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(i)Present the descriptive statistics of the variables remuneration, rank and studnum. Comment on the means and measures of dispersion of the variables.

(ii)Estimate the following simple regression model of remuneration on rank.

Write down the sample regression function and interpret the coefficient estimates.

(iii)Now estimate the following simple regression model with a log-log specification,

Report your regression results in a sample regression function. Interpret the estimated coefficient of log(rank). Is the sign of this estimate what you expect it to be?

(iv)A model that relates the remuneration to the university’s ranking and number of students is:

Report your results in a sample regression function. What can you conclude regarding comparison of the goodness of fit of this regression model versus the regression model in part (ii)?

(v)Now re-estimate the equation in (iv) but using the log of each variable. That is, estimate the model,

(vi)Using the estimated model in (v), test whether rank has a negative effect on remuneration at 1% level of significance.

(vii)Add the variables grademp and gradstudy to the log-log equation in (v) and estimate the following model.

Test whether either of these variables grademp and gradstudy are individually significant at 1% level? Test if they are jointly significant at 5% level?

(viii)Test the overall significance of the model you estimated in part (vii) at 1% level of significance.

(ix)Suppose you want to test whether the Vice Chancellors of the universities located in Victoria are paid higher compared to those in other states. Specify a regression model which will enable you to test such a hypothesis using the model in (v) as a base. Report your results in a sample regression function and perform the hypothesis test at 5% level of significance. What would you infer?

### Remuneration

From the descriptive statistics that capture the given sample data, it is evident that the average remuneration package for the Vice Chancellor is \$801,270. The median value is slightly higher at \$ 805,000 which implies that 50% of the Vice Chancellors in the given data would have salary equal to or lower than \$ 805,000. However, the mode value is still higher at \$ 895,000. The non-coincidence of the central tendency measures is an indication that the probability distribution in the given case would not be normal. Also, there would be presence of negative skew having a slight distortionary effect on the average value. The standard deviation for the salaries is low to medium indicating that while disparities do exist they do not tend to be too extreme. The IQR values highlights that 50% of the values i.e. from 25% to 75% would lie within this compensation interval of about \$230,000 (Fehr and Grossman, 2003).

Rank – The average rank of the University in the given sample is 336.81. The median rank is however significantly greater at 376 which implies that 50% of the sample universities have a rank lower than or equal to 376. The deviation between the median and mode values is indicative of the presence of some universities which have a considerably low rank which is responsible for distortion of the mean. The various measures of dispersion highlight that the dispersion in rank of the universities in the given sample is quite high considering that there are universities at both the lower end as well as the higher end (Liberman, et. al., 2013).

Studnum – The average students studying in a university are about 38,410. The average figure is clearly distorted owing to presence of universities with very high intake of students owing to which the median value is comparatively much smaller at 29,214. Hence, the median is a better indicator of the average number of students in a university rather than the mean. Further, the measures of dispersion suggest that there is very high amount of dispersion that is seen in the given variable for the sample data provided (Harmon, 2011).

• Simple regression model

Remuneration = dependent variable

Rank = Independent variable

The intercept coefficient for the above repression is 1032.54 which tends to highlight the salary level of the Vice Chancellor when the rank is zero. Since the rank theoretically cannot be zero, the predicted value of Vice Chancellor’s remuneration cannot exceed the intercept value.

Further, the slope coefficient for the above relation is negative which implies that as rank increases by 1, there is a decrease in the remuneration of the Vice Chancellor by \$687. Thus poorer the rank of the University, the lower would be the expected remuneration of the Vice Chancellor.

• Simple regression model  by taking log-log specification

The estimated coefficient of the log (rank) is -0.165.  The sign of the coefficient is on expected lines since it would be expected that Universities which are higher on the rankings would have the Vice Chancellors drawing higher remuneration. The above model tends to validate this logical conclusion since there is a drop in the predicted remuneration as the rank of the University tends to increase in magnitude (Liberman, et.al., 2013).

• Regression model in relation to the remuneration of university ranking based on the number of students are highlighted below (Fehr and Grossman, 2003):

Remuneration = dependent variable

Rank, studnum = Independent variable

In order to compare the goodness of fit, it makes sense to compare the R2 values for the two models. In the regression model highlighted in part(ii), the R2 value was 0.335. In comparison, the R2 value for the model illustrated above is 0.5208. It is apparent that with the introduction of the additional independent variable (i.e. Studnum), there has been an improvement in the predictive power of the model (Shi and Tao, 2008). As a result, it would be fait to conclude that goodness of fit is superior for this model as compared to that in part (ii).

• Regression model in relation to the remuneration of university ranking based on the number of students on a log-log specification are highlighted below:

The elasticity of remuneration with regards to studnum is 0.148 which implies that when studnum increases by 1%, the corresponding increase in remuneration would be 0.148%.  Hypothesis test would be used to ascertain if this is significant at 1% or not (Lehman and Romano, 2006).

Null Hypothesis: βstadnum = 0

Alternative Hypothesis: βstadnum ≠ 0

The t statistic as obtained in the output attached above = 2.77 with a corresponding p value of 0.009. Since the p value is lower than given significance level of 0.01, hence the null hypothesis would be rejected and alternative hypothesis would be accepted. Hence, this implies that the elasticity with regards to studnum is significant at 1% significance level since the slope coefficient cannot be assumed to be zero thus proving the significance (Fehr and Grossman, 2003).

