part2_granger

8/4/2019 part2_granger

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Granger Causality

Granger Causality

VAR


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Granger Causality(Time Series Econometrics, J. Hamilton, p302)

If past X contains useful information(in addition to the information in pastY) to predict future Y, we say Xgranger causes Y.

Note that Granger-causality may ormay not indicate causal effect of x on

y (could you think of someexamples?)


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Linear Bivariate Granger CausalityTests

Z fails to Granger-causes y if

MSE[E(xt|It-1)] = MSE[E(xt|Jt-1)]

Here It-1 contains past information on Y and Z

while Jt-1 contains past information on Y only.

Grangers test

Regression: xt=c+xt-1+yt-1+ut Test H0: =0

Sims test

Regression: yt=c+hyt-1+bxt-1+dxt+1+vt Test H0: d=0


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The Stock Price-Volume Relation(Hiemstra and Jones, 1994)

Whether knowing past stock pricemovements improves short-run forecasts ofmovements in trading volume (vice versa)?

People may use changes in equity prices ortrade volume to infer new information in theequity market. Hence, changes in past equityprices or trade volume may affect future pricesand volume. This implies bi-directional price-

volume relationship.


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Data

Daily Dow Jones tock returns andpercentage changes in New yorkStock Exchange trading volume.

Be careful of structural change in themodel

Due to a structural break at the end of

1946, the causality tests are conductedover the 1915 to 1946 and 1947 to 1990periods.


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What if the time series are notstationary?

A typical way to transform anonstationary time series intostationary ones is to take differences

A series xt is integrated of order d(wecall it I(d) process) if the series becomesstationary after differencing dtimes.


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Traditional (Linear) GrangerCausality

The trade volume series appearsnonstationary, suggesting thatdifferencing is needed to make thevolume series stationary.

The traditional linear Granger testdetects unidirectional Granger

causality from stock prices to tradingvolume.


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Improved Granger-Causality Test

Traditional Granger causality testsmight overlook a significant non-linear relation between stock returnsand trading volume.


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Nonlinear Granger Causality Test

Why nonlinear? When the assumption of representative-

agent in traditional trading model is relaxed(i.e. heterogeneous agents are allowed),

nonlinear dynamics become typical in theprice-volume relation.

An example: yt=yt-L*xt-M+t

Findings: Significant nonlinear bidirectionalGranger causality between stock prices andtrading volume in both sample periods.


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Vector Autoregressive (VAR) Model

VAR models simultaneously estimates theinterrelationship between more than oneendogenous variables.

Yt=AYt-1+Ut

Y is a vector containing different variables.

A is a coefficient Matrix.

U is the corresponding vector of residuals, whichhave nonzero cross correlations.


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An Example

Y=(y z)

A=(; )

Oryt=+yt-1+zt-1+ut

zt=+yt-1+zt-1+vt


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Impulse Response

The dynamics of a VAR model can bevisualized by impulse response diagrams.

Given a stable VAR model (like stationarity)

Impose a one-time shock to the system (theshock can be on any endogenous variables)

Trace out the deviation of all endogenousvariables from their equilibrium values in thefollowing time periods.


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Why VAR?

VARs have been used primarily inmacroeconomics to capture therelationship between importanteconomic variables.

VAR model provides a natural

framework to test the GrangerCausality.


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Estimating the VAR Model

Yt=AYt-1+Ut

Under certain conditions on thestructure of A and the property of U,

the multi-equation VAR model can besolved for best estimates of A.

What is structural VAR?

If the structure of A is chosen based oneconomic theory, then we call this VARmodel a structural VAR.


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The Dynamic Effects of GovernmentSpending and Taxes on Output (Blanchardand Perotti, 2002)

Theory: Why and how shouldgovernment spending and taxesaffect aggregate outputs?

Keynesian: A positive effect ofgovernment spending on privateconsumption

Neoclassical: Both increases in taxes andincreases in government spending havea negative effect on private investment


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Data

Quarterly National Income andProduct Accounts

Quarterly Treasury Bulletin. Whole sample is 1947:1 to 1997:4

Due to structural breaks before 1960,only the sample after 1960:1 isselected.


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The Model

Y=(T,G,X) is a vector containing thelogarithms of quarterly taxes,spending, and GDP.


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The Model (continued)

U=(et,eg,ex) is a vector of residuals. Theyrepresent unexpected movements in taxes,spending, and GDP.

t=a1x+a2eg+et

g=b1x+b2et+eg

x=c1t+c2g+ex

Where et, eg, ex are the mutually uncorrelatedstructural shocks that we want to recover

We are interested in the coefficients c1 and c2.


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Identification Strategy

Determine a1 and b1 The two coefficients capture the effects of

economic activity on taxes and spending underexisting fiscal policy rules

No automatic feedback from economic activity togovernment expenditure is identified, so b1=0. Independent estimates suggest a1is around 2.

Construct residuals t=t-a1x and g=g, usethem as instruments for the third equation,

estimating c1 and c2. Then we make (non-structural) assumption

on a2 and b2 to estimate them.


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Main Findings

Effect on Aggregate Output

Positive government spending shockshave a positive effect on output

Positive tax shocks have a negativeeffect on output

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