Chapter multiple Regression Analysis: Inference Deadline



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Chapter 4- Group work- Gasimov Bashir, Kazimov Aykhan, Alkhanov Nijat


CHAPTER 4. Multiple Regression Analysis: Inference

Deadline: 04.12.2021

Theoretical Exercises (Please provide very short and concise answers to each question)

Group work: Gasimov Bashir, Kazimov Aykhan, Alkhanov Nijat

  1. While testing statistical significance on the association between an independent variable and dependent variable, greater t-statistic value (or t-ratio) means ______

While testing statistical significance on the association between an independent variable and dependent variable, greater t-statistic value (or t-ratio) means more likely to reject the null hypothesis.

  1. In simple linear regression, most often we perform a two-tail test of the population slope β1 to determine whether there is sufficient evidence to infer that a linear relationship exists. Please state what is our null hypothesis in this case?

The null hypothesis is stated as β=0.

  1. Which statistic can be used to test hypotheses about a single population parameter?

t-Statistic is a statistic that can be used to test hypotheses about a single population parameter.

  1. Please explain the meaning of the Restricted Model and Unrestricted Model in hypothesis testing.

In hypothesis testing, the model obtained after imposing all of the restrictions required under the null hypothesis is Restricted Model.

In hypothesis testing, the model that has no restrictions placed on its parameters is Unrestricted Model.



  1. Besides the estimated regression coefficient and appropriate t statistic, what else is needed to construct a confidence interval for a regression coefficient?

The standard error of the regression coefficient is needed to construct a confidence interval for a regression coefficient.

  1. Practical Significance, which is defined as the practical importance of an estimate and is measured by its sign and magnitude, as opposed to its statistical significance, is also called _______.

Practical Significance, which is defined as the practical importance of an estimate and is measured by its sign and magnitude, as opposed to its statistical significance, is also called Economic Significance.

  1. Calculate the general t statistic if estimate is equal to 3.5, hypothesized value is equal to 3.1 and the standard error is equal to 0.8.



  1. In hypothesis testing, what is the value against which a test statistic is compared to determine whether the null hypothesis is rejected.

In hypothesis testing, Critical value is the value against which a test statistic is compared to determine whether or not the null hypothesis is rejected.

  1. High (but not perfect) correlation between two or more independent variables is called ________.

High (but not perfect) correlation between two or more independent variables is called multicollinearity.

  1. A rule used to construct a random interval so that a certain percentage of all data sets, determined by the confidence level, yields an interval that contains the population value: Confidence Interval (CI).



  1. What is Auxiliary Regression?

Auxiliary Regression is a regression used to compute a test statistic—such as the test statistics for heteroskedasticity and serial correlation—or any other regression that does not estimate the model of primary interest.

  1. What is t statistic and F statistic?

t-Statistic is a statistic that can be used to test hypotheses about a single population parameter.

F statistic is a statistic used to test multiple hypotheses about the parameters in a multiple regression model.



  1. While testing statistical significance on the association between an independent variable and dependent variable, to reject null hypothesis, the confidence level must be at least: 0.9



  1. The significance level of a test is: _____________________.

The significance level of a test is the probability of rejecting the null hypothesis when it is true.

  1. What is the primary goal of hypothesis testing (t-test)?

Primary goal of hypothesis testing (t-test) is to identify whether sample regression function parameters can be generalized to population parameters.

  1. Please provide the formula to calculate the general t-statistic.



  1. Is the following statement TRUE or FALSE?

The F statistic is always nonnegative as SSR of restricted model is never smaller than SSR of unrestricted model.”- TRUE.

  1. Please fill in the relevant word: F-test is used to test ___ significance.

F-test is used to test joint significance.

  1. An F statistic for testing the equality of regression parameters across different groups (say, men and women) or time periods (say, before and after a policy change) is called ______ statistic.

An F statistic for testing the equality of regression parameters across different groups (say, men and women) or time periods (say, before and after a policy change) is called Chow statistic.

  1. Calculate and interpret confidence intervals for a slope parameter – b1 at 1% level of significance. Note that critical value of t-statistic is 2.57, and standard error of that coefficient is 0.358. If b1=2.085 and show the impact of entrance score over GPA of students, comment on statistical significance of the relationship according to calculated confidence intervals.

