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The Role of Extralegal Factors In Determining Criminal Case Disposition Steven Garber, Steven Klepper, and Daniel Nagin INTRODUCTION The major participants in the criminal justice process exercise substantial discretion. An issue that has received considerable attention is the degree to which the existence of such discretion results in systematic inequities in the disposition of criminal cases. In particular, numerous empirical studies have examined the extent to which members of racial minority groups and/or disadvantaged social classes are treated more harshly because of their race or socioeconomic status. Most of the empirical studies on case disposition use regression and related statistical techniques. Correlations between outcomes of the various processing stages in the criminal justice system and measured case and defendant characteristics are examined. Discrimination is analyzed by testing for an empirical association between extralegal characteristics, such as race and socioeconomic status, and various decisions in the criminal justice system, holding constant observable, legally relevant case and defendant characteristics. A fundamental problem with this approach is that many of the important factors affecting case disposition are We thank Alfred Blumstein, Jacqueline Cohen, and Franklin Fisher for their helpful comments. 129
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130 extremely difficult to measure. In particular, the seriousness of an offense and the quality of the evidence, perhaps the two most important factors affecting case disposition, involve important elements for which researchers typically can observe no data. When seriousness and case quality are correlated with race and social status, the techniques currently being employed yield biased estimates of the effects of extralegal factors on case disposition. This possibility is particularly troublesome for studies of discrimination in the criminal justice system because even if discrimination is present and of sufficient concern to warrant reform, extralegal factors are undoubtedly of secondary importance in explaining variations in case disposition. Under these circum- stances, biases attributable to measurement error may dominate the estimated effects of extralegal factors. Thus inferences about the incidence of discrimination based on standard regression techniques may be seriously distorted and are unlikely to provide a reliable basis for policy reform. One response to this problem is to measure more accurately the primary determinants of case disposition. However, the inherent unobservability of a number of the components of the primary determinants suggests strongly that this strategy is unlikely to resolve the ambiguities that plague existing studies. We propose an alternative approach known as structural equation modeling. It involves explicit mathematical representation of the fundamental mechanisms believed to generate the data. For the study of discrimination in the criminal justice system, it involves modeling the fundamental relation- ships linking observable case outcomes to both their observable and unobservable causes. If a sufficient number of decisions affected by the unobservable principal determinants of case disposition are observed, it is possible to control fully for forces that cannot be observed. It is then possible in principle to make inferences about the extent of discrimination that are not distorted by the inevitable lack of accurate measurements of the primary determinants of case disposition. The methods we propose are relatively complex, but we know of no simpler way to control for the effects of unobservable variables. The paper is organized as follows. First we review nine recent and influential empirical studies of discrimination in the criminal justice system. The major
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131 purpose of the review is to provide motivation and background for the discussion that follows. In the next section we discuss statistical implications of the impossibility of measuring accurately the primary determinants of criminal case disposition. We then illustrate the import of these statistical issues by presenting alternative interpretations of various results reported in the literature. We then illustrate the proposed approach by presenting a structural equation model of criminal case disposition. We model nine decisions affecting the criminal process, taking explicit account of the measurement difficulties discussed. The next section is a discussion of the estimability of the parameters of our illustrative structural model and an example that illustrates how the effects of unobserved variables can be estimated. We then provide a heuristic discussion indicating how our illustrative model aids in the effort to obtain less ambiguous data summaries. In the next section we indicate briefly how future studies might take account simultaneously of the measurement issues emphasized here and the sample selection issues discussed in Klepper et al. (in this volume). The next section is a brief discussion of the trade-offs in specifying alternative structural models of the criminal justice system. The final section contains concluding remarks. EMPIRICAL STUDIES OF DISCRIMINATION