Sanjeev Sridharan, Supreme Court of Virginia, Virginia Criminal Sentencing
Commission, 533 Harrow Road, Richmond, VA 23225, (804) 371-7727 .
CAN VECTOR-AUTOREGRESSION METHODOLOGY HELP IN AN UNDERSTANDING
OF THE CAUSAL DYNAMICS OF THE UNEMPLOYMENT-CRIME RELATIONSHIP?
We apply a dynamic time-series method
called Vector-autoregression (VAR) to
studying the dynamic linkages between
unemployment and crime rates in Virginia.
The promise of the VAR methodology lies
in its ability to provide information on the
dynamic and the feedback properties in
systems of variables (Sims, 1980).
We studied the dynamic relationship
between unemployment and crime for
monthly data from Virginia for the period
between January |983 to December
1992. The basic system studied consisted
of the following variables:
Virginia Unemployment Rates (VAUN)
Property Crimes Rates: property
crimes studied included burglaries
(BURG), larcenies (LARC) and
robberies (ROB).
¢ Total part | and part 2 arrest rates
(TTLAR)
e Jail Crowding (CROWD): Jail crowding
was defined as the percentage of jail
capacity that was occupied by local
responsible jail inmates in any given
month.
We confined our study to property crimes
because the literature suggests that the
linkages between unemployment and
crime are more likely to occur for
property crimes (Cantor and Land, 1985).
Similarly, as we were focusing on crimes of
lesser seriousness we restricted our study
to jails.
I
We focusssed on the following research
questions:
(1) What were the causal properties of the
above system? In other words, which
variables could be considered “causes” of
other variables in the system?
(2) VAR methodology explores the
dynamic linkages in the system by
addressing the following question: How
would a shock (innovation) in one of the
variables affect the other variables in the
system over time ?
CAUSAL CONCEPTS IN THE VAR.
FRAMEWORK:
Vector-autoregression are multivariate
generalizations of the Granger-Causality
framework (Granger, 1969). There are
four causal notions that are basic in the
Granger-Causality framework: Causality,
Feedback, Instantaneous Causality and
Causality Lag.
Definitions:
Let U, be the “information accumulated in
the Universe since time t-|. U,- Y;
denotes all information apart from the
specified series Y,” (Granger, 1969, p.
428)2 Let A, represent the set of past
values, and A,, represent the set of past
and present values.
Causality:
If VarQX/U) < Var(XU-Y),), we say that
Y is causing X; this is denoted by Y, =>
X, “We say that Y, is causing X,, if we are
better able to predict X. using all variable
information than if the information apart
from Y, had been used” (ibid, p. 428).
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Feedback:
If Var(X/U,) < Var(X(U-Y),)
and Var(Y/U,) < Var(¥KU-X)p)
then we say that feedback is occuring.
More generally, “feedback is said to occur
when X, is causing Y, and also Y, is causing
X" (bid. , p. 428).
Instantaneous Causality:
If (VarO</Up, Yop) < Var(X/U,)
then we say that instantaneous causality
Y, => X, is occurring.
“In other words, the current value of X, is
better ‘predicted’ if the present value of Y,
is included in the prediction than if it is not”
(ibid., p. 429).
Causality Lag:
“We define the (integer) causality ‘lag m’ to
be the least value of k such that Var(X/U-
Y(k)) < Var(X/U-Y(k+ 1)). Thus knowing
the values Y,j, j = 0,1,2...m-1, will be of
no help in improving the prediction of X,"
(ibid., p. 429).
Definition of VAR:
Let y, be am x | matrix of random
variables. A VAR of order p (denoted by
VAR(p)) is defined as:
Xe =A FU :
AL) = Ay + AQL + ASL? +... ASL?”
where A, Ap,...A, are mxm matrices of
coefficients and u, is a white noise vector
process with the following properties:
E(u.) = 0
E(uu,’) = Z fors=t
E(uu, ) = 0 fors#t
is the variance-covariance matrix of u,.
u,' is the transpose of u,.
Methodology:
The lag-length to determine the order of
the VAR is fixed using the modified
likelihood-ratio test (Sims, 1980).
Multivariate generalizations of granger
causality tests are used to study the causal
relationships . These tests are known as
Block-exogeneity tests (Freeman et al.,
1989).
Impulse response functions study the
response of the system to “innovations”
(random shocks) in each of the
endogenous variable. Impulse response
functions can be used to simulate the
effects of policy-intervention. As an
example, consider Freeman et al. (1989,
pp. 848-849)":
The VAR approach to public policy
analysis treats governmental decision
making in some respects as “reactive”
and in other respects “unpredictable.”
