**Transcript of Cyberseminar**

**HERC Econometrics with Observational Data**

**Propensity Scores**

**Presenter: Ciarin Phibbs, PhD**

**November 6, 2013**

*This is an unedited transcript of this session. As such, it may contain omissions or errors due to sound quality or misinterpretation. For clarification or verification of any points in the transcript, please refer to the audio version posted at www.hsrd.research.va.gov/cyberseminars/catalog-archive.cfm or contact HERC at herc@va.gov.*

**Dr. Phibbs:**Alright.  My name is Ciaran Phibbs.  I’m one of the economists at the HERC, Health Economics Resource Center, and the focus of today’s topic is on the independent, the right hand side variables.  I’ll note that the genesis of this course was really several of us talking about problems that we encountered that people were—things that weren’t being properly addressed when we reviewing articles or reviewing grant proposals as the basis of it.  Since then we have expanded it some to include some other topics.  

I’ll just note that if you have questions—how you know to ask them—and Risha is, one of the other economists here at HERC, is monitoring those.  She’ll interrupt me if she thinks it’s a clarifying question; otherwise, if it can be deferred to the end we will defer it to the end.

Regression models make several assumptions about the independent variables.  Some of them are fairly strong assumptions, and violating these assumptions can cause problems sometimes, big problems.  The purpose of this talk is to examine some of the more common problems and some of the methods of fixing them.  

This is not all encompassing.  There are things that we do not cover, but it’s—the purpose is to address some of these issues.  Some of these things are not covered or well covered in a standard school of public health, masters level econometric class.  Some classes do cover these, but there’s a lot of variability in them.  

These are things you really need to know to be doing regression analysis, and they’re not always covered as well in the basic classes.  The topics that we’re going to cover today fall into four groups.  Heteroskedasticity, clustering of observations, functional form, and testing for multicollinearity—and actually how to deal with multicollinearity.  

First of all, heteroskedasticity.  The standard regression model where your dependent—you have an independent variable, a dependent variable Yi that is a function of a constant, some Matrix βX of independent variables and error terms.  We are assuming that the error terms are independent of the Xi.  One common pattern that you see is that the size of the error terms will vary systematically with X are common—there’s a couple of common examples that the error terms get much bigger.  There’s a lot of heteroskedasticity as the X increases.  

The classic economic example is that if your dependent variable is consumption spending and your independent variable is income, that as income increases the errors increase.  You think of it intuitively as low incomes—everybody spends all their income.  Income and consumption spending are very highly correlated.  As your income goes up you have more discretion to not spend and save, and how much you spend versus save can vary more.    The size of the error terms is going up systematically with income.  That’s the classic example of heteroskedasticity.

In health care there are real problems with heteroskedasticity in terms of modeling costs, and those are actually the focus of separate lectures in this series.  I won’t go into that in detail.  I’m just going to talk about it in general.  Remember that you need to fix it, and to note that with heteroskedasticity what it affects is the standard errors.  The parameter estimates are unbiased.  You’re getting an accurate estimate, but your standard errors are going to be biased.  

The thing I want to note here is that there is a very simple way to fix this.  That is that the robust option in Stata uses Huber-White sandwich estimators to correct standard errors, so you get standard errors that are correct.  There’s—and other statistical packages will include this option.  It is not unfortunately a standard option in SAS, but you can probably get SAS to do it somehow.  I don’t know, because I’ve always just gone straight to Stata.  

There are other things that you can do prior to this method being developed.  You would have—if you had detected heteroskedasticity you would need to do some sort of a transform of the variables.  Use log(X) instead of X as the right hand side variable, weighted least squares, their other format.  The use of those has gone down significantly with the advent of this simple method to correct them.   

Alright, so I—just simple easy to fix.  Make sure you do it.  One interesting thing that one can do if you’re using a package like Stata is to run it without using the robust option and the standard errors.  Then repeat it using the robust option for your standard errors, and that will tell you the extent of the problem.  It can be informative in terms of your data, and it may point you in other directions. 

