6462:. When some of the covariates are endogenous so that instrumental variables estimation is implemented, simple expressions for the moments of the estimator cannot be so obtained. Generally, instrumental variables estimators only have desirable asymptotic, not finite sample, properties, and inference is based on asymptotic approximations to the sampling distribution of the estimator. Even when the instruments are uncorrelated with the error in the equation of interest and when the instruments are not weak, the finite sample properties of the instrumental variables estimator may be poor. For example, exactly identified models produce finite sample estimators with no moments, so the estimator can be said to be neither biased nor unbiased, the nominal size of test statistics may be substantially distorted, and the estimates may commonly be far away from the true value of the parameter.
2085:
6433:
about the effect of smoking on health. If taxes affect health through channels other than through their effect on smoking, then the instruments are invalid and the instrumental variables approach may yield misleading results. For example, places and times with relatively health-conscious populations may both implement high tobacco taxes and exhibit better health even holding smoking rates constant, so we would observe a correlation between health and tobacco taxes even if it were the case that smoking has no effect on health. In this case, we would be mistaken to infer a causal effect of smoking on health from the observed correlation between tobacco taxes and health.
2141:
2129:
2073:
6413:(ATE). Imbens and Angrist (1994) demonstrate that the linear IV estimate can be interpreted under weak conditions as a weighted average of local average treatment effects, where the weights depend on the elasticity of the endogenous regressor to changes in the instrumental variables. Roughly, that means that the effect of a variable is only revealed for the subpopulations affected by the observed changes in the instruments, and that subpopulations which respond most to changes in the instruments will have the largest effects on the magnitude of the IV estimate.
1272:
4637:
2155:). The relationship between attending the tutoring program and GPA may be confounded by a number of factors. Students who attend the tutoring program may care more about their grades or may be struggling with their work. This confounding is depicted in the Figures 1–3 on the right through the bidirected arc between Tutoring Program and GPA. If students are assigned to dormitories at random, the proximity of the student's dorm to the tutoring program is a natural candidate for being an instrumental variable.
6429:, and Baker (1995) note, a problem is caused by the selection of "weak" instruments, instruments that are poor predictors of the endogenous question predictor in the first-stage equation. In this case, the prediction of the question predictor by the instrument will be poor and the predicted values will have very little variation. Consequently, they are unlikely to have much success in predicting the ultimate outcome when they are used to replace the question predictor in the second-stage equation.
954:
4241:
200:). Correlation between smoking and health does not imply that smoking causes poor health because other variables, such as depression, may affect both health and smoking, or because health may affect smoking. It is not possible to conduct controlled experiments on smoking status in the general population. The researcher may attempt to estimate the causal effect of smoking on health from observational data by using the tax rate for tobacco products (
1267:{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}={\frac {\operatorname {cov} (x,\alpha +\beta x+u)}{\operatorname {var} (x)}}\\&={\frac {\operatorname {cov} (x,\alpha +\beta x)}{\operatorname {var} (x)}}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}}=\beta ^{*}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}},\end{aligned}}}
5705:
4632:{\displaystyle {\begin{aligned}{\widehat {\beta }}_{\mathrm {GMM} }&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(X^{\mathrm {T} }Z)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y\\&={\widehat {\beta }}_{\mathrm {IV} }\end{aligned}}}
204:) as an instrument for smoking. The tax rate for tobacco products is a reasonable choice for an instrument because the researcher assumes that it can only be correlated with health through its effect on smoking. If the researcher then finds tobacco taxes and state of health to be correlated, this may be viewed as evidence that smoking causes changes in health.
5434:
3542:
2927:
4109:
6416:
For example, if a researcher uses presence of a land-grant college as an instrument for college education in an earnings regression, she identifies the effect of college on earnings in the subpopulation which would obtain a college degree if a college is present but which would not obtain a degree if
2158:
However, what if the tutoring program is located in the college library? In that case, Proximity may also cause students to spend more time at the library, which in turn improves their GPA (see Figure 1). Using the causal graph depicted in the Figure 2, we see that
Proximity does not qualify as an
235:
After much deliberation, Wright decided to use regional rainfall as his instrumental variable: he concluded that rainfall affected grass production and hence milk production and ultimately butter supply, but not butter demand. In this way he was able to construct a regression equation with only the
6432:
In the context of the smoking and health example discussed above, tobacco taxes are weak instruments for smoking if smoking status is largely unresponsive to changes in taxes. If higher taxes do not induce people to quit smoking (or not start smoking), then variation in tax rates tells us nothing
2310:
Finally, suppose that
Library Hours does not actually affect GPA because students who do not study in the library simply study elsewhere, as in Figure 4. In this case, controlling for Library Hours still opens a spurious path from Proximity to GPA. However, if we do not control for Library Hours
227:
Wright attempted to determine the supply and demand for butter using panel data on prices and quantities sold in the United States. The idea was that a regression analysis could produce a demand or supply curve because they are formed by the path between prices and quantities demanded or supplied.
71:
results. A valid instrument induces changes in the explanatory variable (is correlated with the endogenous variable) but has no independent effect on the dependent variable and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on
6401:
may differ from the effect in a given subpopulation. For example, the average effect of a job training program may substantially differ across the group of people who actually receive the training and the group which chooses not to receive training. For these reasons, IV methods invoke implicit
231:
The problem was that price affected both supply and demand so that a function describing only one of the two could not be constructed directly from the observational data. Wright correctly concluded that he needed a variable that correlated with either demand or supply but not both – that is, an
151:
The instrument cannot be correlated with the error term in the explanatory equation, conditionally on the other covariates. In other words, the instrument cannot suffer from the same problem as the original predicting variable. If this condition is met, then the instrument is said to satisfy the
4697:
One computational method which can be used to calculate IV estimates is two-stage least squares (2SLS or TSLS). In the first stage, each explanatory variable that is an endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model, including both
4965:
2084:
6470:
The assumption that the instruments are not correlated with the error term in the equation of interest is not testable in exactly identified models. If the model is overidentified, there is information available which may be used to test this assumption. The most common test of these
5160:
5700:{\displaystyle \beta _{\text{2SLS}}=({\widehat {X}}^{\mathrm {T} }{\widehat {X}})^{-1}{\widehat {X}}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}Y.}
3350:
2266:
Now, suppose that we notice that a student's "natural ability" affects his or her number of hours in the library as well as his or her GPA, as in Figure 3. Using the causal graph, we see that
Library Hours is a collider and conditioning on it opens the path Proximity
2720:
6417:
a college is not present. This empirical approach does not, without further assumptions, tell the researcher anything about the effect of college among people who would either always or never get a college degree regardless of whether a local college exists.
1609:(the exclusion restriction), then IV may identify the causal parameter of interest where OLS fails. Because there are multiple specific ways of using and deriving IV estimators even in just the linear case (IV, 2SLS, GMM), we save further discussion for the
228:
The problem was that the observational data did not form a demand or supply curve as such, but rather a cloud of point observations that took different shapes under varying market conditions. It seemed that making deductions from the data remained elusive.
1621:
IV techniques have been developed among a much broader class of non-linear models. General definitions of instrumental variables, using counterfactual and graphical formalism, were given by Pearl (2000; p. 248). The graphical definition requires that
2140:
3763:
3929:
1724:
5271:
926:
58:
are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term (endogenous), in which case
3301:
1888:
4832:
6135:
4843:
2128:
3877:
This expression collapses to the first when the number of instruments is equal to the number of covariates in the equation of interest. The over-identified IV is therefore a generalization of the just-identified IV.
5054:
640:
2072:
251:(2001) present a survey of the history and uses of instrumental variable techniques. Notions of causality in econometrics, and their relationship with instrumental variables and other methods, are discussed by
3872:
6441:
The strength of the instruments can be directly assessed because both the endogenous covariates and the instruments are observable. A common rule of thumb for models with one endogenous regressor is: the
6479:, is based on the observation that the residuals should be uncorrelated with the set of exogenous variables if the instruments are truly exogenous. The Sargan–Hansen test statistic can be calculated as
959:
5427:
5043:
3537:{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }X\beta +(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }e\rightarrow \beta }
559:
491:
5732:
is numerically identical to the expression displayed above. A small correction must be made to the sum-of-squared residuals in the second-stage fitted model in order that the covariance matrix of
2922:{\displaystyle {\widehat {\beta }}_{\mathrm {OLS} }=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }y=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }(X\beta +e)=\beta +(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }e}
4246:
4170:
7123:
Bound, J.; Jaeger, D. A.; Baker, R. M. (1995). "Problems with
Instrumental Variables Estimation when the Correlation between the Instruments and the Endogenous Explanatory Variable is Weak".
3585:
in the underlying process which generates the data, the appropriate use of the IV estimator will identify this parameter. This works because IV solves for the unique parameter that satisfies
1437:
684:
1346:
801:
Consider for simplicity the single-variable case. Suppose we are considering a regression with one variable and a constant (perhaps no other covariates are necessary, or perhaps we have
4747:
144:
The instrument must be correlated with the endogenous explanatory variables, conditionally on the other covariates. If this correlation is strong, then the instrument is said to have a
6359:
3205:
2984:
4970:
In the second stage, the regression of interest is estimated as usual, except that in this stage each endogenous covariate is replaced with the predicted values from the first stage:
2379:
6458:
When the covariates are exogenous, the small-sample properties of the OLS estimator can be derived in a straightforward manner by calculating moments of the estimator conditional on
2231:. However, if we control for Library Hours by adding it as a covariate then Proximity becomes an instrumental variable, since Proximity is separated from GPA given Library Hours in
5316:
7469:
3921:
3621:
3583:
3342:
847:
438:
7392:
Wooldridge, J. (1997): Quasi-Likelihood
Methods for Count Data, Handbook of Applied Econometrics, Volume 2, ed. M. H. Pesaran and P. Schmidt, Oxford, Blackwell, pp. 352–406
1938:
1753:
4202:
4104:{\displaystyle {\widehat {\beta }}_{\mathrm {GMM} }=(X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y}
2261:
2229:
2120:
1792:
3652:
2305:
5165:
This method is only valid in linear models. For categorical endogenous covariates, one might be tempted to use a different first stage than ordinary least squares, such as a
1632:
2285:
2197:
2177:
343:
5866:
5823:
4687:
1373:
1302:
133:
is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the explanatory equation but is correlated with the
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5186:
855:
6507:
6387:
5750:
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3055:
2689:
2588:
2564:
1393:
405:
7152:
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6193:
5343:
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2669:
2642:
2615:
2542:
2515:
2484:
2451:
2418:
2034:; a proxy of such cause may also be used, if it satisfies conditions 1–5. The exclusion restriction (condition 4) is redundant; it follows from conditions 2 and 3.
3216:
5975:
6273:
6253:
6233:
6213:
6161:
6035:
6015:
5995:
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5909:
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1607:
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1507:
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385:
365:
305:
285:
216:, best known for his excellent description of the production, transport and sale of vegetable and animal oils in the early 1900s in the United States. In 1945,
6402:
assumptions on behavioral response, or more generally assumptions over the correlation between the response to treatment and propensity to receive treatment.
6610:
1815:
4755:
4960:{\displaystyle {\widehat {X}}=Z{\widehat {\delta }}={\color {ProcessBlue}Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }}X={\color {ProcessBlue}P_{Z}}X.\,}
6043:
6710:
Leigh, J. P.; Schembri, M. (2004). "Instrumental
Variables Technique: Cigarette Price Provided Better Estimate of Effects of Smoking on SF-12".
5155:{\displaystyle \beta _{\text{2SLS}}=\left(X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}X\right)^{-1}X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}Y.}
7376:
7354:
7332:
7298:
7259:
7006:
Wooldridge, J. (2010). Econometric
Analysis of Cross Section and Panel Data. Econometric Analysis of Cross Section and Panel Data. MIT Press.
6991:
6877:
802:
4698:
exogenous covariates in the equation of interest and the excluded instruments. The predicted values from these regressions are obtained:
6513:) from the OLS regression of the residuals onto the set of exogenous variables. This statistic will be asymptotically chi-squared with
80:
387:
is a matrix, usually with a column of ones and perhaps with additional columns for other covariates. Consider how an instrument allows
88:
3801:
2058:
are a representation of this structure, and the graphical definition given above can be used to quickly determine whether a variable
2018:
These conditions do not rely on specific functional form of the equations and are applicable therefore to nonlinear equations, where
564:
7150:
Stock, J.; Wright, J.; Yogo, M. (2002). "A Survey of Weak
Instruments and Weak Identification in Generalized Method of Moments".