VAR modelers almost always treat
policy variables as endogenous in that
they place them on both the left- and
right-hand sides of their equation
systems. At the same time, VAR
modelers understand that policies
fluctuate in seemingly random ways, or
that governments make novel, surprise
decisions. These decisions presumably
can be represented as econometric
innovations in the policy variables. The
impact of these shocks or “policy
innovations” presumably are reflected in
the respective, recursive moving
average responses of the system.
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RESULTS:
Sims likelihood-ratio test indicated that
eight lags would be better than four lags.
However, given the large number of
coefficients involved in estimating VAR
models with eight lags, we restricted the
number of lags to four.
The significance levels of the F-tests for the
block exogeneity tests are presented in
table |. At the 0.01 level of significance
VAUN is exogenous to the system. At the
0.05 level of significance, TTLAR is a
predictor of VAUN. At the 0.05 level of
significance, VAUN is a predictor of ROB
and BURG. TTLAR is a statistically
significant predictor of ROB and BURG
(p< 0.01). Significant predictors of TTLAR
include VAUN (p<0.05), LARC(p<0.01),
ROB(p<0.05) and BURG (p<0.05).
BURG is a significant predictor of CROWD.
(p< 0.05). We find feedback effects
between TTLAR and each of the following
variables: VAUN, ROB and BURG.
Impulse Response Functions help in an
understanding of the direction of the
above relationships (figure | plots the
response of TTLAR to shocks in the
variables of the system). An increase in
arrest rates leads to a drop in
unemployment rates . This decrease is
strongest 3 months after the shock in
arrest rates. An increase in
unemployment rate is accompanied by
small increases in the burglary and robbery
rates. For robberies, the effect is strongest
7 months after a shock to the
unemployment rate; for burglaries the
response is strongest 3 months after the
shock. A shock in the arrest rates initially
leads to drops in robbery and burglary
rates; however, after 6 months such a
shock leads to an increase in burglary and
robbery rates. An increase in the
unemployment rate results in a decrease in
arrest rates. This effect is strongest eight
months after the initial shock (see figure |).
Similar impulse response functions were
obtained for the other variables.
DISCUSSION/CONCLUSION:
We recommend an application of VAR
methodology in criminal justice settings
with a few cautions: VAR have been
criticized for their robustness properties
(Runkle, 1987).° In addition, there are
problems with the interpretation of
impulse response functions (Lutkepohl,
1993).5 However, despite these
problems, we still recommend VAR
methodology for the following reasons:
(|) they have the potential to provide
information on the feedback processes in
criminal justice. The concept of feedback
has yet to be fully appreciated in criminal
justice settings.
(2) More broadly, the VAR framework has
the potential to help the theorist as well as
the policy-maker focus on the “dynamics”
of relationships between variables in
criminal justice.
Table |: Block-exogeneity tests : Significance of F-
tests
Equation | Var. Larc Reo Burg tar Crows,
Vanacie
Vaun 0.00% | 0.12 0.03" 0.03* 0.03" 0.18
bere 0.05 0.03* | 0.02° 0.04* 0.co** | 0.45
Rob 0.13 0.48 0.02" 0.60 0.04 0.45
Burg 0.33 0.02" | 0.71 0.00%" | 0.02° 0.04"
Tear O.ci* 0.21 0.00" | 0.00% | 0.COF* | 0.96
Crowd 0.57 0.80 0.33 0.50 0.58 0.00%*
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Figure t* Response of TTLAR to One S.D Innovations
Response of TTUAR‘b VAUN Response TLAR to LARC
" “
7 "
. V\
a a
| ae =
3 0
a a a
Response of TTLAR 9 ROB Rescate 2! TLAR io BURG
Filles
a a TTT TIT TT.
Resconse ol TTLAR 19 TTLAR Response of TTLAR to CROWD
»
10.
°
Tras ee ee ae
' Sims, CA. (1980). Macroeconomics and Reality.
Econometrica. 48: 1-40.
? Cantor, D. and Land, K.C. (1985).
Unemployment and Crime Rates in the Post-World
War Il United States: A Theoretical and Empirical
Analysis. American Sociological Review. 50: 317-
332.
> Granger, C.W,. (1969). Investigating Causal
Relations by Econometric Models and Cross-
Spectral Methods. Econometrica 37: 424-438.
* Freeman, J... Williams, |.T., and Lin, T.-m.
(1989). Vector Autoregression and the Study of
Politics. American Journal of Political Science. 33:
842:877.
* Runkle, D. (1987). Vector Autoregression and
Reality. Journal of Business and Economic Statistics.
5: 437:442
° Lutkepohl, H. (1993). Intreduction to Multiple
Time Series Analysis, Springer-Verlag, Berlin.
S\u
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