This brings me to a general plan I don’t have the slide for, but I want to make, and that is that before you start delving into running regression you really need to carefully explore and understand your data.  We’ll probably come back to this, but it’s an important point.  Carefully examine your data: all the independent variables.  Look at them with correlations; look for patterns; look at them in terms of partitioning the data; and really understand your data before you start running regressions.  It can be very informative.  You could detect problems and in terms of cleaning up your data, and it may also guide your analysis.

Clustering is something that we actually encounter fairly often in health care.  Coming back to our standard regression model—it assumes that the error terms are uncorrelated.  This may not always be the case.  We frequently run—have—when we’re looking at health care applications have variables that are different levels or units of observation, so—  

**Moderator:**Ciaran.

**Dr. Phibbs**- yes?

**Moderator:**I’m sorry to break in.  Your audio is fading in and out a little bit.  Is there any way to pull your microphone a little closer?

**Dr. Phibbs**Okay.  It may have drifted away.  Is that better?

**Moderator:**That is significantly better.  Thank you.

**Dr. Phibbs**Okay.  The head set just drifted away.

**Moderator:**Yeah.

**Dr. Phibbs**Okay, so a classic example of this is you’re running a regression.  You’ve got a whole bunch of discharge abstracts.  You’re looking at in-patient care, and the patients are clustered within hospitals.  The problem here is that—so to continue on this example where xi  is a patient level variable and x2 is a hospital level variable.  In reality you may have many more of those, but the regression in this case is going to assume that there are as many hospitals as patients.  

If you remember back to your basic regression course that as your sample slice goes up your standard errors get smaller, because the regression thinks that things are more precise.  What’s happening in this case is that it is assuming that each—for each that you actually have a separate hospital for each of these observations.  On β2 here—that hospital level variable—the parameter estimates are unaffected, but the standard errors are going to be too small.

Now I’m going to reiterate this clustering has zero effect on the parameter estimate.  All that it affects is the standard error, and when you have this clustering it is going to make the standard errors too small.  There are several ways of addressing this.  Generalized Estimating Equations can be used.  There are also formal hierarchical models which take—model the structure in series—can be used.  I’m not going to talk about hierarchical models here, because that is beyond the scope.  It would be a whole lecture in and of itself, but that is something that may need to be considered when you have this type of structured data.

One thing that I will note is that in Stata there is a cluster option which uses the same Huber-White correction, but it’s to correct the standard errors.  It’s what—basically what you do is you give a command—it’s an option in the regressions where you say cluster equals.  You tell it what the cluster is—in this case hospital ID—and that will, it will correct the standard errors for you.

In terms of GEE versus the Stata method is that they yield essentially the same result.  For formal hierarchical structures modeling versus just estimating the results and correcting for the clustering by either GEE or the Stata cluster method, you will—the answer can be very similar.  It can be very different.  It depends on the structure of the data and the nature of the relationships.  It is also—will vary by how the data varied and how much the structure matters.

Compared to hierarchical linear modeling, sometimes the answers are very similar, especially with bigger samples.  Other times they are not.  It just depends on the nature of your data.  You can sometimes infer this, but again I am only just alluding to HLM.  I’m going to move on and talk about the clustering.  

Quite an example of how clustering matters: I had a project a while ago that’s non VA relevant that looked at neonatal intensive care units and looked at patient volume at the NICUs and also the level of intensive care.  The neonatologies formally has a structure in terms of the degree of higher level NICUs or more advanced on mortality.  This was published in the New England Journal a while ago.

I apologize to the VA audience for not using a VA example, but this is one where I carefully examined this issue, and so it is actually a good teaching example in this perspective.  The failure to—I also want to note that as you will see it’s easy to fix this with the cluster option Stata.  If you’re looking through the literature, especially older literature, you see failure to correct for this all the time.  The correction can be minor or very large.  It depends on the number of clusters and how those are relative to the number of observations.  With big samples the effects can be fairly small.  

