Bayesian Data Analysis - Stan or JAGS Programming Language - Patriot Bank - Coefficients Regression and Correlation - Statistics Assessment Answer

Assessment Task:

Bayesian Data Analysis Statistics Assessment Answer

Task Description

You must apply Bayesian data analysis methods You may use either Stan or JAGS programming language to model your solutions. 

You may use any reference with one exception. The exception is that you may not consult with, or accept help from, any person (other than your instructor).  Ideally, submit your solution on an RMarkdown Word document.  If you are unable to create an RMarkdown Word document, submit your R scripts and, if you’d like, your RMD file.  Even if you are successful in creating an RMarkdown Word document, you may submit your R scripts.  

1.  You’re in the wealth management division of the Patriot Bank. Your client just asked you to estimate “the beta” (i.e., the statistical association with the ‘market”) of her favourite stock, Acme.  Your bank’s convention is to use daily returns of the S&P 500 as the surrogate of the market, and the daily returns of 4-week US T-bills as the risk-free rate. 

Your coworker has downloaded more than 2 years of daily returns from Acme and the S&P 500, as well as the closing price of 4-week US T-bills. She partially prepared the data by subtracting the daily T-bill rates from the both the Acme and S&P 500 return; thus, leaving you the set of excess daily returns for both Acme and the S&P 500 (see the “acme.csv” file). The returns are reflected as decimals (not in terms of percentages). 

a. Perform a qualitative exploratory analysis of the data.
b. Perform a regression of Acme’s daily excess returns as a function of the S&P 500’s daily excess returns. Report your results of Acme’s beta.
c. Perform diagnostic checks. At a minimum, produce a trace plot, pairs plot, and density plots. Submit those plots.
d. Perform posterior predictive checks to validate your model.
e. Create a scatterplot of the excess returns of Acme as a function of the excess returns of S&P 500. 

(1)  Add a linear regression function (i.e., a straight line) to your plot that represents the average statistical relationship between the firms’ daily returns.
(2)  Add text of the math model of the linear regression function.
 

2. You are a staff member of the Office of the Chief Risk Officer in your bank. In the past quarter, your bank issued 60,801 auto loans. You’ve noticed that nearly 3 per cent of those loans went bad. However, the bad-loan rate is not uniformly distributed among your borrowers.  Your boss wants to better understand the distribution of bad-loan rates by age group. Traditionally, your bank has categorized borrowers’ ages in four groups: 21-30; 31-36; 36-48; and over 48. 

Perform a logistics regression to determine the probability of a bad loan by age group 1 through 4. Report your results. Display the by-age bad-loan probabilities as a boxplot.  

3. You are managing a mutual fund. You are assessing the performance of two portfolios: Portfolio A and Portfolio B.  Each portfolio comprise 15 funds. Each fund has a market value of $1 million. Thus, the funds are equally weighted.  The annual excess log-returns of each fund are reported in the “funds returns.csv” worksheet.)
Analyze the portfolios’ annual performance to assess the difference, if any, of the average excess log-returns between the two portfolios.
Present your summary statistics from your analysis, including your inference about the weight of evidence in support of your hypothesis of a difference.  In other words, state whether the evidence suggests that the average means are different. If you infer a difference, how strong is the support for your inference?
Also, compute the Sharpe ratio of both portfolios.  (Recall that a Sharpe ratio is a common performance measure for assessing portfolios.  For our non-finance colleagues in the course, the Sharpe ratio of a portfolio is defined as follows: SR = excess returns/standard deviation. We should prefer the portfolio with a higher Sharpe ratio. Why? Because that portfolio earns a higher return per unit of risk.) 

4. You are the regional manager of a brokerage firm. You oversea five district offices. Each district office has 25 sell-side analysts evaluating companies for future earnings growth and other investment criteria. You have collected data of each analyst’s weekly Bloomberg terminal time (y) and associated data: x1, the number of years of overall experience; and x2, the number of years with your firm. See the “terminal time.csv” worksheet.

Given the repeated data for each district, you assume the data generating process is a normal distribution with varying intercepts – a multi-level Gaussian model. 

a. Analyze these data to infer the following:

(1)  the coefficients of the x1 and x2 explanatory variables
(2) the coefficients of the varying intercepts
(3) the standard deviation estimator of the overall population
(4) the standard deviation of the distribution of varying intercepts

b. Assess the following:

(1) whether time in the industry is a plausible predictor of time on the Bloomberg terminal.
(2)  whether time with the firm is a plausible predictor of time on the Bloomberg terminal.

c. Produce diagnostic plots. At a minimum

(1) trace plots of the beta coefficients
(2) trace plots of the varying intercept coefficients
(3) trace plots of the population and varying intercepts standard deviation
(4) pairs plot and stan plot of the posterior “beta” coefficients. 
(5) stan plot of the varying intercept coefficients.

d. Assess the validity of your model with posterior predictive check graphics. 

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