I N THE CRIMINAL JUSTICE SYSTEM This section reviews nine recent and influential studies on the incidence of discrimination in the criminal justice system. The studies are of three kinds: studies of the choice of sentence given conviction, studies of case disposition] given arrest andVor indictment, and one longitudinal study of forcible sex offenses from arrest through sentencing. Some of the studies analyze samples combining dissimilar offenses, whereas others concentrate on specific offenses ranging from theft to murder. We first discuss the studies of sentence given conviction and case disposition given arrest. This is followed by a review of studies on the various stages preceding sentencing, beginning with the conviction process and working backward to the choice of plea, release on and setting of bail, choice of legal
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132 representation, charge, and the decision to prosecute. Since only LaFree (1980a) examines separately the stages preceding sentencing, it was necessary to supplement the nine studies with a few additional studies of the stages preceding sentencing. We conclude with a summary of the major findings of the various studies. Case Disposition and Sentencing Studies The various studies of case disposition and sentencing focus on a small number of common forces. They include: (1) Seriousness of the offense. Nearly all the studies include a measure of the charge to control for seriousness of the offense. The exceptions are Farrell and Swigert (1978), Swigert and Farrell (1977), and Chiricos and Waldo (1975), which concentrate on specific offenses. Some of the studies also try to use characteristics of the offense to control more completely for seriousness. For example, Lizotte (1977:569) takes account of such factors as whether the defendant resisted arrest, the number of defendants, the sobriety of the defendant, injury to the victim, and the value of property taken. For forcible sex offenses, LaFree (1980a) considers such factors as whether a weapon was used and the type of offense (i.e., rape or attempted rape). (2) Prior record. All of the studies include a variable to represent the criminal history of the defendant. Measures of prior record range from a dummy variable indicating whether the defendant was ever arrested to the total of the maximum statutory penalties of the defendant's prior convictions. (3) Type of legal representation. A number of studies examine the choice of legal representation, distinguishing no attorney, a public defender, and privately retained counsel. The choice of legal representation is expected to affect primarily the probability of conviction and the sentence resulting from a plea bargain. Legal representation is not generally viewed as affecting sentence if the defendant is convicted at trial, although Tiffany et al. (1975) include a measure of legal representation in their sentencing study of defendants convicted at trial.
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133 (4) Release on bail. A number of the studies include a variable indicating whether the defendant was released on bail. Being out on bail is expected to improve the defendant's ability to develop an effective defense, which is expected to be helpful both at trial and in plea bargaining. (5) Type of conviction. Some of the studies that combine guilty pleas and trial convictions include an additive dummy variable denoting whether the defendant pleaded guilty. In studies of case disposition, a plea of guilty is generally expected to lead to a worse outcome in that it precludes acquittal. In studies of convictions, guilty pleas are generally assumed to be the result of plea bargains and hence are expected to result in lighter sentences, ceteris paribus. (6) Miscellaneous factors. Some of the studies include the age of the defendant, whether the defendant is employed, and the type of county (urban versus rural) in which the defendant is convicted. (7) Discriminatory or extralegal factors. Various characteristics of the defendant that are not legally relevant are included in all the studies. They include race, socioeconomic status (SES), sex, and the racial composition of the victim-defendant dyed. In addition, Clarke and Koch (1976) use the average income in the Census tract in which the defendant resides as a measure of the defendant's income, while other studies use SES as a proxy for income. Swigert and Farrell (1977) distinguish a characteristic they label "normal primitive" to denote particularly lower class, black defendants who are (stereotypically) thought to be disposed toward violent behavior. The conclusions of the various studies of final case outcome can be summarized as follows. First, virtually all the studies that include a variable measuring the charge found that the seriousness of the offense is the most important factor affecting case outcome. This is most evident for studies that analyze only convictions. Second, all the studies conclude that the prior record of the defendant is important. Third, all the studies that include a variable denoting whether the defendant makes bail infer that it is an important factor in case outcome. Fourth, most of the studies that include legal representation found that it affects case outcome, but the nature of this effect varies considerably among the
