Econometrics of money and nance Lecture nine: multivariate ...

Econometrics of money and financeLecture nine: multivariate modeling I

Zongxin Qian

School of Finance, Renmin University of China

November 4, 2014

Revisit the example from monetary economics

yt = a1yt−1 + a2Etyt+1 − a3(Rt − Etπt+1) + e1t (1)

πt = b1πt−1 + b2Etπt+1 + b3yt + e2t (2)

Rt = φπEtπt+1 + e3t (3)

I We estimated the Phillips curve as a single equation in thelast lecture

I We can also estimate the three-equation model together.

The monetary economics example in matrix form

I Recall that the three equation model can be reduced to:

yt = a1yt−1 + a2Etyt+1 − a3(φπ − 1)Etπt+1 + e1t − a1e3t(4)

πt = b1πt−1 + b2Etπt+1 + b3yt + e2t (5)

I In matrix form, this is

Yt = A0 + A1EtYt+1 + A2Yt−1 + Ut (6)

I Can you write out Ai s?

Solution of the Matrix form

I The matrix form of the model has a unique solution undercertain condition (Blachard-Khan condition).

I The solution has a vector autoregression (VAR) form.However, the solution method is at the master’s level.

I To proceed, I make the following simplifying assumptionEtYt+1 = Yt−1, Etπt+1 = πt−1.

I In addition, I modify the Taylor rule as Rt = φππt + e3t .

I Now it is easy to show that the solution is a SVARB0Yt = B1 + B2Yt−1 + ut . Can you tell the parametermatrices?

I If B0 is invertible, the SVAR can be written as a restrictedVAR Yt = B−10 B1 + B−10 B2Yt−1 + B−10 ut .

The unrestricted VAR

I For now, let’s forget about the restriction imposed by thestructural model.

I The unrestricted VAR can be estimated by either OLS orMLE, just as the single equation AR model.

I Note that the consequence of not imposing a valid restrictionis efficiency loss.

I However, the structural restriction might be wrong and leadto biased estimates and we don’t want to test the restrictionswith a wrong model!!!

I This tradeoff is at the center of various debates amongeconomists, particularly in the field of monetary economics.

Estimation of the unrestricted VAR

I The density function of a k dimensional multivariate normaldistribution is (2π)−

k2 |Ω|− 1

2 exp[− 1

2 (X − µ)′Ω−1(X − µ)].

I Consider the VAR(1): Yt = BYt−1 + Ut , Ut ∼ (0, Ω).

I Set Y0 = 0, Y1 ∼ (0, Ω); Yt |Yt−1 ∼ (BYt−1, Ω) for t > 2.

I Following the same procedure of AR(1) estimation, we obtainl(B, Ω) = − kT

2 ln(2π) + T2 ln|Ω−1| −

12Σt=T

t=1 [(Yt −BYt−1)′Ω−1(Yt − BYt−1)]

I Maximization of the log-likelihood function with respect to Bis equivalent to minimizeΣt=Tt=1 [(Yt − BYt−1)′Ω−1(Yt − BYt−1)]

I This is a GLS estimator. But it will be equivalent to OLS if allequations in the VAR have a same set of regressors (Zellner,1962).

Statistical identification v.s. economic identification

I Economists are more interested in economic identificationwhich means distinguishing between different economicstories.

I For this reason, structural models with restrictions fromtheoretical models are popular these days.

I However, as we have mentioned, a wrong restriction may biasthe estimation.

I Moreover, restriction imposed on a misspecified model doesnot generate reliable results even if the restriction itself isvalid.

I For this reason, statistical identification is important. Toavoid problems with complicated structural models, theliterature has offered various solutions.

Solution I: the LSE approach-test, test, test

I Start with a general dynamic reduced-form model.

I Test whether the residuals follow a normal i.i.d.

I Testing down and simplifying dynamics.

I Test for exogeneity and reduce dimensionality.

I Identify and estimate the structural model.