The example I’m going to show you—I have almost 50,000 patient observations and over 200 hospitals.  There are ten years of data with repeat observations.   Just to show—this table here is pulling up some but not all of the parameter estimates from the main table.  There are—so Heidi, how do I get the pointer again?

**Moderator:**At the top of the screen next to the draw there is that arrow.

**Dr. Phibbs**Arrow.  Okay, okay.  Got it.  Okay, and then I can—so basically what you’re seeing here, and I haven’t put all of them in, is the level of the NICU in terms of 1, 2, to 3B.  The reference group here is just for completeness is the really big tertiary centers that are treating lots of patients.  It’s the combination of the level and the number that they’re treating.

Just to look at the first one here: you see that the odds ratio that we reported in the paper—that won’t change the confidence interval here.  This is corrected for clustering, and you can see here that the unadjusted is slightly tighter.  It didn’t matter much.  What we had here in terms of why this didn’t change very much is that these are hospitals that are treating less than ten infants.  The difference between the number of observations and the number of hospitals with—the number of patients in a hospital is small, so you wouldn’t expect much change.

Then if you look down here you see that the changes are relatively moderate even though they are—these are big units.  These are units that are treating between 50 and 100 patients a year.  The unadjusted is tighter, but the change is not huge.  Again I also alluded to the fact, because these were big units.  There’s lots and lots of observations.  That observation is going to be relatively precisely estimated to start with, so the clustering won’t have much of an effect.  

An example here is one where this is medium level units that were treating a relatively large number of patients.  One of the relevant things here is that there were only about five of these types of hospitals.  Here the effect on the standard error was fairly large, and it went from—the unadjusted said that this relationship was significant.  In reality that relationship is not statistically significant.  It actually changed the results for this particular group of hospitals.

Again overall the effects are relatively small, and that is because in general these are large samples.  There are other applications where the results can be quite dramatic.  If you have relatively small samples and a lot of clustering of your data your standard errors can be quite different, and failing to cluster for them can result in a big change in your results.

Functional form is another thing that frequently is ignored or not adequately addressed when people are doing regression analysis.  The βX—I’ve lost my pointer.  I’m trying to get the pointer back here, and it isn’t letting me.  Oh, well.  We’re okay.  The βX assumes that each variable has a linear relationship with Y.  In health care for a lot of the variables we use, we’re using binary variables, so that’s not an issue here, because it’s just a shift of the intercept.  If you are trying to control for age, for example, and there are lots of other things like that where the variable is a continuous or at least an integer variable with a fair bit of variability.  It may not have a linear relationship.  

Age is something that is a classic case of having a non-linear relationship for many health conditions.  The risk of you having that condition or of age increasing the chance of mortality for the entire range of middle age is essentially a flat relationship where age has no effect.  Then it starts to go up slowly.  Then it accelerates as you age, so that is a non-linear relationship.  If you just put age in as an aging years in your model you’re going to have a mis-specified model, and you’re trying to model something with a linear relationship that is not linear.

The bottom line: you need to check and check carefully the functional form for every non-binary variable in your model.  While there are formal tests for model specification, some of which may be exposed to in classes, in my opinion and in the opinion of others as well these tests don’t really show what you’re looking at.  Again, this is the idea of understanding your data carefully.  

There’s a simple way to do this that—one could say, “Okay, well, we’d like to use the age example.”  You could say, “Okay, well, I’m going to do Log age, and I’m going to try all these other functions and see what fits best.”  Before you do that what I would recommend is you can use dummy variables to carefully examine the functional form and guide how you move.  Look at the distribution of the variable, the age, and I’m going to continue to use age in this case as my example, and create a set of dummy variables in reasonably small intervals.

You can do this with no excluded category, and then you just have to run the model with no intercept.  How many—continue the age example—how many variables you would—how you would define those categories will depend on how big a sample you are in the age spread.  You have to have an adequate number of observations in each category.  Carefully look at the frequency distribution of age, slice it into a bunch of small groups, and then run the model.

