How many instrumental variables

Students entering school later can leave school after a shorter period of learning because compulsory schooling depends on the age of the student 16 or 17, according to the state. Hence, males born early in the year receive less schooling than males born later in the year do.

Footnote 2 This situation reflects a good exogenous instrument in which schooling is correlated to institutional rules. It has no causal effects on earnings because the month of birth is random Angrist and Krueger , p. Therefore, Angrist and Kruger used the quarter of the birth year for estimating the unbiased effects of schooling on earnings.

Interestingly, the effect did not differ significantly from estimates based on classical OLS models showing small but significant effects of schooling on earnings.

The estimation of the causal path between age and cognitive proficiency is problematic because of several factors, such as grade retention, delayed entry, and entry grade acceleration. In such situations, the assigned relative age i. On one hand, because most students enter school on time and are never retained, the relative age is correlated with the actual age.

There are no explanations for why students born at different times of the year are not more or less smart than others. Birth dates and institutional rules among others were also used recently in other large-scale assessment studies.

In estimating the relationship between cognitive skills and earnings, Hanushek et al. The first set, defined for all PIAAC countries, contained two variables: years of schooling and parental education.

They argued that these variables could be used as instruments because they influenced skill development but were determined before the individual entered the labor market.

However, the authors were skeptical about these instruments, pointing out that family background may exert direct effects on earnings, and ability may show intergenerational persistence Hanushek et al. Therefore, this set is a good example of the utilization of bad instruments, where assumptions necessary for IVs are not likely to hold. In such a situation causal inference is likely to be biased.

The second approach is both interesting and credible. Because of the limitations of the data availability, the second approach was used in the US sample only. Hanushek et al. This strategy was also used in other contexts e. The individual minimum school-leaving age was used as an instrument for skills. Because this information was related with skill level but not earnings it seemed to provide a good instrument that confirms a positive relationship between cognitive skills and income.

Another interesting use of IVs that emerged from the variable of institutional rules is a study by Angrist and Lavy Estimating the causal effect of class size on scholastic achievement could be complex because classroom size is correlated with many unobservable characteristics, such as popularity of the school, quality of teaching, quality of management, available resources, etc.

The use of this instrument is based on the bureaucratic rules that set the maximum number of students in a classroom. In this scenario, the instrument is the variable that concerns whether the school must create additional classrooms in order to avoid the maximum number of students in a classroom. This variable is negatively correlated with class size, but it is hard to anticipate the causal relationship between the instrument and student achievement as all schools were obligated to follow the rule regardless of size, quality of teaching, or location.

The IVs estimates show that reducing class size induces a significant and substantial increase in test scores. Currie and Yelowitz introduced a very interesting application of IV, which was based on institutional rules. Estimates using the OLS regression can be biased because the participants in the program might have unmeasured traits that contribute to poor housing outcomes and poor academic performance of their children. To overcome this problem, the authors used an IVs technique that was based on the institutional rules that were mandatory for the participants.

According to these rules, a family with two same-sex children would be eligible for a two-bedroom apartment, whereas a family with opposite-sex children would be eligible for a three-bedroom apartment. Consequently, households with one boy and one girl are far more likely to be in public housing than are households with two boys or two girls because sex composition affects the size of the subsidy for which the family is eligible.

The gender composition of the family appears to be a valid instrument because it correlates with the participation in program. However, as claimed by the authors, who reviewed the literature that investigated the relationship between sex composition and educational attainment, gender composition showed no causal links with outcomes. Using the IVs technique, Currie and Yelowitz found that the participants in the public housing program lived in better material conditions, and their children were less likely to be left behind than households that did not participate in the program, which was not consistent with probably biased OLS estimates.

The last example of an instrumental variable based on institutional rules refers to the work of Angrist and Lavy This study was subsequently replicated by Machin et al. The IVs technique was applied to a situation similar to experimental settings. In their analysis, Angrist and Lavy aimed to estimate the effect of using technology in teaching on student achievement. Although simple OLS estimates might not yield a causal effect because the scholastic achievements of students and the use of computer technology might correlate with unobservable factors i.