Normality test

I The JB (Jarque-Bera) test test for the null hypothesis ofnormality:

I The JB test statistics is N6 (S

2 + (K−34 )2), S is the skewness,K is the kurtosis

I Under the null hypothesis of normality, the JB statistics followa χ2 distribution with 2 degrees of freedom

I Eviews provide the JB test for residuals after estimation.View/Residual Diagnostics/Histogram-Normality

Normality test in Eviews

Normality test for a multivariate model

I The JB (Jarque-Bera) test has multivariate versions.

I The complications is that the residuals can be correlated.That is, the residual vector u ∼ N(0,C ), where is notdiagonal.

I To perform the test we first orthogonalize the residuals so thatthe transformed residuals no longer correlate with each other.

I One can use either the Cholesky (Ltkepohl) or triangle(Doornik and Hansen) decomposition to achieve this.

I Eviews can do these things for us (see next two slides).

Normality test for a VAR model in Eviews

Normality test for a VAR model in Eviews

VAR Residual Normality TestsOrthogonalization: Residual Correlation (Doornik-Hansen)Null Hypothesis: residuals are multivariate normalDate: 11/26/13 Time: 20:03Sample: 1 993Included observations: 991

Component Skewness Chi-sq df Prob.

1 -0.286836 13.27030 1 0.00032 0.131291 2.865489 1 0.0905

Joint 16.13578 2 0.0003

Component Kurtosis Chi-sq df Prob.

1 14.69512 1202.215 1 0.00002 6.284798 222.8960 1 0.0000

Joint 1425.111 2 0.0000

Component Jarque-Bera df Prob.

1 1215.485 2 0.00002 225.7615 2 0.0000

Joint 1441.247 4 0.0000

Serial correlation test

I The Breusch-Godfrey test test for the null hypothesis of noserial correlation:

I The original regression isyt = c + a1yt−1 + a2yt−2 + · · ·+ apyt−p + ut ,ut = φ1ut−1 + φ2ut−2 ++ · · ·+ φput−m + et , et is i. i. d.

I The orginal regression generates a series of residuals. Usingthose data, we can run an auxiliary regression as followsyt = d + b1yt−1 + b2yt−2 + · · ·+ bpyt−p + θ1ut−1 +θ2ut−2 + · · ·+ θput−m + wt

I The LM statistics is (T −m)R2, which follows a χ2

distribution with m degrees of freedom

I When the sample size is small the F test for the jointsignificance of uts is better than the χ2 test

Serial correlation test in Eviews

Eviews provide the LM test for residuals after estimation.View/Residual Diagnostics/Serial Correlation LM Test

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 3.654696 Prob. F(4,155) 0.0071Obs*R-squared 13.96215 Prob. Chi-Square(4) 0.0074

Serial correlation test (multivariate) in EviewsThere is also a multivariate version of the LM test. In Eviews, it isView/Residual Diagnostics/Serial Correlation LM Test

VAR Residual Serial Correlation LM TestsNull Hypothesis: no serial correlation at lag order hDate: 11/26/13 Time: 20:28Sample: 1 993Included observations: 991

Lags LM-Stat Prob

1 7.60051 0.10742 10.12839 0.03833 5.181823 0.26914 9.272124 0.05465 2.448518 0.65396 2.846312 0.58397 2.639085 0.61998 1.592152 0.81029 3.806246 0.4329

10 1.180014 0.881411 0.708716 0.950212 1.271123 0.8663

Probs from chi-square with 4 df.

Granger Causality/block exogeneity test

Consider a bivariate VAR of M1 growth and CPI inflation:(mt

πt

)=

(c1c2

)+

(b111 b112b121 b122

)(mt−1πt−1

)+

(b211 b212b221 b222

)(mt−2πt−2

)+

(v1tv2t

)

I Test for the null b112 = b212 = 0 is a test for whether inflationis a Granger cause of money growth.

I Test for the null b121 = b222 = 0 is a test for whether moneygrowth is a Granger cause of inflation.

I However, the name cause is misleading. The test cannot tellthe causal relationship. It is a test for forecasting power.

