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Problemset Shrouded Fees

Authors: Tobias Krabel (main author) Sebastian Kranz (some modifications)

library(RTutor)
rps.dir = "D:/libraries/RTutorShroudedFees/RTutorShroudedFees/inst/ps" 
setwd(rps.dir)
ps.name = "Shrouded Fees"; sol.file = paste0(ps.name,"_sol.Rmd")
libs = c("ggplot2","ggthemes", "dplyr", "grid", "scales", "foreign", "sandwich", "texreg") # character vector of all packages you load in the problem set
#name.rmd.chunks(sol.file,only.empty.chunks=FALSE)

create.ps(sol.file=sol.file, ps.name=ps.name, user.name=NULL,libs=libs, extra.code.file = "The Impact of Shrouded Fees_extracode.r", var.txt.file = "shroudedvar.txt")


setwd("D:/libraries/RTutor/examples")
show.ps(user.name="John Doe", ps.name=ps.name, load.sav=FALSE, launch.browser=TRUE, sample.solution=TRUE, is.solved=TRUE, rps.dir=rps.dir)

Welcome to the interactive offline R tutorial about the AER paper "The Impact of Shrouded Fees: Evidence From a Natural Experiment in the Indian Mutual Funds Market" by Santosh Anagol and Hugh Hoikwang Kim. The paper studies how two policy changes in India affect the number of and investments in Indian funds.

In this problem set, you will gradually analyze the main points of the paper.

Exercise 1.1 -- Summary Statistics of Funds

To get first insights into the data, you will do summary statistics on the initial amounts of money new Indian mutual funds collected and on the number of new funds started.

a) Getting the data

In the directory of your problem set, you should have saved the file "fund_data". Use the command 'read.table()' to load it. Assign the result to a variable called fund. To proceed, press the 'check' button to check whether your solution is correct.

< info "read.table()"

'dat = read.table("tablename")' loads the data with the name "tablename" and assigns it to dat. It's important to use quotation marks!

>

fund = read.table("fund_data")
#< hint
display("fund = read.table() is the basis of the command. Now, the only thing you have to do is entering the name of the data you want to load. Don't forget to use quotation marks.")
#>

. continue

You can look at the data by pressing the 'data' button above. It contains information on Indian equity funds. Anagol and Kim focus on this type of fund as its major clientele consists of retail investors, our subjects of interest.

< info "Variables of Interest 1"

The variables we are mainly interested in right now are: - new_open_no = the number of new open-end funds started in month t. - new_closed_no = the number of new closed-end funds started in month t. - new_open_amt = the amount of money allocated by new open-end schemes in millions of 2009 U.S. dollars. - new_closed_amt = the amount of money collected by new closed-end schemes in millions of 2009 U.S. dollars.

>

< info "Closed- versus Open-End Funds"

Open-end funds typically allow investors to retrieve and invest money at any point of time. Closed-end funds are not allowed to allocate new money once they collected an initial corpus of money. Liquidity in these funds is slightly more restricted but closed-end fund schemes still offer regular redemption windows (many funds offer daily, weekly or monthly redemption).

> info

b) Summarizing the data

We want to create a summary statistics that shows the monthly average amounts of money the closed and open form of equity funds collects per year (in millions of 2009 U.S. dollars) and the number of new funds started on average per year.

< info "group_by() and summarise()"

group_by(dat, column1, column2, ...)

groups the data dat by all given columns. This is useful when we want to apply functions on dat "by" given columns.

summarize(dat, var1 = fun(col), var2 = ..., ...)

summarizes data "dat" by applying function "fun" on column "col". The result is stored in the new column var1. If you want to apply many functions and generate many columns, you have to separate the commands with a comma.

>

Using the function group_by, let fundg be the grouped fund data, grouped by year.

#< task
library(dplyr)
#>
fundg = group_by(fund, year)
#< hint
display("Use the command 'group_by()'. The first parameter you have to assign is the data whose columns you want to group. Behind the data, assign the name of the column you want to group by.")
#>

Now use summarise to create a summary funds of fundg with the following four columns (in this order): - Open.amt = a variable in which you store the mean of na.omit(new_open_amt), - Closed.amt = a variable in which you store the mean of na.omit(new_closed_amt), - Open.no = the mean of na.omit(new_open_no), and - Closed.no = the mean of na.omit(new_closed_no).

< info "na.omit()"

As there may be missing values (NAs) in these four columns and R cannot automatically ignore these NAs (an error message would be the result), we have to use the function 'na.omit()' to only take the values of each column which are not missing values.

>

funds = summarize(fundg, 
  Open.amt = mean(na.omit(new_open_amt)),
  Closed.amt = mean(na.omit(new_closed_amt)),
  Open.no = mean(na.omit(new_open_no)),
  Closed.no = mean(na.omit(new_closed_no))
)
#< hint
display("'funds = summarize(fundg, Open.amt = mean(na.omit(new_open_amt)), Closed.amt = mean(na.omit(new_closed_amt)), Open.no = mean(na.omit(new_open_no)), Closed.no = mean(na.omit(new_closed_no)))' is the solution.")
#>

With a natural experiment in the Indian funds market, Anagol and Kim study the impact of two law changes (in 2006 and 2008) on the number of new funds started and the money they collect. They want to find out if closed-end funds have exploited a law, that has forced them to shroud fees, by charging higher fees. Furthermore, they test whether closed-end funds became significantly more popular than open-end funds after that law change because investors did not realize the total amount of fees they have to pay to closed-end funds. It's important to note that open-end funds were forced to unshroud fees during the same period. Take a look at funds. Focusing on fund starts, we can see a pattern that suggests that closed-end funds thrived between 2006 and 2008 compared to the other periods, whereas open-end funds did not significantly became more popular. Moreover, it appears that closed-end funds also allocated more new money during that period compared to the periods before and after.

