The Classical Model Multicollinearity and Endogeneity
What Information Does OLS use? • Kennedy (1981, 2002 and 2003) presents Ballentine Diagrams – these show how OLS uses information to estimate regression coefficients – and, where the variance (precision) of those estimated coefficients comes from
• See his 2002 paper on the web: http://www.amstat.org/publications/jse/v10n 1/kennedy.html
OLS Ballentine • Where’s Cov(X,Y)? • What is its value? • Where’s epsilon?
Ballentines with 2 Regressors • • • • •
size of circles gives variance. overlap gives identification of estimator. size of overlap gives variance of estimator. where’s the error term? 3 possible estimators of coefficients: – (x1,y) overlap, (x2,y) overlap (use x1,x2 overlap twice; – divide the (x1,x2) overlap – disgard the (x1,x2) overlap
• What happens if x1,x2 have higher covariance?
Leaving Out the Constant • Compare the formula for the slope with a constant: X − X )(Y − Y ) ( ∑ ˆ β = ∑ (X − X ) i
• To that without a constant βˆ1 =
∑ (X Y ) ∑ (X ) i
i i 2 i
Violating Assumption 6: Cov( X 1 , X 2 ) ≠ 0 •
Recall we assume that no independent variable is a perfect linear function of any other independent variable. – If a variable X1 can be written as a perfect linear function of X2 , X3 , etc., then we say these variables are perfectly collinear. – When this is true of more than one independent variable, they are perfectly multicollinear.
Perfect multicollinearity presents technical problems for computing the least squares estimates. – Example: suppose we want to estimate the regression: Yi = β0 + β1X1i + β2X2i + εi where X1 = 2X2 + 5. That is, X1 and X2 are perfectly collinear. Whenever X2 increases by one unit, we see X1 increase by 2 units, and Y increase by 2β1 + β2 units. It is completely arbitrary whether we attribute this increase in Y to X1, to X2, or to some combination of them. If X1 is in the model, then X2 is completely redundant: it contains exactly the same information as X1 (if we know the value of X1, we know the value of X2 exactly, and vice versa). Because of this, there is no unique solution to the least squares minimization problem. Rather, there are an infinite number of solutions. – Another way to think about this example: β1 measures the effect of X1 on Y, holding X2 constant. Because X1 and X2 always vary (exactly) together, there’s no way to estimate this.
Imperfect Multicollinearity •
It is quite rare that two independent variables have an exact linear relationship – it’s usually obvious when it does happen: e.g., the “dummy variable trap”
However it is very common in economic data that two (or more) independent variables are strongly, but not exactly, related – in economic data, everything affects everything else
X1i = α0 + α1X2i X1i = α0 + α1X2i + ζi where ζi is a stochastic error term
Examples of economic variables that are strongly (but not exactly) related: – income, savings, and wealth – firm size (employment), capital stock, and revenues – unemployment rate, exchange rate, interest rate, bank deposits
• • •
Thankfully, economic theory (and common sense!) tell us these variables will be strongly related, so we shouldn’t be surprised to find that they are ... But when in doubt, we can look at the sample correlation between independent variables to detect imperfect multicollinearity When the sample correlation is big enough, Assumption 6 is “almost” violated
Consequences of Multicollinearity • Least squares estimates are still unbiased • recall that only Assumptions 1-3 of the CLRM (correct specification, zero expected error, exogenous independent variables) are required for the least squares estimator to be unbiased • since none of those assumptions are violated, the least squares estimator remains unbiased
Consequences of Multicollinearity • The least squares estimates will have big standard errors • this is the main problem with multicollinearity • we’re trying to estimate the marginal effect of an independent variable holding the other independent variables constant. • But the strong linear relationship among the independent variables makes this difficult – we always see them move together • That is, there is very little information in the data about the thing we’re trying to estimate • Consequently, we can’t estimate it very precisely: the standard errors are large
Consequences of Multicollinearity • • • • • •
The computed t-scores will be small think about what 1 & 2 imply for the sampling distribution of the least squares estimates: the sampling distribution is centered over the true parameter values (the estimates are unbiased) but the sampling variance is very large (draw a picture) thus we’re “likely” to obtain estimates that are “far” from the true parameter values because the estimates are very imprecise, we have a difficult time rejecting any null hypothesis, because we know that even when the null is true, we could find an estimate far from the hypothesized value from the formula for the t statistic: ˆ
β k − β 0k t= se( βˆk )
we see that because the standard error is big, the t statistic is small (all else equal), and thus we are unlikely to reject H0
Consequences of Multicollinearity • Small changes in data or specification lead to large changes in parameter estimates • adding/deleting an independent variable, or adding/deleting a few observations can have huge effects on the parameter estimates • why? • think about the sampling distribution of the least squares estimates: it’s very spread out around the true coefficient values. In different samples/specifications we are likely to get very different estimates • because there is so little independent variation in the independent variables, the least squares estimator puts a lot of weight on small differences between them. Small changes in sample/specification that affect these small differences (even a little bit) get a lot of weight in the estimates • e.g., if two variables are almost identical in most of the sample, the estimator relies on a few observations where they move differently to distinguish between them. Dropping one of these observations can have a big effect on the estimates.