Granger-Causality test in Eviews

Granger-Causality test in Eviews

VAR Granger Causality/Block Exogeneity Wald Tests

Date: 11/04/14 Time: 13:47

Sample: 1992Q1 2007Q1

Included observations: 58

Dependent variable: M1G

Excluded Chi-sq df Prob.

PI_SA 6.208926 2 0.0448

All 6.208926 2 0.0448

Dependent variable: PI_SA

Excluded Chi-sq df Prob.

M1G 1.681850 2 0.4313

All 1.681850 2 0.4313

Solution II: minimal structural VARs

I Since one big problem for structural models is thatassumptions imposed are too strong, a simple way out is torelax the restrictions.

I Structural VARs which are popular in current studies imposeonly minimal structural assumptions, such as the timing of thestructural shocks.

I Still face a problem of omitted variables? Try a structuralFAVAR model which is a VAR augmented by factor analysis.




I Still face a problem of omitted variables?

Try a structuralFAVAR model which is a VAR augmented by factor analysis.




I Still face a problem of omitted variables? Try a structuralFAVAR model which is a VAR augmented by factor analysis.

The triangular decomposition

I For any positive definitive symmetric matrix Ω, there is aunique decomposition Ω = ADA′.

I A =

1 0 0 . . . 0a21 1 0 . . . 0a31 a32 1 . . . 0

......

... . . ....

an1 an2 an3 . . . 1

and

D =

d11 0 0 . . . 00 d22 0 . . . 00 0 d33 . . . 0...

...... . . .

...0 0 0 . . . dnn

.

The triangular decomposition: details

I Introduce a matrix E1 ≡

1 0 0 . . . 0

−Ω21Ω−111 1 0 . . . 0−Ω31Ω−111 0 1 . . . 0

......

... . . ....

−Ωn1Ω−111 0 0 . . . 1

,

I It can be shown that

E1ΩE ′1 = H ≡

h11 0 0 . . . 00 h22 h23 . . . h2n0 h32 h33 . . . h3n...

...... . . .

...0 hn2 hn3 . . . hnn

.

The triangular decomposition: details continued

I Introduce another matrix E2 ≡

1 0 0 . . . 00 1 0 . . . 00 −h32h−122 1 . . . 0...

...... . . .

...0 −hn2h−122 0 . . . 1

,

I It can be shown that

E2HE′2 = K ≡

k11 0 0 . . . 00 k22 0 . . . 00 0 k33 . . . k3n...

...... . . .

...0 0 kn3 . . . knn

.

The triangular decomposition: details continued

I Carrying on similar procedures, we can find a series of Ei s,such that En−1 . . . E2E1ΩE ′1E

′2 . . . E ′n−1 = D

I So A = (En−1 . . . E2E1)−1

The Cholesky decomposition

I Ω = PP ′, where P ≡ AD1/2

I diag(P) =

√d11 0 0 . . . 00

√d22 0 . . . 0

0 0√d33 . . . 0

......

... . . ....

0 0 0 . . .√dnn

The recursive SVAR

I Consider the following reduced-form VAR

Yt = A1Yt−1 + . . . + ApYt−p + Ut , (7)

where Ut ∼ (0, ΣU) and ΣU is a non-diagonal matrix.

I It can be written as a recursive SVAR

Yt = A1Yt−1 + . . . + ApYt−p +W εt , (8)

where εt ∼ (0, Σε) and Σε = DD ′ is a diagonal matrix.

I W ≡ PD−1 and D = diag(P).

I Alternatively, one can define W ≡ P and Σε = I .

The recursive SVAR

I Elements in Ut cannot be taken as independent shocks sinceΣU is non-diagonal.

I That is why we would like to calculate εt , of which theelements are called structural shocks.

I Defining W ≡ P and Σε = I suggests that all structuralshocks have the same variance. This is quite a strongassumption.

I Defining W ≡ PD−1 and D = diag(P) allows the variancesof structural shocks to be different.

