# Heteroscedasticity

# Introduction

One of the assumptions made about residuals/errors in OLS regression is that the errors have the same but unknown variance. This is known as constant variance or homoscedasticity. When this assumption is violated, the problem is known as heteroscedasticity.

##### Consequences of Heteroscedasticity

- The OLS estimators and regression predictions based on them remains unbiased and consistent.
- The OLS estimators are no longer the BLUE (Best Linear Unbiased Estimators) because they are no longer efficient, so the regression predictions will be inefficient too.
- Because of the inconsistency of the covariance matrix of the estimated regression coefficients, the tests of hypotheses, (t-test, F-test) are no longer valid.

olsrr provides the following 4 tests for detecting heteroscedasticity:

- Bartlett Test
- Breusch Pagan Test
- Score Test
- F Test

## Bartlett Test

Bartlett’s test is used to test if variances across samples is equal. It is sensitive to departures from normality. The Levene test is an alternative test that is less sensitive to departures from normality.

You can perform the test using 2 continuous variables, one continuous and one grouping variable, a formula or a linear model.

#### Use grouping variable

    ols_test_bartlett(hsb, 'read', group_var = 'female')

    ##
    ##     Bartlett's Test of Homogenity of Variances
    ## ------------------------------------------------
    ## Ho: Variances are equal across groups
    ## Ha: Variances are unequal for atleast two groups
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    1
    ##  Chi2          =    0.1866579
    ##  Prob > Chi2   =    0.6657129

#### Using variables

    ols_test_bartlett(hsb, 'read', 'write')

    ##
    ##     Bartlett's Test of Homogenity of Variances
    ## ------------------------------------------------
    ## Ho: Variances are equal across groups
    ## Ha: Variances are unequal for atleast two groups
    ##
    ##         Data
    ##  ---------------------
    ##  Variables: read write
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    1
    ##  Chi2          =    1.222871
    ##  Prob > Chi2   =    0.2687979

## Breusch Pagan Test

Breusch Pagan Test was introduced by Trevor Breusch and Adrian Pagan in 1979. It is used to test for heteroskedasticity in a linear regression model and assumes that the error terms are normally distributed. It tests whether the variance of the errors from a regression is dependent on the values of the independent variables. It is a \(\chi^{2}\) test.

You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables. It includes options to perform multiple tests and p value adjustments. The options for p value adjustments include Bonferroni, Sidak and Holm’s method.

#### Use fitted values of the model

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model)

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##              Data
    ##  -------------------------------
    ##  Response : mpg
    ##  Variables: fitted values of mpg
    ##
    ##        Test Summary
    ##  ---------------------------
    ##  DF            =    1
    ##  Chi2          =    1.429672
    ##  Prob > Chi2   =    0.231818

#### Use independent variables of the model

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model, rhs = TRUE)

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##            Data
    ##  --------------------------
    ##  Response : mpg
    ##  Variables: disp hp wt drat
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    4
    ##  Chi2          =    1.513808
    ##  Prob > Chi2   =    0.8241927

#### Use independent variables of the model and perform multiple tests

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE)

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##            Data
    ##  --------------------------
    ##  Response : mpg
    ##  Variables: disp hp wt drat
    ##
    ##         Test Summary (Unadjusted p values)
    ##  ----------------------------------------------
    ##   Variable           chi2       df        p
    ##  ----------------------------------------------
    ##   disp             1.2355345     1    0.2663334
    ##   hp               0.9209878     1    0.3372157
    ##   wt               1.2529988     1    0.2629805
    ##   drat             1.1668486     1    0.2800497
    ##  ----------------------------------------------
    ##   simultaneous     1.5138083     4    0.8241927
    ##  ----------------------------------------------

#### Bonferroni p value Adjustment

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'bonferroni')

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##            Data
    ##  --------------------------
    ##  Response : mpg
    ##  Variables: disp hp wt drat
    ##
    ##         Test Summary (Bonferroni p values)
    ##  ----------------------------------------------
    ##   Variable           chi2       df        p
    ##  ----------------------------------------------
    ##   disp             1.2355345     1    1.0000000
    ##   hp               0.9209878     1    1.0000000
    ##   wt               1.2529988     1    1.0000000
    ##   drat             1.1668486     1    1.0000000
    ##  ----------------------------------------------
    ##   simultaneous     1.5138083     4    0.8241927
    ##  ----------------------------------------------

