Domestic and Transnational Terrorism

Chapter 5

APPLIED ECONOMETRIC TIME SERIES 4TH ED.

WALTER ENDERS

Figure 5.1 Domestic and Transnational Terrorism

Panel (a): Domestic Incidents in

ci de

nt s

pe r q

ua rt

1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 0

100

150

200

250

300

350

400

Panel (b): Transnational Incidents

in ci

de nt

s pe

r q ua

rt er

1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009 0

An Intervention Model

Consider the model used in Enders, Sandler, and Cauley (1990) to study the impact of metal detector technology on the number of skyjacking incidents:

yt = a0 + a1yt1 + c0zt + ?t, ?a1? < 1 where zt is the intervention (or dummy) variable that takes on the value of zero prior to 1973Q1 and unity beginning in 1973Q1 and ?t is a white-noise disturbance. In terms of the notation in Chapter 4, zt is the level shift dummy variable DL. 0 -0 1 1 - 0 011 i i t i t it i i a y c a aza ? ? ? ? ? ? ? ? ? ? ? Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 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 of ARIMA models. You can use the Perron (1989) test for structural change discussed in Chapter 4. STEP 2: Estimate the various models over the entire sample period, including the effect of the intervention. STEP 3: Perform diagnostic checks of the estimated equations. 4 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 5 0 5 10 15 20 25 30 35 40 (in ci de nt s pe r q ua rt er ) Figure 5.2: Skyjackings Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 6 Figure 5.3: Typical Intervention Functions Panel (a): Pure Jump (a) 1 2 3 4 5 6 7 8 9 10 0.00 0.25 0.50 0.75 1.00 1.25 Panel (c): Gradually Changing (c) 1 2 3 4 5 6 7 8 9 10 0.00 0.25 0.50 0.75 1.00 1.25 Panel (b): Pulse (b) 1 2 3 4 5 6 7 8 9 10 0.00 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 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Table 5.1: Metal Detectors and Skyjackings 7 Pre? Interventi on Mean a1 Impact Effect (c0) Long?Run Effect Transnational {TSt} 3.032 (5.96) 0.276 (2.51) ?1.29 (?2.21) ?1.78 U.S. Domestic {DSt} 6.70 (12.02) ?5.62 (?8.73) ?5.62 Other Skyjackings {OSt} 6.80 (7.93) 0.237 (2.14) ?3.90 (?3.95) ?5.11 Notes: 1. t-statistics are in parentheses 2. The long-run effect is calculated as c0/(1 ? a1) Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. ADLs and Transfer Functions 8 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Transfer Functions yt = a0 + A(L)yt1 + C(L)zt + B(L)?t where A(L), B(L), and C(L) are polynomials in the lag operator L. In a typical transfer function analysis, the researcher will collect data on the endogenous variable {yt} and on the exogenous variable {zt}. The goal is to estimate the parameter a0 and the parameters of the polynomials A(L), B(L), and C(L). Unlike an intervention model,{zt} is not constrained to have a particular deterministic time path. It is critical to note that transfer function analysis assumes that {zt} is an exogenous process that evolves independently of the {yt} sequence. 9 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. The CCVF The cross-correlation between yt and zti is defined to be ?yz(i) ? cov(yt, zti)/(?y?z ) where ?y and ?z = the standard deviations of yt and zt, respectively. The standard deviation of each sequence is assumed to be time independent. Plotting each value of ?yz(i) yields the cross-correlation function (CCF) or cross-correlogram. 10 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Interpreting the CCVF yt = a0 + a1yt1 + C(L)zt + ?t (5.7) The theoretical CCVF (and CCF) has a shape with the following characteristics: All ?yz(i) will be zero until the first nonzero element of the polynomial C(L). A spike in the CCVF indicates a nonzero element of C(L). Thus, a spike at lag d indicates that ztd directly affects yt. 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- covariances will be direct, whereas if 1 < a1 < 0, the decay pattern will be oscillatory. Only the nature of the decay process changes if we generalize equation (5.7) to include additional lags of yti. 11 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 12 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Estimating a Parsimonous ADL 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 of Step 1 to obtain D(L)yt ? yft. STEP 3: Identify plausible candidates for the A(L) function. Regress yt (not yft) on the selected values of {zt} to obtain a model of the form yt = C(L)zt + et STEP 4: Combine the results of Steps 2 and 3 to estimate the full equation. At this stage, you will estimate A(L), and C(L) simultaneously. Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 14 ?0.007 ?0.006 ?0.005 ?0.004 ?0.003 ?0.002 ?0.001 0 0.001 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Lo g Sh ar e Figure 5.5 Italy's Share of Tourism Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 15 Consider two of the equations from the Brookings Quarterly Econometric Model CNF = 0.0656YD - 10.93[PCNF/PC]t-1 + 0.1889[N + NML]t-1 (0.0165 (2.49) (0.0522) CNEF = 4.2712 + 0.1691YD - 0.0743[ALQDHH/PC]t-1 (0.0127) (0.0213) where: CNF = personal consumption expenditures on food YD = disposable personal income PCNF = price deflator for personal consumption expenditures on food PC = price deflator for personal consumption expenditures N = civilian population NML = military population including armed forces overseas CNEF = personal consumption expenditures for nondurables other than food ALQDHH =end-of-quarter stock of liquid assets held by households 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 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 16 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." Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. "St. Louis model" estimated by Anderson and Jordan (1968). 17 Using U.S. quarterly data from 1952 - 1968, they estimated the following reduced-form GNP determination equation: ?Yt = 2.28 + 1.54?Mt + 1.56?Mt-1 + 1.44?Mt-2 + 1.29?Mt-3 + 0.40?Et + 0.54?Et-1 - 0.03?Et-2 - 0.74?Et-3 (5.16) where ?Yt = change in nominal GNP ?Mt = change in the monetary base ?Et = change in "high employment" budget deficit Testing whether the sum of the monetary base coefficients (i.e. 1.54 + 1.56 + 1.44 + 1.29 = 5.83) differs from zero yields a t-value of 7.25. Hence, they concluded that changes in the money base translate into changes in nominal GNP. On the other hand, the test that the sum of the fiscal coefficients (0.40 + 0.54 - 0.03 - 0.74 = 0.17) equals zero yields a t- value of 0.54. According to Anderson and Jordan, the results support "lagged crowding out" in the sense that an increase in the budget deficit initially stimulates the economy. Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Reduced Form 18 Sims (1980) also points out several problems with this type of analysis. Ensuring that there is no feedback between GNP and the money base or the budget deficit. However, the assumption of no feedback is unreasonable if the monetary or fiscal authorities deliberately attempt to alter nominal GNP. As in the thermostat 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 time-series econometrics, changes in GNP would "cause" changes in the money supply. One appropriate strategy would be to simultaneously estimate the GNP determination equation and the money supply feedback rule. Comparing the two types of models, Sims (pp. 14-15, 1980) states: "Because existing large models contain too many incredible restrictions, empirical research aimed at testing competing macroeconomic theories too often proceeds in a single- or few- equation framework. For this reason alone, it appears worthwhile to investigate the possibility of building large models in a style which does not tend to accumulate restrictions so haphazardly. ... It should be feasible to estimate large-scale macromodels as unrestricted reduced forms, treating all variables as endogenous." Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Structural VARs 19 yt = b10 ? b12zt + ?11yt-1 + ?12zt-1 + ?yt zt = b20 ? b21yt + ?21yt-1 + ?22zt-1 + ?zt 0 1 -1t t tBx x ?? ? ?? ? Pre-multiply by B-1 to obtain xt = A0 + A1xt-1 + et 1 1 1 0 0 1 1; ; and- - t t= = eA B A B B ???? ? 112 121 1 1 10t t yt11 12 20t t zt21 22 y yb b= + + z zb b ?? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ?? ? Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. A 1st-Order VAR in Standard Form 20 yt = a10 + a11yt-1 + a12zt-1 + e1t zt = a20 + a21yt-1 + a22zt-1 + e2t e1t = (?yt b12?zt)/(1 b12b21) e2t = (?zt b21?yt)/(1 b12b21) Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. The VAR Structure 21 Consider the following 2-variable 1-lag VAR in standard form: yt = a10 + a11yt-1 + a12zt-1 + e1t zt = a20 + a21y t-1 + a22zt-1 + e2t It is assumed that e1t and e2t are serially uncorrelated but the covariance Eet1e2t need not be zero. If the variances and covariance are time-invariant, we can write the variance/covariance matrix as: where: Var(eit) = ?ii and Cov(e1t,e2t) = ?12 = ?21. 11 22 12 21 = ? ? ?? ? ? ? ? ? ? ? Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Forecasting ? If your data run through period T, it is straightforward to obtain the one-step-ahead forecasts of your variables using the relationship ? ETxT+1 = A0 + A1xT. ? A two-step-ahead forecast can be obtained recursively from ETxT+2 = A0 + A1ETxT+1 = A0 + A1[A0 + A1xT]. ? Since unrestricted VARs are overparameterized, the forecasts may be unreliable. In order to obtain a parsimonious model, many forecasters would purge the insignificant coefficients from the VAR. ? After reestimating the so-called near-VAR model using SUR, it could be used for forecasting purposes. 22 Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. Terrorism and Israeli real per capita GDP 23 1 1 1 1 11 14 1 2 1 2 1 3 1 3 41 44 1 3 1 4 ( ) ... ( ) ... ( ) ... ( ) t t t t t t t t t t t t t t t t GDP GDP c T e A L A L I I c T e EXP EXP c T e A L A L NDC NDC c T e ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ? ?? ? ? ?? ?? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? The aim of the study was to investigate the effects of terrorism (T) on the growth rates of Israeli real per capita GDP (?GDPt), investment (?It), exports (?EXPt), and nondurable consumption (?NDCt). The authors use quarterly data running from 1980Q1 to 2003Q3 so that there are 95 total observations. Copyright © 2015 John, Wiley & Sons, Inc. All rights reserved. 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 terrorism actually ended in 2003Q4 (so that all values of Tj = 0 for j > 2003Q4). Under this assumption, the annual growth rate of GDP was estimated to be 2.5% through 2005Q3. Instead, when they set the values of Tj at the 2000Q4 to 2003Q4 period average, the growth rate of GDP was estimated to be zero. Thus, a steady level of terrorism would have cost the Israeli economy all of its real output gains. In actuality, the largest influence of terrorism was found to be on investment. The impact of terrorism on investment was twice as large as the impact on real GDP.