The recursive SVAR

I W is a lower triangular matrix, so

W εt =

W11ε1t

W21ε1t +W22ε2tW31ε1t +W32ε2t +W33ε3t

...Wn1ε1t +Wn2ε2t +Wn3ε3t + . . . +Wnnεnt

I Therefore, the ordering of endogenous variables in a VAR

matters a lot for the interpretation of its recursive SVARversion.

I The first VAR variable is only contemporaneously affected bystructural shock to itself.

I The second VAR variable is contemporaneously affected bystructural shocks to itself and to the first variable...

Impulse response analysis

I Recall that an AR model can be rewritten as a MA(∞) model.

I Similarly, a VAR model can also be written asYt = µ + ut + ψ1ut−1 + ψ2ut−2 + . . ..

I ∂Yt+h

∂ut= ψh, where the ij ’s element of ψh is the IRF of the jth

shock on variable i in period t + h.

I An important application of the SVAR is IRF analysis.

The recursive SVAR: an example

I Consider a VAR with three variables (πt , yt ,Mt), where Ihave replaced the usual monetary policy variable in athree-equation SVAR, policy interest rate, by the growth rateof aggregate money supply, M2. This is because moneysupply is more important as a policy tool in the Chineseexample we are going to look at.

I The ordering I use means that monetary policy actions haveonly lagged impact on inflation and output.

I Moreover, shocks to output have only lagged impact oninflation.

Impulse responses in the exampleHere we use the ordering (πt , yt ,Mt). We use VAR(2)and samplefrom 2008m10 to 2013m11.

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to INF_SA

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to YGAP

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to M2_SA

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to INF_SA

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to YGAP

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to M2_SA

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to INF_SA

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to YGAP

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to M2_SA

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

Impulse responses in the exampleNow we change the ordering to (yt , πt ,Mt).

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to YGAP

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to INF_SA

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10

Response of YGAP to M2_SA

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to YGAP

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to INF_SA

-.2

.0

.2

.4

.6

1 2 3 4 5 6 7 8 9 10

Response of INF_SA to M2_SA

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to YGAP

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to INF_SA

-0.4

0.0

0.4

0.8

1.2

1 2 3 4 5 6 7 8 9 10

Response of M2_SA to M2_SA


Estimate a VAR in Eviews

Specify your VAR in Eviews

Generate IRFs with your VAR in Eviews

Choose the Cholesky ordering in Eviews

The recursive SVAR: the impact of normalization

I The two specifications W = P and W = PD−1 can generatedifferent results.

I Eviews use W = P. You have to be aware of this when usingEviews.

I In the next slide, we preset an example where the twospecifications generate different results.

I We estimate a bivariate VAR(12) for exogenous tax changeand log output of the US. The sample period is1950q1-2007q4.

I Raw data is taken fromhttps://www.aeaweb.org/articles.php?doi=10.1257/aer.100.3.763

IRF of the bivariate VAR, W = P

-.012

-.008

-.004

.000

.004

.008

.012

.016

2 4 6 8 10 12 14 16 18 20

Response of LGDP to LGDP

-.012

-.008

-.004

.000

.004

.008

.012

.016

2 4 6 8 10 12 14 16 18 20

Response of LGDP to EXOR

-.1

.0

.1

.2

2 4 6 8 10 12 14 16 18 20

Response of EXOR to LGDP

-.1

.0

.1

.2

2 4 6 8 10 12 14 16 18 20

Response of EXOR to EXOR


IRF of the bivariate VAR, W = PD−1

This graph is generated with matlab programing.

3 6 9 12 15 18

0

20

40

60

80

100

Impulse response of exogenous

3 6 9 12 15 18

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Impulse response of gdp

Application of recursive SVAR

I As mentioned before, economic theories usually imply morecomplicated restrictions than just the ordering of variables in arecursive VAR.

I A common practice to test theories is to match the impulseresponses of the structural model to those of a recursive VAR.

I Of course, this approach has limitations.

1. Not all theoretical models have a reduced-form VARrepresentation.

2. Estimation of VAR(∞) requires truncation which can causebias.

3. Some research topics might not have a clear time ordering ofvariables.

Econometrics of money and nance Lecture nine: multivariate ...

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