By the way: if you need more background information about the paper, read the info below.

< info "Background"

The shrouding hypothesis, which is tested in the paper, states that investors differently realize fees depending on the transparency ot the fee structure. If a fee is nontransparent, funds may charge higher fees and still collect more money compared to funds that use more transparent fees because investors are not cognizant of the hidden fee. In other words, funds simply exploit the trait of the nontransparent fee structure and collect more money at the expense of the consumers.

There were two law changes studied in the paper, one on April 4, 2006 (law change 1), the other one on January 31, 2008 (law change 2). The period before law change 1 is called Regime 1, Regime 2 is the period after law change 1 and before law change 2 and Regime 3 is the period after law change 2. Before law change 1, both open-end and closed-end funds could choose which fee they want to use (entry loads or initial issue expenses). Entry loads are fees that are paid in a lump sum fashion right at the beginning of an investment. Investors directly experience that money has been deducted from their investment when they see the first statement. In contrast, initial issue expenses are more shrouded than entry loads because they are deducted over time. That is, for example, if an investor has to pay 6 percent initial issue expenses and invests 100 rupees, 6 rupees are deducted from the initial investment over time. If the fund exists for 3 years, which equals 365*3 = 1,095 days, 6/1,095 = 0.0055 rupees are deducted per day. This suggests that an investor may not distinguish "usual" changes in the net asset value from a reduction of the value of the investment due to the fee. Law change 1: The Indian financial market regulator decided that open-end funds were only allowed to charge entry loads, whereas closed-end funds were only allowed to charge initial issue expenses. This is an interesting policy change as it forces one specific fund form to charge less salient fees. Law change 2: Now, both fund organizations had to charge entry loads.

Under the shrouding hypothesis, we expect closed-end funds to exploit the first law change, to charge higher fees and to collect more money. As funds get higher revenues from fees when more money is invested, our intuition suggestes that more money collected should result in more closed-end funds being started in Regime 2 compared to Regime 1. In Regime 3, we expect the proportion of closed-end funds to revert to levels of Regime 1 as closed-end fund are now forced to use the less shrouded entry loads and thus have to compete with open-end funds.

> info

Exercise 1.2 -- Figures of fund starts and inflows

Numbers are nice, plots are nicer! To visualize the natural experiment Anagol and Kim were studying, an illustrative figure is convenient.

We will check if we can see an effect that matches the predictions under the shrouding hypothesis when we plot fund starts and net flows into funds (we plot net flows to take the inflows to and outflows of existing funds into account).

< info "Variables of Interest 2"

The net flow variables of interest are: - net_flow_open_amt = the net flows to open-end funds calculated as the flows into new and existing open-end funds minus outflows of existing funds = new_open_amt + ex_open_amt - open_red. - net_flow_closed_amt = the net flows to closed-end funds calculated as the flows into new and existing closed-end funds minus outflows of existing funds = new_closed_amt + ex_closed_amt - closed_red.

> info

Data preparation

#< task
# Load fund data
fund = read.table("fund_data")
# Be careful! If you load a data frame, first check if its date variable has still a date class! In our case, R loaded the data frame but changed the date variable's class to 'factor'. As nobody likes factors in this problem set, we have to convert it back to the class 'date' (this is necessary for the plot)
fund$date = as.Date(fund$date)

# We generate the exact integer values of the two law changes' dates to draw two vertical dashed lines. 'as.Date()' converts the string into a date variable, which is in turn necessary for R to convert it into a numeric value
april.2006 = as.numeric(as.Date("2006-04-04")) # Beginning of Regime 2
feb.2008 = as.numeric(as.Date("2008-02-01"))   # Beginning of Regime 3
#>

a) Open-end equity fund starts

The following code generates a plot of open-end equity fund starts, stores it in p2.1 and shows p2.1. I use the function 'qplot()' in the package "ggplot2". Run the code below.

#< task_notest
# load package
library(ggplot2)

# Plot open-end fund starts
p2.1 = qplot(data = fund, x = date, y = new_open_no, geom = "bar", stat = "identity") +                                                 
  geom_vline(aes(xintercept = april.2006), color = "#BB0000", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "#BB0000", linetype = "dashed", size = 1)

# 'geom = "bar"' calls a bar plot
# By default, when used a bar plot, qplot thinks that we want to plot a histogram and counts the values defined in x. 'stat = "identity"' ensures that qplot really displays the numbers in y as bars
# ggplot2 facilitates adding commands by just using the + symbol. In this case, we add vertical lines to illustrate the two law changes.

# Show p2.1
p2.1
#> task

b) Now, use the qplot function from above to plot:

  1. new_closed_no (assign it to p2.2)

  2. net_flow_open_amt (assign it to p2.3)

  3. net_flow_closed_amt (assign it to p2.4)

After you have generated the plots, show all three plots (in the same order you have generated them).