Impulse Responses

Consider a 2-variable model:

The impulse response function is obtained using the moving average representation:

11 12 1 1 1

21 22 2 1 1

( ) ( )

n n

t t i t i t i i

n n

t t i t i t i i

y a i y a i z e

z a i y a i z e

? ? ? ?

? ? ?

? ?

11 1 12 2 1 1 1

21 1 22 2 2 1 1

( ) ( )

n n

t t i t i t i i n n

t t i t i t i i

y b i e b i e e

z b i e b i e e

? ? ? ?

? ? ?

? ?

Impulse Responses: An Example 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 shoc t x(t) y(t) t x(t) y(t) 1 1 0.2 1 0 1 2 0.74 0.34 2 0.2 0.7 3 0.586 0.386 3 0.28 0.53 4 0.487 0.387 4 0.302 0.427 5 0.419 0.369 5 0.297 0.359 6 0.367 0.342 6 0.28 0.311 7 0.325 0.313 7 0.258 0.274 8 0.29 0.284 8 0.235 0.243 9 0.26 0.257 9 0.213 0.217 10 0.233 0.232 10 0.193 0.195 11 0.21 0.209 11 0.174 0.175 12 0.188 0.188 12 0.157 0.157 13 0.17 0.169 13 0.141 0.141 14 0.153 0.152 14 0.127 0.127

The Residuals vs the Pure Shocks

e1t = (?yt – b12?zt)/(1-b12b21) e2t = (?zt – b21?yt)/(1-b12b21)

If we set b12 or b21equal to zero, we can identify the shocks

Identification

e1t = g11?1t + g12?2t e2t = g21?1t + g22?2t

or: et = G?t

If we let var(?1t) = and var(?2t) = , it follows that:

E?1t?2t ?