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134 studies. Clarke and Koch (1976) and Tiffany et al. (1975) conclude that for some types of cases legal representation affects the sentence received, while Hagan (1975), Lizotte (1977), Farrell and Swigert (1978), and Swigert and Farrell (1977) infer that legal represen- tation matters principally through making bail and secondarily through choice of plea. Fifth, type of conviction generally seems to be important: Defendants who plead guilty fare worse on average than those who plead not guilty (Hagan, 1975:541; Farrell and Swigert, 1978:449; Swigert and Farrell, 1977:26) but fare better than defendants who are convicted at trial (LaFree, 1980a:850). The inferences concerning the role of extralegal characteristics differ considerably across the studies. One point of agreement is that if extralegal character- istics affect case outcome, their quantitative signi- ficance is small compared with the other factors discussed above. This view is consistent with Hagan's (1974) review of earlier studies. Most of the studies find a role for some extralegal characteristics, and different characteristics appear to be important in different studies. Swigert and Farrell (1977) and Farrell and Swigert (1978) infer that for murder cases, SES has a significant effect on case outcome, holding constant a number of other factors. LaFree (1980a) found that for forcible sex offenses, cases involving white victims and black defendants are generally treated more harshly. Clarke and Koch (1976) infer that for burglaries and larcenies, defendants with lower incomes are more likely to be imprisoned. They attribute most of this effect to the correlations between income and making bail and income and the choice of legal representation. Tiffany et al. (1975) found that for defendants with no prior record, blacks receive significantly more severe sentences, holding constant a number of other factors. Some studies that did not find a direct role for extralegal characteristics in determining case dispo- sition suggest an indirect role for such factors. Hagan's (1975) results suggest that individuals with lower socioeconomic status in Canada are charged with more serious offenses and that charge directly affects case outcome. Lizotte (1977) infers that race and SES play important roles in determining whether the defendant is released on bail, which in turn has an important effect on case outcome. Only the results of Gibson (1978) and Chiricos and Waldo (1975) suggest a role
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135 neither for race nor SES, although Gibson does present some evidence of racial discrimination on the part of some judges.2 Overall the studies suggest that low-status blacks fare worse in the criminal justice system than other defendants. Below we examine the information provided by the various studies concerning the extent to which this disadvantage is attributable to events prior to the sentencing decision. The Conviction Process LaFree (1980a, 1980b) focuses on factors affecting conviction at trial for forcible sex offenses. None of the other studies focuses directly on the conviction process, although Clarke and Koch t 1 qlh) nr~vi ~" =~m" ~ ~, ,= ~ evidence concerning tne conviction Process tor burglaries . _ . _ _ ~of: ~ _ ~. t · . ~ ana larcenies. 1nalrect evidence about the conviction process is also provided by the studies of Lizotte (1977), Hagan (1975), Farrell and Swigert (1978), and Swigert and Farrell (1977), all of whom examined case outcome following arrest or indictment. The various studies emphasize two types of factors: quality of the evidence and prior record of the defendant. For forcible sex offenses, LaFree (1980b) constructed measures of the quality of the prosecution case and the quality of the defense case. He included other variables, such as misconduct on the part of the defendant and the victim's living arrangement, to proxy for whether the alleged act was voluntary. Lizotte (1977) also recognizes the importance of such factors. He constructed an index that represents the availability of 10 different types of evidence, including such factors as the number of eye witnesses, length of time between arrest and the incident, the recovery of a weapon, etc. (Lizotte, 1977:568-9). Clarke and Koch (1976) also constructed an admittedly crude measure of the quality of the evidence using the length of time elapsed between the offense and the arrest. LaFree (1980b) used a measure of promptness of the report of the offense to the police. Prior record is also cited in some of the studies. LaFree (1980b:843) notes that despite legal procedures intended to conceal from the jury the defendant's prior record, it was often inferred by jurors from other testimony or through the defendant's failure to testify. Clarke and Koch (1976:72) conjecture that prior record