#< task_notest
# Plot closed-end equity fund starts

# Plot open-end equity net flows

# Plot closed-end equity net flows

# Show all plots

#> task
p2.2 = qplot(data = fund, x = date, y = new_closed_no, geom = "bar", stat = "identity") +                                                 
  geom_vline(aes(xintercept = april.2006), color = "#BB0000", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "#BB0000", linetype = "dashed", size = 1)
#< hint
display("Just take the code from above and change the values for y. Don't forget to show the plots.")
#>
p2.3 = qplot(data = fund, x = date, y = net_flow_open_amt, geom = "bar", stat = "identity")+ 
  geom_vline(aes(xintercept = april.2006), color = "#BB0000", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "#BB0000", linetype = "dashed", size = 1) 

p2.4 = qplot(data = fund, x = date, y = net_flow_closed_amt, geom = "bar", stat = "identity") + 
  geom_vline(aes(xintercept = april.2006), color = "#BB0000", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "#BB0000", linetype = "dashed", size = 1)

p2.2
p2.3
p2.4

If you want to display all four plots in one window, you can use the following function. Simply uncomment the code and run the chunk.

#< task_notest
# Show one figure containing all four plots.
multiplot(p2.1, p2.2, p2.3, p2.4, cols = 2)
#> task

c) Look at the different plots to answer the following questions (replace '???' with the right answer):

#< task_notest
# Do you see a particular effect in open-end fund starts and net flows during Regime 2 compared to the other periods??
# sol31 = "???" # Assign "yes" or "no" to sol31 and remove the comment

# How about closed_end fund starts and net flows? Do you see an effect there? 
# sol32 = "???" # Assign "yes" or "no" to sol32 and remove the comment 
#> task

sol31 = "no"
sol32 = "yes"

We observed that new closed-end equity funds have collected more money during Regime 2 compared to Regimes 1 and 3. Furthermore, it seems that closed-end equity funds became popular during Regime 2 but lost some of their appeal in Regime 3. With respect to open-end funds, it appears that there is no remarkable effect on the two outcome variables (starts an net flows).

Exercise 2.1 -- Fee structure

We answered two questions with respect to the shrouding hypothesis: 1. Did closed-end funds become more popular in Regime 2 in comparison with Regimes 1 and 3? 2. Did closed-end funds collect more money in Regime 2 in comparison with Regimes 1 and 3?

Yet, one question is unanswered: Did closed-end funds really charge higher fees than open-end funds during Regime 2?

a) Load "fee_data" and assign it to fee.

fee = read.table("fee_data")

The data contains all equity funds started after law change 1.

< info "Variables of Interest 3"

> info

Remember that only closed-end funds were forced to charge initial issue expenses during Regime 2, whereas open-end funds were forced to charge entry loads.

b) Our goal is to compare the percentage entry loads and initial issue expenses for open and closed funds.

First, use the function filter from the packge dplyr to generate two subsets from fees:

  1. Let open be the subset of fee containing all open fund forms (fund_org == "Open").

  2. Let closed be the subset of fee containing all closed fund forms (fund_org == "Closed").

< info "filter()"

'filter(dat, condition)' filters the data "dat" using a condition with boolean operators, e.g. 'filter(fund, month_year > 555)'

>

#< task_notest
library(dplyr)
fee$date = as.Date(fee$date)

# open = ... enter your code here
# closed = ... enter your code here
#> task
open = filter(fee, fund_org == "Open")
#< add_to_hint
display("Make sure you correctly spell the data frames and the columns. The command 'filter' takes two parameters. 1. the data frame you want to filter and 2. the condition for a row to be in the filtered data. You can copy the condition from above.")
#>
closed = filter(fee, fund_org == "Closed")
#< add_to_hint
display("Make sure you correctly spell the data frames and the columns. The command 'filter' takes two parameters. 1. the data frame you want to filter and 2. the condition for a row to be in the filtered data. You can copy the condition from above.")
#>

You can take a look at fee, open or closed in the data explorer before moving on to the next exercise where we draw some plots.

c) illustrate fees with basic plot functions

In the remainder of this exercise, you sequentially learn more of the basic plot commands in order to compare the fee structure of open and closed funds.

For start, use the basic plot(...) function to plot the entry loads charges by open-end funds. Let x be the opening date and y be the entry loads charged.

plot(x = open$date, y = open$entry_load_nfo)

Hint: You can type colnames(open) to see all column names, and access a column of a data.frame with the $ operator, e.g. open$fund_name.

d) Let us adapt the plot, by starting the y-Axis from 0 and adding nicer labels. Take the plot command from c) and add the following arguments: - xlab="Date" - ylab="fees in %" - ylim=c(0,6.5)

plot(x = open$date, y = open$entry_load_nfo, xlab="Date",ylab="fees in %",ylim=c(0,6.5))

e) Let us add to the plot from d) the initial issue expenses charged by closed-end funds. First copy the plot command. Then use the command 'points()' which has the same syntax as 'plot()'. Let x be the opening date and y be the initial issue expenses charged. Use the data frame closed. Don't forget to call the subset in the parameters x and y! AlsoAdd the parameter col = "red" to the call to make these points red.

plot(x = open$date, y = open$entry_load_nfo, xlab="Date",ylab="fees in %",ylim=c(0,6.5)) 
#< add_to_hint
display("Just copy the plot function from above.")
#>
points(x = closed$date, y = closed$actual_iie, col = "red")
#< add_to_hint
display("Don't forget to call the subset `closed` in the parameters x and y!")
#>

We observe that closed-end funds generally charged higher fees than open-end funds.

f) Finally, add the two vertical lines indicating the two law changes. Use the command abline(v=...) and replace the "..." with the two date variables april.2006 and feb.2008.