6940:
6899:
6788:
6712:
6140:
Balke and Pearl derived tight bounds on ACE and showed that these can provide valuable information on the sign and size of ACE.
4648:
7442:
5352:
4986:
6530:
121:
Explanatory variables that suffer from one or more of these issues in the context of a regression are sometimes referred to as
2319:
We now revisit and expand upon the mechanics of IV in greater detail. Suppose the data are generated by a process of the form
443:
7395:
Terza, J. V. (1998): "Estimating Count Models with
Endogenous Switching: Sample Selection and Endogenous Treatment Effects."
7086:
Heckman, J. (1997). "Instrumental variables: A study of implicit behavioral assumptions used in making program evaluations".
6406:
4655:. Since the parameters are the solutions to a set of linear equations, an under-identified model using the set of equations
496:
3643:
4125:
3626:
Now an extension: suppose that there are more instruments than there are covariates in the equation of interest, so that
2054:
cannot be inferred from data and must instead be determined from the model structure, i.e., the data-generating process.
7464:
6510:
2022:
can be non-additive (see Non-parametric analysis). They are also applicable to a system of multiple equations, in which
7426:
2146:
Figure 4: Proximity qualifies as an instrumental variable, as long as we do not include
Library Hours as a covariate.
1398:
645:
7448:
7432:
7278:
7088:
6869:
4204:
are all squared matrices of the same dimension. We can expand the inverse, using the fact that, for any invertible
3062:
221:
134:
122:
114:
6410:
6369:
The exposition above assumes that the causal effect of interest does not vary across observations, that is, that
1307:
6608:
Bullock, J. G.; Green, D. P.; Ha, S. E. (2010). "Yes, But What's the Mechanism? (Don't Expect an Easy Answer)".
5169:
for the first stage followed by OLS for the second. This is commonly known in the econometric literature as the
4715:
6281:
3151:
2951:
3089:× 2 consisting of a column of constants and one instrumental variable. However, this technique generalizes to
2325:
3097:
being a matrix composed of a constant and 5 instruments. In the discussion that follows, we will assume that
1447:
reflect the underlying causal effect of interest. IV helps to fix this problem by identifying the parameters
6476:
6389:
is a constant. Generally, different subjects will respond in different ways to changes in the "treatment"
5276:
7161:
7050:
6619:
1440:
731:
408:
239:
Formal definitions of instrumental variables, using counterfactuals and graphical criteria, were given by
126:
60:
6895:"Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments"
3899:
3758:{\displaystyle {\widehat {\beta }}_{\mathrm {GMM} }=(X^{\mathrm {T} }P_{Z}X)^{-1}X^{\mathrm {T} }P_{Z}y,}
3588:
3550:
3309:
811:
414:
6683:
1917:
1732:
687:
107:
55:
4175:
2706:
Suppose also that a regression model of nominally the same form is proposed. Given a random sample of
2234:
2202:
2093:
1765:
2290:
1719:{\displaystyle (Z\perp \!\!\!\perp Y)_{G_{\overline {X}}}\qquad (Z\not \!\!{\perp \!\!\!\perp }X)_{G}}
6450:
that the excluded instruments are irrelevant in the first-stage regression should be larger than 10.
2152:
2151:
Suppose that we wish to estimate the effect of a university tutoring program on grade point average (
148:. A weak correlation may provide misleading inferences about parameter estimates and standard errors.
76:
7166:
6624:
7308:
7290:
7055:
7039:
Balke, A.; Pearl, J. (1997). "Bounds on treatment effects from studies with imperfect compliance".
6536:
213:
92:
7199:"Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator"
2270:
2182:
2162:
7364:
7228:
7220:
7179:
7105:
7068:
6764:
6645:
6590:
6582:
4114:
In the just-identified case, we have as many instruments as covariates, so that the dimension of
313:
7324:
5829:
5786:
5266:{\displaystyle ({\widehat {X}}^{\mathrm {T} }{\widehat {X}})^{-1}{\widehat {X}}^{\mathrm {T} }Y}
1351:
1280:
921:{\displaystyle {\widehat {\beta }}={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}}
263:
While the ideas behind IV extend to a broad class of models, a very common context for IV is in
6821:. Arkiv for Mathematic, Astronomi, och Fysik. Vol. 32A. Uppsala: Almquist & Wiksells.
7372:
7350:
7328:
7294:
7255:
7247:
6987:
6873:
6822:
6729:
6637:
4232:
3796:
3014:
term are correlated, however, the OLS estimator is generally biased and inconsistent for
1450:
778:. OLS will simply pick the parameter that makes the resulting errors appear uncorrelated with
264:
68:
7416:
3296:{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y}
7282:
7212:
7171:
7132:
7097:
7060:
7021:
6949:
6918:
6908:
6797:
6756:
6721:
6629:
6574:
6521:
degrees of freedom under the null that the error term is uncorrelated with the instruments.
6482:
6393:. When this possibility is recognized, the average effect in the population of a change in
6372:
5735:
5715:
3040:
2674:
2573:
2549:
1795:
1756:
1378:
390:
217:
51:
6166:
5321:
4658:
3771:
2998:, under certain regularity conditions the second term has an expected value conditional on
2647:
2620:
2593:
2520:
2493:
2462:
2429:
2396:
7420:
6696:
6447:
6426:
3130:
7283:
6539: – Technique for improving the efficiency of estimators in conditional moment models
5954:
2311:
and remove it as a covariate then Proximity can again be used an instrumental variable.
2030:
through several intermediate variables. An instrumental variable need not be a cause of
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370:
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244:
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This specification approaches the true parameter as the sample gets large, so long as
7458:
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6594:
2711:
2700:
2055:
252:
7183:
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5173:, because second-stage IV parameter estimates are consistent only in special cases.
3623:, and therefore hones in on the true underlying parameter as the sample size grows.
2134:
Figure 3: Proximity does not qualify as an instrumental variable given Library Hours
7342:
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7136:
7064:
6725:
6649:
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248:
192:
For example, suppose a researcher wishes to estimate the causal effect of smoking (
39:
35:
1883:{\displaystyle (Z\perp \!\!\!\perp Y_{x})\qquad (Z\not \!\!{\perp \!\!\!\perp }X)}
4827:{\displaystyle {\widehat {\delta }}=(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }X,\,}
6255:, the following constraint, called "Instrumental Inequality" must hold whenever
5346:
2995:
2159:
instrumental variable because it is connected to GPA through the path Proximity
240:
84:
7402:
Wooldridge, J. (2002): "Econometric Analysis of Cross Section and Panel Data",
7175:
6802:
6783:
6662:
5760:
When the form of the structural equations is unknown, an instrumental variable
3081:× 2 matrix composed of a column of constants and one endogenous variable, and
7436:
6130:{\displaystyle {\text{ACE}}=\Pr(y\mid {\text{do}}(x))=\operatorname {E} _{u}.}
31:
6826:
2078:
Figure 1: Proximity qualifies as an instrumental variable given Library Hours
7371:(Fifth international ed.). Mason, OH: South-Western. pp. 490–528.
100:
6733:
6641:
2122:, which is used to determine whether Proximity is an instrumental variable.
2695:
are uncorrelated and that they are drawn from distributions with the same
99:
changes in the dependent variable change the value of at least one of the
2696:
2307:
GPA. As a result, Proximity cannot be used as an instrumental variable.
165:
Informally, in attempting to estimate the causal effect of some variable
6923:
6913:
6894:
2986:
in the introduction (this is the matrix version of that equation). When
7224:
7025:
6768:
6586:
3018:. In this case, it is valid to use the estimates to predict values of
852:
In this case, the coefficient on the regressor of interest is given by
137:
explanatory variables, conditionally on the value of other covariates.
7289:(Sixth ed.). Upper Saddle River: Pearson Prentice-Hall. pp.
7109:
6633:
6443:
1951:
qualifies as an instrument if the given criteria hold conditional on
7216:
7198:
6747:
Epstein, Roy J. (1989). "The Fall of OLS in Structural Estimation".
6578:
6560:
5997:
do not allow for the identification of the average causal effect of
3093:
being a matrix of a constant and, say, 5 endogenous variables, with
17:
7101:
6143:
In linear analysis, there is no test to falsify the assumption the
686:
due to any of the reasons listed above—for example, if there is an
267:. Traditionally, an instrumental variable is defined as a variable
212:
The first use of an instrument variable occurred in a 1928 book by
6561:"Identification and estimation of local average treatment effects"
64:
2490:-th value of an unobserved error term representing all causes of
140:
In linear models, there are two main requirements for using IVs:
6784:"Retrospectives: Who Invented Instrumental Variable Regression?"
3136:
Suppose that the relationship between each endogenous component
3002:
of zero and converges to zero in the limit, so the estimator is
2062:
qualifies as an instrumental variable given a set of covariates
307:
and uncorrelated with the "error term" U in the linear equation
6533: – Statistical methods to correct for endogeneity problems
3867:{\displaystyle P_{Z}=Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }}
3210:
The most common IV specification uses the following estimator:
6819:
Confluence Analysis by Means of Instrumental Sets of Variables
3010:
and the other unmeasured, causal variables collapsed into the
1962:
The equations of interest are "structural", not "regression".
635:{\displaystyle X'(Y-X{\widehat {\beta }})=X'{\widehat {U}}=0}
110:
that affect both the dependent and explanatory variables, or
7254:. Princeton: Princeton University Press. pp. 217–221.
129:
produces biased and inconsistent estimates. However, if an
5422:{\displaystyle P_{Z}^{\mathrm {T} }P_{Z}=P_{Z}P_{Z}=P_{Z}}
5038:{\displaystyle Y={\widehat {X}}\beta +\mathrm {noise} ,\,}
3069:
but (in our underlying model) is not correlated with
3026:, but the estimate does not recover the causal effect of
2457:-th values of the independent variable(s) and a constant,
7323:(Fifth ed.). New York: McGraw-Hill Irwin. pp.
2699:(that is, that the errors are serially uncorrelated and
486:{\displaystyle \operatorname {cov} (X,{\widehat {U}})=0}
7018:
Two-stage predictor substitution for time-to-event data
1304:
is what the estimated coefficient vector would be if
115:
covariates are subject to non-random measurement error
7444:
Econometrics lecture (topic: two-stages least square)
6485:
6375:
6284:
6261:
6241:
6221:
6201:
6169:
6149:
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5917:
5897:
5877:
5832:
5789:
5766:
5738:
5718:
5437:
5355:
5324:
5279:
5189:
5057:
4989:
4846:
4758:
4718:
4661:
4244:
4178:
4128:
3932:
3902:
3804:
3774:
3655:
3591:
3553:
3353:
3312:
3219:
3154:
3043:
2954:
2948:. This equation is similar to the equation involving
2723:
2677:
2650:
2623:
2596:
2576:
2552:
2523:
2496:
2465:
2432:
2399:
2328:
2293:
2273:
2237:
2205:
2185:
2165:
2096:
1920:
1818:
1768:
1735:
1635:
1595:
1575:
1555:
1535:
1515:
1495:
1475:
1453:
1401:
1381:
1354:
1310:
1283:
957:
934:
858:
814:
784:
764:
744:
716:
696:
648:
567:
554:{\displaystyle \min _{\beta }(Y-X\beta )'(Y-X\beta )}
499:
446:
417:
393:
373:
353:
316:
293:
273:
7428:
Econometrics lecture (topic: instrumental variable)
6365:
Interpretation under treatment effect heterogeneity
4165:{\displaystyle X^{\mathrm {T} }Z,Z^{\mathrm {T} }Z}
169:("covariate" or "explanatory variable") on another
7316:
6861:
6676:"Weak Instruments: An Overview and New Techniques"
6501:
6381:
6353:
6267:
6247:
6227:
6215:is discrete. Pearl (2000) has shown that, for all
6207:
6187:
6155:
6129:
6029:
6009:
5989:
5969:
5943:
5923:
5903:
5883:
5860:
5817:
5772:
5744:
5724:
5699:
5421:
5337:
5310:
5265:
5154:
5037:
4959:
4826:
4741:
4681:
4631:
4196:
4164:
4103:
3915:
3866:
3787:
3757:
3615:
3577:
3536:
3336:
3295:
3199:
3049:
2978:
2921:
2683:
2663:
2636:
2609:
2582:
2558:
2536:
2509:
2478:
2445:
2412:
2373:
2299:
2279:
2255:
2223:
2191:
2171:
2114:
1932:
1882:
1786:
1747:
1718:
1601:
1581:
1561:
1541:
1521:
1501:
1481:
1461:
1431:
1387:
1367:
1340:
1296:
1266:
940:
920:
841:
790:
770:
750:
722:
702:
678:
634:
553:
485:
432:
399:
379:
359:
337:
299:
287:that is correlated with the independent variable
279:
6971:. Cambridge, England: Cambridge University Press.