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136 might affect the prosecutor's efforts to convict the defendant. Swigert and Farrell (1977) and Farrell and Swigert (1978) also consider the role of prior record in the conviction process. The findings of the various studies suggest that both the quality of the evidence and the defendant's prior record affect the likelihood of conviction. LaFree (1980a, 1980b) infers that both factors are relevant in forcible sex offenses. Lizotte's (1977) results concerning the role of the quality of the evidence are inconclusive, but he attributes this to the equivocal nature of his index when applied to different types of crimes (Lizotte, 1977:57). Clarke and Koch (1976) found a minor role for the promptness of arrest and an insignificant effect of prior record, although their results are difficult to interpret since cases settled by guilty plea as well as at trial were considered jointly. Only LaFree (1980a) examined the role of the race of the defendant in affecting the likelihood of conviction at trial. He did not find a significant role for race. The Plea Decision A number of the studies examined the choice of plea. None of the studies proposes an explicit theory of the plea bargaining process. The choice of plea is approached in an exploratory fashion, with different researchers examining the role of different factors. LaFree's (1980b) analysis is the most detailed investigation of the plea decision. He found that the amount of evidence assembled by the defense is an important factor in the choice of plea, with the accumulation of more evidence lowering the probability of a guilty plea. Concerning the role of race, he found that black defendants were less likely to plead guilty, regardless of the race of the victim. He was unable to determine whether this is attributable to the attitude of the prosecutor or the defendant or both. Hagan (1975) also analyzed the choice of plea. He concludes that defendants charged with more serious offenses and represented by private counsel are less likely to plead guilty. He found no role for race or SES. The other study that considered the choice of plea is Swigert and Farrell (1977). They conclude that the single most important factor affecting the choice of plea
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137 is the perceived characteristics of the defendant, and those classified as normal primitive are more likely to plead guilty. Release on Bail Three factors are cited as affecting the decision to make bail: the amount of bail, the income of the defendant, and the defendant's legal representation.3 The role of the first two factors is obvious. The role of legal representation is less obvious and differs across the studies that consider it. The importance of bail amount is supported by Lizotte (1977:5711. Clarke and Koch (1976:83) found that the defendant's income is an important factor in the ability to make bail. Lizotte (1977:571) provides evidence of a role for race and SES in making bail, which he interprets as proxies for the defendant's income. Lizotte (1977:572) and Swigert and Farrell (1977:25) found significant but somewhat opposite roles for legal representation. Setting of Bail Only Lizotte (1977) analyzed the setting of bail, although other studies contain speculation concerning the determinants of the bail amount. Lizotte (1977:571) found that seriousness of the offense, the defendant's prior record, and the defendant's legal representation influence the bail amount. Defendants represented by courtroom regulars and public defenders on average are required to post lower bonds. Lizotte (1977:566) offers, but is not able to test, the hypothesis that the quality of the evidence also affects the level of bail. He did not find a significant effect of race or SES on bail amount, although the results of Swigert and Farrell (1977:25) on making bail suggest possible discrimination against "normal primitives" in the determination of the level of bail. Choice of Legal Representation Defendants can choose either no attorney, a public defender, or a private attorney. (Lizotte further
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138 distinguishes between courtroom regular and nonregular private attornies.) This choice was studied by Lizotte (1977), Hagan (1975), Farrell and Swigert (1978), and Swigert and Farrell (1977). Four factors are cited: the seriousness of the offense, the prior record of the defendant, the quality of the evidence, and extralegal characteristics of the defendant. The seriousness of the offense and the prior record of the defendant are included as predictors of the sentence the defendant would receive if convicted. The quality of the evidence is included as a predictor of the probability of conviction. It is expected that the greater the probability of conviction and the more serious the offense, the greater the incentive of the defendant to retain higher-quality legal representation. A private attorney is assumed to mount the best defense and no attorney the worst. The primary extralegal character- istic that is expected to affect the choice of attorney is the defendant's income. Other characteristics of the defendant are included only to proxy for income when income is not observed. The results of the various studies suggest that seriousness of the offense is the most important determinant of choice of attorney (Lizotte, 1977:570; Clarke and Koch, 1976:83; Hagan, 1975:541). None of the researchers was able to measure the quality of the evidence and hence none can test its effect on choice of attorney. However, Clarke and Koch (1976:83) found that case outcome and choice of attorney are highly corre- lated, those who choose no attorney having a much smaller