CHAPTER 11: HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT?

ü  The Nature of Heteroscedasticity

There are several reasons why the variances of ui may be a variable, some of which are as follows:

1.      Following the error-learning models, as people learn, their errors of behavior become smaller over time.

2.      As incomes grow, people have more discretionary income and hence more scope of choice about the disposition of their income.

3.      As data collecting techniques,        is likely to decrease.

4.      Heteroscedasticity can also arise as a result of the presence of outliers.

5.      Another source of heteroscedasticity arises from violating Assumption 9 of CLRM, namely, that the regression model is correctly specified.

6.      Another source of heteroscedasticity is skewness in the distribution of one or more regressors included in the model.

7.      Heteroscedasticity can also arise because of (1) incorrect data transformation and (2) incorrect functional form.

ü  OLS Estimation in the Presence of Heteroscedasticity

var(β₂) =   ∑X_i²

                    (∑X_i²)²

var(β₃) = 

                 ∑X_i²

ü  The Method of Generalized Least Squares

-          Takes such information into account explicitly and is therefore capable of producing estimators that are BLUE.

ü  Difference Between OLS and GLS

OLS:

∑u_i² = ∑(Y_i – β₁ – β₂X_i)²

GLS:

∑w_iu_i² = ∑w_i(Y_i – β₁X_i – β₂X_i)²

ü  Consequence of Using OLS in the Presence of Heteroscedasticity

ü  OLS Estimation Disregarding Heteroscedasticity

In short, if we persist in using the usual testing procedures despite heteroscedasticity, whatever conclusions we draw or inferences we make may be very misleading.

ü  Detection of Heteroscedasticity

ü  Informal Methods

Nature of the Problem. Very often the nature of the problem under consideration suggests whether heteroscedasticity is likely to be encountered.

Graphical Method. If there are no priori or empirical information about the nature of heteroscedasticity, in practice one can do the regression analysis on the assumption that there is no heteroscedasticity and then do the postmortem examination of the residual squared u_i² to see if they exhibit any systematic pattern.

ü  Formal Methods

Park Test. Park formalizes the graphical method by suggesting that        is some function of the explanatory variable X_i.

Glejser Test. After obtaining the residuals u_i from the OLS regression, Glejser suggests regressing the absolute values of u_i on the X variable that is thought to be closely associated with        .

Spearman’s Rank Correlation Test.

r_s = 1 -        ∑d_i²

                   n(n² – 1)

Step 1: Fit the regression to the data on Y and X and obtain the residuals u_i.

Step 2: Ignoring the sign of u_i, that is, taking their absolute value   u_i    , rank both   u_i  and X_i or (Y_i) according to an ascending or descending order and compute the Spearman’s rank correlation coefficient given previously.

Step 3: Assuming that the population rank correlation coefficient ρ_s is zero and n>8, the significance of the sample r_s can be tested by the t test as follows.

t = r_s   n – 2

            1 – r²s

Goldfeld-Quandt Test. This popular method is applicable if one assumes that the heteroscedastic variance     , is positively related to one of the explanatory variable in the regression model.

Step 1: Order or rank the observations according to the values of X_i, beginning with the lowest X value.

Step 2: Omit c central observations, where c is specified a priori, and divide the remaining (n – c) observations into two groups each of (n – c) /2 observations.

Step 3: Fit separate OLS regressions to the first (n – c)/2 observations and the last (n – c)/2 observations and obtain the respective residual sums of squares RSS₁ and RSS₂, RSS₁ representing the RSS from the regression corresponding to the smaller X_i values and RSS₂ that from the larger X_i values.

Step 4: Compute the ratio

λ = RSS₂/df

      RSS₁/df

If u_i are assumed to be normally distributed and if the assumption of homoscedasticity is valid.

Breusch-Pagan-Godfrey Test

Step 1: Estimate

Y_i = β₁ + β₂X_2i + β_kX_k + u_i

by OLS and obtain the residuals u₁, u₂, . . . ,u_n

Step 2: Obtain        = ∑u_i2/n

Step 3: Construct variables p_i defined as

p_i = u_i2/

which is simply each residual squared divided by      .

Step 4: Regress pi thus considered on the z’s as

p_i = α₁ + α₂Z_2i + . . . + α_mZ_mi + v_i

where vi is the residual term of this regression.

Step 5: Obtain the ESS and define

= ½(ESS)

White’s General Heteroscedasticity Test

Step 1: Given the data, we estimate

Y_i = β₁ + β₂X_2i + β₃X_3i + u_i

and obtain the residuals, u_i.

Step 2: We then run the following regression:

u_i2 = α₁ + α₂X_2i + α₃X_3i + α₄X_2i + α₅X_3i + α₆X_2iX_3i + v_i

Step 3: Under the null hypothesis that there is no heteroscedasticity, it can be shown that the sample size (n) times the R² obtained from the auxiliary regression asymptotically follows the chi-square distribution with df equal to the number of regressors in the auxiliary regression. That is,

n   R²  _asy   X²df

Step 4: If the chi-square value obtained in n   R² _asy X²df exceeds the critical chi-square value at the chosen level of significance, conclusion is that there is heteroscedasticity.

ü  Other Tests of Heteroscedasticity

·         Koenker-Bassett (KB) Test

ü  Remedial Measures

ü  When       is Known: The Method of Weighted Least Squares

ü  When       not Known

·         Plausible assumptions about heteroscedasticity pattern

Assumption 1: The error variance is proportional to X_i2

E(u_i²) =       X_i2

Assumption 2: The error variance is proportional to X_i. The square root transformation:

E(u_i²) =         X_i

Assumption 3: The error variance is proportional to the square of the mean value of Y.

E(u_i²) =       [E(Y_i)]²

Assumption 4: A log transformation such as

lnY_i = β₁ + β₂lnX_i + u_i

very often reduces heteroscedasticity when compared with the regression Y_i = β₁ + β₂X_i + u_i