E NDOGENEITY AND I NSTRUMENTAL

V

There are two main procedures used to draw causal inferences from observational data: matching techniques and regression anal-ysis. We focus here on regression analysis for three reasons. First, the advanced panel data methods we describe in the next section

have been developed and are available only in a regression setting.

Second, the existing literature we will analyze in the following chapters is also essentially based on regression analysis. Third, it can be shown that regression analysis and matching methods are both based on the same principle of “selection on observables”, the main difference between the two techniques being the practi-cal procedures used to implement this principle (Angrist and Pischke 2009). Angrist and Pischke also show that the substantive conclusions to which the two procedures lead should generally be the same.

3.2.1ENDOGENEITY AND CONDITIONAL

INDEPENDENCE

In this sub-section, we show under which assumptions it is possi-ble to solve the selection bias issue described in sub-section 3.1.2 by relying on a quasi-experimental setting based on observational data in a regression analysis. More precisely, the conditional inde-pendence assumption can be satisfied if the covariates responsible for the correlation between the treatment variable and the out-come variable can be controlled for. As we already pointed out at the end of sub-section 3.1.3, solving the selection bias allows us to estimate the treatment effect on the treated, but does not pro-vide a solution to the bias related to the heterogeneous causal ef-fects between treatment and control group. In other words, we will show under which conditions it is possible to estimate the ef-fect of union membership on a particular attitude for the individ-uals that are union members during the period under examination.

The results are thus generalizable only to those particular union members and not to the population of wage-earners as a whole, including also non-members. One way to try to generalize the re-sults to the control group is by arguing that the causal effect is indeed homogeneous on all kinds of individuals and in all kinds of unions. This could be theoretically done by showing that the mechanisms leading to the causal effect are the same for every

kind of individual and for every kind of union. In the next para-graphs, in order to make the reading more fluid, we will talk about the causal effect of union membership even though we are indeed talking about the causal effect of union membership only on those actually observed as members.

A quasi-experiment related to the research question of this the-sis can be represented through the following equation:

𝒂𝒕𝒕𝒊𝒕𝒖𝒅𝒆_𝒊 = 𝜶 + 𝜷 𝒖𝒏𝒊𝒐𝒏_𝒊+ 𝝃_𝒊, 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝑵 (𝟑. 𝟕)

where attitude is, for simplicity, an attitude measured in a numeric scale. α represents the intercept term. β is a coefficient that we would like to capture the causal effect of union membership on the dependent variable. Union is a dummy variable representing the union membership status (“Non-member” or “Member”). ξ is an error term including all variables that have an impact on the considered attitude and for which we do not control for. N is the number of individuals composing our population and i a sub-script identifying each of them. We describe thus the procedure directly on population parameters. The additional issues to con-sider with the estimation using actual sample data are described in section 3.4. For simplicity, we also do not include any control var-iables, but the same procedure could be generalized by including them.

The question here is: does the β coefficient represent the causal effect of union membership on the dependent variable? Under which conditions is this coefficient not affected by the selection bias described in subsection 3.1.2? Before answering these ques-tions, we represent (on the next page) the same quasi-experiment in a path model:

Figure 3.1: Path model representing the effect of union membership on a particular attitude

The arrow from the “union” variable to the “attitude” variable represents the causal relationship we would like to estimate. The arrow from the error term “ξ” to the “attitude” variable corre-sponds to the determinants of the considered attitude that are not taken into account in the model. In the path model, the two ques-tions stated before can be translated to: if we estimated the model in equation 3.7 through Ordinary Least Squares (OLS), under which conditions does the β coefficient correspond to the causal effect of union membership on the dependent variable repre-sented in figure 3.1? Can we estimate the causal effect of union membership on the dependent variable represented in figure 3.1 through the β coefficient in equation 3.7? Intuitively, this happens if and only if we are capable of excluding that the relationship un-der examination (represented by the arrow union → attitude) is not disturbed by the presence of other relationships (represented by the two bold arrows) that interfere with it. Formally, this can be stated through the conditional independence assumption in re-gression form (we will see that this assumption is equivalent to the one presented in section 3.1.2):

𝑬(𝝃_𝒊) = 𝑬(𝝃_𝒊 | 𝒖𝒏𝒊𝒐𝒏_𝒊) = 𝟎, 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝑵 (𝟑. 𝟖)

