Write down the lagrange function

Assignment Help Applied Statistics

Reference no: EM131992160

ASSIGNMENT PROJECT -

Question A - Cobb-Douglas function

The Cobb-Douglas function for a new product is given by Y(L, K) = 3L^0.2K^0.8

where L is the number of units of labour and K is the number of units of capital required to produce Y(L, K) units of the product.

A1. In (L, K) space, where labour is on horizontal axis ranging from 0 to 10 and capital is on the vertical axis ranging from 0 to 10, depict all possible combinations of L and K resulting in production levels of 5, 10, 15, and 20 units. [These curves are called isoquants]. In a single graph you should have four curves (one for each production level).

Suppose that each unit of labour costs $25, and each unit of capital costs $50. If $500 has been budgeted for the production of this product, determine how this amount should be allocated in order to maximize production, and find the maximum production.

A2. Write down the Lagrange function.

A3. Solve the problem using Lagrange Multipliers (show your work).

Question B - Stock Market

Introduction - The Capital Asset Pricing Model (CAPM) takes into account the stock's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by β in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. CAPM shows that the cost of equity capital is determined only by beta. Despite its invention in the early 1960s, the CAPM still remains popular due to its simplicity and applicability in a variety of situations.

The CAPM is a model for pricing an individual security or portfolio. The risk of a portfolio comprises systematic risk, also known as undiversifiable risk, and unsystematic risk which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all securities-i.e. market risk. Unsystematic risk is the risk associated with individual assets. Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio (firm-specific risks "average out"). The same is not possible for systematic risk within one market. Depending on the market, a portfolio of approximately 20 securities would be sufficiently diversified.

The beta from a single factor model in the form r_i = α_i + β_ir_m + ε_iis a good approximation to the CAPM beta.

The basic idea is that stocks tend to move together, driven by the same economic forces (the market). Here, the dependent variable, r_i are percentage returns for stock i, and independent variable, r_m are percentage returns for a broad market index.

α_i is the intercept and β_i is the slope of the linear relationship between the stock returns and the market. ε_i are the residual returns that cannot be explained by the market fluctuation (this is your idiosyncratic or firm-specific fluctuations).

In the file Assignment Data.xlsx, tab:ASX200 stocks (Prices), you will find prices for 131 stocks as well as the S&P/ASX 200 Index (a benchmark for the Australian stock market) from January 1, 2014 to December 27, 2017.

1. Pick any 3 securities making sure they are from different industries (full name, industry, sector and sub-sector information are provided in column headings as well as in a separate tab called ASX200 stocks (INFO)).

2. Convert your chosen security prices and the market index into percentage returns. For each asset/index, percentage returns are defined as (price on day (t) - price on day (t-1))/(price on day (t-1)). This will define your returns for the three stocks, r_i, and the market return, r_m.

B1. Perform OLS regression for each stock separately and report regression outputs for the three models from Excel/Matlab including line fit plots and residual plots.

B2. For each stock, discuss the OLS assumptions and violations (if any) based on the results from B1.

B3. Discuss the estimated betas for your three stocks and their statistical significance. Are these betas in line with your expectations? Provide your reasoning. What does it mean if a stock has a beta equal to 1? What does it mean if a stock has a beta equal to zero?

B4. Discuss the measure of fit (R²) of your regressions in B1. Are these R² in line with your expectations? Provide your reasoning. Note that R^2 is the fraction of variation in the dependent variable (the return on a stock/portfolio of stocks) that is explained by movements in the independent variable (the return on the market index).

B5. Construct an equally weighted portfolio consisting of your three chosen stocks (equally weighted portfolio returns are simply the average of individual stock returns in that portfolio, r_p = (r₁ + r₂ + r₃)/3) and find the portfolio beta. Report regression output (including line fit plots and residual plots), assess the OLS assumptions and violations (if any) and discuss the estimated portfolio beta and the measure of fit of your regression. How does the measure of fit for the portfolio compares with the measures of fit for your individual stocks? Comment on portfolio diversification effect using your R²s.

Question C - Population Survey

Introduction - Each month the Bureau of Labor Statistics in the U.S. Department of Labor conducts the "Current Population Survey" (CPS), which provides data on labour force characteristics of the population, including the level of employment, unemployment, and earnings. Approximately 65,000 randomly selected U.S. households are surveyed each month. The sample is chosen by randomly selecting addresses from a database comprised of addresses from the most recent decennial census augmented with data on new housing units constructed after the last census. The exact random sampling scheme is rather complicated (first small geographical areas are randomly selected, then housing units within these areas randomly selected).

The file Assignment Data.xlsx, tab: Young Workers Survey contains the data from the survey. These data are for full-time workers, defined as workers employed more than 35 hours per week for at least 48 weeks in the previous year. Data are provided for workers whose highest educational achievement is (1) a high school diploma, and (2) a bachelor's degree.

Series in Data Set:

FEMALE: 1 if female; 0 if male

YEAR: Year

AHE: Average Hourly Earnings

BACHELOR: 1 if worker has a bachelor's degree; 0 if worker has a high school degree

Use the data in Assignment Data.xlsx, tab: Young Workers Survey to answer the following questions:

C1. Run a regression of average hourly earnings (AHE) on age (Age). Report Excel/Matlab output. What is the estimated effect of Age on earnings? If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

C2. Run a regression of average hourly earnings (AHE) on age (Age), gender (Female), and education (Bachelor). Report Excel/Matlab output. What is the estimated effect of Age on earnings? If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

C3. Are the results from the regression in C2 substantively different from the results in C1 regarding the effects of Age and AHE? Does the regression in C1 seem to suffer from omitted variable bias?

