Heteroscedasticity

Lecture notes about Heteroscedasticity, its consequences, detection, and remedy.

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An application of different statistical methods applied to the economic data used to find empirical relationships between economic data is called Econometrics. In other words, Econometrics is “the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference”.

Econometrics means “Economic Measurement”. Econometrics is the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of statistical inference.

Econometrics can also be defined as the empirical determination of economic laws. Econometrics can be classified as (i) Theoretical Econometrics and (ii) Applied Econometrics.

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Key Points of Heteroscedasticity

The following are some key points about heteroscedasticity. These key points are about the definition, example, properties, assumptions, and tests for the detection of heteroscedasticity (detection of hetero in short).

One important assumption of Regression is that the variance of the Error Term is constant across observations. If the error has a constant variance, then the errors are called homoscedastic, otherwise heteroscedastic. In the case of heteroscedastic errors (non-constant variance), the standard estimation methods become inefficient. Typically, to assess the assumption of homoscedasticity, residuals are plotted.

Heteroscedasticity

  • The disturbance term of OLS regression $u_i$ should be homoscedastic.
  • By Homo, we mean equal, and scedastic means spread or scatter.
  • By hetero, we mean unequal.
  • Heteroscedasticity means that the conditional variance of $Y_i$ (i.e., $var(u_i))$ conditional upon the given $X_i$ does not remain the same regardless of the values taken by the variable $X$.
  • In case of heteroscedasticity $E(u_i^2)=\sigma_i^2=var(u_i^2)$, where $i=1,2,\cdots, n$.
  • In case of Homoscedasticity $E(u_i^2)=\sigma^2=var(u_i^2)$, where $i=1,2,\cdots, n$
    Homoscedasticity means that the conditional variance of $Y_i$ (i.e. $var(u_i))$ conditional upon the given $X_i$ remains the same regardless of the values taken by the variable $X$.
  • The error terms are heteroscedastic, when the scatter of the errors is different, varying depending on the value of one or more of the explanatory variables,
  • Heteroscedasticity is a systematic change in the scatteredness of the residuals over the range of measured values
  • The presence of outliers may be due to (i) The presence of outliers in the data, (ii) incorrect functional form of the regression model, (iii) incorrect transformation of the data, and (iv) missing observations with different measures of scale.
  • The presence of hetero does not destroy the unbiasedness and consistency of OLS estimators.
  • Hetero is more common in cross-section data than time-series data.
  • Hetero may affect the variance and standard errors of the OLS estimates.
  • The standard errors of OLS estimates are biased in the case of hetero.
  • Statistical inferences (confidence intervals and hypothesis testing) of estimated regression coefficients are no longer valid.
  • The OLS estimators are no longer BLUE as they are no longer efficient in the presence of hetero.
  • The regression predictions are inefficient in the case of hetero.
  • The usual OLS method assigns equal weights to each observation.
  • In GLS the weight assigned to each observation is inversely proportional to $\sigma_i$.
  • In GLS a weighted sum of squares is minimized with weight $w_i=\frac{1}{\sigma_i^2}$.
  • In GLS each squared residual is weighted by the inverse of $Var(u_i|X_i)$
  • GLS estimates are BLUE.
  • Heteroscedasticity can be detected by plotting an estimated $u_i^2$ against $\hat{Y}_i$.
  • Plotting $u_i^2$ against $\hat{Y}_i$, if no systematic pattern exists then there is no hetero.
  • In the case of prior information about $\sigma_i^2$, one may use WLS.
  • If $\sigma_i^2$ is unknown, one may proceed with heteroscedastic corrected standard errors (that are also called robust standard errors).
  • Drawing inferences in the presence of hetero (or if hetero is suspected) may be very misleading.
Bruesch-Pagan Test of Heteroscedasticity

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The Breusch-Pagan Test (Numerical Example)

To perform the Breusch-Pagan test for the detection of heteroscedasticity, use the data from the following file Table_11.3.

Step 1:

The estimated regression is $\hat{Y}_i = 9.2903 + 0.6378X_i$

Step 2:

The residuals obtained from this regression are:

$\hat{u}_i$$\hat{u}_i^2$$p_i$
-5.3130728.228730.358665
-8.0687665.104940.827201
6.4980142.224070.536485
0.553390.306240.003891
-6.8244546.573180.591743
1.364471.861770.023655
5.7977033.613330.427079
-3.5801512.817440.162854
0.986620.973420.012368
8.3090869.040850.877209
-2.257695.097150.064763
-1.335841.784460.022673
8.0420164.673910.821724
10.47524109.730661.3942
6.2309338.824510.493291
-9.0915382.655881.050197
-12.79183163.630992.079039
-16.84722283.828793.606231
-17.35860301.321043.828481
2.719557.395950.09397
2.397095.746040.073007
0.774940.600520.00763
9.4524889.349301.135241
4.8857123.870140.303286
4.5306320.526580.260804
-0.036140.001311.66E-05
-0.303220.091940.001168
9.5078690.399441.148584
-18.98076360.269094.577455
20.26355410.611595.217089

The estimated $\tilde{\sigma}^2$ is $\frac{\sum u_i^2}{n} = \frac{2361.15325}{30} = 78.7051$.

Compute a new variable $p_i = \frac{\hat{u}_i^2}{\hat{\sigma^2}}$

Step 3:

Assuming $p_i$ is linearly related to $X_i(=Z_i)$ and run the regression of $p_i=\alpha_1+\alpha_2Z_{2i}+v_i$.

The regression Results are: $\hat{p}_i=-0.74261 + 0.010063X_i$

Step 4:

Obtain the Explained Sum of Squares (ESS) = 10.42802.

Step 5:

Compute: $\Theta = \frac{1}{2} ESS = \frac{10.42802}{2}= 5.2140$.

The Breusch-Pagan test follows Chi-Square Distribution. The $\chi^2_{tab}$ value at a 5% level of significance and with ($k-1$) one degree of freedom is 3.8414. The $\chi_{cal}^2$ is greater than $\chi_{tab}^2$, therefore, results are statistically significant. There is evidence of heteroscedasticity at a 5% level of significance.

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Bruesch-Pagan-Test-of-Heteroscedasticity
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