• Hypothesis test would be used to ascertain if the rank slope coefficient is significant at 1% or not.

Null Hypothesis: βRank = 0

Alternative Hypothesis: βRank ≠ 0

The t statistic as obtained in the output attached above = -3.315 with a corresponding p value of 0.002 Since the p value is lower than given significance level of 0.01, hence the null hypothesis would be rejected and alternative hypothesis would be accepted. Hence, this implies that the elasticity with regards to rank is significant at 1% significance level since the slope coefficient cannot be assumed to be zero thus proving the significance (Shi and Tao, 2008). Further, this implies that negative effect is indeed significant.

• Regression model in relation to the remuneration of university ranking based on the number of students including grademp and gradstudy on a log-log specification are highlighted below    (Fick, 2015):

Hypothesis testing would be used to ascertain whether the variables grademp and gradstudy are significant for not.

Significance test for Grademp

The requisite hypotheses are as highlighted below.

Null Hypothesis: βGrademp = 0

Alternative Hypothesis: βGrademp ≠ 0

The t statistic as obtained in the output attached above = 1.189 with a corresponding p value of 0.243. Since the p value is greater than given significance level of 0.01, hence the null hypothesis would not be rejected and alternative hypothesis would not be accepted. Hence, this implies that the slope coefficient can be assumed to be zero and therefore is not significant.

### Significance test for Gradstudy

The requisite hypotheses are as highlighted below.

Null Hypothesis: βGradstudy = 0

Alternative Hypothesis: βGradstudy ≠ 0

The t statistic as obtained in the output attached above = 2.589 with a corresponding p value of 0.014. Since the p value is greater than given significance level of 0.01, hence the null hypothesis would not be rejected and alternative hypothesis would not be accepted (Eriksson and Kovalainen, 2015). Hence, this implies that the slope coefficient can be assumed to be zero and therefore is not significant.

Also, at 5% significance level, the joint significance of the two variables is not established since the corresponding p value would tend to exceed 0.05.

• Hypothesis test would be used to ascertain the significance of the model indicated in part (vii).

The relevant hypotheses are highlighted below.

Null Hypothesis: All the slope coefficients are insignificant and hence can be assumed as zero.

Alternative Hypothesis: Atleast one of the slope coefficients is significant and hence cannot be assumed to be zero.

The relevant output to perform this test is the ANOVA output as highlighted below.

From the above output, it is apparent that the test statistic F value comes out as 8.5943. The corresponding p value has been computed as 0.0001. Since this p value is lower than the given significance level of 1% or 0.01, hence there is enough evidence available so that the null hypothesis is rejected and alternative hypothesis is accepted (Koch, 2013). As a result, it can be concluded that there is atleast one slope coefficient which is significant and hence non-zero. Hence, it would be appropriate to conclude that the given multiple regression model is significant.

• The relevant regression output in order to find the impact of states on the remuneration of the Vice Chancellors is shown below.

For the above regression, Victoria has been given a dummy variable 1 while all the other states are given a value 0. It is apparent that the coefficient is positive which potentially implies that Vice Chancellors in Universities based out of Victoria are paid higher than their counterparts in other locations.

To check the significance, the following hypothesis would be taken.

Null Hypothesis: βState = 0

Alternative Hypothesis: βstate ≠ 0

The t statistic as obtained in the output attached above = 0.595 with a corresponding p value of 0.556. Since the p value is greater than given significance level of 0.05, hence the null hypothesis would not be rejected and alternative hypothesis would not be accepted (Taylor and Taylor, 2004).  Hence, this implies that the slope coefficient can be assumed to be zero and therefore is not significant. Therefore, there is no significant effect of the location of the university on the remuneration.

## References

Eriksson, P. and Kovalainen, A. (2015) Quantitative methods in business research. 3rd edn. London: Sage Publications.

Fehr, F. H.  and Grossman, G. (2003). An introduction to sets, probability and hypothesis testing. 3rd edn. Ohio: Heath.

Flick, U. (2015) Introducing research methodology: A beginner's guide to doing a research project. 4th edn. New York: Sage Publications.

Harmon, M. (2011) Hypothesis Testing in Excel - The Excel Statistical Master. 7th edn. Florida: Mark Harmon.

Hillier, F. (2006) Introduction to Operations Research. 6th edn. New York: McGraw Hill Publications.

Koch, K.R. (2013) Parameter Estimation and Hypothesis Testing in Linear Models. 2nd edn. London: Springer Science & Business Media.

Lehman, L. E. and Romano, P. J. (2006) Testing Statistical Hypotheses. 3rd edn. Berlin : Springer Science & Business Media.

Lieberman, F. J., Nag, B., Hiller, F.S. and Basu, P. (2013) Introduction To Operations Research. 5th edn. New Delhi: Tata McGraw Hill Publishers.

Shi, N. Z. and Tao, J. (2008) Statistical Hypothesis Testing: Theory and Methods. 3rd edn. Singapore : World Scientific.

Taylor, K. J. and  Cihon, C. (2004) Statistical Techniques for Data Analysis. 2nd edn. Melbourne: CRC Press.

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