Detecting Multicollinearity • • •
It’s important to keep in mind that most economic variables are correlated to some degree – that is, we face some multicollinearity in every regression that we run The question is how much? And is it a problem? We’ve seen one method of detecting collinearity already: look at the sample correlation between independent variables. – rule of thumb: sample correlation > 0.8 is evidence of severe collinearity – problem: if the collinear relationship involves more than 2 independent variables, you may not detect it this way Look at Variance Inflation Factors (VIFs) – regress each independent variable Xji on all the other independent variables – Xji = α0 + α1X1i + α2X2i + αj-1Xj-1,i + αj+1Xj+1,i + αkXki + εi – collect the R2 of this regression (call it Rj2), and compute the VIF: VIF(Xj) = 1 / (1 - Rj2) – Rule of thumb: VIF(Xj) > 5 is evidence of severe multicollinearity
Remedies for Multicollinearity • Get more data – this is always a good idea, and is the best remedy for multicollinearity when it is possible – basically, the multicollinearity problem is just that there’s not enough independent variation in the data to separately identify marginal effects. If we get more data, we got more variation to identify these effects.
• Do nothing – we know our estimates are unbiased – put up with the inflated standard errors
• The text has a couple of other suggestions ... – drop an irrelevant variable – why is it there in the first place? beware omitted variable bias! – transform the multicollinear variables – only if it’s consistent with economic theory & common sense
Violating Assumption 3: Cov(Xi,εi) = 0 • We saw that correlated missing regressors induce bias. • So does biased sample selection and reverse causality. • Consider correlated missing regressors.
Omitted Variables (review) • Suppose the true DGP is: Yi = β0 + β1X1i + β2X2i + εi but we incorrectly estimate the regression model: Yi = β0* + β1*X1i + εi* – example: Y is earnings, X1 is education, and X2 is “work ethic” – we don’t observe a person’s work ethic in the data, so we can’t include it in the regression model • That is, we omit the variable X2 from our model • What is the consequence of this? • Does it mess up our estimates of β0 and β1? – it definitely messes up our interpretation of β1. With X2 in the model, β1 measures the marginal effect of X1 on Y holding X2 constant. We can’t hold X2 constant if it’s not in the model. – Our estimated regression coefficients may be biased – The estimated β1 thus measures the marginal effect of X1 on Y without holding X2 constant. Since X2 is in the error term, the error term will covary with X1 if X2 covaries with X1 .
Omitted Variables (Review) X 1i − X 1 )(Yi − Y ) X 1i − X 1 )( β 0 + β1 X 1i + β 2 X 2i + ε i − β 0 − β1 X 1 − β 2 X 2 − ε ) ( ( ∑ ∑ i i ˆ E β1 = E =E 2 2 ∑ i ( X 1i − X 1 ) ∑ i ( X 1i − X 1 )
∑ ( X 1i − X 1 ) β1 ( X 1i − X 1 ) + β 2 ( X 2i − X 2 ) + ε i − ε = E i 2 X X − ( ) ∑ 1 i 1 i
2 β X − X ∑ i ( X 1i − X ) β 2 ( X 2i − X 2 ) + ε i − ε 1 ∑ i ( 1i 1) = E + 2 2 ∑ ( X 1i − X 1 ) X − X ∑ i ( 1i 1 ) i
∑ ( X 1i − X ) ( X 2i − X 2 ) + ε i − ε = β1 + β 2 E i 2 X − X ( ) ∑ i 1i 1
) = β
∑ i ( X 1i − X 1 ) Cov [ X 1 , X 2 ] β2 = β1 + nCov X , X = + β β [ ] 1 2 1 2 2 Var [ X 1 ] ∑ i ( X 1i − X )
E ∑ i ( X 1i − X )( X 2i − X 2 )
The estimated parameter is biased, with bias linear in the true parameter on the left-out variable, and the covariance of the left-out variable with the included variable.
General Endogeneity Bias • In general, we say that a variable X is endogenous if it is correlated with the model error term. Endogeneity always induces bias. • In the 1 included regressor case, we always have X − X ε − ε ( ) ( ) ∑ i i i E β 1 = β1 + E 2 ∑ ( Xi − X ) i Cov [ X 1 , ε ] = β1 + Var [ X 1 ]
Endogeneity in a Scatter-Plot • Endogeneity is easy to draw. • Consider a 1 variable (with intercept) model • Let the conditional mean of the error term also rise linearly with the included variable • Draw the true regression line and the data • The OLS regression line will pick up both the slope of Y in X and the slope of the conditional mean of the error with respect to X.
General Endogeneity Bias • Endogeneity bias shows up in the Ballentine diagrams. • Correlated missing regressor: x2 is invisible. • What does the OLS estimator do to the (x1,x2) overlap? • more generally, some of what looks like (x1,y) is really the model error term, and not (x1,y).
