Review

Review of Standard Errors (cont.) Problem: we do not know s 2 Solution: estimate s 2 We do not observe the ACTUAL error terms, ei We DO observe the residual, ei

Review of Standard Errors (cont.) Our formula for Estimated Standard Errors relied on ALL the Gauss–Markov DGP assumptions. For this lecture, we will focus on the assumption of homoskedasticity. What happens if we relax the assumption that ?

Heteroskedasticity (Chapter 10.1) The variance of ei is NOT a constant s 2. The variance of ei is greater for some observations than for others.

Heteroskedasticity (cont.) For example, consider a regression of housing expenditures on income. Consumers with low values of income have little scope for varying their rent expenditures. Var(ei ) is low. Wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes. Var(ei ) is high.

Figure 10.1 Rents and Incomes for a Sample of New Yorkers

OLS and Heteroskedasticity What are the implications of heteroskedasticity for OLS? Under the Gauss–Markov assumptions (including homoskedasticity), OLS was the Best Linear Unbiased Estimator. Under heteroskedasticity, is OLS still Unbiased? Is OLS still Best?

OLS and Heteroskedasticity (cont.) A DGP with Heteroskedasticity

OLS and Heteroskedasticity (cont.) The unbiasedness conditions are the same as under the Gauss–Markov DGP. OLS is still unbiased!

OLS and Heteroskedasticity (cont.) To determine whether OLS is “Best” (i.e. the unbiased linear estimator with the lowest variance), we need to calculate the variance of a linear estimator under heteroskedasticity.

OLS and Heteroskedasticity The variance of a linear estimator is OLS minimizes OLS is no longer efficient!

OLS and Heteroskedasticity (cont.) Under heteroskedasticity, OLS is unbiased but inefficient. OLS does not have the smallest possible variance, but its variance may be acceptable. And the estimates are still unbiased. However, we do have one very serious problem: our estimated standard error formulas are wrong!

OLS and Heteroskedasticity (cont.) Implications of Heteroskedasticity: OLS is still unbiased. OLS is no longer efficient; some other linear estimator will have a lower variance. Estimated Standard Errors will be incorrect; C.I.’s and hypothesis tests (both t- and F- tests) will be incorrect.

OLS and Heteroskedasticity (cont.) Implications of Heteroskedasticity OLS is no longer efficient; some other linear estimator will have a lower variance. Can we use a better estimator? Estimated Standard Errors will be incorrect; C.I.’s and hypothesis tests (both t- and F- tests) will be incorrect. If we keep using OLS, can we calculate correct e.s.e.’s?

Tests for Heteroskedasticity Before we turn to remedies for heteroskedasticity, let us first consider tests for the complication. There are two types of tests: Tests for continuous changes in variance: White and Breusch–Pagan tests Tests for discrete (lumpy) changes in variance: the Goldfeld–Quandt test

The White Test The White test for heteroskedasticity has a basic premise: if disturbances are homoskedastic, then squared errors are on average roughly constant. Explanators should NOT be able to predict squared errors, or their proxy, squared residuals. The White test is the most general test for heteroskedasticity.

The White Test (cont.) Five Steps of the White Test: Regress Y against your various explanators using OLS Compute the OLS residuals, e1...en Regress ei2 against a constant, all of the explanators, the squares of the explanators, and all possible interactions between the explanators (p slopes total)

The White Test (cont.) Five Steps of the White Test (cont.) Compute R2 from the “auxilliary equation” in step 3 Compare nR2 to the critical value from the Chi-squared distribution with p degrees of freedom.

The White Test: Example

The White Test The White test is very general, and provides very explicit directions. The econometrician has no judgment calls to make. The White test also burns through degrees of freedom very, very rapidly. The White test is appropriate only for “large” sample sizes.

The Breusch–Pagan Test The Breusch–Pagan test is very similar to the White test. The White test specifies exactly which explanators to include in the auxilliary equation. Because the test includes cross-terms, the number of slopes (p) increases very quickly. In the Breusch–Pagan test, the econometrician selects which explanators to include. Otherwise, the tests are the same.

The Breusch–Pagan Test (cont.) In the Breusch–Pagan test, the econometrician selects m explanators to include in the auxilliary equation. Which explanators to include is a judgment call. A good judgment call leads to a more powerful test than the White test. A poor judgment call leads to a poor test.

The Goldfeld–Quandt Test Both the White test and the Breusch–Pagan test focus on smoothly changing variances for the disturbances. The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. Under homoskedasticity, all subgroups should have the same estimated variances.

The Goldfeld–Quandt Test (cont.) The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. The econometrician must divide the data into h discrete subgroups.

The Goldfeld–Quandt Test (cont.) If the Goldfeld–Quandt test is appropriate, it will generally be clear which subgroups to use.

The Goldfeld–Quandt Test (cont.) For example, the econometrician might ask whether men and women’s incomes vary similarly around their predicted means, given education and experience. To conduct a Goldfeld–Quandt test, divide the data into h = 2 groups, one for men and one for women.

The Goldfeld–Quandt Test (cont.)

Goldfeld–Quandt Test (cont.)

Goldfeld–Quandt Test: An Example Do men and women’s incomes vary similarly about their respective means, given education and experience? That is, do the error terms for an income equation have different variances for men and women? We have a sample with 3,394 men and 3,146 women.

Goldfeld–Quandt Test: An Example (cont.)

Goldfeld–Quandt Test: An Example (cont.)