Equation 3.8 states that the average of the error term is zero and that the explanatory variable under examination is uncorrelated with the error term. The first statement is a condition that can be satisfied by shifting the intercept term. The second statement is much more interesting for our purposes. If a variable satisfies this condition, it is called exogenous. Otherwise, if it shows a correla-tion with the error term, it is said to be endogenous. Why does endogeneity create a problem in estimating the causal impact of a variable? Intuitively, this happens because the coefficient we get through OLS does not represent the impact of union membership alone on the attitude. Indeed, it embodies also the impact of other variables present in the error term. The correlation observed be-tween union membership and the attitude is influenced by rela-tionships other than the studied causal relationship. Hence, the correlation does not represent the sole causal relationship, but the sum of a multiplicity of relationships.

Recapitulating, we are interested into estimating the causal im-pact of the union membership variable on the dependent variable.

In order to know whether equation 3.7 allows us to estimate this causal effect, we must check whether the union membership var-iable is exogenous. If it is, the correlation between union member-ship and the dependent variable corresponds to a causal relation-ship. If it is not the case, in which ways can the union membership variable not be exogenous, i.e. endogenous? Formulate it differ-ently, when is the relationship we are interested in influenced by other ones? For our purposes, there are essentially two cases un-der which the union membership variable (or in general any other

covariates we would like to include in our model) can be consid-ered as endogenous³. They are both represented by the bold ar-rows in figure 3.1 and we describe each one separately.

Figure 3.2 illustrates the first case, the so called problem of

“Omitted Variable Bias” (OVB):

Figure 3.2: Illustration of the “omitted variable bias” issue

A bias related to an omitted variable occurs when a relevant vari-able in the system is included in the error term. A varivari-able is con-sidered relevant if it impacts at the same both the dependent and the independent variable. Its omission creates a bias in the estima-tion of the causal impact of the independent variable on the de-pendent one. In the case of a simple regression model (as in equa-tion 3.7), it can be shown that this bias is equal to (Wooldridge 2010:67):

●

3 In reality, there are at least two additional situations that could lead to the endogeneity of an independent variable. The first one, a measurement error in the explanatory variable, is not an issue in our case. In fact, we are pretty confident that the answers to the union membership status given by the re-spondents are reliable. The other concern, a miss-specification error in the

𝑩𝒊𝒂𝒔 (𝜷) = 𝜸𝒄𝒐𝒗(𝒖𝒏𝒊𝒐𝒏, 𝒐𝒎𝒊𝒕𝒕𝒆𝒅 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆)

𝒗𝒂𝒓(𝒖𝒏𝒊𝒐𝒏) (𝟑. 𝟗)

where γ is the hypothetical coefficient of the omitted variable when included in regression 3.7. It is possible to extend this for-mula to multiple omitted variables. We see that the bias is zero if the variable has either no impact on the dependent variable or it is not correlated with the explanatory variable (thus violating the definition of relevant omitted variable given above). Let us exam-ine the case of a concrete relevant omitted variable in our setting.

For example, if we consider job satisfaction as the dependent var-iable, an omitted variable could be represented by the level of ed-ucation of the individual. In the previous chapter, we have seen that, in Switzerland, individuals with a tertiary education have more chances to be union members than the rest of wage-earners.

It is also reasonable to suppose that individuals with a tertiary ed-ucation enjoy better working conditions and are thus more satis-fied with their job than the rest of the working population. Edu-cation is thus positively correlated with union membership and job satisfaction. Since highly educated individuals are more satis-fied with their jobs and have higher chances to become union members than the rest of the population, according to equation 3.9, measuring a simple correlation between union membership and job satisfaction without controlling for the level of education would lead to a selection bias that overestimates the true causal relationship. Including the education level as a control variable would solve the problem. However, education is certainly not the only omitted variable that can bias the estimation. For example, another omitted variable that would generate a bias could be rep-resented by an innate individual predisposition to challenge the views of the management. We can suppose that such a predispo-sition would be negatively correlated with job satisfaction and pos-itively correlated with union membership. Relying another time on equation 3.9, omitting this variable would lead to a negative bias.

Controlling for this variable would again solve the problem. This strategy is called “selection on observables” and it imitates what

we do in a randomized experiment. Controlling for the variables that create a selection bias leads to eliminate the selection bias.

