Quantitative Methods

- Heteroskedasticity

Heterskedasticity

- OLS assumes homoskedastic error terms. In OLS,

the data are homoskedastic if the error term does

not have constant variance. - If there is non-constant variance of the error

terms, the error terms are related to some

variable (or set of variables), or to case .

The data is then heteroskedastic.

Heteroskedasticity

- Example (from wikipedia, I confessit has

relevant graphs which are easily pasted!) - Note as X increases, the variance of the error

term increases (the goodness of fit gets worse)

Heteroskedasticity

- As you can see from the graph, the b (parameter

estimate estimated slope or effect of x on y)

will not necessarily change. - However, heteroskedasticity changes the standard

errors of the bsmaking us more or less

confident in our slope estimates than we would be

otherwise.

Heteroskedasticity

- Note that whether one is more confident or less

confident depends in large part on the

distribution of the dataif there is relatively

poor goodness of fit near the mean of X, where

most of the data points tend to be, then it is

likely that you will be less confident in your

slope estimates than you would b otherwise. If

the data fit the line relatively well near the

mean of X, then it is likely that you will be

more confident in your slope estimates than you

would be otherwise.

Heteroskedasticity why?

- Learning?either your coders learn (in which case

you have measurement error), or your cases

actually learn. For example, if you are

predicting wages with experience, it is likely

that variance is reduced among those with more

experience.

Heteroskedasticity why?

- Scope of choice some subsets of your data may

have more discretion. So, if you want to predict

saving behavior with wealth?wealthier individuals

might show greater variance in their behavior.

Heteroskedasticity

- Heteroskedasticity is very common in pooled data,

which makes sensefor example, some phenomenon

(i.e., voting) may be more predictable in some

states than in others.

Heteroskedasticity

- But note that what looks like heteroskedasticity

could actually be measurement error (improving or

deteriorating, thus causing differences in

goodness of fit), or specification issues (you

have failed to control for something which might

account for how predictable your dependent

variable is across different subsets of data).

Heteroskedasticity Tests

- The tests for heteroskedasticity tend to

incorporate the same basic idea of figuring out

through an auxiliary regression analysis

whether the independent variables (or case , or

some combination of independent variables) have a

significant relationship to the goodness of fit

of the model.

Heteroskedasticity Tests

- In other words, all of the tests seek to answer

the question Does my model fit the data better

in some places than in others? Is the goodness

of fit significantly better at low values of some

independent variable X? Or at high values? Or

in the mid-range of X? Or in some subsets of

data?

Heteroskedasticity Tests

- Also note that no single test is definitivein

part because, as observed in class, there could

be problems with the auxiliary regressions

themselves. - Well examine just a few tests, to give you the

basic idea.

Heteroskedasticity Tests

- The first thing you could do is just examine your

data in a scatterplot. - Of course, it is time consuming to examine all

the possible ways in which your data could be

heteroskedastic (that is, relative to each X, to

combinations of X, to case , to other variables

that arent in the model such as pooling unit,

etc.)

Heteroskedasticity Tests

- Another test is the Goldfeld-Quandt. The

Goldfeld Quandt essentially asks you to compare

the goodness of fit of two areas of your data. - Disadvantages?you need to have pre-selected an X

that you think is correlated with the variance of

the error term. - G-Q assumes a monotonic relations between X and

the variance of the error term. - That is, is will only work to diagnose

heteroskedasticity where the goodness of fit at

the low levels of X is different than the

goodness of fit of high levels of X (as in the

graph above). But it wont work to diagnose

heteroskedasticity where the goodness of fit in

the mid-range of X is different from the goodness

of fit at both the low end of X and the high end

of X.

Heteroskedasticity Tests

- Goldfeld-Quandt test--steps
- First, order the n cases by the X that you think

is correlated with ei2. - Then, drop a section of c cases out of the

middle(one-fifth is a reasonable number). - Then, run separate regressions on both upper and

lower samples. You will then be able to compare

the goodness of fit between the two subsets of

your data.