I’m going to give another example here for that same NICU data set that I had before where I was looking at the effect of patient volume.  That has a non-linear thing, so what I’m going to do is—you graph out the parameter estimates.  Then this gives you an idea of how to model, or if you’re going to use categorical variables how to define those categorical variables instead of just making an arbitrary limit.  Okay, I’m going to do less than ten or less than fifty.  Look and see—is there a logical point at which those—and for—this is another example where we—if you look at this slide it really is an—it isn’t a straight linear relationship.  

The relationship between volume and mortality is very high initially, declining rapidly, and then it starts to gradually flatten out.  Then it looks like it goes pretty flat.  That is true if you look at it for different levels of care as well, if I do that separately.  In this case what we determined to do because of the complexity of the relationship was we did not use—we used categories instead of trying to piece together some sort of a piece wise function.  

For some applications this may be the best thing to do.  It may be—in the example I’m saying here, we’re just breaking hospitals down into very low volume, moderately low volume, medium volume, higher volume and very high volume.  This was—it was not just volume, but it was volume interacted with level of care was a pretty complex relationship.  It can be very difficult—I mean in this case it was extremely difficult to get a continuous function to accurately predict across the entire range. 

We used categorical variables.  That’s something that one needs to consider that you don’t necessarily—you have to consider the fit of this.  In this case—and I can assert here that if you get things—if I try to model a continuous function across the entire range—if I get it to fit well in the middle it does not fit well on the tails.  If I get it to fit in the tails, it does not fit well in the middle.

I chose in this application to use a set of binary variables.  The other thing that one could do is use a continuous variable but have separate functions for different range.  You have the effect of volume from one to twenty, or whatever, and split it up and have different numbers.  That still might be a non-linear function, but you can let the function be independent over different ranges.  As an aside when you have complex relationships using categorical variables it may be easier to present to medical audiences.

I’m just reshowing that slide before—here I captured that relationship by using—and this doesn’t present all of the categories that we used.  I used a combination of the levels and volume categories to break that relationship up.  If there aren’t any questions about that I’m going to proceed on to multicollinearity.

Which is a problem that rears its head an awful lot.  That is—and this is a simple assumption—what if X1 and X2 are strongly correlated?  If X1 goes up, and X2 goes up, and Y goes up, the regression will have trouble attributing the effect to each variable.  This increases the standard errors.  It can also increase the parameter estimates.  In the extreme you can get a case, and I’ve actually seen examples of this.  Where if X and Y are very highly—if X1 and X2 are very correlated where if you put both in the model—let’s say what happens is that one of them.  Let’s say the true parameter estimate on X1 should have been 0.5, but X1 and X2 are very correlated.  

As you put them both in the model you make a parameter estimate on X1 of 1.5 and a parameter estimate on X2 of -1.0 or something like that.  You’re getting offsetting effects and vigor, and they probably won’t get significant.  That actually can happen with strongly correlated variables.  You have to really carefully examine the structure of your data to make sure that you aren’t having collinearity affect your estimates.

Strong simple correlation—we know you have a problem.  You can actually have problems that can be affecting your parameter estimates and your standard errors that are at more moderate levels of correlation.  The rule of thumb that people try to tell you is if your correlation is less that 0.5 you’re going to be good, but you can actually have problem correlation in terms of collinearity that are affecting the estimates due to lower levels of correlation.  I’ll cycle back to this in a minute in terms of what’s actually causing that. 

The Various Inflation Factor which you can get, have printed out in SAS and Stata and most other packages, regression packages, measures the inflation in the variances of each parameter estimate due to collinearities among the regressors.  SAS also prints out something called the Tolerance which is just the inverse of the Various Inflation Factor.  The rule of thumb that you should apply is if the Variance Inflation Factor for a given parameter is greater than ten it means that that parameter has collinearity problems with some of the other regressors.  You need to investigate to find out what and determine how you’re going to address this.