The problem is that the omitted variables that potentially have an impact on both attitudes and union membership are very often unobserved. The innate predisposition cited before would be very complicated to measure and the bias could not be eliminated by adding the omitted variable in the regression model. We will come back to this point in sub-section 3.3.1, by showing how panel data can solve the problem associated with this type of unobserved heterogeneity.

Figure 3.3 shows the second major source of endogeneity af-fecting our research question, the problem of reverse causality:

Figure 3.3: Illustration of the “reverse causality” issue

As the name implies, this problem arises when the dependent var-iable is suspected itself of having an impact on the independent variable. It creates a bias because, again, the coefficient of the un-ion membership variable in equatun-ion 3.7 would include the impact of union membership and the impact on it from the dependent variable. As an example, consider again the attitude representing job satisfaction. This attitude presents a problem of reverse cau-sality because a low level of job satisfaction is one of the main

reasons to become a union member, hoping for an improvement of the objective working conditions through the union bargaining activity.

3.2.2THE SOLUTION TO THE ENDOGENEITY PROBLEM: INSTRUMENTAL VARIABLES

The endogenous nature of the union membership variable, related to a problem of omitted variable bias or reverse causality, is the regression equivalent of the selection bias described in sub-section 3.1.2. We have seen that, if an omitted variable is observable, it is possible to get rid of the bias it creates by including it as control variable in the regression model. However, especially when the dependent variable is an attitude, the omitted variables that possi-bly bias the estimation may be represented by unobserved param-eters that cannot be controlled for. The reverse causality issue con-tributes to further complicate the estimation.

For these reasons, a classical regression model with a key ex-planatory variable and a series of covariates is not enough to esti-mate the causal effect of union membership on any of the atti-tudes we consider (we will detail in each chapter the type of en-dogeneity problems to consider). In order to appropriately deal with the problem of the endogenous nature of the union mem-bership variable, an instrumental variable for union memmem-bership is needed. An instrument for a given independent variable has to satisfy two conditions:

-1) it has to be correlated with the independent variable -2) it has to be uncorrelated with the error term

Figure 3.4 (on the next page) depicts the features of an instru-ment for the union membership variable:

Figure 3.4: Illustration of the properties of an instrumental variable

In our case, the first assumption states that the instrument has to be correlated with the union membership variable. As we will highlight for the instruments we have constructed in the panel data setting, the higher the correlation, the better.

The second assumption implies that the instrument has to not be correlated with the error term. Equivalently, this means that the instrument can influence the dependent variable only through the union membership variable. There has to not be any other channel through which the instrument affects the dependent var-iable. This assumption is formally untestable and the researcher needs to provide some solid theoretical arguments to motivate its validity.

If one can find an instrument for the union membership vari-able, the IV estimation consists of an application of the method of moments (Wooldridge 2013:409). In order to obtain the causal effect represented by the β coefficient in equation 3.7, one substi-tutes the population parameters with population moments, in this case the covariance with the instrument. Equation 3.7 becomes then:

𝒄𝒐𝒗(𝒂𝒕𝒕𝒊𝒕𝒖𝒅𝒆, 𝑰𝑽) = 𝜷 𝒄𝒐𝒗(𝒖𝒏𝒊𝒐𝒏, 𝑰𝑽) + 𝒄𝒐𝒗(𝜼, 𝑰𝑽) (𝟑. 𝟏𝟎)

Since cov(η, IV) = 0 by definition, the β coefficient representing the causal impact of union membership on the attitude can be ob-tained through:

𝜷 =𝒄𝒐𝒗(𝒂𝒕𝒕𝒊𝒕𝒖𝒅𝒆, 𝑰𝑽)

𝒄𝒐𝒗(𝒖𝒏𝒊𝒐𝒏, 𝑰𝑽) (𝟑. 𝟏𝟏)

In order to better understand why an instrument provides a way to solve the endogeneity problem, it is more useful to illus-trate its functioning through the main procedure used to obtain the IV estimates: 2-stage-least-squares (2SLS). The name is moti-vated by the fact that the procedure involves the use of the OLS estimation in a two-step approach (moreover, it can be shown that every IV estimator can be expressed as a two-step approach).

The first stage of 2SLS consists of an OLS estimation of the union membership variable on the instrument (and on all other exogenous covariates present in the original model):

𝒖𝒏𝒊𝒐𝒏𝒊 = 𝜽 𝑰𝑽𝒊 + 𝜹𝒊, 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝑵 (𝟑. 𝟏𝟐)

where θ is the estimated coefficient of the instrument and δ rep-resents the error term.