probability of conviction and imprisonment. Prior record appears to affect the choice of attorney in Lizotte (1977) and, to a lesser degree, in Swigert and Farrell (1977): Those with more extensive prior records are more likely to choose either no attorney or a private nonregular attorney in Lizotte (1977:570) and a public defender in Swigert and Farrell (1977:23). Hagan's (1975:541) results, however, suggest that the effect of prior record is insignificant. As for extralegal characteristics of the defendant, the results of Clarke and Koch (1976:83) suggest a role for income, and Farrell and Swigert (1978:448) and Swigert and Farrell (1977:23-24) found a role for SES, which they interpret as a proxy for income. In contrast, Hagan (1975:541) and Lizotte (1977:571) found no role for race or SES in the choice of attorney.4
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139 The Charge Decision The charge decision was analyzed in detail only by LaFree (1980a). The role of race (alone) was considered by Hagan (1975). For forcible sex offenses, LaFree (1980a:850) found that the type of offense (i.e., attempted rape or rape), the use of a weapon, and victim preference are important elements of the charge decision. These variables are interpreted primarily as measures of the seriousness of the offense (LaFree, 1980a:852). LaFree (1980a:850) also found that the racial composition of the victim-defendant dyed affects the charge decision; cases involving a black defendant and a white victim led to a more serious charge. Hagan (1975:541) also found that SES is correlated with charge: Individuals with lower SES were charged with more serious crimes. Hagan did not control for other factors (such as seriousness), presumably because of a lack of data. The Decision to Prosecute The only study that focused on the decision to prosecute is LaFree (1980a). He found that for forcible sex offenses, the charge, the presence of witnesses, the use of a weapon, and the defendant's age are important determinants of the decision to prosecute. These findings generally agree with Frase's (1980) detailed investigation of the reasons given by U.S. attornies for dismissals. Frase found that the three factors cited most often for dismissing a case are the seriousness of the offense, the quality of the evidence, and the defendant's prior record. LaFree (1980a:850) also found that the racial composition of the victim-defendant dyed affects th e decision to prosecute; cases involving a white victim and a black defendant less likely to be dismissed. Frase did not consider the role of race or other extralegal characteristics in his study. Sublunary of the Major Findings Virtually all the studies suggest that three factors ar of particular importance in the processing of cases
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173 bargain decision into the model because most convictions result from guilty pleas. Using the plea bargain theory in Klepper et al. ~ . . . (in this volume), an additional unobservable variable representing culpability might be introduced into the model. Another desirable extension of the illustrative structural model would involve use of an explicit theory to structure the suspected correlations between race and seriousness and case quality. In the present formu- lation, for example, a positive estimate Of 611 would ~na~cate the existence of a correlation between race and seriousness but would provide no information concerning the source of such a correlation. Recall from the discussion that the literature contains at least three reasons to expect race and seriousness to be correlated. If the source of the correlation is labeling, as emphasized by Lizotte, then we might interpret this as legislative discrimination. If the economic basis to expect a correlation is operative, we might interpret 611 as reflecting discrimination in the labor market. v~s~ngu~sn~ng among such possibilities would seem to be quite desirable for policy purposes. Finally, we might not be able to observe directly anything about the financial capabilities of defendants. If so, wealth could be modeled as an unobservable. Then variables such as the average income in the census tract containing the residence of the accused might be regarded as a classical proxy and hence an indicator of the wealth of the defendant. When other indicators of wealth are observable (e.g., level of bail, release on bail, choice of attorney) models treating wealth as an unobservable are likely to be identified. Finally, we briefly mention specification issues relating to the fact that some of the indicators we have defined may not be directly observable. In some instances such variables are modeled as direct causes of other indicators or latent variables. An interesting specification issue is whether such variables should be viewed identically in their roles as both indicators and causes. For example, consider Y2, which was defined as an index determining the probability that the victim cooperates with the prosecution. This variable also appears in equation (13) as a factor contributing to case quality. One might interpret Y2 in equation (13) as a dichotomous outcome determined-by the continuous-index version Of Y2. But victim cooperation might be more plausibly viewed as a continuous variable representing _ . . . . . .