5951:. Unlike linear models, however, measurements of
1926:
1925:
1924:
1869:
1868:
1867:
1862:
1861:
1830:
1829:
1828:
1741:
1740:
1739:
1698:
1697:
1696:
1691:
1690:
1647:
1646:
1645:
224:in his dissertation, giving the method its name.
6318:
6309:
6286:
6055:
1947:then the above definitions are modified so that
501:
7153:Journal of the American Statistical Association
7125:Journal of the American Statistical Association
7042:Journal of the American Statistical Association
2671:constant. The econometric goal is to estimate
1432:{\displaystyle \operatorname {cov} (x,u)\neq 0}
679:{\displaystyle \operatorname {cov} (X,U)\neq 0}
6509:(the number of observations multiplied by the
4980:on the predicted values from the first stage:
4642:Reference: see Davidson and Mackinnnon (1993)
2066:. To see how, consider the following example.
1439:in the underlying model that we believe, then
6938:Heckman, J. (2008). "Econometric Causality".
2691:. For simplicity's sake assume the draws of
1969:stands for all exogenous factors that affect
493:(when we minimize the sum of squared errors,
8:
7470:Simultaneous equation methods (econometrics)
7369:Introductory Econometrics: A Modern Approach
6982:Davidson, Russell; Mackinnon, James (1993).
6663:https://www.stata.com/meeting/5nasug/wiv.pdf
6611:Journal of Personality and Social Psychology
6454:Statistical inference and hypothesis testing
5780:can still be defined through the equations:
1805:The counterfactual definition requires that
220:applied the same approach in the context of
7349:. Oxford: Basil Blackwell. pp. 42–67.
6864:Causality: Models, Reasoning, and Inference
6855:
6853:
6851:
6849:
6843:. South-Western, Scarborough, Kanada, 2009.
6782:Stock, James H.; Trebbi, Francesco (2003).
6554:
6552:
3646:(GMM) can be used. The GMM IV estimator is
1341:{\displaystyle \operatorname {cov} (x,u)=0}
236:instrumental variable of price and supply.
4742:{\displaystyle X=Z\delta +{\text{errors}}}
1443:gives an inconsistent estimate which does
7165:
7054:
6922:
6912:
6801:
6623:
6493:
6484:
6374:
6354:{\displaystyle \max _{x}\sum _{y}\leq 1.}
6312:
6299:
6289:
6283:
6260:
6240:
6220:
6200:
6168:
6148:
6091:
6067:
6047:
6045:
6022:
6002:
5982:
5956:
5936:
5916:
5896:
5876:
5857:
5831:
5814:
5788:
5765:
5737:
5717:
5685:
5674:
5673:
5660:
5646:
5635:
5634:
5611:
5610:
5605:
5594:
5593:
5580:
5566:
5555:
5554:
5549:
5538:
5537:
5514:
5513:
5502:
5501:
5491:
5476:
5475:
5468:
5467:
5456:
5455:
5442:
5436:
5413:
5400:
5390:
5377:
5366:
5365:
5360:
5354:
5329:
5323:
5299:
5281:
5280:
5278:
5253:
5252:
5241:
5240:
5230:
5215:
5214:
5207:
5206:
5195:
5194:
5188:
5138:
5132:
5125:
5124:
5111:
5095:
5089:
5082:
5081:
5062:
5056:
5034:
5014:
4997:
4996:
4988:
4956:
4942:
4936:
4921:
4920:
4907:
4893:
4892:
4880:
4866:
4865:
4848:
4847:
4845:
4823:
4810:
4809:
4796:
4782:
4781:
4760:
4759:
4757:
4734:
4717:
4693:Interpretation as two-stage least squares
4660:
4615:
4614:
4603:
4602:
4581:
4580:
4567:
4553:
4552:
4525:
4524:
4511:
4497:
4496:
4476:
4475:
4459:
4445:
4444:
4417:
4416:
4403:
4389:
4388:
4371:
4370:
4357:
4343:
4342:
4322:
4321:
4305:
4291:
4290:
4263:
4262:
4251:
4250:
4245:
4243:
4184:
4183:
4177:
4152:
4151:
4134:
4133:
4127:
4091:
4090:
4077:
4063:
4062:
4045:
4044:
4031:
4017:
4016:
4003:
3989:
3988:
3971:
3970:
3947:
3946:
3935:
3934:
3931:
3907:
3901:
3857:
3856:
3843:
3829:
3828:
3809:
3803:
3779:
3773:
3743:
3732:
3731:
3718:
3705:
3694:
3693:
3670:
3669:
3658:
3657:
3654:
3597:
3596:
3590:
3559:
3558:
3552:
3518:
3517:
3504:
3490:
3489:
3466:
3465:
3452:
3438:
3437:
3417:
3416:
3403:
3389:
3388:
3368:
3367:
3356:
3355:
3352:
3318:
3317:
3311:
3283:
3282:
3269:
3255:
3254:
3234:
3233:
3222:
3221:
3218:
3200:{\displaystyle x_{i}=Z_{i}\gamma +v_{i},}
3188:
3172:
3159:
3153:
3042:
2979:{\displaystyle \operatorname {cov} (X,y)}
2953:
2909:
2908:
2895:
2881:
2880:
2839:
2838:
2825:
2811:
2810:
2790:
2789:
2776:
2762:
2761:
2738:
2737:
2726:
2725:
2722:
2676:
2655:
2649:
2628:
2622:
2601:
2595:
2575:
2551:
2528:
2522:
2501:
2495:
2470:
2464:
2437:
2431:
2404:
2398:
2362:
2346:
2333:
2327:
2292:
2272:
2242:
2236:
2210:
2204:
2184:
2164:
2101:
2095:
2003:is held constant (exclusion restriction).
1919:
1863:
1854:
1838:
1817:
1773:
1767:
1734:
1710:
1692:
1683:
1663:
1658:
1634:
1594:
1574:
1554:
1534:
1514:
1494:
1474:
1454:
1452:
1400:
1380:
1359:
1353:
1309:
1288:
1282:
1213:
1204:
1153:
1097:
1028:
981:
963:
962:
958:
956:
933:
874:
860:
859:
857:
813:
783:
763:
743:
715:
695:
647:
642:.) If the true model is believed to have
615:
614:
589:
588:
566:
504:
498:
463:
462:
445:
419:
418:
416:
392:
372:
352:
315:
292:
272:
6984:Estimation and Inference in Econometrics
5178:Proof: computation of the 2SLS estimator
2617:of a one unit change in each element of
2374:{\displaystyle y_{i}=X_{i}\beta +e_{i},}
1589:(the first stage) but uncorrelated with
1509:, but based on whether another variable
95:model. Such correlation may occur when:
75:Instrumental variable methods allow for
7347:Lectures on Advanced Econometric Theory
6967:Bowden, R.J.; Turkington, D.A. (1984).
6548:
2068:
561:, the first-order condition is exactly
42:and related disciplines, the method of
7248:"Testing Overidentifying Restrictions"
6692:
6681:
6405:The standard IV estimator can recover
1958:The essence of Pearl's definition is:
6986:. New York: Oxford University Press.
6163:is instrumental relative to the pair
5311:{\displaystyle {\widehat {X}}=P_{Z}X}
5133:
5090:
4937:
4881:
3073:. For simplicity, one might consider
1348:. In this case, it can be shown that
7:
3061:that is highly correlated with each
3037:To recover the underlying parameter
2710:observations from this process, the
2424:-th value of the dependent variable,
2046:is unobserved, the requirement that
6275:satisfies the two equations above:
3916:{\displaystyle \beta _{\text{GMM}}}
3616:{\displaystyle Z^{\mathrm {T} }e=0}
3578:{\displaystyle Z^{\mathrm {T} }e=0}
3337:{\displaystyle Z^{\mathrm {T} }e=0}
3113:unspecified. An estimator in which
1943:If there are additional covariates
842:{\displaystyle y=\alpha +\beta x+u}
433:{\displaystyle {\widehat {\beta }}}
7197:Nelson, C. R.; Startz, R. (1990).
6761:10.1093/oxfordjournals.oep.a041930
6088:
5675:
5636:
5612:
5595:
5556:
5539:
5515:
5469:
5367:
5254:
5208:
5126:
5083:
5027:
5024:
5021:
5018:
5015:
4922:
4894:
4811:
4783:
4619:
4616:
4582:
4554:
4526:
4498:
4477:
4446:
4418:
4390:
4372:
4344:
4323:
4292:
4270:
4267:
4264:
4185:
4153:
4135:
4092:
4064:
4046:
4018:
3990:
3972:
3954:
3951:
3948:
3858:
3830:
3733:
3695:
3677:
3674:
3671:
3598:
3560:
3519:
3491:
3467:
3439:
3418:
3390:
3372:
3369:
3319:
3284:
3256:
3238:
3235:
3057:, we introduce a set of variables
2910:
2882:
2840:
2812:
2791:
2763:
2745:
2742:
2739:
2566:is an unobserved parameter vector.
1933:{\displaystyle \perp \!\!\!\perp }
1748:{\displaystyle \perp \!\!\!\perp }
1626:satisfy the following conditions:
25:
6893:Angrist, J.; Krueger, A. (2001).
6466:Testing the exclusion restriction
4689:does not have a unique solution.
4197:{\displaystyle Z^{\mathrm {T} }X}
2256:{\displaystyle G_{\overline {X}}}
2224:{\displaystyle G_{\overline {X}}}
2115:{\displaystyle G_{\overline {X}}}
1787:{\displaystyle G_{\overline {X}}}
6954:10.1111/j.1751-5823.2007.00024.x
6941:International Statistical Review
6900:Journal of Economic Perspectives
6789:Journal of Economic Perspectives
6713:Journal of Clinical Epidemiology
6559:Imbens, G.; Angrist, J. (1994).
5911:are two arbitrary functions and
3145:and the instruments is given by
2944:denote column vectors of length
2300:{\displaystyle \leftrightarrow }
2139:
2127:
2083:
2071:
805:any other relevant covariates):
185:only through its effect on
6531:Control function (econometrics)
6407:local average treatment effects
4837:and save the predicted values:
1847:
1676:
1610:
7137:10.1080/01621459.1995.10476536
7065:10.1080/01621459.1997.10474074
6726:10.1016/j.jclinepi.2003.08.006
6674:Nichols, Austin (2006-07-23).
6342:
6339:
6321:
6305:
6182:
6170:
6121:
6118:
6106:
6100:
6081:
6078:
6072:
6058:
5854:
5842:
5811:
5799:
5488:
5451:
5227:
5190:
4904:
4885:
4793:
4774:
4564:
4545:
4508:
4489:
4486:
4468:
4456:
4437:
4400:
4381:
4354:
4335:
4332:
4314:
4302:
4283:
4074:
4055:
4028:
4000:
3981:
3963:
3840:
3821:
3715:
3686:
3528:
3501:
3482:
3449:
3430:
3400:
3381:
3266:
3247:
2973:
2961:
2892:
2873:
2861:
2846:
2822:
2803:
2773:
2754:
2644:, holding all other causes of
2294:
2274:
2186:
2166:
2038:Selecting suitable instruments
1877:
1848:
1844:
1819:
1707:
1677:
1655:
1636:
1420:
1408:
1329:
1317:
1251:
1245:
1234:
1222:
1191:
1185:
1174:
1162:
1144:
1138:
1127:
1106:
1081:
1075:
1064:
1037:
1019:
1013:
1002:
990:
912:
906:
895:
883:
667:
655:
600:
576:
548:
533:
526:
510:
474:
453:
1:
4651:estimator for the case where
3644:generalized method of moments
2010:should not be independent of
1798:in which all arrows entering
407:to be recovered. Recall that
7016:Lergenmuller, Simon (2017).
6511:coefficient of determination
6473:overidentifying restrictions
6437:Testing for weak instruments
6195:. This is not the case when
5183:The usual OLS estimator is:
4233:Invertible matrix#Properties
4118:is the same as that of
3109:matrix and leave this value
2280:{\displaystyle \rightarrow }
2247:
2215:
2192:{\displaystyle \rightarrow }
2172:{\displaystyle \rightarrow }
2106:
1855:
1778:
1684:
1668:
1375:is an unbiased estimator of
7406:, Cambridge, Massachusetts.
