Time Series and Panel Data Analysis
6.5 Methodology and Discussion on Empirical Results
6.5.2 Co integration test
It is to be noted that before estimating a VECM, cointegration test is carried out between the variables under consideration for the possibility of long-run relationship among them. For doing this the present study utilizes the Johansen (1988) and Johansen and Juselius (1990) multivariate maximum likelihood estimation procedure. Instead of putting in the detailed discussion on these procedures a brief exposition is provided here since there are
number of literatures where these methodologies have been widely discussed (Dickey et al.
1991, Cuthbertson et al. 1992).
The exposition on this technique is done by defining an autoregressive representation having a k-lag vector as follows:
Xt = α+П1Xt-1 + П2Xt-2+ ...+ ПkXt-k+ν t (t=1, 2,...,T) (1)
Here Xt and α refer to a px1 vector of non-stationary I(1) variables and a px1 vector of constant terms respectively. ν t represents px1 vector of white noises with mean zero and finite variance. П 1, П 2... Пk are coefficient matrices .
After reparametrising the above equation (1) one gets the following,
∆Xt = α + Г1 ∆Xt-1 + Г2 ∆Xt-2 + ... + Гk-1 ∆Xt-k+1 + ПXt-k + νt (2) where Г i = -(I - П1 - П2 - ... - П i ) (i=1, 2,...k-1)
and
П = -(I - П 1 - П 2 -...- П k. ) (3)
Through the estimates of Гi and П in equation (2) one can get necessary information on the short- run as well as long-run adjustments to changes in Xt. It is pertinent to note that the prime interest of the Johansen and Juselius methodology lies on the parameter matrix П, since this includes information about the long-run association among the variables in the data vector. The number of cointegrating vectors in the system is determined by the rank r of this matrix П, rk(Il).
To detect the presence of co integration among the variables, co integration test has to be conducted. In this study, the Maximum Likelihood method of Johansen (1988, 1991) test and Johansen & Juselius (1990) been adopted to conduct the co integration test for the macroeconomic variables. These results are placed in Table 6.2. While finding out the co integration two statistics were used, that is, maximum eigen value statistic ( ) and trace statistics ( ). The Johansen-Juselius, (JJ), procedure utilizes these two test statistics to determine the number of co integrating vectors.
The trace test statistic, for the null, hypothesizes that there are at most r number of co integrating vectors. The maximum eigenvalue test statistic constructs the null hypothesis as at most r co integrating vectors, and the alternative hypothesis as r+1 co integrating vectors.
It is to be noted that once variables are proven to be cointegrated, there will also be the existence of a corresponding ECM representation (Engle and Granger 1987). For co integrated model, the significance of the error correction term is tested in VECM.
For VECM specifications, Model I of the present study can be specified in the following manner. (For brevity the specifications for the other model have not been written below).
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Above equation consists of α as the constants, as cointegration vector which is the Error Correction Term (ECT) and β, , , , refer to the estimation parameters.
Since the results revealed that all series of the present study are of the same order of integration as shown in Table 6.1, one can now go for the co integration test.
Selection of lag is of utmost importance before conducting the co integration test. And Johansen test of multivariate co integration test requires specification of sufficient time lags.
Generally AIC and SBIC criteria of lag selection are chosen for determination of optimum lag length. It is worth mentioning that in case of annual data, it is seen that different studies have preferred the maximum lag length to be 2 or 3 (Enders 2004, Tang 2009, Han 2009). In the present study, accordingly, the maximal lag length is taken to be 2 while finding out the optimal lag length taking into account the relatively small sample size and nature of annual data.
Upon placing the variables through the co integration test, in the case of total crime model, the trace test reveals that there are at most two co integration vectors among the variables. Therefore, following the result of trace statistics the null hypothesis of existence of at most two co integrating relationship cannot be rejected at 5% level of significance . But here one faces problem of interpretation of result when there exist multiple co-integration
vectors. It has also been argued that the maximum eigen-value statistics may be preferred over trace statistic as this statistic has the sharper alternative hypothesis in deciding the number of co integration vectors present in a system of non-stationary variables (Enders, 2004) Therefore, in the case of total crime model, the existence of one co integration vector as confirmed by maximum eigenvalue test statistics has been taken in the present study. It is seen that in the case of Burglary-theft crime model, both the trace and the maximum eigenvalue tests led to the same conclusion – the presence of one co integrating vector. From here, one can conclude that, in both the models, the cointegration space is uniquely defined by a single vector (Harris and Sollis, 2003). All this leads one to conclude that there exists a long-run relationship between the series.