The impact of entrance score over GPA is statistically significant at 1%. level of significance.

  1. Consider a given model, log(income) = b0 + b1*log(work_hour) + b2*Exper +u. Note that income denote monthly salary, and work_hour denote average working hours per week while Exper represent total job experience (measured in years) of the individual. If b1=0.209, and standard error of b1 is 0.037, comment on statistical significance of the corresponding association. Note that, critical value of t-test at 5% significance level is 1.96.

The impact of working hour over monthly salary is statistically significant at 5% level of significance level.

  1. Consider the given model, log(LS) = b0 + b1*log(income) + b2*work_hour +b3*age. Note that LS represents life satisfaction of an individual (changing between 5 and 35), income represents monthly income, work_hour shows average weekly working hours, and age shows age of the individual. Null hypothesis is, H0: b1=b2=b3=0. F-test value is 7.82, and probability of F-statistic is 0.0001. According to hypothesis test result: _______

Joint impact of monthly income, average weekly work hour and age over life satisfaction of individuals is statistically significant at 1% level of significance.

  1. Consider the given model, log(LS) = b0 + b1*log(income) + b2*work_hour +b3*age. Note that LS represents life satisfaction of an individual (changing between 5 and 35), income represents monthly income, work_hour shows average weekly working hours, and age shows age of the individual. If estimated coefficient of b3 is 0.003 and standard error of b1 is 0.0013, comment on statistical significance of the association. Note that t-critical value is 2.57 at 99% confidence level, 1.96 at 5% significance level, and 1.67 at 90% significance level. According to hypothesis test result: _____.

The impact of age over life satisfaction of an individual is statistically significant at 5% level of significance.

  1. Consider a given model, log(price) = b0 +b1*distance +b2*room +u. Note that price represents price of houses in Baku city, distance represents distance of each house from the nearest metro station, and room show number of rooms of each house. If estimated coefficient of b1 is 0.012 and standard error of b1 is 0.018, comment on statistical significance of the association. Note that t-critical value is 2.57 at 99% confidence level, 1.96 at 5% significance level, and 1.67 at 90% significance level. According to hypothesis test result: _____

The impact of distance from nearest metro station over price of the house is statistically insignificant.

  1. Consider a given model, log(LS)= b0 +b1*log(income) +b2*log(income)^2 +b3*work_hour +b4*age +u. Note that LS represents life satisfaction of an individual (changing between 5 and 35), income represents monthly income, work_hour shows average weekly working hours, and age shows age of the individual. “^2” display the quadratics. If b1 = 4.039, and b2= - 0.362, and both coefficients are statistically significant at 5% significance level, how much is marginal impact of monthly income over life satisfaction if income equals 350 AZN?

Marginal impact of monthly income over life satisfaction is -0.00202.

  1. Consider a given model, log(LS)= b0 +b1*log(income) +b2*log(income)^2 +b3*work_hour +b4*age +u. Note that LS represents life satisfaction of an individual (changing between 5 and 35), income represents monthly income, work_hour shows average weekly working hours, and age shows age of the individual. “^2” display the quadratics. Suppose that b1>0, and b2<0, and both are statistically significant at 5% significance level. Which one of the following is true about the impact of monthly income over life satisfaction of individuals?

The impact of monthly income over life satisfaction varies across the level of monthly income, with diminishing marginal returns.

  1. Consider a given model, log(LS)= b0 +b1*log(income) +b2*log(income)^2 +b3*work_hour +b4*age +u. Note that LS represents life satisfaction of an individual (changing between 5 and 35), income represents monthly income, work_hour shows average weekly working hours, and age shows age of the individual. “^2” display the quadratics. If b1 = 3.426, and b2= - 0.243, and both coefficients are statistically significant at 5% significance level, how much is the threshold level, and marginal impact of monthly income over life satisfaction if income equals 1200 AZN? Note than log(1200)=7.09.

Threshold level is 1152, Marginal impact is -0.019%.

  1. Please provide the formula to calculate the correlation between X and Y.

The correlation between X and Y can be calculated by dividing the covariance between X and Y by the product of the two standard deviations.



  1. Please describe the procedure for testing joint hypothesis.

When testing joint hypothesis, you should use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.


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