Estimation Statistics MCQs 3

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Estimation Statistics MCQs Quiz covers the topics of Estimate and Estimation for the preparation of exams and different statistical job tests in Government/ Semi-Government or Private Organization sectors. These tests are also helpful in getting admission to different colleges and Universities. The Estimation Statistics MCQs Quiz will help the learner to understand the related concepts and enhance their knowledge too.

This MCQs quiz is about statistical inference. It will help you to understand the basic concepts related to Inferential statistics. This test will also help you to prepare yourself for different exam related to education or jobs.

1. The process of making estimates about the population parameter from a sample is called

 
 
 
 

2. If $\hat{\theta}$ is the estimator of the parameter $\theta$, then $\hat{\theta}$ is called unbiased if:

 
 
 
 

3. The value of a statistic tends towards the value of the population as the sample size increases. What is it said to be?

 
 
 
 

4. The end points of a confidence interval are called

 
 
 
 

5. A set (range) of values calculated from the sample data and is likely to contain the true value of the parameter with some probability is called:

 
 
 
 

6. The end points of a confidence interval are called:

 
 
 
 

7. Estimation can be classified into

 
 
 
 

8. The estimator is said to be ___________ if the mean of the estimator is not equal to the mean of the population parameter.

 
 
 
 

9. The difference between the two end points of a confidence interval is called

 
 
 
 

10. A formula or rule used for estimating the parameter of interest is called:

 
 
 
 

11. For computing the confidence interval about a single population variance, the following test will be used

 
 
 
 

12. The estimate is the observed value of an

 
 
 
 

13. A range (set) of values within which the population parameter is expected to occur is called:

 
 
 
 

14. The numerical value which we determine from the sample for a population parameter is called

 
 
 
 

15. The probability associated with confidence interval is called

 
 
 
 

16. ‘Statistic’ is an estimator and its computer values are called:

 
 
 
 

17. There are two main branches of statistical inference, namely

 
 
 
 

18. The estimate is the observed value of an:

 
 
 
 

19. The process of using sample data to estimate the values of unknown population parameters is called

 
 
 
 

20. A single value used to estimate a population value is called:

 
 
 
 

Statistical inference is a branch of statistics in which we conclude (make wise decisions) about the population parameter by making use of sample information. Statistical inference can be further divided into the Estimation of parameters and testing of the hypothesis.

Estimation is a way of finding the unknown value of the population parameter from the sample information by using an estimator (a statistical formula) to estimate the parameter. One can estimate the population parameter by using two approaches (I) Point Estimation and (ii) Interval Estimation.

Estimation, point estimate and Interval Estimate

In point Estimation, a single numerical value is computed for each parameter, while in interval estimation a set of values (interval) for the parameter is constructed. The width of the confidence interval depends on the sample size and confidence coefficient. However, it can be decreased by increasing the sample size. The estimator is a formula used to estimate the population parameter by making use of sample information.

Estimation Statistics

Online Estimation Statistics MCQs

  • The process of making estimates about the population parameter from a sample is called
  • There are two main branches of statistical inference, namely
  • Estimation can be classified into
  • A formula or rule used for estimating the parameter of interest is called:
  • ‘Statistic’ is an estimator and its computer values are called:
  • The estimate is the observed value of an:
  • The process of using sample data to estimate the values of unknown population parameters is called
  • The numerical value which we determine from the sample for a population parameter is called
  • A single value used to estimate a population value is called:
  • A set (range) of values calculated from the sample data and is likely to contain the true value of the parameter with some probability is called:
  • A range (set) of values within which the population parameter is expected to occur is called:
  • The end points of a confidence interval are called:
  • The probability associated with confidence interval is called
  • The estimator is said to be ________ if the mean of the estimator is not equal to the mean of the population parameter.
  • If $\hat{\theta}$ is the estimator of the parameter $\theta$, then $\hat{\theta}$ is called unbiased if:
  • The value of a statistic tends towards the value of the population as the sample size increases. What is it said to be?
  • For computing the confidence interval about a single population variance, the following test will be used
  • The end points of a confidence interval are called
  • The difference between the two end points of a confidence interval is called
  • The estimate is the observed value of an
Estimation Statistics MCQs

Estimation is a fundamental part of statistics because populations can be very large or even infinite, making it impossible to measure every single member. By using estimation techniques, we can draw conclusions about the bigger picture from a manageable amount of data.