#< task
april.2006 = as.numeric(as.Date("2006-04-04")) # Beginning of Regime 2
feb.2008 = as.numeric(as.Date("2008-02-01"))   # Beginning of Regime 3
#>
#< task_notest
# Copy the plot and points command from above
# First, draw the first law change in April 4, 2006.
# abline(v=...)
# Then, draw the second law change on January 31, 2008.
# abline(v=...)
#> task

plot(x = open$date, y = open$entry_load_nfo, xlab="Date",ylab="fees in %",ylim=c(0,6.5)) 
points(x = closed$date, y = closed$actual_iie, col = "red")
abline(v=april.2006)
abline(v=feb.2008)

g) The code below generates a similar plot using the ggplot2 functions.

First we add to the data frame fee generate a column fee that contains the total fee charged by the fund.

#< task
# Helper function to convert NA to some other value
convert.na =function(x,new.val) {
  x[is.na(x)] = new.val
  x
}
# Compute total fee
fee$total_fee = convert.na(fee$entry_load_nfo,0)+convert.na(fee$actual_iie,0)
#>

Then we generate a plot in which fees of closed and open fund are differentiated by color and shape. We use the package 'ggthemes' to show the plot in a Stata theme.

#< task_notest
library(ggthemes)

# Show plot of fees of open and closed funds use a Stata theme
qplot(data=fee,x=date,y=total_fee, color=fund_org,shape=fund_org,
      size=I(3.5), geom="point",ylab="fee (%)",
      main="Fees of closed funds (iie) vs open funds (entry load)") +
  scale_shape_manual(values=c(19,1)) +
  geom_vline(aes(xintercept = april.2006), color = "black", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "black", linetype = "dashed", size = 1) +
  theme_stata()
#>

We see that in Regime 2 closed funds charge substantial higher fees than open funds.

Unfortunately, our current data set has almost no fee observations for closed funds outside Regime 2. Why???

< info "plot with separated fund type and fee type"

The code below creates a plot that codes the fee type by color and the fund type by shape. If you run it, you will see that we don't have any observations in which closed funds charge entry loads, however.

library(tidyr)
# create separate rows for iie fees and entry_load fees 
df = gather(fee, key=fee_type, value=fee_val, actual_iie, entry_load_nfo)

# set fees to NA if it is zero while total fee is positive
df$fee_val[is.true(df$fee_val==0 & df$total_fee >0)] = NA

# show plot
qplot(data=df,x=date,y=fee_val, color=fee_type,shape=fund_org,
      size=I(3.5), geom="point",ylab="fee (%)",
      main="Fees of closed funds (iie) vs open funds (entry load)") +
  scale_shape_manual(values=c(19,1)) +
  geom_vline(aes(xintercept = april.2006), color = "black", linetype = "dashed", size = 1) +
  geom_vline(aes(xintercept = feb.2008), color = "black", linetype = "dashed", size = 1)

>

While the syntax for ggplot takes some time to learn, it is extremely powerful in creating useful plots.

Economic insight:

Our observations suggest that in Regime 2, closed-end equity funds may have become popular because they could hide fees by using the more shrouded fee structure. In Regime 3, when they were forced to reframe their fee structure from initial issue expenses to less shrouded entry loads, closed-end equity funds may have known that investors will not be willing to pay higher entry loads and stopped creating new funds. From the investors' perspective, they might not offer some additional benefit that justified the extra expense.

The illustrative power of plots provided further insights into the topic. Yet, for a scientific research study, we need more than simple plots to further support our theory. That's when regressions become important.

Exercise 3.1 -- Regression Analysis for Equity Fund Starts

Anagol and Kim statistically analyze the following questions:

  1. Is the proportion of closed-end equity fund starts in Regime 2 significantly higher than in Regime 1?

  2. Did closed-end equity funds collect significantly more money than open-end funds in Regime 2 compared to Regime 1?

Anagol and Kim use a regression based difference-in-difference approach to identify the effect of law change 1 and law change 2. That is, they test whether the difference in the outcome variable between the closed and open form significantly changed from Regime 1 to Regime 2.

fund.long contains all the information needed on equity funds (type_fund == 2) to run the regression on equity fund starts and net flows. We will focus on fund starts, but feel free to regress net flows in an extra R script.

#< task_notest
fund.long = read.table("regression_data")
#> task

Take a look at the table. We will now investigate the determinants of the number of newly started funds (for each type and month) starts using the following linear regression: 

starts = beta0 + beta1*closed_regime2 + beta2*closed_regime3 + beta3*regime2 + beta4*regime3 + beta5*closed + beta6*sensex1_return + beta7   *sensex3_return + beta8*time_trend + epsilon.

The relevant variables used in the regression are explained below.