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174 the enthusiasm with which the victim cooperates. Similar remarks apply to Y6 (i.e., attorney choice), which appears as an indicator in equation (8) and a cause of the trial outcome in equation (9). It seems clear, however, that y3 (i.e., charge) and y5 (i.e., release on bail) are most plausibly viewed as qualitative outcomes of the indexes defined in equations (5) and (7) when they appear as observable causes in equations (8) (i.e., choice of attorney) and (15) (i.e., the case Duality at trial), respectively. A useful paper in this regard is Beckman (1978), which presents an estimation scheme for situations involving endogenous qualitative regressors. Finally, the fact that we observe only partial information about various indicators raises estimation issues, even if these variables do not also appear as causes of other variables. Estimation would require explicit modeling of the links between the observed data and the indicators of the illustrative model. For example, Muthen (1979) discusses latent variable models with qualitative indicators. Maximumrlikelihood techniques, while somewhat complex in this context, are likely to provide a feasible approach to estimation CONCLUDING REMARKS , ~ In our view, the empirical literature on the criminal justice system has evolved in a natural and appropriate way. Total correlations between case disposition and race and YES suggest an alarming degree of discri- mination. Studies aimed at probing the source of these correlations have attempted to control for the effects of legally relevant variables using crude, albeit the best available, proxies for such factors. reviewed are of this variety. The studies we For the most part, they still find evidence of discrimination, although less than is suggested by the total correlations between case disposition and race and SES. The fact that inclusion of legally relevant variables reduces the correlations between case outcomes and variables such as race and SES provides empirical support for theories that predict correlations between legal factors and personal characteristics of defendants. Under such circumstances, failure to control fully for the effects of legally relevant factors implies that inferences about the extent of discrimination are likely to be erroneous. For this reason the techniques
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175 currently being used offer little hope of providing a reliable basis for policy reform. The approach proposed here seems like a logical next step. Satisfactory resolution of the role of extralegal factors in determining criminal case disposition will be difficult. Structural equation mode ;na chest helm hill it is not a panacea. The restrictions in any identified structural model are likely to be controversial. The possibility of compatible indications emerging from a broad range of structural models raises the hope of developing a consensus. NOTES 1. By case disposition we mean the outcome of a case following arrest andVor indictment. The alternative outcomes include acquittal, dismissal, and various types of sentences (given conviction). Case disposition is generally measured by an index that (arbitrarily) is commensurate to the different types of outcomes. 2. Klepper et al. ~ . ~ . . nine squares, these two are the most sensitive to statistical biases due to sample selection. Conse- quently, the extent of discrimination may be particularly underestimated in these two studies because of the specially selected nature of their samples. (in this volume) argue that among the - 3. Other factors such as prior record and extralegal characteristics of the defendant are also cited (for example, Swigert and Farrell, 1977:25; Farrell and Swigert, 1978:447), but it seems these factors are expected to operate through bail amount (only Lizotte holds constant the effect of bail amount). 4. Although Lizotte (1977) used a strange ordering for the quality of different types of legal representation. 5. For convenience, throughout this paper all random variables are expressed as deviations from their means. 6. An estimator is unbiased for a parameter if its mathematical expectation is equal to that parameter. An estimator is consistent if in the limit as the sample size goes to infinity, the estimates are arbitrarily close to the parameter of interest. ~ ~ Strictly speaking,