WHAT TO DO. 1. Sometimes logging the variables can solve the problem WHAT TO DO? 1. Sometimes logging the variables can solve the problem. Sometimes not. 2. Use Generalized Least Squares to estimate the model with heteroscedasticity.

Generalized Least Squares OLS is unbiased, but not efficient. The OLS weights are not optimal. Suppose we are estimating a straight line through the origin: Under homoskedasticity, observations with higher X values are relatively less distorted by the error term. OLS places greater weight on observations with high X values.

Figure 10.2 Homoskedastic Disturbances More Misleading at Smaller X ’s

Generalized Least Squares Suppose observations with higher X values have error terms with much higher variances. Under this DGP, observations with high X ’s (and high variances of e) may be more misleading than observations with low X ’s (and low variances of e). In general, we want to put more weight on observations with smaller si2

Heteroskedasticity with Smaller Disturbances at Smaller X ’s

Generalized Least Squares To construct the BLUE Estimator for bS, we follow the same steps as before, but with our new variance formula. The resulting estimator is “Generalized Least Squares.”

Generalized Least Squares (cont.) In practice, econometricians choose a different method for implementing GLS. Historically, it was computationally difficult to program a new estimator (with its own weights) for every different dataset. It was easier to re-weight the data first, and THEN apply the OLS estimator.

Generalized Least Squares (cont.) We want to transform the data so that it is homoskedastic. Then we can apply OLS. It is convenient to rewrite the variance term of the heteroskedastic DGP as

Generalized Least Squares (cont.) If we know the di factor for each observation, we can transform the data by dividing through by di. Once we divide all variables by di, we obtain a new dataset that meets the Gauss–Markov conditions.

GLS: DGP for Transformed Data

Generalized Least Squares This procedure, Generalized Least Squares, has two steps: Divide all variables by di Apply OLS to the transformed variables This procedure optimally weights down observations with high di’s GLS is unbiased and efficient

Generalized Least Squares (cont.) Note: we derive the same BLUE Estimator (Generalized Least Squares) whether we: Find the optimal weights for heteroskedastic data, or Transform the data to be homoskedastic, then use OLS weights

GLS: An Example We can solve heteroskedasticity by dividing our variables through by di. The DGP with the transformed data is Gauss–Markov. The catch: we don’t observe di. How can we implement this strategy in practice?

GLS: An Example (cont.) We want to estimate the relationship We are concerned that higher income individuals are less constrained in how much income they spend in rent. Lower income individuals cram into what housing they can afford; higher income individuals find housing to suit their needs/tastes. That is, Var(ei ) may vary with income.

GLS: An Example (cont.) An initial guess: di = incomei If we have modeled heteroskedasticity correctly, then the BLUE Estimator is:

TABLE 10.1 Rent and Income in New York

TABLE 10.5 Estimating a Transformed Rent–Income Relationship,

Checking Understanding An initial guess: di = incomei How can we test to see if we have correctly modeled the heteroskedasticity?

Checking Understanding If we have the correct model of heteroskedasticity, then OLS with the transformed data should be homoskedastic. We can apply either a White test or a Breusch–Pagan test for heteroskedasticity to the model with the transformed data.

Checking Understanding (cont.) To run the White test, we regress nR2 = 7.17 The critical value at the 0.05 significance level for a Chi-square statistic with 2 degrees of freedom is 5.99 We reject the null hypothesis.

GLS: An Example Our initial guess: This guess didn’t do very well. Can we do better? Instead of blindly guessing, let’s try looking at the data first.

Figure 10.4 The Rent–Income Ratio Plotted Against the Inverse of Income

GLS: An Example We seem to have overcorrected for heteroskedasticity. Let’s try

TABLE 10.6 Estimating a Second Transformed Rent–Income Relationship,

GLS: An Example Unthinking application of the White test procedures for the transformed data leads to The interaction term reduces to a constant, which we already have in the auxilliary equation, so we omit it and use only the first 4 explanators.

GLS: An Example (cont.) nR2 = 6.16 The critical value at the 0.05 significance level for a Chi-squared statistic with 4 degrees of freedom is 9.49 We fail to reject the null hypothesis that the transformed data are homoskedastic. Warning: failing to reject a null hypothesis does NOT mean we can “accept” it.

GLS: An Example (cont.) Generalized Least Squares is not trivial to apply in practice. Figuring out a reasonable di can be quite difficult. Next time we will learn another approach to constructing di , Feasible Generalized Least Squares.

Review In this lecture, we began relaxing the Gauss–Markov assumptions, starting with the assumption of homoskedasticity. Under heteroskedasticity, OLS is still unbiased OLS is no longer efficient OLS e.s.e.’s are incorrect, so C.I., t-, and F- statistics are incorrect

Review (cont.) Under heteroskedasticity, For a straight line through the origin,

Review (cont.) We can use squared residuals to test for heteroskedasticity. In the White test, we regress the squared residuals against all explanators, squares of explanators, and interactions of explanators. The nR2 of the auxilliary equation is distributed Chi-squared.

Review (cont.) The Breusch–Pagan test is similar, but the econometrician chooses the explanators for the auxilliary equation.

Review (cont.) In the Goldfeld–Quandt test, we first divide the data into distinct groups, and conduct our OLS regression on each group separately. We then estimate s2 for each group. The ratio of two s2 estimates is distributed as an F-statistic.

Review (cont.) Under heteroskedasticity, the BLUE Estimator is Generalized Least Squares To implement GLS: Divide all variables by di Perform OLS on the transformed variables If we have used the correct di , the transformed data are homoskedastic. We can test this property.