C4. Compare the fit of the regression in C1 and C2 using the regression standard errors, R^2 and R ¯^2. Why are the R² and R ¯² so similar in regression C2?

C5. Run a regression of the logarithm of average hourly earnings, ln(AHE) on Age, gender (Female), and education (Bachelor). Report Excel/Matlab output. What is the estimated effect of Age on earnings? What is the estimated effect of Age on earnings? If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

C6. Run a regression of the logarithm of average hourly earnings, ln(AHE) on ln(Age), gender (Female), and education (Bachelor). Report Excel/Matlab output. What is the estimated effect of Age on earnings? What is the estimated effect of Age on earnings? If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

C7. Run a regression of the logarithm of average hourly earnings, ln(AHE) on Age, Age2, gender (Female), and education (Bachelor). Report Excel/Matlab output. What is the estimated effect of Age on earnings? What is the estimated effect of Age on earnings? If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

C8. Do you prefer the regression in C6 to the regression in C5? Explain.

C9. Do you prefer the regression in C7 to the regression in C5? Explain.

C10. Do you prefer the regression in C7 to the regression in C6? Explain.

C11. Plot the regression relation between Age and ln(AHE) from C5, C6, and C7 for males with a high school diploma. Describe the similarities and differences between the estimated regression functions. Would you answer change if you plotted the regression function for females with college degrees?

C12. Run a regression of ln(AHE), on Age, Age2, Female, Bachelor, and the interaction term Female × Bachelor. What does the coefficient on the interaction term measure? Alexis is a 30-year-old female with a bachelor's degree. What does the regression predict for her value of ln(AHE)? Jane is a 30-year-old female with a high school degree. What does the regression predict for her value of ln(AHE)? What is the predicted difference between Alexis's and Jane's earnings? Bob is a 30-year-old male with a bachelor's degree. What does the regression predict for his value of ln(AHE)? Jim is a 30-year-old male with a high school degree. What does the regression predict for his value of ln(AHE)? What is the predicted difference between Bob's and Jim's earnings?

C13. Is the effect of Age on earnings different for men than for women? Specify and estimate a regression that you can use to answer this question.

C14. Is the effect of Age on earnings different for high school graduates than for college graduates? Specify and estimate a regression that you can use to answer this question.

C15. After running all these regressions (and any others that you want to run), summarize the effect of age on earnings for young workers.

Attachment:- Assignment Files.rar

Reference no: EM131992160

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len1992160

5/22/2018 5:59:44 AM

This assignment has several parts. Some parts are technical and designed to help you understand and apply the material in chapters covered in the first part of the semester with the help of financial applications. Econometrics part is designed to help you develop practical econometric skills (e.g., understanding of regression analysis, interpretation of regression results). You will apply regression analysis as done by finance professionals using real world data with applications to financial market and survey data.

5/22/2018 5:59:37 AM

You must type your answers in the spaces provided. Make use of MS Office Equation environment (i.e., in Ribbon: Insert Equation, see examples within this document). You must include this cover sheet. This assignment can be done in groups of up to six (6) students. Your group members do not need to be part of the same tutorial group, but you must nominate a tutor that would represent the majority of your group members. No names may be added to the group lists after the submission apart from the names that appear below at the time you hand-in your assignment.

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Submit only one assignment for each group. DO NOT hand in a separate assignment for each group member. Late assignments will incur a penalty of 5% for each day past the due date up to 5 days. Assignments submitted 5 days after the due date will receive zero mark. The assignment submission boxes are located on Level 5, Building 8 (Chau Chak Wing Building) in the Business Faculty. The assignment boxes are used by many subjects so it is important to identify your assignment CLEARLY with your names and SIDs. Submit your assignment to FINANCE 3 box. Assignments submitted to boxes other than FINANCE 3 will not be picked up, and as a result, will not be graded. It is your responsibility to ensure that you submit your assignment to the correct assignment box. Do not remove this assignment cover. The onus is in you to submit your assignment to the correct subject box by the due date. The deadline to submit your assignment is 5:00pm Friday

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The following group work protocol is required to satisfy the by-laws on assessment tasks of the UTS Business School. Group work and group assignments are an important part of business education. The purpose of group work goes beyond the requirements of the group project itself. Group work is also intended to develop awareness of the dynamics of teamwork and your role in a team environment. In undertaking group work it is expected that each team member will communicate with other team members in a professional and courteous manner. Resolving any conflict or difficulties that arise within the team is also part of the group dynamics that may emerge during the project. It is expected that each team member will take responsibility for managing and reducing any conflict that may arise. If the team members are unable to resolve tensions or conflicts, then this MUST be discussed face-to-face with the lecturer/tutor during the consultation hours as soon as it becomes obvious that a resolution within the team cannot be agreed.

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In the learning environment of group work we are seeking to ensure that you develop and demonstrate both academic skills and important group work skills such as: Commitment to working with others (e.g., undertaking a fair share of the work, sharing ideas, doing the tasks allocated, attending meetings) Collaboration and inclusiveness (e.g., encouraging and supporting others, respecting others; recognising the skills and valuing the contribution of others, helping resolve conflicts) Contribution to establishing and working towards a common outcome (e.g., establishing and supporting team goals, plans, rules, roles, decisions) Please bear in mind that any member of a group reserves the right to leave the group to form/join another group or submit the assignment individually.

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