Lecture Applied econometric time series (4e) - Chapter 5: Multiequation time-series models

This chapter’s objectives are to: Introduce intervention analysis and transfer function analysis, show that transfer function analysis can be a very effective tool for forecasting and hypothesis testing when it is known that there is no feedback from the dependent to the so-called independent variable,

Chapter 5

Applied Econometric Time  Series 4th ed.

Walter Enders

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Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved 2 Figure 5.1 Domestic and Transnational Terrorism

Panel (a): Domestic Incidents

where zt is the intervention (or dummy) variable that takes on the

value of zero prior to 1973Q1 and unity beginning in 1973Q1 and εt is

a white-noise disturbance In terms of the notation in Chapter 4, zt is the level shift dummy variable DL.

Steps in an Intervention Model

• STEP 1: Use the longest data span (i.e., either the pre­ or the postintervention observations) to find a plausible set 

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Figure 5.3: Typical Intervention Functions

Panel (a): Pure Jump

0.25 0.50 0.75 1.00 1.25

Panel (d): Prolonged Pulse

(d)

1 2 3 4 5 6 7 8 9 10 0.00

0.25 0.50 0.75 1.00 1.25

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Interventi on Mean

Pre-a1 Impact

Effect (c0)

Long-Run Effect

Notes:

1. t­statistics are in parentheses

2. The long­run effect is calculated as c0/(1   a1)

ADLs and Transfer Functions

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• Plotting each value of ρyz(i) yields the cross­correlation 

function (CCF) or cross­correlogram. 

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• All spikes decay at the rate a1; convergence implies that the absolute  value of a1 is less than unity. If 0 < a1 < 1, decay in the cross­

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• STEP 1: Estimate the zt sequence and an AR process.

• STEP 2: Identify plausible candidates for C(L)

– Constrict the filtered {yt} sequence by applying the  filter D(L) to each value of {yt}; that is, use the results 

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0 2 4 6 8 10

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Figure 5.5 Italy's Share of Tourism

Log Share

YD = disposable personal income

PC = price deflator for personal consumption expenditures

and: standard errors are in parenthesis.

The remaining portions of the model contain estimates for the other

components of aggregate consumption, investment spending, government

spending, exports, imports, for the financial sector, various price determination equations, …

The Brookings Model

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Are such ad hoc behavioral assumptions consistent with economic theory?

Sims (p.3, 1980) considers such multi-equation models and argues that:

" what 'economic theory' tells us about them is mainly that any variable that appears on the right-hand-side of one of

these equations belongs in principle on the right-hand-side of all of them To the extent that models end up with very

different sets of variables on the right-hand-side of these

equations, they do so not by invoking economic theory, but (in the case of demand equations) by invoking an intuitive

econometrician's version of psychological and sociological

theory, since constraining utility functions is what is involved here Furthermore, unless these sets of equations are

considered as a system in the process of specification, the

behavioral implications of the restrictions on all equations

taken together may be less reasonable than the restrictions on any one equation taken by itself."

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Using U.S. quarterly data from 1952 ­ 1968, they estimated the following  reduced­form GNP determination equation: 

example, if the monetary authority attempts to control the economy by changing the  money base, we can not identify the "true" model. In the jargon of time­series 

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 =  σ σ

σ σ

Forecasting

 If your data run through period T, it is straightforward to 

obtain the one­step­ahead forecasts of your variables using the relationship 

 After reestimating the so­called near­VAR model using 

SUR, it could be used for forecasting purposes. 

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Cost of terrorism

 To forecast the values of xT+2 and beyond, it is necessary 

to know the magnitude of the terrorism variable over the forecast period. Toward this end, they supposed that all 

Impulse Responses: An Example

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x(t) = 0.7*x(t-1) + 0.2*y(t-1) + e1(t) y(t) = 0.2*x(t-1) + 0.7y(t-1) + e2(t) e2(t) = 0.2*e1(t)

1-unit e1 shock 1-unit e2 shock

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2

2 1

2 2

0 0

 e1t =  yt – b12 zt

 e2t =  zt

(T­c)(log | Σ r | ­ log | Σ u | ) 

can be compared to a  2 distribution with degrees of χ

freedom equal to the number of restrictions

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Alternative test criteria are the multivariate generalizations of the AIC and SBC:

Granger­Causality

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Granger causality: If {yt} does not improve the forecasting  performance of {zt}, then {yt} does not Granger­cause {zt}.  

The practical way to determine Granger causality is to 

consider whether the lags of one variable enter into the 

equation for another variable. 

• Generally, you cannot use Granger causality tests 

concerning the effects of a nonstationary variable

• The issue of differencing is important. 

– If the VAR can be written entirely in first differences, hypothesis tests can be performed on any equation or 

responses decay to zero and so the estimated responses are 

-10 0 10 20 30 40 50 60

0 2 4 6 8 10 12 14 16 -20

-10 0 10 20 30 40 50 60

0 2 4 6 8 10 12 14 16 -2

0 2 4 6 8 10

0 2 4 6 8 10 12 14 16 -2

0 2 4 6 8 10

mt mt

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Sims (1986) used a six­variable VAR of quarterly data over the period 

1948Q1 to 1979Q3. The variables included in the study are real GNP (y),  real business fixed investment (i), the GNP deflator (p), the money supply 

pt t

impulse response functions appear to be consistent with the notion that money  supply shocks affect prices, income, and the interest rate.

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Suppose we are interested in decomposing an I(1) 

sequence, say {yt}, into its temporary and permanent  components. Let there be a second variable {zt} that is 

t t

t t

ε ε

• For example, Blanchard and Quah assume that an 

aggregate demand shock has no long­run effect on real 

GNP. In the long run, if real GNP is to be unaffected by the demand shock, it must be the case that the cumulated effect of anε1t shock on the ∆yt sequence must be equal to 

zero. Hence, the coefficients c11(k) must be such that

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1 ( ) (0) ( ) (0) 0

k k=

Figure 5.9 Responses of Real and  Nominal Exchange Rates 

Responses t o t he nominal shock