1. What are the assumptions of MLRM on order to have an unbiased and consistent
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Question
1. What are the assumptions of MLRM on order to have an unbiased and consistent OLS estimator? What about an efficient estimator?
2. What are your suggestions to reduce the variance/Standard errors of OLS estimators? Why do we prefer small standard errors for the estimators?
3. Provide an example of perfect multi-collinearity in a model of your choice. Provide an example of imperfect multi-collinearity.
4. Why when you have large sample size ("n"), R-Squared and Adjusted R-Square are very similar?
5. If we run a regression of income ( dependent variable ) on education (regressor), and parental education is missing, do you thing the beta hat we get is underestimated or overestimated.
Explanation / Answer
1)
a)Linearity in parameters: the dependent variable is a linear function of a set of independent variables and a error component.
b) Error must follow normal with mean 0 and variance 2
c) The conditional variance of the error term is constant in all independent variables. This state is called homoskedasticity.
d) Error term is independently distributed.
e) There should be any relationship between the independent variables.
Choosing OLS is to reduce the sum of squares of deviation and get to get the best linear unbiased estimator. It is necessary that we obtain unbiased estimator because the expected values of the estimated parameters is equal to the true value describing the relationship between the independent and dependent variables. Minimum the variance better is the prediction. OLS gives the minimum variance unbiased estimator over any other estimators.
2)
The standard error in estimator is due to the presence of multicollinearity. It is the problem of relationship between the independent variables. In order to obtain lesser standard error, the problem of multicollinearity must be taken care of.
The smaller standard error means the better is the estimator and better is the prediction. There should be much deviation from the true value.
3)
Perfect multicollinearity is when two independent variables have strong correlation between them. The relation is very evident in this case.
For example: including height in cm and m as two different variables. In such situations, either one of the variables is only needed.
Imperfect multicollinearity is when there is no relationship between the independent variables theoretically but the analysis shows multicollinearity.
For example: there are two variables which doesn’t have relationship with meaning, but one variable increases (decreases) as the other increases (decreases) . Though they are not related, the values are directly proportional. To overcome his problem, you can collect additional data.
4)
R-squared and adjusted R square is dependent on the number of regression coefficients and its mean square error.
R-squared if the coefficient of determination which indicates the percentage of variation in dependent variable is explained by the independent variables. The biggest drawback of R-squared is that as you keep adding independent variables, R-squared increases irrespective of whether the regression coefficient contributes to the model significantly or not. Hence it could give us the wrong results.
On the other hand, adjusted R-squared considers those regression coefficients which contribute to the model significantly. In multiple linear regression model, adjusted R-squared is more reliable.
5)
To predict income, we need several other factors other than education and parental education isn’t a significant factor.
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