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176 consistency of least squares requires additional (but in this context uninteresting) assumptions concerning the sampling process on x* and z. 7. When ~ equals one and ~ is independent of x*, z, and u, x is called a classical measurement of x*. The results discussed here are straightforward extensions of results discussed by Garber and Klepper (1980) for the case of classical measurements. 8. Formally, f = V(£)/V(xlz). 9. This result is due independently to McCallum (1972) and Wickens (19721. For cases in which more than one variable is measured with error, this result does not generalize straightforwardly (see Garber and Klepper, 1980). 10. These conditions follow from faD/8 = fPx~z[Py~x*lzV(ylx*)/py~zlx*v(ylz)] , where P ~x*lz and Py~zlx* are respectively the correlation coefficients of y and x* given z and y and z given x* and Px~z is the correlation coefficient of x and z. This result can be derived by exploiting the relationship between regression coefficients and-the second moments of the respective conditional distri- butions. 11. An especially valuable collection of theoretical and empirical studies is Goldberger and Duncan (1973). Our discussion borrows from Goldberger (1973), the introduc- tory essay in this volume. 12. We consider residential thefts to include the crimes of breaking and entering, petit larceny, grand larceny, second-degree burglary, first-degree burglary, etc. We avoid the plea bargain issue, despite its importance, because of the apparent lack of any widely held views concerning the determinants of the plea bargain decision. Incorporation of especially contro- versial relationships in the illustrative model could seriously compromise our objectives. 13. In the Washington, D.C., superior court about 50 percent of all arrests are rejected at the initial
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177 screening or subsequently nailed by the prosecutor (Forst et al., 1977). 14. We use 0's throughout for coefficients of unobservable variables and B's for coefficients of observable variables. The first subscript of each coefficient refers to the indicator and the second to the respective unobservable or observable exogenous variable. 15. Discrimination is presumed to be on the basis of race and wealth. The two variables together can be interpreted as the primary components of the SES of the defendant. Alternatively, we might have introduced another observable variable to represent the SES of the defendant. The additon of an accurate measure of SES would not alter the model in any fundamental way. We assume discrimination on the basis of race and wealth alone only to simplify the exposition. 16. Typically, the bail is paid by a bail-bonding agency. The accused pays a nonrefundable fee based on the amount of the bail. Equation (7) can be interpreted as describing the decision to pay this fee. 17. Recall that we model only those cases of residential theft disposed by a dismissal or trial verdict. 18. As will become clear with the introduction of equation (15) below, we distinguish between case quality before and at trial to examine the claim that release on bail is an important determinant of the quality of the defense. 19. As will be more clear when equation (13) and the stochastic assumptions pertaining to £1 are introduced, we attempt to use observable variables to control entirely for correlation between xl* and x2*. Education is viewed as an important correlate of each of these variables. 20. In the section on additional specification issues, we consider treating prior record, or more precisely all the determinants of prior record, as known and observable. 21. Since xl*, x2*, X3*, and X4* are never observed, the scales on which these variables are measured are arbitrary. Until these scales are specified, the
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178 magnitudes of their coefficients are trivially indeter- minate. In order to remove such indeterminacies it is necessary to "normalize" these coefficients by directly or indirectly specifying the units of measurement of each of the unobservable variables. Specific normalizations are chosen for analytic convenience. The coefficient of x2* in equation (15) is chosen as one to specify the scale on X4*, given the normalization (presented below), which defines the scale of x2*. 22. Note that y5 was specified as an index variable in equation (7) but as a dichotomous outcome in the present equation. The complications associated with this type of specification issue are discussed in a later section. 23. Inspection of equations (3) through (15) reveals that w77 and V(£4) are not identified. This is obvious since both X4* and U7 appear only in equation (9). Thus in this model randomness in convictions (represented by U7) cannot be distinguished empirically from random influences on the quality of the evidence at trial (represented by E4). This lack of identifiability is of minimal concern since neither all nor V(~4) is of direct interest. 2 4. The second step involves solving for the structural parameters as functions of the reduced-form parameters (this is illustrated by equation (7)). Use of Slutsky's Theorem (see Goldberger, 1964:118-119) establishes that these solutions provide a basis for consistent estimation of the structural parameters. 25. This follows from the fact that the reduced-form disturbances are uncorrelated with the independent variables of the reduced form. 26. These population variances and covariances can be computed quite simply from Table 3-2 since the ui (i = 1, 2, . . . , 9) and ej (j = 1, 2, 3, 4) are mutually uncorrelated. 27. Various symbols (such as Y1, v1, etc.) are redefined here in order to emphasize the analogies between the simplified model employed here and our illustrative model of the criminal justice process .