< info "variables in regression"

The regression specification used is:

$$starts_{it}=\beta_{0}+\beta_{1}closedregime2_{it}+\beta_{2}closedregime3_{it}+\beta_{3}regime2_{it}+\beta_{4}regime3_{it}+\beta_{5}closed_{i}+\beta_{6}sensex1return_{it}+\beta_{7}sensex3return_{it}+\beta_{8}timetrend_t+\epsilon_{it}.$$

where: - starts = the observed number of funds started for type of fund $i\in{open,\, closed}$ in month x year t. - time_trend = a time trend variable that starts with 0 for our first observation, increasing by one for each subsequent month_year. - sensex1_return = the one month lagged return of the BSE Sensex index in month_year t. That is, e.g., the one month lagged return in April 2006 is the actual return observed one month earlier, in March 2006. - sensex3_return = the three month lagged return of the BSE Sensex index in month_year t. That is, e.g., the three month lagged return in April 2006 is the actual return observed three months earlier, in January 2006. - regime2 = a dummy variable that is equal to 1 if the observation is from Regime 2. - regime3 = the Regime 3 dummy. - closed = a closed dummy that equals 1 if the respective fund is a closed-end fund. - closed_regime2 = a so-called interaction between closed and regime2, that only equals 1 if both closed and regime2 are equal to 1. - closed_regime3 = the interaction between the closed and the Regime 3 dummy variables.

> info

We want to regress starts for different sample sizes. One regression shall comprise all months, the other regression shall only comprise the 22 months before Regime 2, the 22 months in Regime 2 and all 20 months available after Regime 3. We do this to see if the regression results improve if we have better comparison groups for Regime 2 (perhaps, some early months in Regime 1 bias our results).

#< task_notest
library(dplyr)
# restricted sample
fund.short = filter(fund.long, month_year >= 533 & month_year <= 598)
#> task

We will collect all regression results and store them in different variables. After that, we will display all results in one table and take a look at it.

a) First, we rund a simple OLS regression on fund starts

Let's try our first regression.

#< task_notest
# OLS Equity fund starts with 120 months sample
reg1 = lm(starts ~ time_trend + 
              sensex1_return + sensex3_return + 
              regime2 + regime3 + 
              closed + 
              closed_regime2 + closed_regime3,
              data = fund.long)
# Show a summary of reg1
summary(reg1)
#> task

Change the code from above to regress equity fund starts with the restricted sample. Store the regression in reg2.

#< task_notest
# OLS Equity fund starts with 64 months sample
# enter your regression here ...
#> task
reg2 = lm(starts ~ time_trend + 
              sensex1_return + sensex3_return + 
              regime2 + regime3 + 
              closed + 
              closed_regime2 + closed_regime3,
              data = fund.short)
#< add_to_hint
display("You just have to change the 'data' parameter to run the regression on the resticted sample fund.short.")
#>

To nicely compare the regression results for the two data sets, we can use the function showreg that is included in RTutor. (It is just a mapper to the useful functions in the package texreg). It shows regressions in a similar format as you see in published articles.

#< task_notest
showreg(list(reg1, reg2),custom.model.names = c("all months","restricted sample"))
#>

We could also show the results as an HTML table:

#< task_notest
showreg(list(reg1, reg2),custom.model.names = c("all months","restricted sample"), output="html", star.symbol="*")
#>

also Latex output is possible. Yet, we will stick with the standard text output in in this problem set.

How can we interpret the different regression coefficients?

In other words, closed_regime2 measures the difference of the dependent variable between closed and open-end funds in Regime 2 minus the difference between closed and open-end funds in Regime 1.

closed_regime2 and closed_regime3 are the coefficients we are mostly interested in, as they show how closed-end funds react on law changes 1 and 2. The main prediction is that the coefficient on closed_regime2 is greater than 0. closed_regime3 should be at least not be significantly greater than closed_regime2, as we expect a decline in fund starts in Regime 3.

The following info box provides you with more details on why the coefficient for the closed_regime2 dummy can be interpreted as a diff-in-diff effect.

< info "Diff-in-diff"

This info will discuss the interpretation of our diff-in-diff estimators.

First of all, it's important to note that we use both regime dummies and a time trend variable in this regression. As a concequence, the regime variables measure the average fund starts during the respective regime, conditional on the time trend effect in that particular regime. In other words, imagine a constant cX for each regime X that is somehow correlated with the time trend. The coefficients on the regimes measure the average fund starts in the regimes minus the respective constant (E.g., regime2 = average open funds in R2 - c2).