Take another Quiz: Estimation Statistics MCQs

R Programming Language

Breusch Pagan Test for Heteroscedasticity (2021)

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The Breusch Pagan test (named after Trevor Breusch and Adrian Pagan) is used to check for the presence of heteroscedasticity in a linear regression model.

Assume our regression model is $Y_i = \beta_1 + \beta_2 X_{2i} + \mu_i$ i.e we have simple linear regression model, and $E(u_i^2)=\sigma_i^2$, where $\sigma_i^2=f(\alpha_1 + \alpha_2 Z_{2i})$,

That is $\sigma_i^2$ is some function of the non-stochastic variable $Z$’s. The $f()$ allows for both the linear and non-linear forms of the model. The variable $Z$ is the independent variable $X$ or it could represent a group of independent variables other than $X$.

Step to Perform Breusch Pagan test

  1. Estimate the model by OLS and obtain the residuals $\hat{u}_1, \hat{u}_2+\cdots$
  2. Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
  3. Run the regression $\frac{e_i^2}{\hat{\sigma^2}}=\beta_1+\beta_2 Z_i + u_i$ and compute the explained sum of squares (ESS) from this regression
  4. Test the statistical significance of $\frac{ESS}{2}$ by $\chi^2$-test with 1 df at the appropriate level of significance ($\alpha$).
  5. Reject the hypothesis of homoscedasticity in favour of heteroscedasticity if $\frac{ESS}{2} > \chi^2_{(1)}$ at the appropriate level of $\alpha$.
Bruesch-Pagan-Test-of-Heteroscedasticity

Note that the

  • The Breusch Pagan test is valid only if $u_i$’s are normally distributed.
  • For k independent variables, $\frac{ESS}{2}$ has ($\chi^2$) Chi-square distribution with k degree of freedom.
  • If the $u_i$’s (error term) are not normally distributed, the White test is used.

If heteroscedasticity is detected, remedies may include using robust standard errors, transforming the data, or employing weighted least squares estimation to adjust for heteroscedasticity.

The Breusch Pagan test is considered a useful tool for detecting the presence of heteroscedasticity in the regression models. The Breusch Pagan Test helps to ensure the validity of statistical inference and estimation.

A sample of Stata output related to the Breusch-Pagan Test for the detection of heteroscedasticity.

Breusch Pagan Test Stata Output

By analyzing the p-value of the chi-squared test statistic from the second regression, one can decide whether to reject the null hypothesis of homoscedasticity. If the p-value is lower than the chosen level of significance (say, 0.05), one has the evidence of heteroscedasticity.

The following are important points that need to be considered while using Breusch Pagan test of Heteroscedasticity.

  • The Breusch-Pagan test can be sensitive to the normality of the error terms. Therefore, It is advisable to check if the residuals are normally distributed before running the Breusch-Pagan test.
  • There are other tests for heteroscedasticity, but the Breusch-Pagan test is a widely used and relatively straightforward option.
Breusch Pagan Test of Heteroscedasticity

References:

  • Breusch, T.S.; Pagan, A.R. (1979). “Simple test for heteroscedasticity and random coefficient variation”. Econometrica (The Econometric Society) 47 (5): 1287–1294.

See the Numerical Example of the Breusch-Pagan Test for the Detection of Heteroscedasticity

R Frequently Asked Questions

Multicollinearity in Regression Models

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The post is about Multicollinearity in Regression Models.

The objective of multiple regression analysis is to approximate the relationship of individual parameters of a dependency, but not of interdependency. It is assumed that the dependent variable $y$ and regressors $X$’s are linearly related to each other (Graybill, 1980; Johnston, 1963; and Malinvaud, 1968). Therefore, inferences depicted from any regression model are

(i) Identify the relative influence of regressors
(ii) Prediction and/or estimation and
(iii) Selection of an appropriate set of regressors for the model.

Multicollinearity in Regression Models

From all these inferences, one of the purposes of the regression model is to ascertain what extent the dependent variable can be predicted by the regressors in the model. However, to draw some suitable inferences, the regressors should be orthogonal, i.e., there should be no linear dependencies among regressors. However, in most of the applications of regression analysis, regressors are not orthogonal, which leads to misleading and erroneous inferences, especially, in cases when regressors are perfectly or nearly perfectly collinear to each other.

Regarding the multicollinearity in Regression, the condition of non-orthogonality is also referred to as the problem of multicollinearity or collinear data, for example, see Gunst and Mason, 1977;  Mason et al., 1975 and Ragnar, 1934). Multicollinearity is also synonymous with ill-conditioning of the $X’X$ matrix.