Causes of Endogeneity • Correlated Missing Regressors • Sample selection – what if you select the sample on the basis of something correlated with epsilon? – For example, if you only look at workers, and exclude non-workers, the latter might have different epsilons from the former.
• Reverse causality – what if Y causes X? Since Y contains epsilon, then variation in epsilon would show up in X.
Endogeneity in Practise • Is there a correlated missing regressor problem in the immigration cohort analysis you did in assignment 2? • What is the correlation that might be a problem? What direction would the bias go? • Is there a sample selection problem in the paper you read for assignment 3? • What is the correlation that might be a problem? What direction would the bias go?
Correcting for Endogeneity • Correlated Missing Regressors – Include the missing data
• Sample Selection – Narrow the generality of interpretation: no longer applies to the population, but rather to the chunk of the population satisfying sample restrictions
• Reverse Causality is tougher
Correcting for Endogeneity • Endogeneity is like pollution in the X. • You need information that allows you to pull the pollution out of the X. • Including missing regressors is like identifying the pollution exactly, so that you can just use the X that is uncorrelated with that pollution. • Alternatively, you could find a part of the variation in X that is unpolluted by construction.
Instrumental Variables • Instruments are variables, denoted Z, that are correlated with X, but uncorrelated with the model error term by assumption or by construction. • Cov(Z,e)=0, so in the Ballentine, Z and the error term have no overlap. • But, (Z,X) do overlap
2-Stage Least Squares • Regress X on Z – generate Xˆ =E[X|Z], the predicted value of X given Z. – This is “clean”. Since Z is uncorrelated with the model error term, so is any linear function of Z.
• Regress Y on Xˆ • This regression does not suffer from endogeneity • But it does suffer from having less variance in its regressor.
Good Instruments • Instrumental variables just push the problem backwards. – Is your instrument actually exogenous? That is, is it really uncorrelated with the model error term?
• Consider regressing wages on education level – education is endogenous: unobserved ability is correlated with both education and wages – instrument would have to be correlated with education level of a person, but not their ability level • compulsory schooling age (Angrist and Krueger); distance to college (Card); presence of an ag-grant college (Moretti); etc.
Experiments • With correlated missing regressors, you can undo the endogeneity problem completely if you are sure that your variable of interest if randomly distributed with respect to all missing regressors – Ballentine with 2 disjoint regressors shows this.
• experimental methods enforce this randomness by construction
Experiments, example • In assignment 3, you read a paper that showed that visible minority men have lower earnings than white men. • But, is the regressor (visible minority) exogenous? – correlated missing regressors that affect earnings could include field-of-study, age, years of schooling, grades, etc
A Field Experiment • Oreopoulous (2010) conducts a field experiment to try to figure out whether results like mine are driven by endogeneity • The whole point is that he can randomize the “treatment” (in this case, having a minority name). – If it is random, then it has no covariance with any missing regressors
What are resume audit studies? •
Field experiments to test how characteristics on resume affect decisions to contact applicant for job interview – –
Bertrand and Mullainathan (2004) –
Resumes with randomly assigned white sounding names receive 1.5 times more callbacks than resumes with black sounding names
Other resume audit studies: – – –
Similarities and differences between applicants are perfectly known Relatively cheap, so sample can be large
Swedish names versus middle-eastern names in Sweden Old versus young (by date of HS graduation) Mother/Father versus single (by extracurricular activities)
This one: – –
First audit study to look at immigrants that differ from natives by potentially more than one factor First resume audit study in Canada
Example: Different experience • Names
Example: Different name
Random Assignment • What if some firms are racist and others are not? – Because the treatment (getting a minority resume) is randomly assigned, the estimated effect is the average of the racist and nonracist firm behaviour. – You could check how the treatment effect (regression estimate) varies with observable firm characteristics. • E.g., Oreopoulous finds bigger effects in smaller firms.
Random Assignment • Random assignment is enough to undo the effects of correlated missing regressor induced endogeneity. • But, it is too much. Angrist and Pischke (2008) Mostly Harmless Econometrics have a nice exposition of the fact that all you need is conditionally random assignment. • That is, if the regressor-of-interest is independent of the missing regressors, conditional on the included regressors, then the OLS coefficient estimate is unbiased.
Conditional Independence • Suppose that you have data on whether patients receive a certain treatment, and on their health outcome. • Suppose that – The treatment effect depends on patient health – Treatments are offered to unhealthy patients – Healthiness is completely explained by age • Then, regressing health outcome on treatment yields endogeneous results, but regressing health outcome on treatment and age does not.
Conditional Independence • Demanding conditional independence feels a lot like asking that all missing regressors be included. • But, it is less than asking for independence. • When designing data, it can be useful: – In the previous example, we don’t need hospitals to assign treatments randomly (which would be unethical if people need the treatment). – We need to assign treatments randomly conditional on the other observables.