I’ve already mentioned this—that simple correlations is not enough.  You need to go beyond that.  At a minimum you need to look at the correlation, at the correlation and the Variance Inflation Factor.  I’m going to give you an example here where—this was a study I did where we’re looking at the effects of nurse staffing on patient outcomes.  We initially tried to—we were including—we had how RN Tenure—how long the nurses had been working on the unit and RN age.  How old the nurses were.  Logically nurses that had been around working on the unit longer are going to be older, and so they were correlated.  

These two—the correlation between tenure and age was only 0.46, below that magic 0.5.  If you look at the Variance Inflation Factor on RN tenure and RN age—and it depended on the subsets in some of the models and so on—but these were running between 18 and 30.  As I will show you  in a minute they were messing up the results.  I’m going to note that—before I go into the details.  That in terms of multicollearity more observations will help, because they aren’t going to be—as long as they’re not perfectly correlated—the variance patterns will help the regression sort out what the effects are that are due to one variable versus the other.

There are ways to reduce, revise the data that reduce the correlation.  You can do this by, for example, to just—instead of using—I could leave age in the model and model tenure by a series of categorical variables.  They’ll still be correlated, but it may reduce the correlation enough so that it is not a problem.  There are other transformations one could make like this.  In this example we actually ended up dropping age from the model, because it was just conceptually and in reality just too problematic. 

I’m going to show you here what happens, is if I run the model only with age I get a statistical parameter estimate that more experienced nurses are associated with better patient outcomes.  I got the same thing when I was looking only at age.  If you put both in the model the parameter estimates for both got smaller and neither was significant.  Again this is where the correlation—and this is what started us thinking, okay, well, something’s strange here because of our prior’s weren’t being matched.  It comes back to the fact that one needs to carefully examine the data to figure out what’s going on and make adjustments as appropriate.

One of the adjustments may be that you can’t include all the variables that you would like to include in your model.  I’m going to come back to the multicollinearity.  Simple correlations—you clearly have a problem, but there are hidden problems.  What is really going on is that regression—when you’re running a multiple regression—you’re in n-space.  The correlation that matters in terms of causing collinearity problems that can increase.  That can affect either the standard errors or the parameter estimates or both, is how the regression is splitting things down in terms of the correlation on each of those regression planes if you will and n-space.

There’s an option in SAS that allows you to look at this in a lot of detail.  I haven’t seen it in other programs.  It probably exists out there somewhere.  It’s simple to do in SAS.  What it is, is that in the standard regression package there’s something called the Collin option.  Which explains how much of the variation in each eigen vector, which is the amount—which is in matrix language that’s the amount of the variances being explained is explained by each variable.

Intuitively this is the correlation in the Nth dimension of the regression.  I will note here that this is an option in SAS only in the basic PROC REG, but, and you’d say, “Well, what do I do if I’m doing a logistic regression?”  Well, the x matrix is identical, so you can run that regression, because this is the diagnostic on your X matrix.  You can use a different—a regression that’s totally not the same regression, logistic versus OLS.  It is okay to use an OLS thing to run this type of a regression diagnostic, because the X matrix is the same.  Therefore the relationships among the X’s is identical.

You don’t want to use that for your final output, but you can use this for your regression diagnostic to see what is happening.  I actually ran something.  This is again another newborn example.  I chose this because we have—intuitively the birth weight of an infant and the gestational age of the infant are very correlated especially when restricted to samples of premature infants, which is what the sample is drawn from.  In this case I have an R squared of over five, 0.5.  I just ran this.  I have the data, ran a simple arbitrary little model where we looked at birth weight gestational age, and I controlled for race.

In interpreting this code to show you what this Collin output looks like—interpreting the condition index, a condition index of more than ten indicates a problem.  A condition index of more than 100 indicates an extreme problem.  Then you look in the thing.  What we have here is the way SAS sorts this out is—it sorts it out by the value of the eigen vectors which are—bigger eigen vector means that that particular variable is explaining more of the variance.  