Heteroskedasticity Tests

- Obtain the residual sum of squares from each

regression (ESS-1 and ESS-2). - Then, calculate GQ, which has an F distribution.

Heteroskedasticity Tests

- The numerator represents the residual mean

square from the first regressionthat is, ESS-1

/ df. The df (degrees of freedom) are n-k-1.

n is the number of cases in that first subset

of data, and k is the of independent variables

(and then, 1 is for the intercept estimate).

Heteroskedasticity Tests

- The denominator represents the residual mean

square from the first regressionthat is, ESS-2

/ df. The df (degrees of freedom) are n-k-1.

n is the number of cases in that second subset

of data, and k is the of independent variables

(and then, 1 is for the intercept estimate).

Heteroskedasticity Tests

- Note that the F test is useful in comparing the

goodness of fit of two sets of data. - How would we know if the goodness of fit was

significantly different across the two subsets of

data? - By comparing them (as in the ratio above), we can

see if one goodness of fit is significantly

better than the other (accounting for degrees of

freedom?sample size, number of variables, etc.) - In other words, if GQ is significantly greater or

less than 1, that means that the ESS-1 / df is

significantly greater or less than the ESS-2 /

df?in other words, we have evidence of

heteroskedasticity.

Heteroskedasticity Tests

- A second test is the Glejser test
- Perform the regression analysis and save the

residuals. - Regress the absolute value of the residuals on

possible sources of heteroskedasticity - A significant coefficient indicates

heteroskedasticity

Heteroskedasticity Tests

- Glejser test
- This makes sense conceptuallyyou are testing to

see if one of your independent variables is

significantly related to the variance of your

residuals.

Heteroskedasticity Tests

- Whites Test
- Regress the squared residuals (as the dependent

variables) on... - All the X variables, all the cross products

(i.e., possible interactions) of the X variables,

and all squared values of the X variables. - Calculate an LM test statistics, which is n

R2 - The LM test statistic has a chi-squared

distribution, with the degrees of freedom

independent variables.

Heteroskedasticity Tests

- Whites Test
- The advantage of Whites test is that it does not

assume that there is a monotonic relationship

between any one X and the variance of the error

termsthe inclusion of the interactions allows

some non-linearity in that relationship. - And, it tests for heteroskedasticity in the

entire modelyou do not have to choose a

particular X to examine. - However, if you have many variables, the number

of possible interactions plus the squared

variables plus the original variables can be

quite high!

Heteroskedasticity Solutions

- GLS / Weighted Least Squares
- In a perfect world, we would actually know what

heteroskedasticity we could expectand we would

then use weighted least squares. - WLS essentially transforms the entire equation by

dividing through every part of the equation with

the square root of whatever it is that one thinks

the variance is related to. - In other words, if one thinks ones variance of

the error terms is related to X1 2, then one

divides through every element of the equation

(intercept, each bx, residual) by X1.

Heteroskedasticity Solutions

- GLS / Weighted Least Squares
- In this way, one creates a transformed equation,

where the variance of the error term is now

constant (because youve weighted it

appropriately). - Note, however, that since the equation has been

transformed, the parameter esimates are

different than in the non-transformed versionin

the example above, for b2, you have the effect of

X2/X1 on Y, not the effect of X2 on Y. So, you

need to think about that when you are

interpreting your results.

Heteroskedasticity Solutions

- However...
- We almost never know the precise form that we

expect heteroskedasticity to take. - So, in general, we ask the software package to

give us Whites Heteroskedastic-Constant

Variances and Standard Errors (Whites robust

standard errors). (alternatively, less commonly,

Newey-West is similar.) - (For those of you who have dealt with

clusteringthe basic idea here is somewhat

similar, except that in clustering, you identify

an X that you believe your data are clustered

on. When I have repeated states in a

databasethat is, multiple cases from California,

etc.I might want to cluster on state (or, if I

have repeated legislators, I could cluster on

legislator. Etc.) In general, its a

recognition that the error terms will be related

to those repeated observationsthe goodness of

fit within the observations from California will

be better than the goodness of fit across the

observations from all states.)