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Part (a)
(i) Collinearity occurs when 2 independent variables are related to each other (William, R., 2015). After adding another variable, wealth (W), into model 3.1, the changes as follows:
Addition of the wealth variable is providing a significant result in the regression model, hence, not much of the problem. Though there is a possibility of results different from theoretical ones, there may be some significant reasons for them as well. As per Stat Trek. (2020), the problem of collinearity can be solved by:
(ii) To know the worth of adding the variable W, the comparison of R-squared, and adjusted R-squared of model 3.1 and model 3.2 has been done. According to CFI. (2015-20) and Bhalla, D. (2014), R-squared measures the variations in the dependent variable due to that in the independent variables of the model, whereas, adjusted R-squared also considers the impact of the number of independent variables in the model. On adding the new variable, the R-squared will either rise or remain the same as before, whereas, the adjusted R-squared will rise when a significant variable, which has a significant impact on the dependent variable will be added, otherwise it can fall and/or become negative.
In the given models, the R-squared values of models 3.1 and 3.2 are the same, whereas, the adjusted R-adjusted values of model 3.2 have increased. This implies that it is worth to add the additional variable, wealth into model 3.1 as it is explaining the variations in the dependent variable significantly (Online Stat, n.d.).
Part (b)
(i) From the given information, the research and the development are a dependent variable whereas, the sales are an independent variable. The sample size is 31 groups (n = 31).
The econometric model will be: RDi = a + b Si, where,
RD = research and development expenditure
S = sales
i = industry grouping (1, 2, …, 31)
There should be a positive impact of sales on R&D expenditures, as more the sales will be, the more income the industry group will have and more expenditure on R&D will occur.
(ii) Using STATA software and the given data:-
The regression model will be:
RD = 4215.823 + 0.1363 S
The model interprets that when S = 0, the RD expenditure will be 4215.823 RM millions, whereas when the sales increase by 1 RM millions, the RD expenditure will be 0.1363 RM millions in the groupings of the industry.
(iii) The problems in the model are:
Testing heteroscedasticity using the Breusch-Pagan test in STATA software,
At the 5% significance level, the null hypothesis will be rejected as the p-value < 0.05. Therefore, there is heteroscedasticity in the data.
(iv) According to Williams, R. (2020), the heteroscedasticity can be resolved by:
Using the STATA for WLS regression:
It can be noticed that the value of the constant has been changed along with the sign. Also, the value of the coefficient, Sales has also increased.
Part (a)
(i) Apart from the small sample size, there is a problem of multicollinearity in the model. Grace-Martin, K. (2008-20) stated that there are 8 ways to detect the multicollinearity, here are:
- The VIF value will be more as the R-squared is high (CFI, 2015-20).
- All the variables individual t-tests are coming out to be insignificant, but significant F-test.
(ii) The problem can be resolved:-
Part (b)
(i) Using the STATA software, the regression model will be:
It = 66668.97 + 1.183862 St, where t = 1, 2, …, 41
I = inventories, and S = sales
The model says, when sales = 0, then there will be inventories of 66668.97 RM million across the years, whereas, change in sales by 1 RM million will lead to 1.183862 RM million increase in inventories across the years.
(ii) Using the STATA software, the Durbin Watson d-test is used to test the autocorrelation in the data, therefore, the d-statistic value (2, 41) = 0.1255528.
From Durbin Watson Significance Tables and Durbin Watson test, it can be concluded that we reject the null hypothesis of no autocorrelation at the significance level of 1%, hence, there is autocorrelation.
(iv) Eliminating the autocorrelation, by transforming the data using the Prais-Winston method (Correcting for Autocorrelation in the residuals using Stata), the new regression model will be:
(v) After transformation, from the previous table calculated above, the transformed d-statistic value comes out to be 1.89272. Using the tables at a 1% significance level, we can conclude that the d-statistic value is greater than the upper bound, and hence, we fail to reject the null hypothesis, and hence, there is no autocorrelation (Biamir, 2016).
Bhalla, D. (2014). Difference between adjusted R-squared and R-squared. Retrieved from Listen Data: https://www.listendata.com/2014/08/adjusted-r-squared.html
Biamir. (2016). Durbin Watson Test. Retrieved from Analytics & Cloud Amir: https://biamir.wordpress.com/2016/04/25/durbin-watson-test/
CFI. (2015-20). What are the adjusted R-squared? Retrieved from CFI: https://corporatefinanceinstitute.com/resources/knowledge/other/adjusted-r-squared/
CFI. (2015-20). What is the Variance Inflation Factor (VIF)? Retrieved from: https://corporatefinanceinstitute.com/resources/knowledge/other/variance-inflation-factor-vif/
Correcting for Autocorrelation in the residuals using Stata. Retrieved from: http://people.brandeis.edu/~cerbil/cocran-orcutt.htm
Durbin-Watson Significance Tables. Retrieved from: https://www3.nd.edu/~wevans1/econ30331/Durbin_Watson_tables.pdf
Durbin-Watson Test. Retrieved from: http://www.math.nsysu.edu.tw/~lomn/homepage/class/92/DurbinWatsonTest.pdf
Grace-Martin, K. (2008-20). Eight ways to detect multicollinearity. Retrieved from The Analysis Factor: https://www.theanalysisfactor.com/eight-ways-to-detect-multicollinearity/
How-to Guide for Stata. Retrieved from: https://methods.sagepub.com/dataset/howtoguide/heteroscedasticity-in-eclsk-1998-stata
Online Stat. (n.d.). Lesson 12: Multicollinearity & other regression pitfalls. Retrieved from: https://online.stat.psu.edu/stat501/book/export/html/981
Prabhakaran, S. (2017). How to detect heteroscedasticity and rectify it? Retrieved from Data science plus: https://datascienceplus.com/how-to-detect-heteroscedasticity-and-rectify-it/
Stat Trek. (2020). Multicollinearity and regression analysis. Retrieved from: https://stattrek.com/multiple-regression/multicollinearity.aspx
The Minitab Blog. (2020). What are the effects of multicollinearity and when can I ignore them? Retrieved from: https://blog.minitab.com/blog/adventures-in-statistics-2/what-are-the-effects-of-multicollinearity-and-when-can-i-ignore-them
William, R. (2015). Multicollinearity. The University of Notre Dame. Retrieved from: https://www3.nd.edu/~rwilliam/stats2/l11.pdf
Williams, R. (2020). Heteroscedasticity. The University of Notre Dame. Retrieved from: https://www3.nd.edu/~rwilliam/stats2/l25.pdf
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