var and causality tests

Multivariate time series modelsVAR’s and Causality Tests

Chapter 12&15 Asteriou (apprx 20-pages)

Chapter 6 Brooks

Chapter 5 Enders

Simultaneous Equations Models

All the models we have looked at thus far have dealt with single dependent variables and estimations of single equations

All of the independent variables have been EXOGENOUS and y has been an ENDOGENOUS variable.

However interdependence is often encountered in economics, where several dependent variables are determined simultaneously

An example from economics to illustrate - the demand and supply of a good: (1)

(2)(3)

where = quantity of the good demanded = quantity of the good supplied

St = price of a substitute good Tt = some variable embodying the state of technology[Note: We’re trying to estimate the parameters (i.e. greek letters)]

Q P S udt t t t Q P T vst t t t Q Qdt st

Qdt

Qst

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 by the market process (price affects quantity and quantity affects price). Another way to think of this: when P changes all we observe is changes in

equilibrium, we don’t know whether this is due to a shift of only demand or only supply or some combination. i.e. we can’t separately estimate the effect of a change in price on demand or on supply

P and Q are endogenous variables (they are determined by the model), while S and T are taken to be exogenous.

We can obtain REDUCED FORM equations corresponding to (4) and (5) by solving equations (4) and (5) for P and for Q (separately).

Simultaneous Equations Models: The Structural Form

Q P S u Q P T v

The steps before we see the maths1. Solve the equations simultaneously to find

an expression for P that does not involve Q This is the reduced form for P

2. Rearrange supply and demand to get them in the form P=…. Then solve simultaneously to get an expression for Q that does not involve P

This is the reduced form for Q

3. We can estimate the reduced forms

Solving to find an expression for P,

(6)

Rearranging (6),

(7)

Obtaining the Reduced Form

P S u P T v

P P T S v u

( ) ( ) ( ) P T S v u

P T Sv u

Rearranging (4) and (5) to get P on the left hand side and then equating them.

Solving to find an expression for Q, (8)

Multiplying (8) through by ,

(9)

Obtaining the Reduced Form (cont’d)

Q S u Q T v

Q Q T S u v

( ) ( ) ( ) Q T S u v

Q T Su v

Q S u Q T v

Obtaining the Reduced Form (cont’d)

(7) and (9) are the reduced form equations for P and Q.

Note that each of the reduced form equations include the error terms from both (4) and (5)

(i.e. μ and υ)

P T Sv u

Q T Su v

But what would happen if we had estimated equations (4) and (5), i.e. the structural form equations, separately using OLS?

Both equations (4) and (5) depend on P. One of the

CLRM assumptions was that the error term should be uncorrelated with the explanatory variables.

It is clear from (7) that P is related to the errors in (4) and

(5) - i.e. it is stochastic. Another way of saying this is that a shock affecting one

equation will have an effect on the second equation via the influence on P!

Simultaneous Equations Bias

This can be seen more clearly in the following general form of a simultaneous equation model.

A) y1t = b10 + b11y2t+b12x2t+b14x4t +e1t

B) y2t = b20 + b21y1t+b23x3t+b24x4t +e2t

Let one of the error terms e1t increase and hold all else constant Then:

This will cause y1t to increase due to equation A In turn y1t will cause y2t to increase (if b21 is positive) due to equation B) if y2t increases in B) then it also increases in A) where it is an explanatory variable. Hence an increase in e1t caused an increase in an explanatory variable in the same equation

(meaning the error and the explanatory variable are correlated with each other!!).

(I have shown this with the arrows: Blue leads to Red leads to Green

Simultaneous Equation Bias

Simultaneous Equations Bias

What would be the consequences for the OLS estimator, , if we ignore the simultaneity?

Application of OLS to structural equations which are part of a simultaneous system will lead to biased and inconsistent estimators

Hence it would not be possible to estimate simultaneous equations validly using OLS.

So What Can We Do? Taking the normal equations (7) and (9), we can rewrite them as

(10)

(11) We CAN estimate equations (10) & (11) using OLS since all the RHS

variables are exogenous [(10) doesn’t have Q while (11) doesn’t have P => feedback has been controlled for!].