As there is no Regime 1 dummy explicitly included in the regression, Regime 1 is the baseline scenario. If both regime2 and regime3 are equal to 0, the observation starts has to be from Regime 1. The constant beta0 measures the average value of the outcome variable conditional on the time trend, if all (dummy) variables are 0 (Imagine that beta0 belongs to a dummy variable that is always equal to 1). Since Regime 1 is the baseline scenario and the closed-end dummy is also equal to zero, beta0 measures the average number of open-end fund starts during Regime 1, conditional on the time-trend. That is, the coefficients on the constant is the average number of open-end fund starts during Regime 1 minus a time trend constant c1. We can further infer that beta5, the coefficient on the closed dummy, measures the additional effect of being a closed-end fund in Regime 1. If closed turns to 1, the number of fund starts increases by beta5, as the effect of open-end fund starts is already included in the constant beta0. In other words, beta5 is the slope of the closed-end dummy variable, or the difference between the average closed-end fund starts and the average open-end fund starts in Regime 1. This value is not conditional on the time trend, as it is a difference between values within a regime, that depend on the same time-trend effect, or constant c1. By taking the difference between two values that have the same constant, this constant cancels out. The coefficient on the Regime 2 dummy variable beta3 measures the effect of the Regime 2 dummy compared to the constant beta0, that is, the number of open-end funds started in Regime 2 minus the number of open-end funds started in Regime 1, conditional on the time trend. Finally, if the Closed x Regime 2 dummy variable equals one, both the Regime 2 dummy and the closed-end dummy variables are equal to 1. From this follows that beta1 measures the additional effect of being both closed and in Regime 2, or in other words, the effect of being closed-end instead of open-end in Regime 2 compared to the effect of being closed-end instead of open-end in Regime 1 (difference-in-difference). The coefficient on the Closed x Regime 3 dummy variable beta2 follows an equivalent reasoning.

> info

Are the results significant? What is the qualitative relationship between the covariates and the outcome?

Answer the following questions by assigning the right answer to the variables and uncommenting the answers:

#< task_notest
# Take a look at the regression result that uses all months. 

# 1. Compared to Regime 1, does the average number of open-end equity fund starts increase or decrease in Regimes 2 and 3?
# a31 = "???" # add "increase" oder "decrease" here ...

# 2. Is the coefficient on closed_regime2 significant?
# a32 = "???" # add "yes" or "no" here ...

# 3. Does the proportion of closed-end funds diminish after Regime 2? 
# a33 = "???" # add "yes" or "no" here ...
# QUESTION DOES NOT SEEM TO MAKE SENSE
# WHY NOT ASK ABOUT QUANTITATIVE EFFECTS?
#> task
a31 = "decrease"
#< hint
display("Take a look at the regime2 and regime3 dummies.")
#>
a32 = "yes"
#a33 = "yes"
#< hint
display("Take a look at the estimate of closed_regime3. As its value is lower than the estimate of closed_regime2, we can infer that ...?")
#>

b) Correcting standard errors

In an evaluation study, the assumption that the variance of an error term is constant might be incorrect. Sometimes, it is not realistic to suppose that the observations are always influenced by the same error. If the variance of the error term changes throughout the sample, we speak of heteroskedasticity. We would like to correct the standard errors because we are reluctant to celebrate significant coefficients whithout at least using heteroskedasticity robust standard errors.

There is an econometrics haiku that funnily explains the effect of robust standard errors (see Angrist and Pischke (2008), pp. 6 and 7).

     T-stat looks too good.
     Use robust standard errors-
     significance gone.

R does not automatically adjust its standard errors for heteroskedasticity. You can use the argument robust of showreg to show robust standard errors, the argument robust.type specifies the type of robust standard error. For an overview about the types of robust standard errors, you can start with the help of vcovHC in the package 'sandwich' or read Hayes and Cai (2007).

#< task_notest
showreg(
  list(reg1, reg1, reg2, reg2), 
  stars = c(0.001,0.01,0.05,0.1),                                                                 
  custom.model.names = c('All months','All months robust', 'Restricted', 'Restricted robust'),
  robust=c(FALSE,TRUE,FALSE,TRUE), type="HC3"
)
#>

You should see that heteroskedasticity does not affect the values of the estimates. Allowing the variance to be variable just changes the standard errors' values. In our case, they increase for the closed_regime2 coefficient.

c) Poisson regression

If the dependent variable Y is a count variable (a variable which takes on non-negative integer values) we can use a Poisson regression model assuming that the dependent variable Y has a Poisson distribution (see Wooldridge (2006)). In case of fund starts, it seems to be plausible that this variable is a count. We can apply a Poisson regression using the 'glm()' command and adding 'family = "Poisson"'.

#< task_notest
# Column 3 of Table 3
reg3 = glm(starts ~ time_trend + 
               sensex1_return + sensex3_return + 
               regime2 + regime3 + 
               closed + 
               closed_regime2 + closed_regime3,
               data = fund.long,
               family = "poisson")

# show the detailed results 
summary(reg3)
#> task

Let reg4 be the result of the Poisson regression model above for the limited sample infund.short.

#< task_notest
# Column 4 of Table 3
#>
reg4 = glm(starts ~ time_trend + 
               sensex1_return + sensex3_return + 
               regime2 + regime3 + 
               closed + 
               closed_regime2 + closed_regime3,
               data = fund.short,
               family = "poisson")

Note: Using a Poisson model changes the interpretation of the coefficients. The numbers in the regression tables are the percentage changes when multiplied with 100. That is, e.g., the coefficient on the Closed x Regime 2 dummy constitutes a 437 percent change of the difference between closed-end and open-end fund starts.

d) Comparing the 4 regressions

Now use the function showreg to show all 4 regressions reg1-reg4. Use robust standard errors for all 4 regressions of type HC3. Pick well explaining custom.model.names.