As you get down and you see this bottom row that we have a condition index of 18.  That’s problematic.  You look, and you see that it is problematic because in that eigen vector most of it is being explained by gestational age in terms of the loading on that eigen vector.  You have a lot of it also being explained by birth weight,  Those are collinear, and that is what is causing the problem.

Alright, so when you have this collinearity—I mentioned before that you could drop one of the variables.  I can fix this here.  What I can do is put—instead of using continuous variables I can use dummy variables in small intervals, in this case 100g intervals.  I estimated separate dummies for singleton males, singleton females, and multiple births.  Then I used gestational age in two-week intervals.  When I did this—and then I rerun the collinear diagnostics, and it fixed the problem.   

As I said there’s still some correlation, but because of using whole series—complex series of dummies—the maximum condition index when I do this drops to below 8.0, and my model improved.  My model estimates are—the predictive accuracy of my logistic models, the area under the RLC squared curve, also went up.  This is just a graph of what that looks like.  Again this is a situation of breaking things up to—by using dummy variables is one way.  

It could be other function—I used dummies in this case.  You can use other functional transformations to break up the degree of collinearity and get it so that you get it down to accessible levels.  Again it is just—oh, I can’t emphasize this enough.  You have to know your data.  Carefully explore your data.  Don’t just blindly run the regression.  Know what the assumptions are, and carefully test for them.  

Think through—what are the relationships that are going on in my data?  What do I need to test for when I look at it?  Do I have things that don’t make sense?  When I run the regressions—if you have results that don’t make sense is—the first thing is: oh, okay, well, I must have been wrong.  Well, maybe I should explore my data a little more carefully and make sure that it’s not just something in my data, and that my hypothesis is still right.  

This is not data mining.  This is just carefully understanding your data and making sure that your—the analysis that you are running is consistent with the somewhat strong assumptions that underlie regression analysis.  Remember that the assumptions that underlie regression analysis change somewhat depending on the model that you were doing.  Although most of them in terms of the X matrix which we were talking about here are pretty consistent across them.  

It’s an old text, but it’s a useful one.  Belsley, Kuh, and Welsch on Regression Diagnostics.  There are newer texts, but I went to graduate school a long time ago, so that’s the reference that I’m putting out there.  I’d be happy to answer any questions if people have questions.  We didn’t get any questions, so we went through this fairly quickly.  

The next lecture is—in the series—is next week by me on limited dependent variables.  That’s logistic regression but then moving beyond logistic regression for other such things as count data models or choice models, and that’s just a highlight.  I’m happy to answer any questions if somebody has a question.

**Moderator:**Ciaran, at this time it looks like we have no questions.  

**Moderator:**No, we have three questions.  Risha, if you’re not able to see them you need to switch to the presenter view.  [Cross talk 40:07].

**Moderator:**Oh, okay, thanks.  I see them now.  One person is asking, Ciaran, what is the strongly correlated cutoff?  What would be the cutoff for strong correlation?

**Dr. Phibbs**I’m not sure what that—the cutoff I mean there are guidelines in terms of multicollinearity.  You need to look at your Variance Inflation Factors.  You need to look at the numbers coming out of the regression diagnostics.  If you have a Variance Inflation Factor over ten, or you have a condition index greater than ten in this Collin diagnostics, then you have a problem.  You know that you have a problem.

**Moderator:**Okay.  One other person is asking in order to solve multicollinearity why not use PCA and then regress on the new model?  PCA isn’t defined, but I’m assuming this means Principal Components Analysis.

**Dr. Phibbs**That’s a possibility, but then you are, by doing that you are suppressing information.  Principal Components Analysis is essentially a summary.  I mean it is a way—if you have a problem it is a way if you want to force those in, but then you don’t have the ability to make—you don’t have the ability to make any interpretive inferences from that.  You can say, “Okay, the regression coefficient on the first principal component is 0.7.”  Well, what does that mean?  You don’t know.  