But ... we probably don’t care what the values of the coefficients are; what we wanted were the original parameters in the structural equations - , , , , , .

Can We Retrieve the Original Coefficients from the ’s?Short answer: sometimes.This is known as the identification problem

Avoiding Simultaneous Equations Bias

P T S 10 11 12 1

Q T S 20 21 22 2

Identification:

We re-wrote:

as

and

as

So: if we ‘know’ the π’s it

may be possible to solve these equations to find what we care about i.e. α, β, etc. – but not always!!

P T Sv u

Q T Su v

P T S 10 11 12 1

Q T S 20 21 22 2

etc,, 1110

VAR’s

Vector Autoregressive (VAR) Models

Identification of exogenous and endogenous variables was heavily criticised by Sims(1980).

According to Sims if there is simultaneity among a number of variables, then all these variables should be treated in the same way.

All variables should be treated as endogenous This means that in its general reduced form each equation has

the same set of regressors which leads to the development of VAR models

VARs have been often advocated as an alternative to large scale simultaneous equations structural models

• 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(k)

where u1t and u2t are uncorrelated white noise error terms

• This is a bivariate VAR(k) since there are 2 variables and the longest lag is k• The analysis could be extended to a VAR(k) model with g variables and g

equations but still k lags.

• (Note: when looking back at previous slides remember Var is used for Variance!

Vector Autoregressive Models

y y y y y u

y y y y y ut 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

... ...

... ...

• 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 using matrix notation as

or even more compactly as

yt = 0 + 1 yt-1 + ut

21 21 22 21 21

Vector Autoregressive Models: Notation and Concepts

y y y u

y y y ut 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

ut

t

t

t

t

t

1

2

10

20

11 11

21 21

1 1

2 1

1

2

• This model can be extended to the case where there are k lags of each variable in each equation and g endogenous variables:

yt = 0 + 1 yt-1 + 2 yt-2 +...+ k yt-k + ut

g1 g1 gg g1 gg g1 gg g1 g1

• We can also extend this to the case where the model includes first difference terms and cointegrating relationships (a VECM) – we’ll see this later .

Vector Autoregressive Models: Notation and Concepts (cont’d)

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. For OLS to be used efficiently it is also required that there is the same lag length in each equation- Forecasts are often better than “traditional structural” models.

Problems with VAR’s- VAR’s are a-theoretical (as are ARMA models)- 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 so need a lot of data- Do we need to ensure all components of the VAR are stationary?- How do we interpret the coefficients?

Vector Autoregressive Models Compared with Structural Equations Models

Does 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. This cannot be estimated using OLS

y y y u

y y y ut 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 ut 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

ut

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

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-10 + B-11 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. In general the approach is to estimate the VAR equation in standard form

1

122

12 1

2

10

20

11 11

21 21

1 1

2 1

1

2

y

y

y

y

u

ut

t

t

t

t

t

Choosing the Optimal Lag Length for a VAR

An important decision as too few leg will mean the model is misspecified and too many lags will waste degrees of freedom.

To check lag length begin with the longest feasible length given degrees of freedom considerations.

Can use single equation and multi equation information criteria The AIC and SBC can be applied to each equation in the

VAR

Recall that in both the AIC and SBC increasing the lag length will reduce the RSS at the ‘expense’ of increasing k.

k= number of parameters in the equation, which equals g.p +1 where g is the number of variables, p is the number of lags of each variable and 1 is for the intercept term

Choose the AIC or SBC with the lowest value

nkeT

RSSAIC /2)( nke

T

RSSSBC /)(

Choosing the Optimal Lag Length for a VAR

Multivariate versions of the information criteria are available. These can

be defined as:

MAIC = T ln│∑│ + 2N

MSBC = T ln│∑│ + Nln(T)

where │∑│is the determinant of the var/cov matrix of residuals μi in the standard VAR system.

N is the total number of parameters estimated 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).