The problem is to identify the unobserved values of ?1t and ?2t from the regression residuals e1t and e2t.

2 1?

2 2?

2 1

2 2

0 0? ?

? ? ?

? ? ? ? ? ?

Identification 2

If we knew the four values g11, g12 g13 and g14 we could obtain all of the structural shocks for the regression residuals. Of course, we do have some information about the values of the gij. Consider the variance/covariance matrix of the regression residuals:

Eee’ = ? 11 12

21 22

? ? ? ? ? ?

Sims Recursive Ordering

Sims recursive ordering restricts on the primitive system such that the coefficient b21 is equal to zero. Writing (5.17) and (5.18) with the constraint imposed yields

yt = b10 b12zt + g11yt1 + g12zt1 + eyt

zt = b20 + g21yt1 + g22zt1 + ezt

Similarly, we can rewrite the relationship between the pure shocks and the regression residuals given by (5.22) and (5.23) as e1t = ?yt b12?zt e2t = ?zt

Sims Recursive Ordering

e1t = ?yt b12?zt e2t = ?zt

so that var(e1) = 2 2 212y zb? ?? (5.31)

var(e2) = 2z? (5.32) cov(e1, e2) = b12 2z? (5.33)

Hence, it must be the case that:

Eetet’ = EG?t?t’G ‘

Since Eetet’ = ? and E?t?t’ = ??, it follows that:

11 12

21 22

‘G G? ? ? ? ? ? ?

? ?? ? ? ?

2 2 11 12 11 12 11 21 12 22

2 2 21 22 11 21 12 22 21 22

g g g g g g g g g g g g

? ? ? ?

? ?? ?? ? ? ? ?? ? ? ?? ? ? ?

In general you must fix (n2 n)/2 elements for exact identification

Hypothesis Tests

Let ?u and ?r be the variance/covariance matrices of the unrestricted and restricted systems, respectively. Then, in large samples:

(T-c)(log | ? r | – log | ? u | )

can be compared to a ?2 distribution with degrees of freedom equal to the number of restrictions.

Model Selection Criteria

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

AIC = T log | ? |+ 2 N SBC = T log | ? | + N log(T)

Where | ? | = determinant of the variance/covariance matrix of the residuals and N = total number of parameters estimated in all equations.

Granger-Causality

Granger causality: If {yt} does not improve the forecasting performance of {zt}, then {yt} does not Granger-cause {zt}. The practical way to determine Granger causality is to consider whether the lags of one variable enter into the equation for another variable.

Block Exogeneity

Block exogeneity restricts all lags of wt in the yt and zt equations to be equal to zero. This cross-equation restriction is properly tested using the likelihood ratio test. Estimate the yt and zt equations using lagged values of {yt}, {zt}, and {wt} and calculate ?u. Reestimate excluding the lagged values of {wt} and calculate ?r. Form the likelihood ratio statistic:

(T-c)(log | ?r | – log | ?u |

This statistic has a chi-square distribution with degrees of freedom equal to 2p (since p lagged values of {wt} are excluded from each equation). Here c = 3p + 1 since the unrestricted yt and zt equations contain p lags of {yt}, {zt}, and {wt) plus a constant.

To Difference or Not to Difference

Recall a key finding of Sims, Stock, and Watson (1990): If the coefficient of interest can be written as a coefficient on a stationary variable, then a t-test is appropriate.

You can use t-tests or F-tests on the stationary variables. You can perform a lag length test on any variable or any set of

variables 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 any set of equations using t-tests or F-tests.

It is possible to write the VAR in first differences if the variables are I(1) and are not cointegrated. If the variables in question are cointegrated, the VAR cannot be written in first differences

If the I(1) variables are not cointegrated and you use levels:

Tests lose power because you estimate n2 more parameters (one extra lag of each variable in each equation).