showreg(list(reg1,reg2,reg3,reg4),
        stars = c(0.001,0.01,0.05,0.1),                                                                 
        custom.model.names = c('OLS (all)','OLS (subset)','Poisson (all)','Poisson (subset)'),
        robust=c(TRUE), type="HC3"
        )
#< test_arg
ignore.arg = c("stars","custom.model.names")
allow.extra.arg = TRUE
#>

The effect of closed_regime2 is significant and positive, just as we expected. That means, that significantly more closed-end than open-end funds started during Regime 2. This effect increases with a more restricted sample. The effect of closed_regime3 is at least not significantly positive for the full sample, so that it seems plausible that the proportion of close-end funds decreased after Regime 2. The latter effect suggests that investors did not invest in closed-end funds with generally higher fees because they received some special unobservable benefit. It's more plausible that the fee structure (more shrouded initial issue expenses versus less shrouded entry loads) was decisive. In Regime 2, closed-end funds exploited the fact that investors weren't cognizant of the higher initial issue expenses they were paying. In Regime 3, closed-end funds stopped creating new funds as they realized that investors will not be willing to pay higher entry loads.

Exercise 3.2 -- Exploring the Difference of the Difference

In Exercise 3.1, I already explained the interpretation of the regression coefficients. Now, it is time to show you that some of the coefficients are in fact independent of the time trend.

Again, take a look at fund.long. To show you the interpretation of the coefficients on the closed dummy, the closed_regime2 dummy and the closed_regime 3 dummy, we will first calculate the average number of funds started for each regime and each fund form.

First, I will generate a column regime1 that is equal to one if both regime2 and regime3 are 0, using the command 'mutate()'. I assign the preliminary result to fund.long1

#< settings
import.var = list("3.1"=c("fund.long","reg1"))
#> settings
#< task_notest
library(dplyr)
fund.long1 = mutate(fund.long, regime1 = !regime2 & !regime3)
#> task

a) It´s your turn! Generate a new column regime in fund.long1 with the command 'mutate()'. It shall be calculated as follows: regime = 1regime1 + 2regime2 + 3*regime3 Store the result in fund.long2

#< task_notest
# fund.long2 = mutate()
#> task
fund.long2 = mutate(fund.long1, regime = 1*regime1 + 2*regime2 + 3*regime3)
#< hint
display("The syntax of the mutate command is similar to the command I used as an example. Don't forget to use `fund.long1`.")
#>

b) The following code generates a new column fund_form in fund.long2 that is equal to "closed" if the fund is a closed-end fund, and "open" if the fund is an open-end fund. Uncomment the code and run it.

#< task_notest
#fund.long3 = mutate(fund.long2, fund_form = ifelse(closed, "closed", "open"))
#> task
fund.long3 = mutate(fund.long2, fund_form = ifelse(closed, "closed", "open"))

c) Now, let's calculate the averages of starts by fund_form and regime. We use our two old friends: 1. Group fund.long3 by fund_form and regime (in this order). Let the result be fund.longg 2. Summarize fund.longg to fund.longs. Gererate one column average_starts = mean(starts). (Note: Ignore the "*". I use it to highlight the variables.) Show fund.longs afterwards

#< task_notest
# 1. group fund.long3
# ...
# 2. summarize fund.longg
# ...
#> task
fund.longg = group_by(fund.long3, fund_form, regime)
fund.longs = summarize(fund.longg, average_starts = mean(starts))

d) Get a piece of paper and a pencil, it's time for mathematics :-). Here are the coefficients of regression 1.

#< task_notest
showreg(list(reg1), single.row=TRUE)
#> task

We focus on the coefficients on closed. closed_regime2 and closed_regime3. I use the numbers from fund.longs to illustrate that these coefficients really measure differences(-in-differences). - coef on closed = avg closed starts in regime1 - avg open starts in regime1 = 0.03 - 1.81 = - 1.78 - coef on closed_regime2 = (avg closed starts in regime2 - avg open starts in regime2) - (avg closed starts in regime1 - avg open starts in regime1) = (2.05 - 1.82) - (0.03 - 1.81) = 0.23 + 1.78 = 2.01 - coef on closed_regime3 = (avg closed starts in regime3 - avg open starts in regime3) - (avg closed starts in regime1 - avg open starts in regime1) = (0.00 - 2.15) - (0.03 - 1.81) = - 2.15 + 1.78 = - 0.37

Exercise 4 -- Alternative Argument - Performance Comparison

Looking at the results of our analysis, we could say: "The results are consistent with the shrouding argument". However, "being consistent" does not automatically mean "proving". Suppose another scientist argues: "Closed-end equity funds might become more popular during Regime 2 because they simply yielded higher returns than open-end funds that overcompensated the higher fees they charged". Since this is a plausible argument, we have to defend our own theory by showing that this argument cannot explain the popularity of closed-end funds during Regime 2.

a) We will check whether closed-end equity funds outperformed open-end equity funds. Therefore, we load the table "3-Factor_data" and assign it to factor.dat. The data contains information on the performance of closed and open-end equity funds started in Regime 2.

