 1 6 1 4 0, 5 5 3 7 0, 8 9 2 2 0, 7 2 2 2 0, 0 0 0 0
? Chris Brooks 2002,陈磊 2004
6-51Variance Decompositions for the
Property Sector Index Residuals
? Ordering for Variance Decompositions and Impulse Responses:
– Order I,PROPRES,DIVY,UNINFL,UNEM,SPREAD,SIR
– Order II,SIR,SPREAD,UNEM,UNINFL,DIVY,PROPRES.
E xpl a i n e d by i nn o v a t i o n s i n
S IR D IV Y S P R E A D U N E M U N IN F L P R O P R E S
M o n t h s a h e a d I II I II I II I II I II I II
1 0,0 0,8 0,0 38,2 0,0 9,1 0,0 0,7 0,0 0,2 100, 0 51,0
2 0,2 0,8 0,2 35,1 0,2 12,3 0,4 1,4 1,6 2,9 97,5 47,5
3 3,8 2,5 0,4 29,4 0,2 17,8 1,0 1,5 2,3 3,0 92,3 45,8
4 3,7 2,1 5,3 22,3 1,4 18,5 1,6 1,1 4,8 4,4 83,3 51,5
12 2,8 3,1 15,5 8,7 15,3 19,5 3,3 5,1 17,0 13,5 46,1 50,0
24 8,2 6,3 6,8 3,9 38,0 36,2 5,5 14,7 18,1 16,9 23,4 22,0
? Chris Brooks 2002,陈磊 2004
6-52
Impulse Responses and Standard Error Bands
for Innovations in Dividend Yield
Innov at ions in D ividend Y ie lds
-0.06
-0.04
-0.02
0
0, 0 2
0, 0 4
0, 0 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Ste ps Ahea d
? Chris Brooks 2002,陈磊 2004
6-53
? P350 图 6-1
Impulse Responses and Standard Error Bands
for Innovations in unexpected inflation
? Chris Brooks 2002,陈磊 2004
6-54
Innov at ions in t he T- Bil l Y ie ld
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
S t eps Ah ea d
Impulse Responses and Standard Error Bands
for Innovations in the Treasury Bill Yield
