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Multicollinearity and Regression Analysis - Question 3

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:

  • The R-squared value remained the same as model 3.1, whereas, adjusted R-squared value has increased significantly.
  • The value of the constant and the coefficient value of literacy rate have fallen, whereas income's coefficient value is the same.
  • Standard error values of constant and the literacy rates coefficient have increased significantly too.

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:

  • Regressing collinear variables separately with the dependent variable.
  • Omit correlated variables in the regression model, since income is a part of wealth by definition, therefore, only wealth as a variable can be used in place of both.
  • Make another variable, a linear combination of wealth and income, and use that variable in the regression model.

(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:

  • The sample size is small.
  • The value of R-squared and adjusted R-squared is quite low, more independent variables are required to explain the variations in the dependent variable in the model.
  • The major problem is of heteroscedasticity, the variance of the error terms is not constant, and there is a correlation between the independent variable and the error terms. Studying the graph of residual terms against the fitted values, we can conclude that there is no consistency in the data (Prabhakaran, S., 2017).

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 robust standard errors in the regression model.
  • Doing estimation using the weighted least squares method.

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.

Multicollinearity and Regression Analysis - Question 4

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:-

  • By doing individual regression of each independent variable on the dependent one.
  • Combing the collinear variables and then using the regression model.
  • Eliminate the highly correlated variable from the regression model.

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).

References for Variance Inflation Factor Analysis

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

Remember, at the center of any academic work, lies clarity and evidence. Should you need further assistance, do look up to our Accounting and Finance Assignment Help

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