Muhammad Imdad Ullah

Let us start with the nature of heteroscedasticity.

The assumption of homoscedasticity (equal spread, equal variance) is

$$E(u_i^2)=E(u_i^2|X_{2i},X_{3i},\cdots, X_{ki})=\sigma^2,\quad 1,2,\cdots, n$$

The above Figure shows that the conditional variance of $Y_i$ (which is equal to that of $u_i$), conditional upon the given $X_i$, remains the same regardless of the values taken by the variable $X$.

The Figure shows that the conditional value of $Y_i$ increases as $X$ increases. The variance of $Y_i$ is not the same, there is heteroscedasticity.

$$E(u_i^2)=E(u_i^2|X_{2i},X_{3i},\cdots, X_{ki})=\sigma_i^2$$

Nature of Heteroscedasticity

The nature of heteroscedasticity refers to the violation of the assumption of homoscedasticity in linear regression models. In the case of heteroscedasticity, the errors have unequal variances for different levels of the regressors, which leads to biased and inefficient estimators of the regression coefficients. There are several reasons why the variances of $u_i$ may be variable:

• Following the error-learning models, as people learn, their error of behavior becomes smaller over time or the number of errors becomes more consistent. In such cases, $\sigma_i^2$ is expected to decrease.
• As income grows, people have more discretionary income (income remaining after deduction of taxes) and hence more scope for choice about disposition (برتاؤ، قابو) of their income. Similarly, companies with larger profits are generally expected to show greater variability in their dividend (کمپنی کا منافع) policies than companies with lower profits.
• As data collecting techniques improve $\sigma_i^2$ is likely to decrease. For example, Banks having sophisticated data processing equipment are likely to commit fewer errors in the monthly or quarterly statements of their customers than banks without such equipment.
• Heteroscedasticity can also arise as a result of the presence of outliers. The inclusion or exclusion of such an observation, especially if the sample size is small, can substantially (معقول حد تک، درحقیقت) alter the results of regression analysis.
• The omission of variables also results in the problem of Heteroscedasticity. Upon deleting the variable from the model the researcher would not be able to interpret anything from the model.
  \item Heteroscedasticity may arise from the violation of the assumption of CLRM that the model is correctly specified.
• Skewness in the distribution of one or more regressors is another source of heteroscedasticity. For example, income is uneven.
• Incorrect data transformation (ratio or first difference), and incorrect functional form (linear vs log-linear) are also the source of heteroscedasticity.
• The problem of heteroscedasticity is likely to be more in cross-sectional data than in time series data.

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The pose is about a general discussion and an introduction to heteroscedasticity.

Introduction Heteroscedasticity and Homoscedasticity

The term heteroscedasticity refers to the violation of the assumption of homoscedasticity in linear regression models (LRM). In the case of heteroscedasticity, the errors have unequal variances for different levels of the regressors, which leads to biased and inefficient estimators of the regression coefficients. The disturbances $u_i$ in the Classical Linear Regression Model (CLRM) appearing in the population regression function should be homoscedastic; that is they all have the same variance.

In short words, heteroscedasticity means different (or unequal), and the Greek word Skodastic means spread (or scatter). Homoscedasticity means equal spread and heteroscedasticity means unequal spread.

Effect on the Var-Cov Matrix of the Error Terms:
The Var-Cov matrix of errors is

$$E(uu’) = E(u_i^2)=Var(u_i^)=\begin{pmatrix}
\sigma^2 & 0 & \cdots & 0\\ 0 & \sigma^2 & \vdots & 0\\ \vdots & \vdots & \vdots & \vdots\\ 0&0&\ddots &\sigma^2
\end{pmatrix}=\sigma^2 I_n,$$

where $I_n$ is an $n\times n$ identity matrix.

In the presence of heteroscedasticity, the Var-Cov matrix of the residuals will no longer be constant.

$$E(uu’)= E(u_i^2)=Var(u_i^)==\begin{pmatrix}
\sigma_1^2 & 0 & 0 & \cdots & 0 \\0 & \sigma^2_2 & 0 & \cdots & 0 \\ 0 & 0 & \sigma^2_3 & \cdots & 0 \\ 0 & 0 & 0 &\ddots & \sigma_n^2
\end{pmatrix}$$

The Var-Cov matrix of the OLS estimators $\hat{\beta}$ is

\begin{align*}
Cov(\hat{\beta}) &= E\left[(\hat{\beta}-\beta)(\hat{\beta}-\beta)’ \right]\\
&=E\left[[(X’X)^{-1}X’u][(X’X)^{-1}X’u]’ \right]\\
&=E\left[(X’X)^{-1}X’uu’X(X’X)^{-1} \right]\\
&=(X’X)^{-1}X’E(uu’)X(X’X)^{-1}\\
&=(X’X)^{-1}X’\Omega X (X’X)^{-1}
\end{align*}

The following are questions when we are concerned with heteroscedasticity:

• What is the nature of heteroscedasticity?
• What are the consequences of heteroscedasticity?
• How does one detect/ Test Heteroscedasticity?
• What are the remedial measures of heteroscedasticity?

That’s all about some basic introduction to heteroscedasticity.

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