Tests of Heteroscedasticity

To detect the presence of heteroscedasticity using the Park Glejser test, consider the following data.

Year	1992	1993	1994	1995	1996	1997	1998
Yt	37	48	45	36	25	55	63
Xt	4.5	6.5	3.5	3	2.5	8.5	7.5

The step-by-step procedure for conducting the Park Glejser test:

Step 1: Obtain an estimate of the regression equation

$$\hat{Y}_i = 19.8822 + 4.7173X_i$$

Obtain the residuals from this estimated regression equation:

Residuals

-4.1103

-2.5450

8.6071

1.9657

-6.6756

-4.9797

7.7377

Take the absolute values of these residuals and consider it as your dependent variables to perform the different functional forms suggested by Glejser.

Step 2: Regress the absolute values of $\hat{u}_i$ on the $X$ variable that is thought to be closely associated with $\sigma_i^2$. We will use the following function forms.

Sr. No.	Functional Form	Results
1)	$|\hat{u}_t| = \beta_1 + \beta_2 X_i +v_i$	$|\hat{u}_i| = 5.2666-0.00681X_i,\quad R^2=0.00004$ $t_{cal} = -0.014$
2)	$|\hat{u}_t| = \beta_1 + \beta_2 \sqrt{X_i} +v_i$	$|\hat{u}_i| = 5.445-0.0962X_i,\quad R^2=0.000389$ $t_{cal} = -0.04414$
3)	$|\hat{u}_t| = \beta_1 + \beta_2 \frac{1}{X_i} +v_i$	$||\hat{u}_i| = 4.9124+1.3571X_i,\quad R^2=0.00332$ $t_{cal} = -0.12914$
4)	$|\hat{u}_t| = \beta_1 + \beta_2 \frac{1}{\sqrt{X_i}} +v_i$	$\hat{u}_i| = 4.7375+1.0428X_i,\quad R^2=0.00209$ $t_{cal} = 0.10252$

Since none of the residual regression is significant, therefore, the hypothesis of heteroscedasticity is rejected. Therefore, we can say that there is no relationship between the absolute value of the residuals ($u_i$) and the explanatory variable $X$.

How to perform White General Heteroscedasticity?

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R Data Analysis

Data is taken from the Economic Survey of Pakistan 1991-1992. The data file link is at the end of the post “Goldfeld-Quandt Test Example for the Detection of Heteroscedasticity”.

Read about the Goldfeld-Quandt Test in detail by clicking the link “Goldfeld-Quandt Test: Comparison of Variances of Error Terms“.

Goldfeld-Quandt Test Example

For an illustration of the Goldfeld-Quandt Test Example, the data given in the file should be divided into two sub-samples after dropping (removing/deleting) the middle five observations.

Sub-sample 1 consists of data from 1959-60 to 1970-71.

Sub-sample 2 consists of data from 1976-77 to 1987-1988.

The sub-sample 1 is highlighted in green colour, and sub-sample 2 is highlighted in blue color, while the middle observation that has to be deleted is highlighted in red.

The Step-by-Step Procedure to Conduct the Goldfeld Quandt Test

Step 1: Order or Rank the observations according to the value of $X_i$. (Note that observations are already ranked.)

Step 2: Omit $c$ central observations. We selected 1/6 observations to be removed from the middle of the observations. 

Step 3: Fit OLS regression on both samples separately and obtain the Residual Sum of Squares (RSS) for each sub-sample.

The Estimated regression for the two sub-samples are:

Sub-sample 1: $\hat{C}_1 = 1010.096 + 0.849 \text{Income}$

Sub-sample 2: $\hat{C}_2 = -244.003 + 0.88067 \text{Income}$

Now compute the Residual Sum of Squares for both sub-samples.

The residual Sum of Squares for Sub-Sample 1 is $RSS_1=2532224$

The residual Sum of Squares for Sub-Sample 2 is $RSS_2=10339356$

The F-Statistic is $ \lambda=\frac{RSS_2/n_2}{RSS_1/n_1}=\frac{10339356}{2532224}=4.083$

The critical value of $F(n_1=10, n_2=10$ at a 5% level of significance is 2.98.

Since the computed F value is greater than the critical value, heteroscedasticity exists in this case, that is, the variance of the error term is not consistent, rather it depends on the independent variable, GNP.

Your assignment is to perform the Goldfeld-Quandt Test Example using any statistical software and confirm the results.

Download the data file by clicking the link “GNP and consumption expenditure data“.

Learn about White’s Test of Heteroscedasticity

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The post is about Heteroscedasticity Residual Plot.

Heteroscedasticity and Heteroscedasticity Residual Plot

One of the assumptions of the classical linear regression model is that there is no heteroscedasticity (error terms have constant error terms) meaning that ordinary least square (OLS) estimators are (BLUE, best linear unbiased estimator) and their variances are the lowest of all other unbiased estimators (Gauss Markov Theorem).

If the assumption of constant variance does not hold then this means that the Gauss Markov Theorem does not apply. For heteroscedastic data, regression analysis provides an unbiased estimate of the relationship between the predictors and the outcome variables.

As we have discussed heteroscedasticity occurs when the error variance has non-constant variance.  In this case, we can think of the disturbance for each observation as being drawn from a different distribution with a different variance.  Stated equivalently, the variance of the observed value of the dependent variable around the regression line is non-constant. 

We can think of each observed value of the dependent variable as being drawn from a different conditional probability distribution with a different conditional variance. A general linear regression model with the assumption of heteroscedasticity can be expressed as follows

\begin{align*}
y_i & = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \cdots + \beta_p X_ip + \varepsilon_i\\
Var(\varepsilon_i)&=E(\varepsilon_i^2)\\
&=\sigma_i^2; \cdots i=1,2,\cdots, n
\end{align*}

Note that we have a $i$ subscript attached to sigma squared.  This indicates that the disturbance for each of the $ n$ units is drawn from a probability distribution that has a different variance.

If the error term has non-constant variance, but all other assumptions of the classical linear regression model are satisfied, then the consequences of using the OLS estimator to obtain estimates of the population parameters are:

• The OLS estimator is still unbiased
• The OLS estimator is inefficient; that is, it is not BLUE
• The estimated variances and covariances of the OLS estimates are biased and inconsistent
• Hypothesis tests are not valid

Detection of Heteroscedasticity Residual Plot

The residual for the $i$th observation, $\hat{\varepsilon_i}$, is an unbiased estimate of the unknown and unobservable error for that observation, $\hat{\varepsilon_i}$. Thus the squared residuals, $\hat{\varepsilon_i^2} $, can be used as an estimate of the unknown and unobservable error variance,  $\sigma_i^2=E(\hat{\varepsilon_i})$. 

One can calculate the squared residuals and then plot them against an explanatory variable that you believe might be related to the error variance.  If you believe that the error variance may be related to more than one of the explanatory variables, you can plot the squared residuals against each one of these variables.  Alternatively, you could plot the squared residuals against the fitted value of the dependent variable obtained from the OLS estimates.  Most statistical programs (software) have a command to do these residual plots.  It must be emphasized that this is not a formal test for heteroscedasticity.  It would only suggest whether heteroscedasticity may exist.

Below there are residual plots showing the three typical patterns. The first plot shows a random pattern that indicates a good fit for a linear model. The other two plot patterns of residual plots are non-random (U-shaped and inverted U), suggesting a better fit for a non-linear model, than a linear regression model.

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