5712:The resulting estimator of
3891:in the just-identified case
3638:. This is often called the
3129:matrices is referred to as
2026:(and other factors) affect
738:yield the causal impact of
338:{\displaystyle Y=X\beta +U}
173:("dependent variable"), an
7486:
7176:10.1198/073500102288618658
7089:Journal of Human Resources
6870:Cambridge University Press
6803:10.1257/089533003769204416
5861:{\displaystyle y=f(x,u)\,}
5818:{\displaystyle x=g(z,u)\,}
1902:stands for the value that
1549:. If theory suggests that
1368:{\displaystyle \beta ^{*}}
1297:{\displaystyle \beta ^{*}}
222:errors-in-variables models
6841:Introductory Econometrics
6411:average treatment effects
5752:is calculated correctly.
1984:should be independent of
1940:stands for independence.
6421:Weak instruments problem
3642:case. In this case, the
2590:is the causal effect on
1462:{\displaystyle {\beta }}
72:the dependent variable.
7397:Journal of Econometrics
7246:Hayashi, Fumio (2000).
6817:Reiersøl, Olav (1945).
5756:Non-parametric analysis
4704:Regress each column of
4647:There is an equivalent
232:instrumental variable.
27:Technique in statistics
7399:(84), pp. 129–154
7365:Wooldridge, Jeffrey M.
6969:Instrumental Variables
6749:Oxford Economic Papers
6691:Cite journal requires
6503:
6502:{\displaystyle TR^{2}}
6383:
6382:{\displaystyle \beta }
6355:
6269:
6249:
6229:
6209:
6189:
6157:
6131:
6031:
6011:
5991:
5971:
5945:
5925:
5905:
5885:
5862:
5819:
5774:
5746:
5745:{\displaystyle \beta }
5726:
5725:{\displaystyle \beta }
5701:
5423:
5339:
5312:
5267:
5156:
5039:
4961:
4828:
4743:
4683:
4633:
4198:
4166:
4105:
3917:
3868:
3789:
3759:
3617:
3579:
3538:
3338:
3297:
3201:
3051:
3050:{\displaystyle \beta }
3006:and consistent. When
2980:
2923:
2712:ordinary least squares
2685:
2684:{\displaystyle \beta }
2665:
2638:
2611:
2584:
2583:{\displaystyle \beta }
2560:
2559:{\displaystyle \beta }
2538:
2511:
2480:
2447:
2414:
2375:
2301:
2281:
2257:
2225:
2193:
2173:
2116:
1934:
1884:
1788:
1749:
1720:
1603:
1583:
1563:
1543:
1523:
1503:
1483:
1463:
1433:
1389:
1388:{\displaystyle \beta }
1369:
1342:
1298:
1268:
942:
922:
843:
792:
772:
752:
724:
704:
680:
636:
555:
487:
434:
401:
400:{\displaystyle \beta }
381:
361:
339:
301:
281:
127:ordinary least squares
103:("reverse" causation),
61:ordinary least squares
56:controlled experiments
50:) is used to estimate
44:instrumental variables
6504:
6384:
6356:
6270:
6250:
6230:
6210:
6190:
6188:{\displaystyle (X,Y)}
6158:
6132:
6032:
6012:
5992:
5972:
5946:
5926:
5906:
5886:
5863:
5820:
5775:
5747:
5727:
5702:
5424:
5340:
5338:{\displaystyle P_{Z}}
5313:
5268:
5157:
5040:
4962:
4829:
4744:
4684:
4682:{\displaystyle Z'v=0}
4634:
4199:
4167:
4106:
3918:
3869:
3790:
3788:{\displaystyle P_{Z}}
3760:
3618:
3580:
3539:
3339:
3298:
3202:
3052:
2981:
2924:
2686:
2666:
2664:{\displaystyle y_{i}}
2639:
2637:{\displaystyle X_{i}}
2612:
2610:{\displaystyle y_{i}}
2585:
2570:The parameter vector
2561:
2539:
2537:{\displaystyle X_{i}}
2512:
2510:{\displaystyle y_{i}}
2481:
2479:{\displaystyle e_{i}}
2448:
2446:{\displaystyle X_{i}}
2415:
2413:{\displaystyle y_{i}}
2391:indexes observations,
2376:
2302:
2282:
2258:
2226:
2194:
2174:
2117:
1935:
1885:
1789:
1750:
1721:
1604:
1584:
1564:
1544:
1529:is uncorrelated with
1524:
1504:
1489:is uncorrelated with
1484:
1469:not based on whether
1464:
1434:
1390:
1370:
1343:
1299:
1269:
943:
923:
844:
793:
773:
753:
730:separately—then this
725:
705:
681:
637:
556:
488:
435:
402:
382:
362:
340:
302:
282:
196:) on general health (
154:exclusion restriction
125:. In this situation,
81:explanatory variables
7309:Gujarati, Damodar N.
7285:Econometric Analysis
6483:
6373:
6282:
6259:
6239:
6219:
6199:
6167:
6147:
6044:
6021:
6001:
5981:
5955:
5935:
5915:
5895:
5875:
5830:
5787:
5764:
5736:
5716:
5435:
5353:
5322:
5277:
5187:
5171:forbidden regression
5055:
4987:
4844:
4756:
4716:
4659:
4242:
4176:
4126:
3930:
3900:
3802:
3772:
3653:
3589:
3551:
3351:
3310:
3217:
3152:
3041:
2952:
2721:
2675:
2648:
2621:
2594:
2574:
2550:
2521:
2494:
2463:
2430:
2397:
2326:
2291:
2271:
2235:
2203:
2183:
2163:
2094:
1918:
1816:
1766:
1733:
1633:
1617:Graphical definition
1593:
1573:
1553:
1533:
1513:
1493:
1473:
1451:
1399:
1379:
1352:
1308:
1281:
955:
932:
856:
812:
782:
762:
742:
714:
694:
646:
565:
497:
444:
415:
391:
371:
351:
314:
291:
271:
177:is a third variable
79:estimation when the
52:causal relationships
7465:Regression analysis
6914:10.1257/jep.15.4.69
6537:Optimal instruments
6409:(LATE) rather than
5970:{\displaystyle Z,X}
5617:
5561:
5372:
5345:is a symmetric and
3344:in the true model:
2453:is a vector of the
1859:
1688:
928:. Substituting for
690:which affects both
7319:Basic Econometrics
7279:Greene, William H.
7049:(439): 1172–1176.
6860:Pearl, J. (2000).
6499:
6477:Sargan–Hansen test
6379:
6351:
6317:
6304:
6294:
6265:
6245:
6225:
6205:
6185:
6153:
6127:
6027:
6007:
5987:
5967:
5941:
5931:is independent of
5921:
5901:
5881:
5858:
5815:
5770:
5742:
5722:
5697:
5601:
5545:
5419:
5356:
5335:
5308:
5263:
5152:
5144:
5101:
5035:
4957:
4948:
4928:
4824:
4739:
4679:
4629:
4627:
4194:
4162:
4101:
3913:
3864:
3785:
3755:
3613:
3575:
3534:
3334:
3293:
3197:
3047:
2976:
2919:
2681:
2661:
2634:
2607:
2580:
2556:
2534:
2507:
2476:
2443:
2410:
2371:
2297:
2277:
2253:
2221:
2189:
2169:
2112:
2050:be independent of
1995:should not affect
1930:
1880:
1784:
1745:
1716:
1599:
1579:
1559:
1539:
1519:
1499:
1479:
1459:
1429:
1385:
1365:
1338:
1294:
1264:
1262:
938:
918:
839:
788:
768:
748:
720:
700:
676:
632:
551:
509:
483:
430:
397:
377:
357:
335:
297:
277:
146:strong first stage
7378:978-1-111-53439-4
7356:978-0-631-14956-9
7334:978-0-07-337577-9
7300:978-0-13-600383-0
7261:978-0-691-01018-2
6993:978-0-19-506011-9
6879:978-0-521-89560-6
6308:
6295:
6285:
6268:{\displaystyle Z}
6248:{\displaystyle g}
6228:{\displaystyle f}
6208:{\displaystyle X}
6156:{\displaystyle Z}
6070:
6050:
6030:{\displaystyle Y}
6010:{\displaystyle X}
5990:{\displaystyle Y}
5944:{\displaystyle U}
5924:{\displaystyle Z}
5904:{\displaystyle g}
5884:{\displaystyle f}
5773:{\displaystyle Z}
5510:
5484:
5464:
5445:
5289:
5249:
5223:
5203:
5065:
5005:
4874:
4856:
4768:
4737:
4611:
4259:
3943:
3910:
3797:projection matrix
3666:
3364:
3230:
2734:
2250:
2218:
2109:
1977:is held constant.
1906:would attain had
1858:
1781:
1687:
1671:
1602:{\displaystyle u}
1582:{\displaystyle x}
1562:{\displaystyle z}
1542:{\displaystyle u}
1522:{\displaystyle z}
1502:{\displaystyle u}
1482:{\displaystyle x}
1255:
1195:
1148:
1085:
1023:
971:
941:{\displaystyle y}
916:
868:
791:{\displaystyle X}
771:{\displaystyle Y}
751:{\displaystyle X}
723:{\displaystyle Y}
703:{\displaystyle X}
623:
597:
500:
471:
427:
380:{\displaystyle X}
360:{\displaystyle Y}
300:{\displaystyle X}
280:{\displaystyle Z}
265:linear regression
108:omitted variables
83:(covariates) are
16:(Redirected from
7477:
7445:
7429:
7382:
7360:
7338:
7322:
7304:
7288:
7266:
7265:
7243:
7237:
7236:
7194:
7188:
7187:
7169:
7147:
7141:
7140:
7120:
7114:
7113:
7083:
7077:
7076:
7058:
7036:
7030:
7029:
7013:
7007:
7004:
6998:
6997:
6979:
6973:
6972:
6964:
6958:
6957:
6935:
6929:
6928:
6926:
6916:
6890:
6884:
6883:
6867:
6857:
6844:
6839:Wooldridge, J.:
6837:
6831:
6830:
6814:
6808:
6807:
6805:
6779:
6773:
6772:
6744:
6738:
6737:
6707:
6701:
6700:
6694:
6689:
6687:
6679:
6671:
6665:
6660:
6654:
6653:
6634:10.1037/a0018933
6627:
6605:
6599:
6598:
6556:
6508:
6506:
6505:
6500:
6498:
6497:
6388:
6386:
6385:
6380:
6360:
6358:
6357:
6352:
6316:
6303:
6293:
6274:
6272:
6271:
6266:
6254:
6252:
6251:
6246:
6234:
6232:
6231:
6226:
6214:
6212:
6211:
6206:
6194:
6192:
6191:
6186:
6162:
6160:
6159:
6154:
6136:
6134:
6133:
6128:
6096:
6095:
6071:
6068:
6051:
6048:
6036:
6034:
6033:
6028:
6016:
6014:
6013:
6008:
5996:
5994:
5993:
5988:
5976:
5974:
5973:
5968:
5950:
5948:
5947:
5942:
5930:
5928:
5927:
5922:
5910:
5908:
5907:
5902:
5890:
5888:
5887:
5882:
5867:
5865:
5864:
5859:
5824:
5822:
5821:
5816:
5779:
5777:
5776:
5771:
5751:
5749:
5748:
5743:
5731:
5729:
5728:
5723:
5706:
5704:
5703:
5698:
5690:
5689:
5680:
5679:
5678:
5668:
5667:
5659:
5655:
5651:
5650:
5641:
5640:
5639:
5616:
5615:
5609:
5600:
5599:
5598:
5588:
5587:
5579:
5575:
5571:
5570:
5560:
5559:
5553:
5544:
5543:
5542:
5520:
5519:
5518:
5512:
5511:
5503:
5499:
5498:
5486:
5485:
5477:
5474:
5473:
5472:
5466:
5465:
5457:
5447:
5446:
5443:
5428:
5426:
5425:
5420:
5418:
5417:
5405:
5404:
5395:
5394:
5382:
5381:
5371:
5370:
5364:
