The inference about the parameters of regressionmodel and autocorrelation is a challenging task in the presence of heteroskedasticity. In this scenario the heteroskedastic consistent covariance matrix estimators (HCCMEs) and variance ratio (VR) tests are widely used methods to obtain valid inference about regression parameters and autocorrelation respectively. For this purpose, several methods have been suggested in literature. But these methods generally work well when the sample size is sufficiently large and heteroskedasticity level is not very high. In this thesis, we have studied these methods for linear regression model in small samples with high level of heteroskedasticity. In first part of this thesis, we suggest a new heteroskedastic consistent (HC) covariance matrix estimator which takes into account the effect of leverage observations and has better approximation of its true asymptotic distribution. We point out that the basic motivation behind this new modified HC estimator is to provide an estimator which does not require any user specified values. In terms of bias and mean squared error (MSE) a Monte Carlo simulation study provided evidence that this new estimator has better small sample properties over some existing estimators. Real life example also evaluated the finite sample behavior in comparison to those existing estimators. In the second part we suggest heteroskedastic consistent covariance matrix estimator, HC6d, which is based on deviance measure. We have studied the finite sample behavior of the test statistic based on this new HC estimator. We compare its performance with other HC estimators namely HC1, HC3 and HC4m, which are also used in case of leverage observations. Extensive simulation are used to study the effect of various levels of heteroskedasticity on the performance of the quasi tests based on HC estimators. Results showed that the test statistic based on new suggested estimator has better asymptotic approximation and less size distortion in small samples especially when high level heteroskedasticity is present in the data. It has been known that the autocorrelation test do not perform well in presence of heteroskedasticity and variance break case. The remedies to this have been suggested by Jeong and Kang (2012), Shim et al. (2006) and Kim (2006). In the last part of the thesis, we suggested three weighted variance ratio test to test the autocorrelation in presence of heteroskedasticity and variance break. We show through monte carlo simulation that new suggested tests perform well in small samples and are comparable with other tests in terms of size and much better in terms of power properties and also when lag length is misspecified.