This dissertation concerns with some estimations of heteroskedastic models and tests for heteroskedasticity. We reconsider the minimum norm quadratic unbiased estimation (MINQUE) to obtain an alternative estimator of variance-covariance matrix in heteroskedastic models. We derive the analytical expressions for the mean square errors (MSE) of White's (1980), MacKinnon and White's (1985) and MINQU estimators, and perform a numerical comparison. Our results suggest that the bias of White's estimator is the most severely affected by the point of leverage of the regression design. An unbalanced design matrix also seems to increase the variance of MINQUE, which in turn increases the probability of producing negative estimates. The finite sample behavior of the t-statistics is investigated through simulated confidence interval coverage probabilities of the regression coefficients. Although MINQUE generally has the largest MSE, it performs relatively well in terms of the coverage probabilities. We also apply MINQUE for a class of autoregressive conditional heteroskedasticity (ARCH) models. We show the close connection between MINQUE and the maximum likelihood estimator (MLE) for a p th-order ARCH. The relation suggests that MINQUE is a more general procedure than the MLE. Also, the use of MINQUE may be thought of as a way to provide some kind of "adjustment" to the small sample bias of the MLE. Based on the martingale properties of ARCH, we prove the consistency and asymptotic normality of MINQUE. Our analysis shows that, for the p th-order ARCH, MINQUE attains asymptotic efficiency of MLE. To illustrate our procedure we estimate Engle's (1983) model using MINQUE. Our diagnostic checks on this model indicate that MINQUE performs slightly better than the Engle's MLE. Finally, this dissertation concerns with model specification tests. We apply MINQUE to provide a simple modification to the White's (1980) test for heteroskedasticity. We show that the two tests are asymptotically equivalent, and our simulation reveals that they also have very similar finite sample performances. We consider another modification based on eigenvalues. The asymptotic distribution of the modified test under the null hypothesis is derived. Our comparative study shows that the eigenvalue-based test has improved finite sample power properties.