5349:matrix, so that
5344:
5342:
5341:
5336:
5334:
5333:
5318:and noting that
5317:
5315:
5314:
5309:
5304:
5303:
5291:
5290:
5282:
5272:
5270:
5269:
5264:
5259:
5258:
5257:
5251:
5250:
5242:
5238:
5237:
5225:
5224:
5216:
5213:
5212:
5211:
5205:
5204:
5196:
5161:
5159:
5158:
5153:
5145:
5143:
5142:
5131:
5130:
5129:
5119:
5118:
5110:
5106:
5102:
5100:
5099:
5088:
5087:
5086:
5067:
5066:
5063:
5044:
5042:
5041:
5036:
5030:
5007:
5006:
4998:
4966:
4964:
4963:
4958:
4949:
4947:
4946:
4929:
4927:
4926:
4925:
4915:
4914:
4899:
4898:
4897:
4876:
4875:
4867:
4858:
4857:
4849:
4833:
4831:
4830:
4825:
4816:
4815:
4814:
4804:
4803:
4788:
4787:
4786:
4770:
4769:
4761:
4748:
4746:
4745:
4740:
4738:
4735:
4688:
4686:
4685:
4680:
4669:
4649:under-identified
4638:
4636:
4635:
4630:
4628:
4624:
4623:
4622:
4613:
4612:
4604:
4594:
4587:
4586:
4585:
4575:
4574:
4559:
4558:
4557:
4538:
4531:
4530:
4529:
4519:
4518:
4503:
4502:
4501:
4482:
4481:
4480:
4467:
4466:
4451:
4450:
4449:
4430:
4423:
4422:
4421:
4411:
4410:
4395:
4394:
4393:
4377:
4376:
4375:
4365:
4364:
4349:
4348:
4347:
4328:
4327:
4326:
4313:
4312:
4297:
4296:
4295:
4275:
4274:
4273:
4261:
4260:
4252:
4203:
4201:
4200:
4195:
4190:
4189:
4188:
4171:
4169:
4168:
4163:
4158:
4157:
4156:
4140:
4139:
4138:
4110:
4108:
4107:
4102:
4097:
4096:
4095:
4085:
4084:
4069:
4068:
4067:
4051:
4050:
4049:
4039:
4038:
4023:
4022:
4021:
4011:
4010:
3995:
3994:
3993:
3977:
3976:
3975:
3959:
3958:
3957:
3945:
3944:
3936:
3922:
3920:
3919:
3914:
3912:
3911:
3908:
3873:
3871:
3870:
3865:
3863:
3862:
3861:
3851:
3850:
3835:
3834:
3833:
3814:
3813:
3794:
3792:
3791:
3786:
3784:
3783:
3764:
3762:
3761:
3756:
3748:
3747:
3738:
3737:
3736:
3726:
3725:
3710:
3709:
3700:
3699:
3698:
3682:
3681:
3680:
3668:
3667:
3659:
3622:
3620:
3619:
3614:
3603:
3602:
3601:
3584:
3582:
3581:
3576:
3565:
3564:
3563:
3543:
3541:
3540:
3535:
3524:
3523:
3522:
3512:
3511:
3496:
3495:
3494:
3472:
3471:
3470:
3460:
3459:
3444:
3443:
3442:
3423:
3422:
3421:
3411:
3410:
3395:
3394:
3393:
3377:
3376:
3375:
3366:
3365:
3357:
3343:
3341:
3340:
3335:
3324:
3323:
3322:
3302:
3300:
3299:
3294:
3289:
3288:
3287:
3277:
3276:
3261:
3260:
3259:
3243:
3242:
3241:
3232:
3231:
3223:
3206:
3204:
3203:
3198:
3193:
3192:
3177:
3176:
3164:
3163:
3056:
3054:
3053:
3048:
3022:given values of
2985:
2983:
2982:
2977:
2928:
2926:
2925:
2920:
2915:
2914:
2913:
2903:
2902:
2887:
2886:
2885:
2845:
2844:
2843:
2833:
2832:
2817:
2816:
2815:
2796:
2795:
2794:
2784:
2783:
2768:
2767:
2766:
2750:
2749:
2748:
2736:
2735:
2727:
2690:
2688:
2687:
2682:
2670:
2668:
2667:
2662:
2660:
2659:
2643:
2641:
2640:
2635:
2633:
2632:
2616:
2614:
2613:
2608:
2606:
2605:
2589:
2587:
2586:
2581:
2565:
2563:
2562:
2557:
2543:
2541:
2540:
2535:
2533:
2532:
2516:
2514:
2513:
2508:
2506:
2505:
2485:
2483:
2482:
2477:
2475:
2474:
2452:
2450:
2449:
2444:
2442:
2441:
2419:
2417:
2416:
2411:
2409:
2408:
2380:
2378:
2377:
2372:
2367:
2366:
2351:
2350:
2338:
2337:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2262:
2260:
2259:
2254:
2252:
2251:
2243:
2230:
2228:
2227:
2222:
2220:
2219:
2211:
2198:
2196:
2195:
2190:
2178:
2176:
2175:
2170:
2143:
2131:
2121:
2119:
2118:
2113:
2111:
2110:
2102:
2087:
2075:
1939:
1937:
1936:
1931:
1889:
1887:
1886:
1881:
1873:
1860:
1856:
1843:
1842:
1793:
1791:
1790:
1785:
1783:
1782:
1774:
1754:
1752:
1751:
1746:
1725:
1723:
1722:
1717:
1715:
1714:
1702:
1689:
1685:
1675:
1674:
1673:
1672:
1664:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1568:
1566:
1565:
1560:
1548:
1546:
1545:
1540:
1528:
1526:
1525:
1520:
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1468:
1466:
1465:
1460:
1458:
1438:
1436:
1435:
1430:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1364:
1363:
1347:
1345:
1344:
1339:
1303:
1301:
1300:
1295:
1293:
1292:
1273:
1271:
1270:
1265:
1263:
1256:
1254:
1237:
1214:
1209:
1208:
1196:
1194:
1177:
1154:
1149:
1147:
1130:
1098:
1090:
1086:
1084:
1067:
1029:
1024:
1022:
1005:
982:
973:
972:
964:
947:
945:
944:
939:
927:
925:
924:
919:
917:
915:
898:
875:
870:
869:
861:
848:
846:
845:
840:
797:
795:
794:
789:
777:
775:
774:
769:
757:
755:
754:
749:
729:
727:
726:
721:
709:
707:
706:
701:
688:omitted variable
685:
683:
682:
677:
641:
639:
638:
633:
625:
624:
616:
613:
599:
598:
590:
575:
560:
558:
557:
552:
532:
508:
492:
490:
489:
484:
473:
472:
464:
439:
437:
436:
431:
429:
428:
420:
406:
404:
403:
398:
386:
384:
383:
378:
366:
364:
363:
358:
344:
342:
341:
336:
306:
304:
303:
298:
286:
284:
283:
278:
214:Philip G. Wright
21:
7485:
7484:
7480:
7479:
7478:
7476:
7475:
7474:
7455:
7454:
7443:
7427:
7421:Daniel McFadden
7413:
7389:
7379:
7363:
7357:
7341:
7335:
7313:Porter, Dawn C.
7307:
7301:
7277:
7274:
7272:Further reading
7269:
7262:
7245:
7244:
7240:
7217:10.2307/2938359
7196:
7195:
7191:
7167:10.1.1.319.2477
7149:
7148:
7144:
7122:
7121:
7117:
7085:
7084:
7080:
7038:
7037:
7033:
7015:
7014:
7010:
7005:
7001:
6994:
6981:
6980:
6976:
6966:
6965:
6961:
6937:
6936:
6932:
6892:
6891:
6887:
6880:
6859:
6858:
6847:
6838:
6834:
6816:
6815:
6811:
6781:
6780:
6776:
6746:
6745:
6741:
6709:
6708:
6704:
6690:
6680:
6673:
6672:
6668:
6661:
6657:
6625:10.1.1.169.5465
6607:
6606:
6602:
6579:10.2307/2951620
6558:
6557:
6550:
6546:
6527:
6489:
6481:
6480:
6468:
6456:
6439:
6423:
6371:
6370:
6367:
6280:
6279:
6257:
6256:
6237:
6236:
6217:
6216:
6197:
6196:
6165:
6164:
6145:
6144:
6087:
6042:
6041:
6019:
6018:
5999:
5998:
5979:
5978:
5953:
5952:
5933:
5932:
5913:
5912:
5893:
5892:
5873:
5872:
5828:
5827:
5785:
5784:
5762:
5761:
5758:
5734:
5733:
5714:
5713:
5710:
5709:
5681:
5669:
5642:
5630:
5629:
5625:
5624:
5589:
5562:
5533:
5532:
5528:
5527:
5500:
5487:
5454:
5438:
5433:
5432:
5409:
5396:
5386:
5373:
5351:
5350:
5325:
5320:
5319:
5295:
5275:
5274:
5239:
5226:
5193:
5185:
5184:
5180:
5179:
5134:
5120:
5091:
5077:
5076:
5072:
5071:
5058:
5053:
5052:
4985:
4984:
4938:
4916:
4903:
4888:
4842:
4841:
4805:
4792:
4777:
4754:
4753:
4714:
4713:
4695:
4662:
4657:
4656:
4645:
4644:
4626:
4625:
4601:
4592:
4591:
4576:
4563:
4548:
4536:
4535:
4520:
4507:
4492:
4471:
4455:
4440:
4428:
4427:
4412:
4399:
4384:
4366:
4353:
4338:
4317:
4301:
4286:
4276:
4249:
4240:
4239:
4179:
4174:
4173:
4147:
4129:
4124:
4123:
4086:
4073:
4058:
4040:
4027:
4012:
3999:
3984:
3966:
3933:
3928:
3927:
3903:
3898:
3897:
3896:Developing the
3893:
3892:
3890:
3886:
3852:
3839:
3824:
3805:
3800:
3799:
3775:
3770:
3769:
3739:
3727:
3714:
3701:
3689:
3656:
3651:
3650:
3640:over-identified
3592:
3587:
3586:
3554:
3549:
3548:
3513:
3500:
3485:
3461:
3448:
3433:
3412:
3399:
3384:
3354:
3349:
3348:
3313:
3308:
3307:
3278:
3265:
3250:
3220:
3215:
3214:
3184:
3168:
3155:
3150:
3149:
3144:
3131:just-identified
3039:
3038:
2950:
2949:
2904:
2891:
2876:
2834:
2821:
2806:
2785:
2772:
2757:
2724:
2719:
2718:
2673:
2672:
2651:
2646:
2645:
2624:
2619:
2618:
2597:
2592:
2591:
2572:
2571:
2548:
2547:
2524:
2519:
2518:
2497:
2492:
2491:
2466:
2461:
2460:
2433:
2428:
2427:
2400:
2395:
2394:
2358:
2342:
2329:
2324:
2323:
2317:
2289:
2288:
2269:
2268:
2238:
2233:
2232:
2206:
2201:
2200:
2181:
2180:
2161:
2160:
2147:
2144:
2135:
2132:
2123:
2097:
2092:
2091:
2088:
2079:
2076:
2040:
2006:The instrument
1991:The instrument
1980:The instrument
1965:The error term
1916:
1915:
1901:
1834:
1814:
1813:
1794:stands for the
1769:
1764:
1763:
1731:
1730:
1706:
1659:
1654:
1631:
1630:
1619:
1613:section below.
1591:
1590:
1571:
1570:
1551:
1550:
1531:
1530:
1511:
1510:
1491:
1490:
1471:
1470:
1449:
1448:
1397:
1396:
1377:
1376:
1355:
1350:
1349:
1306:
1305:
1284:
1279:
1278:
1261:
1260:
1238:
1215:
1200:
1178:
1155:
1131:
1099:
1088:
1087:
1068:
1030:
1006:
983:
974:
953:
952:
930:
929:
899:
876:
854:
853:
810:
809:
780:
779:
760:
759:
740:
739:
734:procedure will
712:
711:
692:
691:
644:
643:
606:
568:
563:
562:
525:
495:
494:
442:
441:
413:
412:
389:
388:
369:
368:
349:
348:
312:
311:
289:
288:
269:
268:
261:
210:
163:
28:
23:
22:
15:
12:
11:
5:
7483:
7481:
7473:
7472:
7467:
7457:
7456:
7453:
7452:
7440:
7424:
7412:
7411:External links
7409:
7408:
7407:
7400:
7393:
7388:
7385:
7384:
7383:
7377:
7361:
7355:
7339:
7333:
7305:
7299:
7273:
7270:
7268:
7267:
7260:
7238:
7211:(4): 967–976.
7189:
7160:(4): 518–529.
7142:
7115:
7102:10.2307/146178
7096:(3): 441–462.
7078:
7056:10.1.1.26.3952
7031:
7008:
6999:
6992:
6974:
6959:
6930:
6885:
6878:
6845:
6832:
6809:
6796:(3): 177–194.
6774:
6739:
6720:(3): 284–293.
6702:
6693:|journal=
6666:
6655:
6618:(4): 550–558.
6600:
6573:(2): 467–476.