For a VAR in levels, tests for Granger causality conducted on the I(1) variables do not have a standard F distribution. If you use first differences, you can use the standard F distribution to test for Granger causality.

When the VAR has I(1) variables, the impulse responses at long forecast horizons are inconsistent estimates of the true responses. Since the impulse responses need not decay, any imprecision in the coefficient estimates will have a permanent effect on the impulse responses. If the VAR is estimated in first differences, the impulse responses decay to zero and so the estimated responses are consistent.

Seemingly Unrelated Regressions

Different lag lengths yt = a11(1)yt-1 + a11(2)yt-2 + a12zt-1 + e1t zt = a21yt-1 + a22zt-1 + e2t

Non-Causality yt = a11yt-1 + e1t zt = a21yt-1 + a22zt-1 + e2t

Effects of a third variable yt = a11yt-1 + a12zt-1 + e1t zt = a21yt-1 + a22zt-1 + a23wt + e2t

Responses to

Figure 5.8 Impulse Responses of Terrorism

R es

po ns

es o

Domestic

Transnational

Domestic

Transnational

0 2 4 6 8 10 12 14 16 -20

-10

0 2 4 6 8 10 12 14 16 -20

-10

0 2 4 6 8 10 12 14 16 -2

Sims Bernamke

21 23

1 0 0 1

0 0 1

ytyt

mtmt

rt rt

e e g g e

? ? ?

? ?? ? ? ? ? ?? ? ? ?? ? ?? ? ? ? ? ?? ? ? ?? ? ? ? ? ?

Sims Structural VAR

21 23 24

31 36

41 43 46

51 53 54 56

1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1

0 0 0 0 0 1

t rt

t mt

ytt

ptt

t ut

t it

b r b b b m

yb b = pb b b

b b b b u i

? ? ? ? ? ?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?

Sims (1986) used a six-variable 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 as measured by M1 (m), unemployment (u), and the treasury bill rate (r).

Note that it is Overidentified

rt = 71.20mt + ert (5.59) mt = 0.283yt + 0.224pt 0.0081rt + emt (5.60) yt = 0.00135rt + 0.132it + eyt (5.61) pt = 0.0010rt + 0.045yt 0.00364it + ept (5.62) ut = 0.116rt 20.1yt 1.48it 8.98pt + eut (5.63) it = eit (5.64)

Sims views (5.59) and (5.60) as money supply and demand functions, respectively. In (5.59), the money supply rises as the interest rate increases. The demand for money in (5.60) is positively related to income and the price level and negatively related to the interest rate. Investment innovations in (5.64) are completely autonomous. Otherwise, Sims sees no reason to restrict the other equations in any particular fashion. For simplicity, he chooses a Choleski-type block structure for GNP, the price level, and the unemployment rate. The impulse response functions appear to be consistent with the notion that money supply shocks affect prices, income, and the interest rate.

Blanchard-Quah

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 affected by the same two shocks. The BMA representation is:

11 12 1

21 22 2

( ) ( ) ( ) ( )

y L LC C= z L LC C

? ?

?? ? ? ? ? ? ? ? ? ? ? ?

? ? ? ?? ?

1 1 2

1 2 2

var( ) cov( , ) 1 0 cov( , ) var( ) 0 1

= = ? ? ? ? ? ? ?

? ? ? ? ? ? ? ? ?

? ?? ?

The Long-run resrtiction Assume that one of the shocks has a temporary effect on

the {yt} sequence. It is this dichotomy between temporary and permanent

effects that allows for the complete identification of the structural innovations from an estimated VAR.

For example, Blanchard and Quah assume that an aggregate demand shock has no long-run 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 an?1t shock on the ?yt sequence must be equal to zero. Hence, the coefficients c11(k) must be such that

11 1 0

( ) 0t k k=

k =c ? ?

11 0

( ) 0 k=

k =c ?

Since this must be true for all realizations

Recall that:

e1t = c11(0)e1t + c12(0)e2t

e2t = c21(0)e1t + c22(0)e2t

The four restrictions

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