The authors decided to use the IVs technique, which was possible because the Israeli state lottery funded a large-scale computerization program in elementary and middle schools. Participation in the program was not random but defined by a set of rules that allowed program participation.

The authors claimed that the participation rules were not systematically associated with the student outcomes. This situation created an instrument that was not causally connected with the outcome, but it did relate to the usage of computer technologies. By using this instrument, the authors estimated that the causal effect of computer usage on student achievements did not significantly differ from zero.

Another source of instrumental variables is the deviation from cohort trends. This type of instrument was first used by Hoxby to investigate the causal effect of class size and class composition on student achievement. Similarly, if there are more females in the cohort than expected, some students in the cohort will have a peer group that has more females than is typical. On the other hand, such surprises are correlated with class size and composition.

As the IV, Hoxby used predicting models for annual enrollment and the prediction error for a given school in a given year.

The results showed that class size did not have a statistically significant effect on student achievement; however, class composition did. For instance, both boys and girls performed better in reading when they were in classes that had larger proportions of girls. They used school fixed effects Footnote 3 to account for between-school sorting, and IVs for within-school sorting and estimating causal effects.

As the instrument for the class size, they used the average class size at different grade levels in a particular school. This IV was highly correlated with actual class size, but after controlling for school fixed effects, there was little evidence that grade and average class size would affect student performance.

The third potential source of IVs is based on economic theory, which stipulates that, holding other things constant, the lower the cost of enrollment, the higher the ensuing attainment Murnane and Willett , p. Here we need to assume that people are rational actors that calculate their costs and potential benefits. Costs might be defined by several variables, such as money, effort, and time spent commuting. However, in the context of educational research, the distance between the individual and educational institution has been used the most often as an instrumental variable.

Neal used proxies for geographic proximity to Catholic schools as an exogenous source of variation in attendance at these high schools by estimating its effect on test scores.

Dee used the distance to college as the IV to determine the effect of college attainment on civic participation. From a methodological point of view, these three studies are very similar. Dee wanted to test whether college attainment affects civic participation. The main problem with his goal is that the explanatory variable—educational attainment—is not endogenous.

Participants have selected their own levels of educational attainment rather than having it assigned randomly. In utilizing the IVs technique, it must be assumed that there is no causal relationship between distance to college and civic participation and that no unobservable factors are related to both civic participation and geographic placement of the colleges.

This assumption was apparently too strong to hold. For instance civic minded parents might live in more developed areas with larger number of colleges. After controlling for this rich set of control variables, the assumption that conditional geographic placement around the colleges was distributed exogenously appeared convincing.

By using the IVs technique and a rich set of covariates, Dee showed that educational attainment had large and statistically significant effects on civic participation. Classical regression analysis uses the OLS estimation relationship between the predictor and the outcome.

In Fig. This parameter is simply computed as the ratio of the shared part of the variance of the variables X and Y to the total variance of the predictor, which yields the classical OLS linear estimate of the relationship that under the condition of exogeneity might be interpreted in terms of causality.

Venn diagrams representing the OLS and IVs approaches used to estimate causal relationships between two variables Murnane and Willett , pp. When the predictor is endogenous, but an exogenous instrument is found, the IVs technique could be applied. In the IVs estimation, the estimate is based only on the part of variance that we argue is exogenous, that is, as identified by our instrument.

Therefore, to estimate the relationship between X and Y, we use only the part of the variation that is common to our variables of interest and the instrument. This is depicted on the right side of Fig.

This is achieved by regressing the endogenous predictor on the IVs and additional covariates:. In Stage 1, the model instruments and covariates must be exogenous not correlated with first stage residuals. However, the covariates do not have to comply with the assumption that there is only an indirect influence on the outcome variable.