Select the model with the lowest MAIC or MSBC

Diagnostic Checking

Once we have estimated a VAR(p) standard form model we can check the residuals from the equation to see if they are white noise error terms, in particular to see if the residuals are serially correlated Apply the Lagrange Multiplier test of serial

corelation. LM~χ2q where q is the number of lags

you wish to test in the μt series

Apply the Jarque-Bera test for normality. JB~χ22

Causality

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):

This VAR could be written out to express the individual equations as

We might be interested in testing the following hypotheses, and their implied restrictions on the parameter matrices:

t

t

t

t

t

t

t

t

t

t

u

u

y

y

y

y

y

y

y

y

2

1

32

31

2221

1211

22

21

2221

1211

12

11

2221

1211

20

10

2

1

tttttttt

tttttttt

uyyyyyyy

uyyyyyyy

2322231212222212112221121202

1321231112212211112121111101

Block Significance and Causality Tests (cont’d)

Each of these four joint hypotheses can be tested within the F-test framework, since each set of restrictions contains only parameters drawn from one equation.

These tests are 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 both sets of lags are not significant in either equation then y1 and y2 are

independent

Hypothesis Implied Restriction1. Lags of y1t do not explain current y2t

21 = 0 and 21 = 0 and 21 = 02. Lags of y1t do not explain current y1t



22 = 0 and 22 = 0 and 22 = 0

Vector Autoregression (VAR) Vector Autoregression (VAR) AnAn ExampleExample

Granger causality tests – which have been interpreted as testing, for example, the validity of the monetarist proposition that autonomous variations in the money supply have been a cause of output fluctuations.

Granger-causality requires that lagged values of variable A are related to subsequent values in variable B, keeping constant the lagged values of variable B and any other explanatory variables

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger CausalityGranger Causality

In a regression analysis, we deal with the dependence of one variable on other variables, but it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or direction of influence.

In the relationship between GDP and Money supply, the often asked question is whether GDP M or M GDP. Since we have two variables, we are dealing with bilateral causality.

The structural VAR equations are given as:

yttttt

mttttt

ymmy

ymym

1221212

1121111

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger CausalityGranger Causality

We can distinguish four cases:

Unidirectional causality from M to GDP Unidirectional causality from GDP to M Feedback or bilateral causality Independence

Assumptions: Stationary variables for GDP and M Number of lag terms, can be established using AIC/SBC Error terms are uncorrelated

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger Causality Example – Estimation (t-test)Granger Causality Example – Estimation (t-test)

mt does not Granger cause yt relative to an information set consisting of past m’s and y’s iff 21 = 0.yt does not Granger cause mt relative to an information set consisting of past m’s and y’s iff 12 = 0.

In a model with one lag, as in our example, a t-test can be used to test the null hypothesis that one variable does not Granger cause another variable. In higher order systems, an F-test is used.

tttt

tttt

ymy

ymm

2122121

1112111

1. Regress current GDP on all lagged GDP terms but do not include the lagged M variable (restricted regression). From this, obtain the restricted residual sum of squares, RSSR.

2. Run the regression including the lagged M terms (unrestricted regression). Also get the residual sum of squares, RSSUR.

3. The null hypothesis is Ho: i = 0, that is, the lagged M terms do not belong in the regression.

5. If the computed F > critical F value at a chosen level of significance, we reject the null, in which case the lagged m belong in the regression. This is another way of saying that m causes y.

Vector Autoregression (VAR)Vector Autoregression (VAR)Granger Causality – Estimation (F-test)Granger Causality – Estimation (F-test)

)/(

/)(

knRSS

mRSSRSSF

UR

URR

The Sims Causality Test

Sims (1980) put forward an alternative test for causality

The key to this test is that it is not possible for the future to cause the present

Suppose we want to find out does y cause m. For Sims causality testing consider the following model

ttttt

ttttt

mymy

yymm

2123122121

1113112111

The Sims Causality Test

We can test if yt causes mt by testing if 13=0. If 13≠ 0 then yt+1 does not cause mt as the future cannot cause the present

Again this hypothesis can be tested using a t test if there is 1 lead term. If there are more than one lead terms then an F test is applied to jointly test the hypothesis that they are all equal to zero

Overview of course:

var and causality tests

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