6547:
6545:
6542:
6541:
6540:
6534:
6526:
6523:
6496:
6492:
6488:
6467:
6464:
6455:
6452:
6438:
6435:
6422:
6419:
6378:
6366:
6363:
6362:
6361:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6323:
6320:
6315:
6311:
6307:
6302:
6298:
6292:
6288:
6264:
6244:
6224:
6204:
6184:
6181:
6178:
6175:
6172:
6152:
6138:
6137:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6105:
6102:
6099:
6094:
6090:
6086:
6083:
6080:
6077:
6074:
6066:
6063:
6060:
6057:
6054:
6037:, denoted ACE
6026:
6006:
5986:
5966:
5963:
5960:
5940:
5920:
5900:
5880:
5869:
5868:
5856:
5853:
5850:
5847:
5844:
5841:
5838:
5835:
5825:
5813:
5810:
5807:
5804:
5801:
5798:
5795:
5792:
5769:
5757:
5754:
5741:
5721:
5708:
5707:
5696:
5693:
5688:
5684:
5677:
5672:
5666:
5663:
5658:
5654:
5649:
5645:
5638:
5633:
5628:
5623:
5620:
5614:
5608:
5604:
5597:
5592:
5586:
5583:
5578:
5574:
5569:
5565:
5558:
5552:
5548:
5541:
5536:
5531:
5526:
5523:
5517:
5509:
5506:
5497:
5494:
5490:
5483:
5480:
5471:
5463:
5460:
5453:
5450:
5441:
5416:
5412:
5408:
5403:
5399:
5393:
5389:
5385:
5380:
5376:
5369:
5363:
5359:
5332:
5328:
5307:
5302:
5298:
5294:
5288:
5285:
5262:
5256:
5248:
5245:
5236:
5233:
5229:
5222:
5219:
5210:
5202:
5199:
5192:
5181:
5177:
5176:
5175:
5163:
5162:
5151:
5148:
5141:
5137:
5128:
5123:
5117:
5114:
5109:
5105:
5098:
5094:
5085:
5080:
5075:
5070:
5061:
5046:
5045:
5033:
5029:
5026:
5023:
5020:
5017:
5013:
5010:
5004:
5001:
4995:
4992:
4968:
4967:
4955:
4952:
4945:
4941:
4935:
4932:
4924:
4919:
4913:
4910:
4906:
4902:
4896:
4891:
4887:
4884:
4879:
4873:
4870:
4864:
4861:
4855:
4852:
4835:
4834:
4822:
4819:
4813:
4808:
4802:
4799:
4795:
4791:
4785:
4780:
4776:
4773:
4767:
4764:
4733:
4730:
4727:
4724:
4721:
4694:
4691:
4678:
4675:
4672:
4668:
4665:
4640:
4639:
4621:
4618:
4610:
4607:
4600:
4597:
4595:
4593:
4590:
4584:
4579:
4573:
4570:
4566:
4562:
4556:
4551:
4547:
4544:
4541:
4539:
4537:
4534:
4528:
4523:
4517:
4514:
4510:
4506:
4500:
4495:
4491:
4488:
4485:
4479:
4474:
4470:
4465:
4462:
4458:
4454:
4448:
4443:
4439:
4436:
4433:
4431:
4429:
4426:
4420:
4415:
4409:
4406:
4402:
4398:
4392:
4387:
4383:
4380:
4374:
4369:
4363:
4360:
4356:
4352:
4346:
4341:
4337:
4334:
4331:
4325:
4320:
4316:
4311:
4308:
4304:
4300:
4294:
4289:
4285:
4282:
4279:
4277:
4272:
4269:
4266:
4258:
4255:
4248:
4247:
4193:
4187:
4182:
4161:
4155:
4150:
4146:
4143:
4137:
4132:
4112:
4111:
4100:
4094:
4089:
4083:
4080:
4076:
4072:
4066:
4061:
4057:
4054:
4048:
4043:
4037:
4034:
4030:
4026:
4020:
4015:
4009:
4006:
4002:
3998:
3992:
3987:
3983:
3980:
3974:
3969:
3965:
3962:
3956:
3953:
3950:
3942:
3939:
3906:
3894:
3888:
3887:collapses to β
3884:
3882:
3881:
3880:
3860:
3855:
3849:
3846:
3842:
3838:
3832:
3827:
3823:
3820:
3817:
3812:
3808:
3795:refers to the
3782:
3778:
3766:
3765:
3754:
3751:
3746:
3742:
3735:
3730:
3724:
3721:
3717:
3713:
3708:
3704:
3697:
3692:
3688:
3685:
3679:
3676:
3673:
3665:
3662:
3612:
3609:
3606:
3600:
3595:
3574:
3571:
3568:
3562:
3557:
3545:
3544:
3533:
3530:
3527:
3521:
3516:
3510:
3507:
3503:
3499:
3493:
3488:
3484:
3481:
3478:
3475:
3469:
3464:
3458:
3455:
3451:
3447:
3441:
3436:
3432:
3429:
3426:
3420:
3415:
3409:
3406:
3402:
3398:
3392:
3387:
3383:
3380:
3374:
3371:
3363:
3360:
3333:
3330:
3327:
3321:
3316:
3304:
3303:
3292:
3286:
3281:
3275:
3272:
3268:
3264:
3258:
3253:
3249:
3246:
3240:
3237:
3229:
3226:
3208:
3207:
3196:
3191:
3187:
3183:
3180:
3175:
3171:
3167:
3162:
3158:
3140:
3046:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2930:
2929:
2918:
2912:
2907:
2901:
2898:
2894:
2890:
2884:
2879:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2842:
2837:
2831:
2828:
2824:
2820:
2814:
2809:
2805:
2802:
2799:
2793:
2788:
2782:
2779:
2775:
2771:
2765:
2760:
2756:
2753:
2747:
2744:
2741:
2733:
2730:
2680:
2658:
2654:
2631:
2627:
2604:
2600:
2579:
2568:
2567:
2555:
2545:
2531:
2527:
2504:
2500:
2473:
2469:
2458:
2440:
2436:
2425:
2407:
2403:
2392:
2382:
2381:
2370:
2365:
2361:
2357:
2354:
2349:
2345:
2341:
2336:
2332:
2316:
2313:
2296:
2287:Library Hours
2276:
2249:
2246:
2241:
2217:
2214:
2209:
2188:
2179:Library Hours
2168:
2149:
2148:
2145:
2138:
2136:
2133:
2126:
2124:
2108:
2105:
2100:
2089:
2082:
2080:
2077:
2070:
2039:
2036:
2016:
2015:
2004:
1989:
1978:
1963:
1929:
1923:
1897:
1891:
1890:
1879:
1876:
1872:
1866:
1853:
1850:
1846:
1841:
1837:
1833:
1827:
1824:
1821:
1780:
1777:
1772:
1744:
1738:
1727:
1726:
1713:
1709:
1705:
1701:
1695:
1682:
1679:
1670:
1667:
1662:
1657:
1653:
1650:
1644:
1641:
1638:
1618:
1615:
1598:
1578:
1569:is related to
1558:
1538:
1518:
1498:
1478:
1457:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1384:
1362:
1358:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1291:
1287:
1275:
1274:
1259:
1253:
1250:
1247:
1244:
1241:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1212:
1207:
1203:
1199:
1193:
1190:
1187:
1184:
1181:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1152:
1146:
1143:
1140:
1137:
1134:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1096:
1093:
1091:
1089:
1083:
1080:
1077:
1074:
1071:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1027:
1021:
1018:
1015:
1012:
1009:
1004:
1001:
998:
995:
992:
989:
986:
980:
977:
975:
970:
967:
961:
960:
937:
914:
911:
908:
905:
902:
897:
894:
891:
888:
885:
882:
879:
873:
867:
864:
850:
849:
838:
835:
832:
829:
826:
823:
820:
817:
787:
767:
747:
719:
699:
675:
672:
669:
666:
663:
660:
657:
654:
651:
631:
628:
622:
619:
612:
609:
605:
602:
596:
593:
587:
584:
581:
578:
574:
571:
550:
547:
544:
541:
538:
535:
531:
528:
524:
521:
518:
515:
512:
507:
503:
482:
479:
476:
470:
467:
461:
458:
455:
452:
449:
426:
423:
396:
376:
356:
346:
345:
334:
331:
328:
325:
322:
319:
296:
276:
260:
257:
209:
206:
181:which affects
162:
159:
158:
157:
149:
119:
118:
111:
104:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7482:
7471:
7468:
7466:
7463:
7462:
7460:
7451:by Mark Thoma
7450:
7446:
7441:
7438:
7434:
7430:
7425:
7422:
7418:
7415:
7414:
7410:
7405:
7401:
7398:
7394:
7391:
7390:
7386:
7380:
7374:
7370:
7366:
7362:
7358:
7352:
7348:
7344:
7343:Sargan, Denis
7340:
7336:
7330:
7326:
7321:
7320:
7314:
7310:
7306:
7302:
7296:
7292:
7287:
7286:
7280:
7276:
7275:
7271:
7263:
7257:
7253:
7249:
7242:
7239:
7234:
7230:
7226:
7222:
7218:
7214:
7210:
7206:
7205:
7200:
7193:
7190:
7185:
7181:
7177:
7173:
7168:
7163:
7159:
7155:
7154:
7146:
7143:
7138:
7134:
7130:
7126:
7119:
7116:
7111:
7107:
7103:
7099:
7095:
7091:
7090:
7082:
7079:
7074:
7070:
7066:
7062:
7057:
7052:
7048:
7044:
7043:
7035:
7032:
7027:
7023:
7019:
7012:
7009:
7003:
7000:
6995:
6989:
6985:
6978:
6975:
6970:
6963:
6960:
6955:
6951:
6947:
6943:
6942:
6934:
6931:
6925:
6920:
6915:
6910:
6906:
6902:
6901:
6896:
6889:
6886:
6881:
6875:
6871:
6866:
6865:
6856:
6854:
6852:
6850:
6846:
6842:
6836:
6833:
6828:
6824:
6820:
6813:
6810:
6804:
6799:
6795:
6791:
6790:
6785:
6778:
6775:
6770:
6766:
6762:
6758:
6755:(1): 94–107.