Table 6.2
Johansen and Juselius Co-integration Test PANEL 1: Total Crime
No of obs : 32 Sample : 1977-2008
Trend : Constant Lags : 1
maximum rank Ltrace 5% critical value Lmax 5% critical value
r=0 97.39 68.52 48.52 33.46
r=1 48.87 47.21 27.00 27.07
r=2 21.86* 29.68 16.23 20.97
r=3 5.63 15.41 5.54 14.07
PANEL 2: Burglarytheft Crime No of obs : 32
Sample : 1977-2008
Trend : Constant Lags : 1
maximum rank Ltrace 5% critical value Lmax 5% critical value
r=0 85.45 68.52 39.80 33.46
r=1 45.65* 47.21 22.47 27.07
r=2 23.18 29.68 16.89 20.97
r=3 6.29 15.41 5.64 14.07
Source: Author‘s calculation using STATA
Thus, from the above two panels it is seen that the Johansen co-integration test confirms the existence of a unique long run relationship among the variables, namely, total crime and burglary plus theft , economic factors like per capita net state domestic product and inflation, and deterrence factors like presence of police and number of arrests by police.
As the existence of co-integration between variables suggests a long term relationship among the variables under consideration the VEC model can be applied. The estimated equations for different crime models after Johansen normalization are shown below (Johnston and Dinardo, 1997).
Model 1: Total Crime
lcrimerate= 0.45 lcpi -0.64 lpcy+ 1.31 larrest- 1.53 lpolice +8.68
Model 2: Burglarytheft Crime
lburglarytheftrate= 1.03 lcpi – 2.41lpcy+ 2.38 larrest- 3.12 lpolice +22.52
All the estimated explanatory variables are found to significant at 1%
significance level for both the specifications.
Since all the variables are in natural logarithms the estimated coefficients are interpreted as elasticity. The co-integrating regressions result suggests that for the long-run equation, income per capita has a negative relationship with crime in the state. It is seen that increase in per capita income by 1% there would be decrease in total crime rate by 0.64%. It is quite expected that when the general economic prosperity rises there is every possibility that the crime goes down as the legal income level is higher there by increasing the cost of criminal activity. As has been noted by Fajnzylber et al (2002) economic prosperity through growth generally leads to variations in the opportunity cost of crime with the assumption that the increased activities actually have greater influence in the legal sector of the economy. All this results in decline in crime with the improvement in job availability or rising wage level.
It was seen in the trend analysis chapter four of the present study that the increased level of income per capita had led to decrease in crimes.
The long run relationship between inflation and criminal activity is positively related as expected since the rise in prices reduces the real legitimate earning for the people, thereby making illegitimate earning through criminal activity more lucrative as was obvious in essays of Becker(1968) and Ehrlich(1973). For 1% increase in inflation, total crime rate is increased by 0.45%. This coefficient was found to be significant at 1% level of significance. In several
studies the association between inflation and crime rates have been indicated (Allen,1996, Omotor, 2009, Torruam et al.,2014,Seals, 2007)).