The presence of interdependence or the lack of independence is signified by high-order inter-correlation ($R=X’X$) within a set of regressors (Dorsett et al, 1983; Farrar and Glauber1967; Gunst and Mason, 1977; Mason et al., 1975). The perfect multicollinearity situation is a pathological extreme and it can easily be detected and resolved by dropping one of the regressors causing multicollinearity (Belsley et al., 1980). In the case of perfect multicollinearity, the regression coefficients remain indeterminate and their standard errors are infinite. Similarly, perfectly collinear regressors destroy the uniqueness of the least square estimators (Belsley et al., 1980; and Belsley, 1991).

Many explanatory variables (regressors/ predictors) are highly collinear, making it very difficult to infer the separate influence of collinear regressors on the response variable ($y$), that is, estimation of regression coefficients becomes difficult because coefficient(s) measures the effect of the corresponding regressor while holding all other regressors as constant. The problem of not perfect multicollinearity is extremely hard to detect (Chatterjee and Hadi, 2006) as it is not a specification or modeling error it is a condition of deficit data (Hadi and Chatterjee, 1988). On the other hand, the existence of multicollinearity has no impact on the overall regression model and associated statistics such as $R^2$, $F$-ratio, and $p$-value.

Multicollinearity does not also lessen the predictive or reliability of the regression model as a whole, it only affects the individual regressors (Koutsoyiannis, 1977). Note that, multicollinearity refers only to the linear relationships among the regressors, it does not rule out the nonlinear relationships among them. To draw suitable inferences from the model, the existence of (multi)collinearity should always be tested when examining a data set as an initial step in multiple regression analysis. On the other hand, high collinearity is rare, but some degree of collinearity always exists.

Multicollinearity in Linear Regression Models

A distinction between collinearity and multicollinearity should be made. Strictly speaking, multicollinearity usually refers to the existence of more than one exact linear relationship among regressors, while collinearity refers to the existence of a single linear relationship. However, multicollinearity refers to both of the cases nowadays.

There are many methods for the detection/ testing of multi(collinearity) among regressors. However, these methods can destroy the usefulness of the model, since relevant regressor(s) may be removed by these methods. Note that, if there are two predictors then it is sufficient to detect the problem of collinearity using pairwise correlation. However, to check the severity of the collinearity problem, VIF/TOL, eigenvalues, or other diagnostic measures can be used.

For further details about “Multicollinearity in Regression Models” see:

  • Belsley, D., Kuh, E., and Welsch, R. (1980). Diagnostics: Identifying Influential Data and Sources of Collinearity. John Willey & Sons, New York. chap. 3.
  • Belsley, D. A. (1991). A Guide to Using the Collinearity Diagnostics. Computer Science in Economics and Management, 4(1), 3350.
  • Chatterjee, S. and Hadi, A. S. (2006). Regression Analysis by Example. Wiley and Sons, 4th edition.
  • Dorsett, D., Gunst, R. F., and Gartland, E. C. J. (1983). Multicollinear Effects of Weighted Least Squares Regression. Statistics & Probability Letters, 1(4), 207211.
  • Graybill, F. (1980). An Introduction to Linear Statistical Models. McGraw Hill.
  • Gunst, R. and Mason, R. (1977). Advantages of examining multicollinearities in regression analysis. Biometrics, 33, 249260.
  • Hadi, A. and Chatterjee, S. (1988). Sensitivity Analysis in Linear Regression. John Willey & Sons.
  • Imdadullah, M., Aslam, M. and Altaf, S. (2916) mctest: An R Package for Detection of Collinearity Among Regressors
  • Imdadullah, M., Aslam, M. (2016). mctest: An R Package for Detection of Collinearity Among Regressors
  • Johnston, J. (1963). Econometric Methods. McGraw Hill, New York.
  • Koutsoyiannis, A. (1977). Theory of Econometrics. Macmillan Education Limited.
  • Malinvaud, E. (1968). Statistical Methods of Econometrics. Amsterdam, North Holland. pp. 187192.
  • Mason, R., Gunst, R., and Webster, J. (1975). Regression Analysis and Problems of Multicollinearity. Communications in Statistics, 4(3), 277292.
  • Ragnar, F. (1934). Statistical Consequence Analysis by means of complete regression systems. Universitetets Økonomiske Instituut. Publ. No. 5.

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