6754:
6750:
6743:
6740:
6735:
6731:
6727:
6723:
6719:
6715:
6714:
6706:
6703:
6698:
6685:
6677:
6670:
6667:
6664:
6659:
6656:
6651:
6647:
6643:
6639:
6635:
6631:
6626:
6621:
6617:
6613:
6612:
6604:
6601:
6596:
6592:
6588:
6584:
6580:
6576:
6572:
6568:
6567:
6562:
6555:
6553:
6549:
6543:
6538:
6535:
6532:
6529:
6528:
6524:
6522:
6520:
6517: −
6516:
6512:
6494:
6490:
6486:
6478:
6475:, called the
6474:
6465:
6463:
6461:
6453:
6451:
6449:
6445:
6436:
6434:
6430:
6428:
6420:
6418:
6414:
6412:
6408:
6403:
6400:
6396:
6392:
6376:
6364:
6348:
6345:
6336:
6333:
6330:
6327:
6324:
6313:
6300:
6296:
6290:
6278:
6277:
6276:
6262:
6242:
6222:
6202:
6179:
6176:
6173:
6150:
6141:
6124:
6115:
6112:
6109:
6103:
6097:
6092:
6084:
6075:
6064:
6061:
6052:
6040:
6039:
6038:
6024:
6004:
5984:
5964:
5961:
5958:
5938:
5918:
5898:
5878:
5851:
5848:
5845:
5839:
5836:
5833:
5826:
5808:
5805:
5802:
5796:
5793:
5790:
5783:
5782:
5781:
5767:
5755:
5753:
5739:
5719:
5694:
5691:
5686:
5682:
5670:
5664:
5661:
5656:
5652:
5647:
5643:
5631:
5626:
5621:
5618:
5606:
5602:
5590:
5584:
5581:
5576:
5572:
5567:
5563:
5550:
5546:
5534:
5529:
5524:
5521:
5507:
5504:
5495:
5492:
5481:
5478:
5461:
5458:
5448:
5439:
5431:
5430:
5429:
5414:
5410:
5406:
5401:
5397:
5391:
5387:
5383:
5378:
5374:
5361:
5357:
5348:
5330:
5326:
5305:
5300:
5296:
5292:
5286:
5283:
5260:
5246:
5243:
5234:
5231:
5220:
5217:
5200:
5197:
5174:
5172:
5168:
5149:
5146:
5139:
5135:
5121:
5115:
5112:
5107:
5103:
5096:
5092:
5078:
5073:
5068:
5059:
5051:
5050:
5049:
5031:
5011:
5008:
5002:
4999:
4993:
4990:
4983:
4982:
4981:
4979:
4975:
4971:
4953:
4950:
4943:
4939:
4933:
4930:
4917:
4911:
4908:
4900:
4889:
4882:
4877:
4871:
4868:
4862:
4859:
4853:
4850:
4840:
4839:
4838:
4820:
4817:
4806:
4800:
4797:
4789:
4778:
4771:
4765:
4762:
4752:
4751:
4750:
4731:
4728:
4725:
4722:
4719:
4711:
4707:
4703:
4699:
4692:
4690:
4676:
4673:
4670:
4666:
4663:
4654:
4650:
4643:
4608:
4605:
4598:
4596:
4588:
4577:
4571:
4568:
4560:
4549:
4542:
4540:
4532:
4521:
4515:
4512:
4504:
4493:
4483:
4472:
4463:
4460:
4452:
4441:
4434:
4432:
4424:
4413:
4407:
4404:
4396:
4385:
4378:
4367:
4361:
4358:
4350:
4339:
4329:
4318:
4309:
4306:
4298:
4287:
4280:
4278:
4256:
4253:
4238:
4237:
4236:
4234:
4230:
4227:
4223:
4219:
4215:
4211:
4207:
4191:
4180:
4159:
4148:
4144:
4141:
4130:
4121:
4117:
4098:
4087:
4081:
4078:
4070:
4059:
4052:
4041:
4035:
4032:
4024:
4013:
4007:
4004:
3996:
3985:
3978:
3967:
3960:
3940:
3937:
3926:
3925:
3924:
3904:
3883:Proof that β
3879:
3875:
3853:
3847:
3844:
3836:
3825:
3818:
3815:
3810:
3806:
3798:
3780:
3776:
3752:
3749:
3744:
3740:
3728:
3722:
3719:
3711:
3706:
3702:
3690:
3683:
3663:
3660:
3649:
3648:
3647:
3645:
3641:
3637:
3633:
3629:
3624:
3610:
3607:
3604:
3593:
3572:
3569:
3566:
3555:
3531:
3525:
3514:
3508:
3505:
3497:
3486:
3479:
3476:
3473:
3462:
3456:
3453:
3445:
3434:
3427:
3424:
3413:
3407:
3404:
3396:
3385:
3378:
3361:
3358:
3347:
3346:
3345:
3331:
3328:
3325:
3314:
3290:
3279:
3273:
3270:
3262:
3251:
3244:
3227:
3224:
3213:
3212:
3211:
3194:
3189:
3185:
3181:
3178:
3173:
3169:
3165:
3160:
3156:
3148:
3147:
3146:
3143:
3139:
3134:
3132:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3076:
3072:
3068:
3065:component of
3064:
3060:
3044:
3035:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2997:
2993:
2989:
2970:
2967:
2964:
2958:
2955:
2947:
2943:
2939:
2935:
2916:
2905:
2899:
2896:
2888:
2877:
2870:
2867:
2864:
2858:
2855:
2852:
2849:
2835:
2829:
2826:
2818:
2807:
2800:
2797:
2786:
2780:
2777:
2769:
2758:
2751:
2731:
2728:
2717:
2716:
2715:
2714:estimator is
2713:
2709:
2704:
2702:
2701:homoskedastic
2698:
2694:
2678:
2656:
2652:
2629:
2625:
2602:
2598:
2577:
2553:
2546:
2529:
2525:
2502:
2498:
2489:
2471:
2467:
2459:
2456:
2438:
2434:
2426:
2423:
2405:
2401:
2393:
2390:
2387:
2386:
2385:
2368:
2363:
2359:
2355:
2352:
2347:
2343:
2339:
2334:
2330:
2322:
2321:
2320:
2314:
2312:
2308:
2264:
2244:
2239:
2212:
2207:
2156:
2154:
2142:
2137:
2130:
2125:
2103:
2098:
2086:
2081:
2074:
2069:
2067:
2065:
2061:
2057:
2056:Causal graphs
2053:
2049:
2045:
2037:
2035:
2033:
2029:
2025:
2021:
2013:
2009:
2005:
2002:
1998:
1994:
1990:
1987:
1983:
1979:
1976:
1972:
1968:
1964:
1961:
1960:
1959:
1956:
1954:
1950:
1946:
1941:
1927:
1921:
1913:
1909:
1905:
1900:
1896:
1874:
1870:
1864:
1851:
1839:
1835:
1831:
1825:
1822:
1812:
1811:
1810:
1808:
1803:
1802:are cut off.
1801:
1797:
1775:
1770:
1761:
1759:
1742:
1736:
1711:
1703:
1699:
1693:
1680:
1665:
1660:
1651:
1648:
1642:
1639:
1629:
1628:
1627:
1625:
1616:
1614:
1612:
1596:
1576:
1556:
1536:
1516:
1496:
1476:
1455:
1446:
1442:
1426:
1423:
1417:
1414:
1411:
1405:
1402:
1382:
1360:
1356:
1335:
1332:
1326:
1323:
1320:
1314:
1311:
1289:
1285:
1257:
1248:
1242:
1239:
1231:
1228:
1225:
1219:
1216:
1210:
1205:
1201:
1197:
1188:
1182:
1179:
1171:
1168:
1165:
1159:
1156:
1150:
1141:
1135:
1132:
1124:
1121:
1118:
1115:
1112:
1109:
1103:
1100:
1094:
1092:
1078:
1072:
1069:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1034:
1031:
1025:
1016:
1010:
1007:
999:
996:
993:
987:
984:
978:
976:
968:
965:
951:
950:
949:
935:
909:
903:
900:
892:
889:
886:
880:
877:
871:
865:
862:
836:
833:
830:
827:
824:
821:
818:
815:
808:
807:
806:
804:
803:partialed out
799:
785:
765:
745:
737:
733:
717:
697:
689:
673:
670:
664:
661:
658:
652:
649:
629:
626:
620:
617:
610:
607:
603:
594:
591:
585:
582:
579:
572:
569:
545:
542:
539:
536:
529:
522:
519:
516:
513:
505:
480:
477:
468:
465:
459:
456:
450:
447:
424:
421:
410:
394:
374:
367:is a vector.
354:
332:
329:
326:
323:
320:
317:
310:
309:
308:
294:
274:
266:
258:
256:
254:
250:
246:
242:
237:
233:
229:
225:
223:
219:
218:Olav Reiersøl
215:
207:
205:
203:
199:
195:
190:
188:
184:
180:
176:
172:
168:
160:
155:
150:
147:
143:
142:
141:
138:
136:
132:
128:
124:
116:
112:
109:
105:
102:
98:
97:
96:
94:
90:
86:
82:
78:
73:
70:
66:
62:
57:
53:
49:
45:
41:
37:
33:
19:
7403:
7396:
7387:Bibliography
7368:
7346:
7318:
7284:
7252:Econometrics
7251:
7241:
7208:
7204:Econometrica
7202:
7192:
7157:
7151:
7145:
7131:(430): 443.
7128:
7124:
7118:
7093:
7087:
7081:
7046:
7040:
7034:
7017:
7011:
7002:
6983:
6977:
6968:
6962:
6945:
6939:
6933:
6924:1721.1/63775
6907:(4): 69–85.
6904:
6898:
6888:
6868:. New York:
6863:
6840:
6835:
6818:
6812:
6793:
6787:
6777:
6752:
6748:
6742:
6717:
6711:
6705:
6684:cite journal
6669:
6658:
6615:
6609:
6603:
6570:
6566:Econometrica
6564:
6518:
6514:
6472:
6469:
6459:
6457:
6446:against the
6440:
6431:
6424:
6415:
6404:
6398:
6394:
6390:
6368:
6142:
6139:
5870:
5759:
5711:
5273:. Replacing
5182:
5170:
5167:probit model
5164:
5048:which gives
5047:
4977:
4973:
4972:
4969:
4836:
4709:
4705:
4701:
4700:
4696:
4652:
4646:
4641:
4228:
4225:
4221:
4217:
4213:
4209:
4205:
4119:
4115:
4113:
3923:expression:
3895:
3876:
3767:
3639:
3635:
3634:matrix with
3631:
3627:
3625:
3546:
3305:
3209:
3141:
3137:
3135:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3074:
3070:
3066:
3058:
3036:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
2999:
2996:uncorrelated
2991:
2987:
2945:
2941:
2937:
2933:
2931:
2707:
2705:
2692:
2569:
2487:
2454:
2421:
2388:
2383:
2318:
2309:
2265:
2157:
2150:
2063:
2059:
2051:
2047:
2043:
2041:
2031:
2027:
2023:
2019:
2017:
2011:
2007:
2000:
1996:
1992:
1985:
1981:
1974:
1970:
1966:
1957:
1952:
1948:
1944:
1942:
1911:
1907:
1903:
1898:
1894:
1892:
1806:
1804:
1799:
1757:
1728:
1623:
1620:
1444:
1276:
851:
800:
735:
347:
262:
238:
234:
230:
226:
211:
201:
197:
193:
191:
186:
182:
178:
174:
170:
166:
164:
153:
145:
139:
130:
120:
74:
47:
43:
40:epidemiology
36:econometrics
29:
7423:'s textbook
7026:10852/57801
6948:(1): 1–27.
6444:F-statistic
3547:As long as
2517:other than
1760:-separation
1755:stands for
411:solves for
241:Judea Pearl
89:error terms
7459:Categories
7437:Mark Thoma
7020:(Thesis).
6544:References
6425:As Bound,
5347:idempotent
3063:endogenous
2315:Estimation
2090:Figure 2:
1809:satisfies
1611:Estimation
440:such that
243:in 2000.
175:instrument
135:endogenous
131:instrument
123:endogenous
106:there are
101:covariates
93:regression
85:correlated
77:consistent
32:statistics
7404:MIT Press
7233:119872226
7162:CiteSeerX
7051:CiteSeerX
6827:793451601
6620:CiteSeerX
6595:153123153
6377:β
6346:≤
6334:∣
6297:∑
6098:
6065:∣
5740:β
5720:β
5662:−
5582:−
5508:^
5493:−
5482:^
5462:^
5440:β
5287:^
5247:^
5232:−
5221:^
5201:^
5113:−
5060:β
5009:β
5003:^
4974:Stage 2:
4909:−
4872:^
4869:δ
4854:^
4798:−
4766:^
4763:δ
4729:δ
4609:^
4606:β
4569:−
4513:−
4461:−
4405:−
4359:−
4307:−
4257:^
4254:β
4212:matrices
4122:. Hence,
4079:−
4033:−
4005:−
3941:^
3938:β
3905:β
3845:−
3720:−
3664:^
3661:β
3532:β
3529:→
3506:−
3477:β
3454:−
3405:−
3362:^
3359:β
3271:−
3228:^
3225:β
3179:γ
3121:are both
3045:β
2959:
2897:−
2868:β
2853:β
2827:−
2778:−
2732:^
2729:β
2679:β
2578:β
2554:β
2353:β
2295:↔
2275:→
2248:¯
2216:¯
2187:→
2167:→
2107:¯
1928:⊥
1922:⊥
1871:⊥
1865:⊥
1832:⊥
1826:⊥
1779:¯
1743:⊥
1737:⊥
1700:⊥
1694:⊥
1669:¯
1649:⊥
1643:⊥
1456:β
1424:≠
1406:
1383:β
1361:∗
1357:β
1315:
1290:∗
1286:β
1243:
1220:
1206:∗
1202:β
1183:
1160:
1136:
1122:β
1116:α
1104:
1073:
1053:β
1047:α
1035:
1011:
988:
969:^
966:β
904:
881:
866:^
863:β
828:β
822:α
671:≠
653:
621:^
595:^
592:β
583:−
546:β
540:−
523:β
517:−
506:β
469:^
451:
425:^
422:β
395:β
327:β
87:with the
7367:(2013).
7345:(1988).
7315:(2009).
7281:(2008).
7184:14793271
7073:18365761
6734:15066689
6642:20307128
6525:See also
4976:Regress
4702:Stage 1:
4667:′
4653:m < k
3636:M > K
3085:to be a
3077:to be a
3030:on
3004:unbiased
2697:variance
1857:⧸
1686:⧸
611:′
573:′
530:′
255:(2008).
7449:YouTube
7433:YouTube
7417:Chapter
7225:2938359
6769:2663184
6650:7913867
6587:2951620
2486:is the
2420:is the
2199:GPA in
253:Heckman
249:Krueger
245:Angrist
208:History
161:Example
7375:
7353:
7331:
7327:–736.
7297:
7293:–353.