#### Sidak p value Adjustment

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'sidak')

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##            Data
    ##  --------------------------
    ##  Response : mpg
    ##  Variables: disp hp wt drat
    ##
    ##           Test Summary (Sidak p values)
    ##  ----------------------------------------------
    ##   Variable           chi2       df        p
    ##  ----------------------------------------------
    ##   disp             1.2355345     1    0.7102690
    ##   hp               0.9209878     1    0.8070305
    ##   wt               1.2529988     1    0.7049362
    ##   drat             1.1668486     1    0.7313356
    ##  ----------------------------------------------
    ##   simultaneous     1.5138083     4    0.8241927
    ##  ----------------------------------------------

#### Holm’s p value Adjustment

    model <- lm(mpg ~ disp + hp + wt + drat, data = mtcars)
    ols_test_breusch_pagan(model, rhs = TRUE, multiple = TRUE, p.adj = 'holm')

    ##
    ##  Breusch Pagan Test for Heteroskedasticity
    ##  -----------------------------------------
    ##  Ho: the variance is constant
    ##  Ha: the variance is not constant
    ##
    ##            Data
    ##  --------------------------
    ##  Response : mpg
    ##  Variables: disp hp wt drat
    ##
    ##           Test Summary (Holm's p values)
    ##  ----------------------------------------------
    ##   Variable           chi2       df        p
    ##  ----------------------------------------------
    ##   disp             1.2355345     1    0.7990002
    ##   hp               0.9209878     1    0.3372157
    ##   wt               1.2529988     1    1.0000000
    ##   drat             1.1668486     1    0.5600994
    ##  ----------------------------------------------
    ##   simultaneous     1.5138083     4    0.8241927
    ##  ----------------------------------------------

## Score Test

Test for heteroskedasticity under the assumption that the errors are independent and identically distributed (i.i.d.). You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables.

#### Use fitted values of the model

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_score(model)

    ##
    ##  Score Test for Heteroskedasticity
    ##  ---------------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: fitted values of mpg
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    1
    ##  Chi2          =    0.5163959
    ##  Prob > Chi2   =    0.4723832

#### Use independent variables of the model

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_score(model, rhs = TRUE)

    ##
    ##  Score Test for Heteroskedasticity
    ##  ---------------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: disp hp wt qsec
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    4
    ##  Chi2          =    2.039404
    ##  Prob > Chi2   =    0.7285114

#### Specify variables

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_score(model, vars = c('disp', 'hp'))

    ##
    ##  Score Test for Heteroskedasticity
    ##  ---------------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: disp hp
    ##
    ##         Test Summary
    ##  ----------------------------
    ##  DF            =    2
    ##  Chi2          =    0.9983196
    ##  Prob > Chi2   =    0.6070405

## F Test

F Test for heteroskedasticity under the assumption that the errors are independent and identically distributed (i.i.d.). You can perform the test using the fitted values of the model, the predictors in the model and a subset of the independent variables.

#### Use fitted values of the model

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_f(model)

    ##
    ##  F Test for Heteroskedasticity
    ##  -----------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: fitted values of mpg
    ##
    ##       Test Summary
    ##  -------------------------
    ##  Num DF     =    1
    ##  Den DF     =    30
    ##  F          =    0.4920617
    ##  Prob > F   =    0.4884154

#### Use independent variables of the model

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_f(model, rhs = TRUE)

    ##
    ##  F Test for Heteroskedasticity
    ##  -----------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: disp hp wt qsec
    ##
    ##       Test Summary
    ##  -------------------------
    ##  Num DF     =    4
    ##  Den DF     =    27
    ##  F          =    0.4594694
    ##  Prob > F   =    0.7647271

#### Specify variables

    model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars)
    ols_test_f(model, vars = c('disp', 'hp'))

    ##
    ##  F Test for Heteroskedasticity
    ##  -----------------------------
    ##  Ho: Variance is homogenous
    ##  Ha: Variance is not homogenous
    ##
    ##  Variables: disp hp
    ##
    ##       Test Summary
    ##  -------------------------
    ##  Num DF     =    2
    ##  Den DF     =    29
    ##  F          =    0.4669306
    ##  Prob > F   =    0.631555