In the second stage, the predicted values of union membership obtained in the first step are plugged into the original model in lieu of the union membership variable. The β coefficient estimated in this second stage gives the unbiased causal effect:

𝒂𝒕𝒕𝒊𝒕𝒖𝒅𝒆_𝒊 = 𝜶 + 𝜷 𝒖𝒏𝒊𝒐𝒏̂ _𝒊 + 𝝇_𝒊, 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝑵 (𝟑. 𝟏𝟑)

with ς representing again the error term.

Why does the procedure work? Since the union membership variable is endogenous, the goal of the first step is to extract an exogenous counterpart of it by exploiting the instrument. This ex-ogenous version is represented by the predicted values from the first stage regression and they are, by construction, uncorrelated with error term. In fact, since the instrument is uncorrelated with the error term in the original model, the predicted values, repre-senting the part of union membership explained by the instru-ment, must be themselves uncorrelated with the error term in the original model. This exogenous version of the union membership variable, plugged into the original model, does not suffer anymore from the problem of being endogenous and it allows estimating the desired causal effect.

3.2.3LOCAL AVERAGE TREATMENT EFFECTS (LATE):

COOLING DOWN ABOUT INSTRUMENTAL VARIABLES The truth about instrumental variables is less bright than what we described in the previous paragraphs. Recent insights on instru-mental variables have led to reconsider the scope of the conclu-sions made on the basis of IV estimators (Imbens and Angrist 1994). We give here only a brief description of the issues related to the so called “Local Average Treatment Effects” (LATE). A more detailed account of it can be found in Angrist and Pischke (2009), fourth chapter.

When the causal effect under examination cannot be consid-ered homogeneous across individuals, as it seems to be the case for the attitudinal impact of union membership (cf. discussion in sub-section 3.1.3), using an instrumental variable estimator leads to identify the causal effect only for a particular sub-population of the individuals in the treatment group. In the words of Angrist and Pischke (2009), an instrumental variable can be thought as an engine that initiates a first causal step that affects the instrumented

variable which, in turn, produces the true causal effect on the de-pendent variable under examination. This description is in line with the representation in figure 3.4 and with the 2SLS procedure described above. Now, the question is: who are the individuals affected by the instrumental variable? These are called “compli-ers” and are individuals that experience a change in the instru-mented variable in line with the correlation that the instrumental variable shows with it.

To make things more clear, we cite one of the leading examples used by Angrist and Pischke. If we were interested into estimating the effect of military service on the income of veterans, since the veteran status is clearly endogenous with respect to income, we may use as instrument the draft lottery number assigned to each American male eligible for military service. Since these numbers were randomly assigned, they do not have any direct link with in-come, but they are clearly correlated with conscription. Individuals with low numbers were those drafted for military service. The population of compliers is in this case composed of the individuals that served the US Army because of a low draft number and the IV estimation would lead us to estimate the average treatment ef-fect on them. However, this instrument would not tell us anything about the effect of military service for those that enrolled volun-tarily in the Army. In our case, with a dummy instrument (0,1) for union membership that is positively correlated with it, the sub-population of compliers would be represented by those individu-als that would be non-members when the instruments takes the value 0 and that would be members when the instrument takes the value 1. The causal effect estimated with such an instrumental var-iable gives the treatment effect for the sole population of

To make things more clear, we cite one of the leading examples used by Angrist and Pischke. If we were interested into estimating the effect of military service on the income of veterans, since the veteran status is clearly endogenous with respect to income, we may use as instrument the draft lottery number assigned to each American male eligible for military service. Since these numbers were randomly assigned, they do not have any direct link with in-come, but they are clearly correlated with conscription. Individuals with low numbers were those drafted for military service. The population of compliers is in this case composed of the individuals that served the US Army because of a low draft number and the IV estimation would lead us to estimate the average treatment ef-fect on them. However, this instrument would not tell us anything about the effect of military service for those that enrolled volun-tarily in the Army. In our case, with a dummy instrument (0,1) for union membership that is positively correlated with it, the sub-population of compliers would be represented by those individu-als that would be non-members when the instruments takes the value 0 and that would be members when the instrument takes the value 1. The causal effect estimated with such an instrumental var-iable gives the treatment effect for the sole population of