In the estimation, the instruments and covariates are treated exactly the same. The difference is caused by the assumptions and treatment of these variables in Stage 2. The role of the instruments finishes at Stage 1 of 2SLS two-stage least squares, see below. The estimation covariates should be added to the equations in Stage 2 as predictors because they are directly related to the outcome, adding them prevents the omitted variables problem.

The inclusion of covariates in the Stage 1 model helps to fulfill the assumption that there is no direct relationship of the instrument to the outcome. Stage 1 of the model could include as many instruments as possible in order to isolate as much exogenous variation as possible from the endogenous predictor and to increase the precision of the estimates.

However, this guideline refers only to strong instruments. If the instruments are weak, that is, the instruments are weakly correlated with the endogenous predictors, it might be beneficial to use only the strongest instruments.

Weak instruments might introduce a serious bias into estimates, particularly when the number of instruments is large Bound et al. In Stage 1, the model might include not only one instrumented predictor and the instrument s referring to it but a set of instruments and instrumented predictors, depending on the formulation required in Stage 2. However, for clarity, one endogenous instrumented predictor is formulated in Stage 2 in this example:. Sets of covariates Z on both stages of estimation should be the same.

On the one hand covariates not included in the Stage 1 but included in Stage 2 are likely to be correlated with Stage 1 residuals, which will bias all estimates in Stage 2. On the other hand covariates not included in the Stage 2 but included in Stage 1 might bring omitted variables problem to Stage 2 estimation Baltagi , p. Operationally, estimates for the IVs regression are not obtained by the two OLS regressions in Stages 1 and 2 because this procedure results in the biased estimation of standard errors.

Instead, statistical software e. Asymptotically, all listed estimators have the same properties. They are asymptotically unbiased i. Because a portion of the variation used in estimating of causal effect in IV is substantially smaller than in OLS, only the exogenous part of the variation identified by the instrument is used to determine the precision of the estimators.

Consequently, in the IVs approach, standard errors are larger than those generated in OLS estimations. IVs estimates do not provide the average treatment effect ATE , which could be interpreted as the expected average causal effect of the treatment or condition. However, they do provide the local average treatment effect LATE , which means that the IVs technique estimates the average causal effect on those affected by the instrument Kleibergen and Zivot , pp.

In studies that used proximity to relevant educational institution, the LATE effect was only informative for participants whose decision to enroll was influenced by the distance to the institution, but it was not informative for those that were insensitive to distance while deciding whether to enroll.

Several questions could be posed to address this issue. The first question concerns whether there is reason to believe that the effects estimated using IVs might be heterogeneous, that is, differ among the groups of respondents.

If the heterogeneity of the effect could not be clearly rejected, additional questions could be asked. How large is the group that is affected by the instrument? If the group covers the majority of the population, the problem of generalizability might not to be important. Another question about representatives might be asked.

If there is no reason to believe that the individuals affected by the instrument are substantially different from the entire population i. This section presents an example of the IVs technique using large-scale assessment data.

We focus on the question of whether having neighborhood friends in the same classroom might affect learning. Most previous studies showed that children who have friends perform better at school than those who do not have friends Bandura et al.

The results showed that effects of friends were mostly indirect but substantial. These effects were shown to determine motivation Nelson and DeBacker and behavior Wentzel and Caldwell ; Wilkinson et al. In this paper, we refer to Polish data and students in the first grade of upper-secondary school grade In Poland, most students finish non-selective lower-secondary school at 16 years of age and then must choose an upper-secondary school.

The upper-secondary school system is selective, and students are streamed into different courses of study. The move from lower-secondary to upper-secondary school is accompanied by a drastic change in the learning environment, including classroom peers, teachers, and the location of the school.

It seems rational to assume that having friends in the new classroom might be beneficial for the adaptation process and that friends can indirectly affect learning outcomes. In other words, we want to check whether learning gains of students are determined by the fact of having neighborhood friends in the same classroom.