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179 28. These features are the ones that enable us to estimate the various mixtures of structural parameters appearing in the reduced-form disturbances of our model. 29. To see how these expressions are computed, consider for example equation (19d). Since all variables are assumed to have zero means, the covariance of Y1 and Y2 can be computed as E(yl~y2) = E(8lz + Dlx* + Ul)(02Z + 02x + u2) Equation (19d) then follows from the assumptions that z, x*, ul, and u2 are independent and V(x*) = 1. 30. The apparently trivial nature of equation (19;) merely reflects the fact that since z is observable its variance can be estimated directly. 31. This expression for 81 results from the use of equations (led), (lee), (19f), (leg), (19h), (19i), and (19j). It can be verified straightforwardly by substitution of these equations into equation (20) . 32. For example, consider how we checked that y54 is identified. Table 3-1 indicates that the coefficients of Z3 in the reduced-form equations for y5 and y4 are respectively Y541042023 and \042623 Division of t former coefficient by the latter provides a solution for r 54 and thus a basis for consistently estimating y54. 33. The assumption that r~1 exists says merely that values of the x* and z variables and the values of the structural disturbances uniquely determine the values of the indicators. 34. It might be argued that race would enter into this decision: Perhaps anticipation of racial discrimination would affect the accused's decision concerning testi- fying. If that is the case, however, this indicator would still be quite valuable. Suppose that defendants anticipate discrimination in the way described by the other structural equations. In that case race will affect the testimony decision through its effect on expected sentence. This would provide other restrictions that would be useful in disentangling the various structural parameters associated with race.
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180 35. Dealing with sample selection requires the use of a specific distribution because the observed sample is viewed as a random sample from a truncated distribution. Assuming normality per se is not required. 36. The expressions in equation (23) correspond to those in equation (19) and are derived using results reported in Johnson and Kotz (1970:81-83; 1972:70). 37. Perhaps the easiest way to verify this is to solve the expressions in equation (23) for the unconditional moments given by the expressions in equation (19). As reported above, knowledge of these moments is sufficient to identify all of the structural parameters. A particularly helpful fact in checking identification here is that a = V(yl) can be consistently estimated by the sample variance of Y1 computed from all of the observations on Y1. 38. Although it does complicate the form of these moments. 39. Formally, they can be eliminated by substitution of y4 from equation (6) into equation (7) and substitution of Ye from equation (10) into equation (11). 40. We did not invoke these restrictions in the illustrative model because incorporation of controversial assumptions would compromise the major purpose of this paper. 41. Formally, this involves assuming that the variance of £3 (see equation (14)) is zero. The attractiveness of such an assumption certainly depends on the extensiveness of the available information concerning prior record. 42. Note, however, that one would not want to delete equations representing decisions resulting in sample selections because it is precisely the structure provided by these equations that allows us to correct the sample selection bias.
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Representative terms from entire chapter: