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| # 3135 May Intensive: Final Examination. | ||
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| # You may use all materials presented in class, as well as online resources. | ||
| # You may not, however, assist other students in this course with their answers. | ||
| # You are bound by NYUSPS Policy on Academic Integrity and Plagiarism. | ||
| # Submit to NYU Classes as: [NetID]_Final.R. | ||
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| library(quantmod) | ||
| library(stargazer) | ||
| library(fredr) | ||
| library(tidyr) | ||
| library(PASWR2) | ||
| library(MASS) | ||
| library(repmis) | ||
| library(latex2exp) | ||
| library(dplyr) | ||
| library(ggplot2) | ||
| library(tidyverse) | ||
| library(RCurl) | ||
| library(haven) | ||
| library(forecast) | ||
| library(depmixS4) | ||
| library(mFilter) | ||
| fredr_set_key('30e6ecb242a73869e11cb35f6aa3afc3') # My key, please don't abuse. | ||
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| getwd() | ||
| setwd("C:/Users/erikl/OneDrive/Documents/NYU_CUSP/Summer2020/REDA") | ||
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| # Question 1. | ||
| # Consider the following sequence of numbers, called x1: 6 2 9 7 7 1 9 9 6 5 6 1 4 7 9 1 0 5 5 3 | ||
| # Using R, calculate the mean, variance and standard deviation of x1. | ||
| # Consider another sequence of numbers, called x2: 9 5 8 1 9 2 2 6 9 8 5 1 0 2 7 4 9 7 0 9 | ||
| # Using R, calculate the covariance and correlation between x1 and x2. | ||
| # Using R, run the bivariate linear model that regresses x2 on x1. | ||
| # Based on a null hypothesis of no relationship, what is the 95% confidence interval on the slope coefficient? | ||
| # Would you reject or fail to reject the null hypothesis at this level of confidence? | ||
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| x1 <- c(6, 2, 9, 7, 7, 1, 9, 9, 6, 5, 6, 1, 4, 7, 9, 1, 0, 5, 5, 3) | ||
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| mean(x1) | ||
| var(x1) | ||
| sd(x1) | ||
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| x2 <- c( 9, 5, 8, 1, 9, 2, 2, 6, 9, 8, 5, 1, 0, 2, 7, 4, 9, 7, 0, 9) | ||
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| cov(x1, x2) | ||
| cor(x1, x2) | ||
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| ols = lm(x1 ~ x2) | ||
| stargazer(ols, type="text", title="OLS Results", single.row=TRUE, | ||
| ci=TRUE, ci.level=0.95) | ||
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| ## Slope CI at 95%: 0.105 (-0.296, 0.506) | ||
| ## We would FAIL to reject the null hypothesis because the zero slope is included in the CI, showing no relation between x1 and x2. | ||
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| # Question 2. | ||
| # For a publicly-traded stock of your choice, download data on the adjusted closing | ||
| # stock price, as well as the market (or exchange) on which it trades between 2006 and 2017. | ||
| # Calculate the daily returns for both the stock and the market on which it trades. | ||
| # Generate graphs of closing adjusted prices for both. | ||
| # Generate graphs of returns over time for both. | ||
| # Generate histograms of returns for both. | ||
| # Using a bivariate linear regression, estimate the CAPM presented in class. | ||
| # Assuming the EMH, test the basic CAPM hypotheses using your data. | ||
| # Integrate the Fama-French 5 Factor data presented in class. | ||
| # Estimate the Fame-French 3-facotr and 5-factor models. | ||
| # Describe how your results change as you move from the 3- to the 5-factor model, addressing changes in the adjusted beta, | ||
| # as well as changes in conclusions regarding hypothesis testing of the adjusted beta. | ||
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| getSymbols(c('GS','^NYA'), from="2006-01-01", to="2017-12-31") | ||
| data = merge(as.zoo(GS$GS.Adjusted), as.zoo(NYA$NYA.Adjusted)) | ||
| names = c("GS", "NYA") | ||
| colnames(data) = names | ||
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| data.level = as.xts(data) ## Levels | ||
| data.returns = diff(log(data.level), lag=1) ## Log returns | ||
| data.returns = na.omit(data.returns) ## Dump missing values | ||
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| plot(GS$GS.Adjusted, main="Goldman Sachs Closing Prices", col = "darkgreen") | ||
| plot(NYA$NYA.Adjusted, main="NYSE Closing Prices", col = "darkgreen") | ||
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| plot(data.returns$GS, main="Goldman Sachs Daily Returns", col = "darkgreen") | ||
| plot(data.returns$NYA, main="NYSE Daily Returns", col = "darkgreen") | ||
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| hist(data.returns$GS, breaks=100, col="darkblue", freq=F, | ||
| main="Histogram of GS Daily Returns (2006-2017)", xlab="Daily Returns", | ||
| xlim=c(-.2, .2)) | ||
| abline(v=0, col="red") | ||
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| hist(data.returns$NYA, breaks=100, col="darkblue", freq=F, | ||
| main="Histogram of GS Daily Returns (2006-2017)", xlab="Daily Returns", | ||
| xlim=c(-.2, .2)) | ||
| abline(v=0, col="red") | ||
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| plot.ts(data.returns$NYA, data.returns$GS, pch=16, | ||
| col="darkblue", main="CAPM Data", xlab="Returns of NYSE", | ||
| ylab="Returns of GS") ## time series plot in R | ||
| grid(lw = 2) | ||
| abline(lm(data.returns$GS ~ data.returns$NYA), col="red") ## I added the best fit line so the graph looks similar to that presented in class for Apple.library(ggplot2) | ||
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| capm.ols = lm(data.returns$GS ~ data.returns$NYA) | ||
| stargazer(capm.ols, type="text", title="CAPM Results", single.row=TRUE, | ||
| ci=TRUE, ci.level=0.99) | ||
| ## Results | ||
| ##NYA 1.350*** (1.291, 1.409) | ||
| ##Constant 0.0001 (-0.001, 0.001) | ||
| ## Assuming EMH, our NULL HYPOTHESIS: A == 0 && B == 1, ALT HYPOTHESIS: A != 0 || B != 1, where A is the constant and B is the slope | ||
| ## Using a 99% confidence interval we REJECT the null hypothesis | ||
| ## Although Zero is within the CI for Alpha, Beta's CI doesn't include a slope of 1, indicating that the GS stock is aggressive with no excess returns. | ||
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| ff5f = read_dta("FF5F.dta") # Data are stored in multiple formats. | ||
| ff5f$mktrf = ff5f$mktrf / 100 | ||
| ff5f$smb = ff5f$smb / 100 | ||
| ff5f$hml = ff5f$hml / 100 | ||
| ff5f$rmw = ff5f$rmw / 100 | ||
| ff5f$cma = ff5f$cma / 100 | ||
| ff5f$rf = ff5f$rf / 100 | ||
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| a = xts(x=ff5f, order.by = ff5f$date) | ||
| b = GS$GS.Adjusted | ||
| b = diff(log(b), lag=1) | ||
| b = na.omit(b) | ||
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| merged_data = merge(a, b, join='right') ## Merge into single time-series dataset | ||
| names = c("date", "mktrf", "smb", "hml", "rmw", "cma", "rf", "gs") | ||
| colnames(merged_data) = names | ||
| merged_data = merged_data[, colnames(merged_data) != "date"] # Narrow dataframe | ||
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| ff3f.ols = lm('gs - rf ~ mktrf + smb + hml', data = merged_data) | ||
| stargazer(ff3f.ols, type="text", title="FF3F Results: GS", single.row=TRUE, | ||
| ci=TRUE, ci.level=0.95) | ||
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| ff5f.ols = lm('gs - rf ~ mktrf + smb + hml + rmw + cma', data = merged_data) | ||
| stargazer(ff5f.ols, type="text", title="FF5F Results: GS", single.row=TRUE, | ||
| ci=TRUE, ci.level=0.95) | ||
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| ## Moving from 3-factor to 5-factor we see that the adjusted beta is still above (not even within CI), so again we see that | ||
| ## GS stock is aggresive with no excess returns, so we REJECT the null hypothesis. | ||
| ## Should be noted that GS stock isn't as aggresive once we move from 3-factor to 5-factor | ||
| ## No chances in our conclusion | ||
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| # Question 3. | ||
| # Read in the Staten Island sales data presented in class. | ||
| # Suppose your focus was not interpretation but prediction, and you wanted to introduce non-linearities in the model. | ||
| # Create square and cubed variables for age. (A cube is x^3.) | ||
| # Implement your approach using R and characterize your results as you think appropriate. | ||
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| SI_Sales = read_dta("SI Sales.dta") | ||
| SI_Sales$age2 = SI_Sales$age ^2 | ||
| SI_Sales$age3 = SI_Sales$age ^3 | ||
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| cor(SI_Sales) | ||
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| model = lm('price ~ unit_size + land_size + age + age2 + age3', data = SI_Sales) | ||
| stargazer(model, type="text", title="Model", single.row=TRUE, | ||
| ci=TRUE, ci.level=0.95, digits = 2) | ||
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| ## From our results we see that as time progresses, initially age will be part of the appreciation of the house. | ||
| ## But eventually the age of the house becomes a liability as Age2 and Age3 start to overtake Age and negatively affect the value of a house | ||
| ## But we should also note that Age2 Age3 have nonnegative values in their CI, | ||
| ## perhaps showing a chance that some homes will keep appreciating regardless of the house (e.g. value of land outstrips decay of the house) | ||
| ## We introduced unit-size and land_size since they're correlated to age | ||
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| # Question 4. | ||
| # Use the stock data from Question 2. | ||
| # Using the Autocorrelation Function (ACF) in the library "forecast", do you find that returns are autocorrelated? | ||
| # Calculate daily volatility of returns. | ||
| # Is volatility autocorrelated? | ||
| # Are these results consistent with the strong form of the EMH? | ||
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| getSymbols(c('GS'), from="2006-01-01", to="2017-12-31") | ||
| data = merge(as.zoo(GS$GS.Adjusted)) | ||
| names = c("GS") | ||
| colnames(data) = names | ||
| data.level = as.xts(data) ## Levels | ||
| data.returns = diff(log(data.level), lag=1) ## Log returns | ||
| data.returns = na.omit(data.returns) ## Dump missing values | ||
| data.returns$GSvol = sqrt(data.returns$GS ** 2) | ||
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| acf(data.returns$GS, lag.max=5, type=c("correlation"), | ||
| main="Autocorrelation of GS Returns") | ||
| ## For GS we don't see previous days affecting today's stock daily returns, so no autocorrelation for daily returns | ||
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| acf(data.returns$GSvol, lag.max=10, | ||
| type=c("correlation"), main="Autocorrelation of GS Volatility") | ||
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| ## GS stock Volatility shows that previous days do have an impact on today's stock volatility, so yes it's autocorrelated for volatility | ||
| ## But there's no autocorrelation when it comes to GS stock daily returns | ||
| ##These results are in line with the strong form of EMH | ||
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| # Question 5. | ||
| # Download quarterly US GDP data from FRED as far back as you are able to go. | ||
| # Graph the data. | ||
| # Using the HP Filter, decompose these data into cyclical and trend components. | ||
| # Graph your results. | ||
| # How would you interpret the results given economic history? | ||
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| gdp = fredr('GDP', observation_start = as.Date("1947-01-01")) | ||
| plot(gdp$date, gdp$value, pch=16, col="darkblue", main="GDP", | ||
| xlab="Date", ylab="GDP ($Billions") ## time series plot in R | ||
| lines(gdp$date, gdp$value, col="darkblue") | ||
| grid(lw=2) | ||
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| hp = hpfilter(gdp$value, freq=129600, type="lambda", drift=FALSE) | ||
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| plot(gdp$date, hp$trend, pch=16, col="blue", | ||
| xlab = "Date", ylab = "Trend GDP", | ||
| main="De-Seasonalized Trend") | ||
| lines(gdp$date, hp$trend, col="blue") | ||
| grid(lw=2) | ||
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| plot(gdp$date, hp$cycle, pch=16, col="blue", | ||
| xlab = "Date", ylab = "Seasonal GDP Changes", | ||
| main="Seasonal Cyclicality") | ||
| lines(gdp$date, hp$cycle, col="blue") | ||
| grid(lw=2) | ||
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| ## From my results the GDP trends up. | ||
| ## But its seasonality has recently been more wild, with huge swings in growth and steep declines | ||
| ## The results point to the fact that in this new millenium, steep economic collapse has been followed by strong economic growth | ||
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| # Extra Credit. | ||
| # Ingest the 2000 Census Data for the state of New Jersey. | ||
| # Using the naming conventions in class, rename the variables. | ||
| # Plot the longitude and latitude of the centroids in New Jersey. | ||
| # Are they consistent with the population clusters of New Jersey? | ||
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| url = "http://www2.census.gov/geo/docs/maps-data/data/gazetteer/census_tracts_list_34.txt" | ||
| data = read.csv(url, header=TRUE, sep='\t') | ||
| names = c('usps', 'geo', 'pop', 'hu', 'land', 'water', | ||
| 'landSqmi', 'waterSqmi', 'lat', 'long') | ||
| colnames(data) = names | ||
| plot(data$long, data$lat, pch=16, col="blue", | ||
| main="The Garden Stte by Census Centroid", xlab="Longitude", ylab="Latitude") | ||
| grid(lw=2) | ||
| ## Yes these pop clusters are consistent to what we would expect | ||
| ## namely population clusters near NYC and Philadelphia and between that corridor | ||
| ## plus the clusters along the beach like Atlantic City |