The data used in the present analysis were collected from a national extension of the PISA study. This system is called a structural simultaneous equation system since y1 and y2 are simultenously determined. The regressor y2 depends on y1 through the second equation. As y1 is directly dependent on u1, the regressor y2 is also correlated with u1 and hence endogenous in the first equation. Assuming that u1 and u2 are uncorrelated, then The above equation system is also described as reversed causality because the dependent variable y1 has a feedback effect on the regressor y2.

In the above example z2 and z1 are straightforward instruments for IV estimation of the first and second equation, respectively. An instrumental variable, Z is uncorrelated with the disturbance e but is correlated with X e. If we assume a situation where an experimenter implemented a randomized experiment where the participants are preschool children, in which the treatment is Watching Sesame Street TV Program, and the outcome of interest is score on letter recognition test.

In this experiment, watching itself cannot be randoimzed but only encouragement to watch the show can be randomly assigned. Taking advantage of the randomization of encouragement, could esitmate a causal effect of watching for at least some of the people in the study. As shown above in the below, the children in the trial could be categorized according to their compliance status.

Ignorability of the instrument: The instrument should be randomized or conditionally randomized with respect to the outcome and treatment variables. Nonzero associaiton between IV and treatment variable: The instrument must have an effect on the treatment. Monotonicity: Assume that there were no children who would watch if they were not encouraged but who would not watch if they were encouraged no defiers. Exclusion restriction: The instrument has no direct effect on the outcome, except indirectly through the treatment.

The Intent-to-treat effect ITT in the hypothetical table above for the 10 observations is an average of the effects for the 4 induced watchers, along with 6 zeros corresponding to the encouragement effects for the always takers and never takers:. The effect of watching Sesame Street for the complier is 8. This ratio is called the Wald estimate. But two-stage least squares is a more general estimation strategy with a regression framework, which allows for controlling covariates.

Article Navigation. August 01 This Site. Google Scholar. Gary Solon Gary Solon. Author and Article Information. Atsushi Inoue. North Carolina State University.

Gary Solon. Received: March 21 Accepted: September 12 Online Issn: For these nonlinear IV methods, theoretical results exist under very restricted assumptions which do not cover the possible frameworks of real data. Overall, in the context of binary outcome several simulation studies investigate mainly 2-stage IV methods with the first step being linear and the second step being logistic as in [ 12 ].

A few articles concern double probit models Chapman et al. Very few address the comparison of GMM and 2-stage approaches and none study GMM, 2-stage double logistic using the prescription preference-based instrumental variables.

These comparisons are based on the estimation of the regression coefficient of the exposure varible in nonlinear logistic model. Our numerical comparison of the methods involves several scenarios with different confounding levels and different instrument strengths for which computation formulas are established in the context of dichotomous outcome and exposure.

We recall the general formula of the covariance matrix for the two-step estimation methods and give the corresponding expression for two-stage nonlinear least squares method in the context of logistic regressions. The paper is organized as follows: we specify the model and describe the methods of estimation that will be analysed.

Then we describe the simulation design, the criteria for evaluating the performances of the methods and the results of our simulations. The final sections discuss the results and make some concluding remarks. The function F. Without a confounding variable, all observed regressors are exogenous.

In this case, the true model is written. We will denote 3 the conventional model. If this model is adjusted to data in the presence of an unobserved confounder, the estimated coefficients would lead to a bias with a level depending on the confounding level. As the confounder X u is not independent of treatment, the residuals of the conventional model are associated with the treatment.

A common strategy is to consider another regression model that links the endogenous variable with others. Equation 2 defines the auxiliary model that predicts treatment T as a function of covariables X 1 , X 2 , the confounder X u and another variable Z.

Variable Z denotes the instrumental variable or instrument related to the treatment, i. The bias due to the unobserved confounder can significantly be reduced by means of the two-stage regression model using a valid instrument. As defined by Johnston and colleagues [ 14 ] and Greenland [ 15 ], a valid instrument must not be correlated with an unobserved confounder or with the error term in the true model 1.