That’s okay if you’re just—if these factors are just things that you’re trying to control for and not the variables of interest.  If you’re interested in the parameter estimates, then you can’t do that.  It will depend on what’s collinear and how that fits with what you’re trying to do.  For those that may not understand principle components it essentially is a way of trying to understand how a series of variables explains the variance.  The first principle component explains the most, and the second principle component explains the second most, etcetera.

You could put some combination of those in.  That is a way to do—when you have two or three variables that you’re just trying to control for that are collinear, and you want to make sure that all of them are controlled for.  That is a way to do that.  As I said I’m not going to list all of the possible ways to do it, but then you can’t make any inferences on the contribution of each of the components of the Principle Components.

**Moderator:**Great.  Thanks, Ciaran.  Our third question is asking about different ways to deal with the collinear variables.  This person is asking instead of immediately [audio cuts out 43:11] collinear could you use an alternative approach in which you test the joint hypothesis if the coefficients for age and tenure, for example, are zero.  He is saying that if X tested the joint hypothesis might find that the pair of coefficients is statistically significant.  If that were in fact the case, would it be appropriate to retain both of those two variables, tenure and age, in the final equation?

**Moderator:**Okay, well, the problem here is that they were—these two variables were so collinear that you were—I talked about the offsetting effects where they were—the regression wasn’t able to sort—there wasn’t sufficient number of observations to allow us to sort out the effects of each.  It was actually—as I showed in that example it was making the parameter estimates of both smaller.  The combined parameter estimates were both—it was clearly a situation where, because it was biasing not just the standard errors but the parameter estimates that in a meaningful way, that we chose not to do that.  Yes, you can use F-test to test for the joint significance, but in this case, given how it was affecting the parameter estimates, that F-test would probably be not significant as well.

**Moderator:**Okay, and one last question actually goes back to heteroskedasticity.  It’s requested that you please go over the example for heteroskedasticity that were in your slides.

**Dr. Phibbs**Okay, well, I didn’t have an—I didn’t have an example per se.  What I talked about, and I’m going back here, is that I said—the example that I talked about was if I was modeling consumption spending and the independent variable was income that the error in that prediction goes up as your income goes up.  If you’re at, in general, at low levels of income, as your income increases you spend it all.  If you don’t have enough to eat, and you get some more money you’re going to buy more food.  You’re going to spend it.  As your income goes up where you aren’t having essential needs, you’re not spending all of your additional income.  You’re going to start to save some or invest it, donate it to charity, whatever—you’re not going to be spending it all.

As the income goes up, the propensity of different people to do that varies, so that the error in the regression model is going to— the variances in the, or the size of those EIs, goes up as income goes up.  That’s sort of an intuitive one where basically as the value goes up the size of the error term goes up, and that’s a heteroskedastic error term then.

**Moderator:**Okay, thanks Ciaran.  That looks like we’ve exhausted all four questions, and there’s no others.

**Dr. Phibbs**Okay.

**Moderator:**Okay.  Ciaran, did you have any final remarks you wanted to make before we close things out today?

**Dr. Phibbs**Just to repeat my appeal to understand your data.  Look at your data carefully, and make sure that you’re testing the assumptions before you run your regression models.  If you’re uncertain, ask for help.

**Moderator:**Fantastic.  Thank you.  For the audience once again—just a reminder.  We are having the next session in the series one week from today.  I actually just sent the registration information out to everyone.  It should hopefully be hitting your email boxes within the next fifteen minutes or so.  If you’re not already registered, please take a moment and fill that out, and we will see you back here one week from today.  

As I am closing the session out today, you will have a feedback form pop up on your screen.  We would very much appreciate if you would take a few moments to fill that out.  We do read through all of your feedback and try to incorporate it into current and upcoming sessions.  Thank you everyone for joining us for today’s HERC Econometric Session, and we hope to see you at a future session.  Thank you.

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