In the present study, the expected sign is seen in the estimated coefficient of one of the deterrent variables, namely, number of civil police. As seen in the estimated equation of the total crime the more the number of police the less number crime is committed. Thus presence of police is an important deterrence factor against crime. The estimated elasticity is found to be quite high which shows that 1% increased in civil police man would lead to reduction in burglary-theft crime rate by 3.12% and total crime rate by 1.53%. Often there is public hue and cry to raise the members of police as a measure to counteract the rising rate of crime. It is widely believed that greater the presence of police in the streets and crime infested areas, better is the crime prevention because the show of police patrol as well as the quick investigation of criminal cases would act as deterrent to committing crime. Negative relationship may point to the fact that the more the cases registered there are demands for recruitment or deployment of more police which would have a deterrence effect on crimes and finally that would mean less crimes being committed as Levitt (2002) showed this in case of USA. Accordingly, one can come across several analytical studies by taking empirical data to examine the linkage between crimes and numbers of police available in the law enforcement department (Levitt 1997;Marvell and Moody,1996; Nagin,1998)
However the behavior of the variable arrest needs more explanation. The sign of the coefficient is found to be different than what is normally expected since the positive sign of the coefficient of variable arrest indicate that there is positive association between crimes and arrest of criminals by police. It may be mentioned that the fear of incarceration through imprisonment can act also deterrence factor as it raises cost of crime. Thus incarceration has both the deterrence and incapacitation effects. Though there should be an inverse relationship between incarceration and crime, the association does not seem to be that straight forward and obvious. This is due to ‗hardening effect‘ or adverse peer group reaction that the criminals go through due to incarceration which may result in more participation in crimes (Chen and Shapiro , 2004). Thus the extent of crime reduction through incarceration would depend on the resultant effect of these opposing forces. Dills et al (2008) observed that arrest rates and crime correlate positively, contrary to what is expected. This positive correlation does not prove the arrest probability has no deterrent effect, nor does it imply that higher
arrests increase crime. Also it is seen in the chapter four of this study on trend analysis that most of the arrest didn‘t result in incapacitation of offenders in the jail because of poor investigation leading to low rate of charge sheet and conviction. When crime rate increases there is rise in work pressure on the criminal justice system impinging on the effectiveness and efficiency of the police and the court system which has to suffer from delay in their sentences with more number of charged offenders. (Han, 2009). This means that though figures of arrest increase, the lack of concomitant incapacitation would lead to lack of desired effect on the crime.
Also, it may be noted that a number of empirical studies have questioned how the authorities and the prevention policies can combat crime better. Different variables have been tested, such as the growth of police force (De Oliveira, 2003), the money spent for the appropriate equipment (Imrohoroglu et al. 2000), people who have been arrested (Corman et al. 1987, Corman & Mocan, 2000), convicted (Pudney et al. 2000; Funk & Kugler, 2003) or sentenced to imprisonment (Levitt, 1996). The results are still ambiguous, but it seems that the possibility of sentencing and conviction are more effective ways for crime prevention than the others. That is because, in most cases, criminal actions are not always connected with arrests, and arrests do not always lead to convictions and imprisonments. However, since the sample size in the present study is not a big one there must be caution in accepting the conclusion here.
It is to be noted that the sets of elasticity of the coefficients of estimated equations are found to be different for different crime models which means that the dependent variables respond in different degrees to the changes in explanatory variables.
Specifically, as seen in the above estimated models, economic variables (income and price index) have a larger effect on burglary and theft in comparison to total crimes, as the former type of crimes are property related offences. Even in case of deterrence factors like police presence and arrests, one can see the higher degree of impact by these factors on burglary- theft crime.
It would be worthwhile to proceed to the short run relationships between the different variables and their adjustments towards the long run equilibrium relationship. A look at the error correction terms for both the models shows that though they are positive,
they are insignificant. This might be the result of the lack of sufficient lags arising from paucity of data that might result in insignificance (Raja and Ullah 2013, Miller 1991). Also it is argued that although the coefficient on the error correction term must be negative for the stability of the long run equilibrium relationship for a bivariate model, it need not be the case for a multivariate model due to the interplay of numerous feedback mechanisms between the different variables in the models (Ansari and Ahmed, 2007;Rousseau and Wachtel,1998;
Enders,1995). It is also seen that inflation does not play a role in influencing crime in the short run. This is expected, as it takes some time for the purchasing power of an individual to be reduced as a result of an onset of inflation (Tang, 2009). On the other hand, police presence has a significant impact (at 10% level of significance) on the total crime rate in the short run. A rise in police presence significantly reduces the crime rate in the short run, which is fully according to expectations.