Formally, we assume that the instrument Z meets the following assumptions:. As already proved in the simple case of a linear model for which the functions F and r are equal to identity in Eqs. Finding a strong instrument is then a crucial step in all procedures of instrumental variable estimation. An instrumental variable can be determined in many ways, provided that it meets the assumptions listed above. One of the problems is to find a valid instrument with a reasonable strength.

The strength of an instrumental variable can be defined as resulting from the level of its association with the endogenous treatment. As such, it could be quantified by using the correlation coefficient between the treatment and the related instrument. In the wide range of pharmacoepidemiologic applications, we can summarize the various instrumental variables in three categories:. Geographical variation. Proximity to the care provider can positively influence access to treatment of a patient compared to others who live far away from health services.

To account for this difference between patients, some researchers see [ 16 ] consider the distance between a patient and a care provider as an instrumental variable.

Although this seems realistic as there is no direct association between this distance and the occurrence of disease, the presence or absence of health services can be associated with some socioeconomic characteristics. The latter are often considered as unmeasured confounders that call into question the suitability of using this instrument.

Calendar time. The use of calendar time as an instrument in pharmacoepidemiology often relies on the occurrence of an event that could change the attitude of the physician or patient regarding a treatment.

This could be a change in guidelines for example or a change due to the arrival of a new drug on the market. The time from that event to the date of treatment defines the calendar time which clearly affects the outcome of the treatment since the change in physician or patient attitude will be more pronounced immediately after the event has occurred than later.

An example of use of calendar time as an instrument can be found in [ 17 ]. The most often used instrumental variables in pharmacoepidemiology are preference-based [ 18 — 20 ]. The issue is to compare the effectiveness of two treatments T 1 and T 2 when the assignment of treatment to the patient is not randomized.

This is the case in observational studies where the prescriber of the treatment the physician introduces an effect that influences the outcome via the prescribed treatment. This effect results in the instrumental variable that reflects the influence of care-providers on the patient-treatment relationship.

Those instrumental variables and some others are presented in a more detailed form in [ 23 ], [ 24 , 25 ] or in [ 26 ] with enlightening discussion on their validity. This corresponds to the empirical estimator of the probability for a physician to prefer the treatment of interest. The two-stage residual inclusion method is a modified version of the two-stage least squares 2SLS method used to estimate the parameters in linear models with instrumental variables. As mentioned by Greene [ 5 ], the first stage of the 2SLS method predicts the endogenous variable the treatment here using the instruments and other covariables Eq.

In the second stage, the endogenous variable is just replaced by its prediction from the first stage. This method is called two-stage predictor substitution 2SPS when the first stage is nonlinear. This method also generalizes to the nonlinear models i. The rationale of this approach can be intrinsically related to the form of the true model: sometimes, the prediction equation of the outcome includes the error term of the auxiliary regression.

An example is the case when the confounder is the only source of error in the auxiliary regression as considered in [ 27 ].

It is based on the classical assumption. This assumption does not hold in general because the confounder X u is unobserved i. Typically, w corresponds to the vector of exogenous and endogenous variables with endogenous regressors replaced by their corresponding instruments. Using the law of iterated expectation, the last condition implies. In turn, the GMM minimizes the quadratic form. Some alternative procedures are also suggested by Hansen and colleagues [ 29 ]. The properties of the 2SRI are addressed by Terza in [ 27 ] when the residual also acting as unobserved confounder in the first-stage regression at Eq.

The residual from the first-stage is indeed an unknown function of an unobserved confounder. Then there is a bias that depends on the form of this unknown function when one applies the 2SRI method to the structural model at Eqs. The derivation of the covariance matrix of 2SRI estimator follows a two-step regression covariance matrix of the form. The GMM is a well documented estimation procedure.

Both in linear and nonlinear models, several results on the estimator have already been established. In the literature on econometric analysis, the nonlinear GMM with an instrumental variable has received particular attention. In the pioneering work by Amemiya [ 30 ], the author demonstrated the consistency and derived the asymptotic distribution of the nonlinear two-stage least squares estimator NL2SLS. This result provided an important insight into how to handle nonlinear models with endogeneity.