7258:
7231:
7223:
7182:
7164:
7110:146178
7108:
7071:
7053:
6990:
6876:
6825:
6767:
6732:
6648:
6640:
6622:
6593:
6585:
6427:Jaeger
5871:where
4736:errors
3768:where
2932:where
2384:where
2042:Since
1893:where
1729:where
1277:where
948:gives
259:Theory
69:biased
7419:from
7229:S2CID
7221:JSTOR
7180:S2CID
7106:JSTOR
7069:S2CID
6765:JSTOR
6646:S2CID
6591:S2CID
6583:JSTOR
4231:(see
3632:T × M
3630:is a
3101:is a
2544:, and
1999:when
1973:when
1910:been
1796:graph
1395:. If
91:in a
67:give
65:ANOVA
54:when
7373:ISBN
7351:ISBN
7329:ISBN
7295:ISBN
7256:ISBN
6988:ISBN
6874:ISBN
6823:OCLC
6730:PMID
6697:help
6638:PMID
6448:null
6235:and
5977:and
5891:and
5444:2SLS
5064:2SLS
4224:) =
4216:and
4208:-by-
4172:and
3117:and
2994:are
2990:and
2940:and
1914:and
1762:and
710:and
247:and
189:.
113:the
63:and
18:2SLS
7447:on
7435:by
7431:on
7325:711
7291:314
7213:doi
7172:doi
7133:doi
7098:doi
7061:doi
7022:hdl
6950:doi
6919:hdl
6909:doi
6798:doi
6757:doi
6722:doi
6630:doi
6575:doi
6397:on
6310:max
6287:max
6049:ACE
6017:on
4749:):
4712:, (
4708:on
4235:):
4220:, (
3909:GMM
3885:GMM
2956:cov
2703:).
2153:GPA
1445:not
1441:OLS
1403:cov
1312:cov
1240:var
1217:cov
1180:var
1157:cov
1133:var
1101:cov
1070:var
1032:cov
1008:var
985:cov
901:var
878:cov
758:on
736:not
732:OLS
650:cov
502:min
448:cov
409:OLS
30:In
7461::
7311:;
7250:.
7227:.
7219:.
7209:58
7207:.
7201:.
7178:.
7170:.
7158:20
7156:.
7129:90
7127:.
7104:.
7094:32
7092:.
7067:.
7059:.
7047:92
7045:.
6946:76
6944:.
6917:.
6905:15
6903:.
6897:.
6872:.
6848:^
6794:17
6792:.
6786:.
6763:.
6753:41
6751:.
6728:.
6718:57
6716:.
6688::
6686:}}
6682:{{
6644:.
6636:.
6628:.
6616:98
6614:.
6589:.
6581:.
6571:62
6569:.
6563:.
6551:^
6349:1.
6319:Pr
6069:do
6056:Pr
4222:AB
3889:IV
3874:.
3133:.
3125:×
3105:×
3034:.
2936:,
2263:.
1955:.
798:.
48:IV
38:,
34:,
7439:.
7381:.
7359:.
7337:.
7303:.
7264:.
7235:.
7215::
7186:.
7174::
7139:.
7135::
7112:.
7100::
7075:.
7063::
7028:.
7024::
6996:.
6956:.
6952::
6927:.
6921::
6911::
6882:.
6829:.
6806:.
6800::
6771:.
6759::
6736:.
6724::
6699:)
6695:(
6678:.
6652:.
6632::
6597:.
6577::
6519:k
6515:m
6495:2
6491:R
6487:T
6460:X
6399:y
6395:x
6391:x
6343:]
6340:)
6337:z
6331:x
6328:,
6325:y
6322:(
6314:z
6306:[
6301:y
6291:x
6263:Z
6243:g
6223:f
6203:X
6183:)
6180:Y
6177:,
6174:X
6171:(
6151:Z
6125:.
6122:]
6119:)
6116:u
6113:,
6110:x
6107:(
6104:f
6101:[
6093:u
6089:E
6085:=
6082:)
6079:)
6076:x
6073:(
6062:y
6059:(
6053:=
6025:Y
6005:X
5985:Y
5965:X
5962:,
5959:Z
5939:U
5919:Z
5899:g
5879:f
5855:)
5852:u
5849:,
5846:x
5843:(
5840:f
5837:=
5834:y
5812:)
5809:u
5806:,
5803:z
5800:(
5797:g
5794:=
5791:x
5768:Z
5695:.
5692:Y
5687:Z
5683:P
5676:T
5671:X
5665:1
5657:)
5653:X
5648:Z
5644:P
5637:T
5632:X
5627:(
5622:=
5619:Y
5613:T
5607:Z
5603:P
5596:T
5591:X
5585:1
5577:)
5573:X
5568:Z
5564:P
5557:T
5551:Z
5547:P
5540:T
5535:X
5530:(
5525:=
5522:Y
5516:T
5505:X
5496:1
5489:)
5479:X
5470:T
5459:X
5452:(
5449:=
5415:Z
5411:P
5407:=
5402:Z
5398:P
5392:Z
5388:P
5384:=
5379:Z
5375:P
5368:T
5362:Z
5358:P
5331:Z
5327:P
5306:X
5301:Z
5297:P
5293:=
5284:X
5261:Y
5255:T
5244:X
5235:1
5228:)
5218:X
5209:T
5198:X
5191:(
5150:.
5147:Y
5140:Z
5136:P
5127:T
5122:X
5116:1
5108:)
5104:X
5097:Z
5093:P
5084:T
5079:X
5074:(
5069:=
5032:,
5028:e
5025:s
5022:i
5019:o
5016:n
5012:+
5000:X
4994:=
4991:Y
4978:Y
4954:.
4951:X
4944:Z
4940:P
4934:=
4931:X
4923:T
4918:Z
4912:1
4905:)
4901:Z
4895:T
4890:Z
4886:(
4883:Z
4878:=
4863:Z
4860:=
4851:X
4821:,
4818:X
4812:T
4807:Z
4801:1
4794:)
4790:Z
4784:T
4779:Z
4775:(
4772:=
4732:+
4726:Z
4723:=
4720:X
4710:Z
4706:X
4677:0
4674:=
4671:v
4664:Z
4620:V
4617:I
4599:=
4589:y
4583:T
4578:Z
4572:1
4565:)
4561:X
4555:T
4550:Z
4546:(
4543:=
4533:y
4527:T
4522:Z
4516:1
4509:)
4505:Z
4499:T
4494:Z
4490:(
4487:)
4484:Z
4478:T
4473:Z
4469:(
4464:1
4457:)
4453:X
4447:T
4442:Z
4438:(
4435:=
4425:y
4419:T
4414:Z
4408:1
4401:)
4397:Z
4391:T
4386:Z
4382:(
4379:Z
4373:T
4368:X
4362:1
4355:)
4351:Z
4345:T
4340:X
4336:(
4333:)
4330:Z
4324:T
4319:Z
4315:(
4310:1
4303:)
4299:X
4293:T
4288:Z
4284:(
4281:=
4271:M
4268:M
4265:G
4229:A
4226:B
4218:B
4214:A
4210:n
4206:n
4192:X
4186:T
4181:Z
4160:Z
4154:T
4149:Z
4145:,
4142:Z
4136:T
4131:X
4120:Z
4116:X
4099:y
4093:T
4088:Z
4082:1
4075:)
4071:Z
4065:T
4060:Z
4056:(
4053:Z
4047:T
4042:X
4036:1
4029:)
4025:X
4019:T
4014:Z
4008:1
4001:)
3997:Z
3991:T
3986:Z
3982:(
3979:Z
3973:T
3968:X
3964:(
3961:=
3955:M
3952:M
3949:G
3859:T
3854:Z
3848:1
3841:)
3837:Z
3831:T
3826:Z
3822:(
3819:Z
3816:=
3811:Z
3807:P
3781:Z
3777:P
3753:,
3750:y
3745:Z
3741:P
3734:T
3729:X
3723:1
3716:)
3712:X
3707:Z
3703:P
3696:T
3691:X
3687:(
3684:=
3678:M
3675:M
3672:G
3628:Z
3611:0
3608:=
3605:e
3599:T
3594:Z
3573:0
3570:=
3567:e
3561:T
3556:Z
3526:e
3520:T
3515:Z
3509:1
3502:)
3498:X
3492:T
3487:Z
3483:(
3480:+
3474:X
3468:T
3463:Z
3457:1
3450:)
3446:X
3440:T
3435:Z
3431:(
3428:=
3425:y
3419:T
3414:Z
3408:1
3401:)
3397:X
3391:T
3386:Z
3382:(
3379:=
3373:V
3370:I
3332:0
3329:=
3326:e
3320:T
3315:Z
3291:y
3285:T
3280:Z
3274:1
3267:)
3263:X
3257:T
3252:Z
3248:(
3245:=
3239:V
3236:I
3195:,
3190:i
3186:v
3182:+
3174:i
3170:Z
3166:=
3161:i
3157:x
3142:i
3138:x
3127:K
3123:T
3119:Z
3115:X
3111:K
3107:K
3103:T
3099:X
3095:Z
3091:X
3087:T
3083:Z
3079:T
3075:X
3071:e
3067:X
3059:Z
3032:y
3028:X
3024:X
3020:y
3016:β
3012:e
3008:X
3000:X
2992:e
2988:X
2974:)
2971:y
2968:,
2965:X
2962:(
2946:T
2942:e
2938:y
2934:X
2917:e
2911:T
2906:X
2900:1
2893:)
2889:X
2883:T
2878:X
2874:(
2871:+
2865:=
2862:)
2859:e
2856:+
2850:X
2847:(
2841:T
2836:X
2830:1
2823:)
2819:X
2813:T
2808:X
2804:(
2801:=
2798:y
2792:T
2787:X
2781:1
2774:)
2770:X
2764:T
2759:X
2755:(
2752:=
2746:S
2743:L
2740:O
2708:T
2693:e
2657:i
2653:y
2630:i
2626:X
2603:i
2599:y
2530:i
2526:X
2503:i
2499:y
2488:i
2472:i
2468:e
2455:i
2439:i
2435:X
2422:i
2406:i
2402:y
2389:i
2369:,
2364:i
2360:e
2356:+
2348:i
2344:X
2340:=
2335:i
2331:y
2245:X
2240:G
2213:X
2208:G
2104:X
2099:G
2064:W
2060:Z
2052:U
2048:Z
2044:U
2032:X
2028:Y
2024:X
2020:U
2014:.
2012:X
2008:Z
2001:X
1997:Y
1993:Z
1988:.
1986:U
1982:Z
1975:X
1971:Y
1967:U
1953:W
1949:Z
1945:W
1912:x
1908:X
1904:Y
1899:x
1895:Y
1878:)
1875:X
1852:Z
1849:(
1845:)
1840:x
1836:Y
1823:Z
1820:(
1807:Z
1800:X
1776:X
1771:G
1758:d
1712:G
1708:)
1704:X
1681:Z
1678:(
1666:X
1661:G
1656:)
1652:Y
1640:Z
1637:(
1624:Z
1597:u
1577:x
1557:z
1537:u
1517:z
1497:u
1477:x
1427:0
1421:)
1418:u
1415:,
1412:x
1409:(
1336:0
1333:=
1330:)
1327:u
1324:,
1321:x
1318:(
1258:,
1252:)
1249:x
1246:(
1235:)
1232:u
1229:,
1226:x
1223:(
1211:+
1198:=
1192:)
1189:x
1186:(
1175:)
1172:u
1169:,
1166:x
1163:(
1151:+
1145:)
1142:x
1139:(
1128:)
1125:x
1119:+
1113:,
1110:x
1107:(
1095:=
1082:)
1079:x
1076:(
1065:)
1062:u
1059:+
1056:x
1050:+
1044:,
1041:x
1038:(
1026:=
1020:)
1017:x
1014:(
1003:)
1000:y
997:,
994:x
991:(
979:=
936:y
913:)
910:x
907:(
896:)
893:y
890:,
887:x
884:(
872:=
837:u
834:+
831:x
825:+
819:=
816:y
786:X
766:Y
746:X
718:Y
698:X
674:0
668:)
665:U
662:,
659:X
656:(
630:0
627:=
618:U
608:X
604:=
601:)
586:X
580:Y
577:(
570:X
549:)
543:X
537:Y
534:(
527:)
520:X
514:Y
511:(
481:0
478:=
475:)
466:U
460:,
457:X
454:(
375:X
355:Y
333:U
330:+
324:X
321:=
318:Y
295:X
275:Z
202:Z
198:Y
194:X
187:X
183:Y
179:Z
171:Y
167:X
156:.
117:.
46:(
20:)
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