#< task_notest
# Get the relevant data
factor.dat = read.table("3-Factor_data")
factor.dat$date = as.Date(factor.dat$date)

library(dplyr)
factor.dat = select(factor.dat, date, open_monthly_raw_return,
    closed_monthly_raw_return, mean_sensex_monthly, monthly_bill,
    closed_excess, open_excess, market_excess, cmo_return,  hml, smb)
#> task

b) Let's do some summary statistics first to make ourselves familiar with the data. Summarize factor.dat to factors. Generate the following variables: - open_raw = the average of open_monthly_raw_return, - closed_raw = the average of closed_monthly_raw_return, - open_marketadj = the average of open_monthly_raw_return - mean_sensex_monthly, and - closed_marketadj = the average of closed_monthly_raw_return - mean_sensex_monthly.

< info "Variables of Interest 4"

> info

factors = summarize(factor.dat, open_raw = mean(open_monthly_raw_return),
                    closed_raw = mean(closed_monthly_raw_return),
                    open_marketadj = mean(open_monthly_raw_return - mean_sensex_monthly),
                    closed_marketadj = mean(closed_monthly_raw_return - mean_sensex_monthly))
#< hint
display("You can directly apply the mean function on the difference of two columns. E.g., mean(col1-col2) calculates the average difference between col1 and col2. open_marketadj = mean(open_monthly_raw_return - mean_sensex_monthly) is one part of the solution.")
#>

x_marketadj is the average return of a fund form controlled for marked movements. Take a look at the result. According to the summary, does it seem plausible that closed-end funds outperformed open-end funds during Regime 2?

#< task_notest
# a41 = "???" # add "yes" or "no"
#> task
a41 = "no"

We don't have a t-value that enables us to assess whether the differences are significant. One option is to use a t-test, yet, we are running a regression to test for significance.

c) We will use both the 1-factor CAPM and the 3-factor model by Fama and French (1993) to regress cmo_return (see below) and store the results in reg1f and reg3f, respectively.

< info "Regression on Returns"

What does cmo_return mean? - cmo_return = closed-excess minus open-excess, used as the performance of closed-end funds compared to open-end funds. - closed_excess = close-end funds' excess return calculated as raw return minus risk-free rate (monthly_bill). - open_excess = open-end funds' excess return calculated as the raw return minus risk-free rate.

"Why using these models? As we know, a common assumption in asset pricing is that each asset is somehow correlated with the market and has consequently a market related return (see Fama and French (1993)). However, we want to show that the closed-end funds' idiosyncratic return during Regime 2 are not significantly higher than the open-end funds' idiosyncratic return. To find the idiosyncratic return of an asset, you can use the 1-factor (CAPM) or 3-factor model.

1-factor and 3-factor model The 1-factor model is based on the CAPM. The regression specification is excess_return_i = alpha + beta * market_excess_return with the idiosyncratic average excess return alpha.

Fama and French (1993) extended the 1-factor model to a 3-factor model to increase its explanatory power. Besides an overall market factor, they use two company related factors: one factor related to firm size and one factor related to book-to-market equity. To calculate these factors, Anagol and Kim used non-financial firms listed on the Indian Bombay Stock Exchange. The regression specification is excess_return_i = alpha + beta1 * market_excess_return + beta2 * SMB + beta3 * HML with - SMB (Small minus big): the average of the value-weighted return on big firms (market equity greater than the median market equity) minus the value-weighted return on small firms (market equity smaller than or equal to the median market equity) for each month. - HML (high minus low): the average of the value-weighted return on "high" firms (upper 30% book-to-market equity) minus the value-weighted return on "low" firms (lower 30% book-to-market equity) for each month.

SMB: For each year t, we calculate the median market equity (med(ME), based on the market equity at the end of January at year t) of all non-financial firms and the 30% quantile (q30) and the 70% quantile (q70) of the proportion of book equity to market equity (BEME, based on book equity and market equity at year t-1) of all non-financial firms. After that, we categorize each firm according to its relative position in January. In year t, it is "big", if its market equity (ME) is greater than med(ME), and "small", if its ME is smaller than or equal to med(ME). In addition, in year t, it is "high", "medium" or "low" if its BEME is greater than q70, between q30 and q70 or lower than q30, respectively.

According to the 2 ME and 3 BEME categories, we can form 6 different portfolios: - big firms with high BEME (B/H), - big firms with medium BEME (B/M), - big firms with low BEME (B/L), - small firms with high BEME (S/H), - small firms with medium BEME (S/M) and - small firms with low BEME (S/L).

For each portfolio, we calculate the monthly value-weighted returns (weighted according to ME). If we want to calculate the SMB factor, we now deduct the monthly return of each of the big portfolios from the return of the respective small portfolios and compute the average of this differences. So SMB is calculated as 1/3 * [ (S/H - B/H) + (S/M - B/M) + (S/L - B/L) ].

HML: We deduct the monthly return of the low portfolios (only low big and low small) from the monhtly return of the high portfolios (only high big and high small) and compute the average of this differences. So HML is calculated as 1/2 * [ (B/H - B/L) + (S/H - S/L) ].

Note: Although firms are categorized on an annually basis, SMB and HML contain monthly returns.

> info

#< task_notest
reg1f = lm(data = factor.dat, cmo_return ~ market_excess)

reg3f = lm(data = factor.dat, cmo_return ~ market_excess + hml + smb)

library(texreg)
showreg(list(reg1f, reg3f), digits = 4,
          stars = c(0.001,0.01,0.05,0.1),
          custom.model.names = c("1-Factor Closed minus Open (1)", 
                                  "3-Factor Closed minus Open (2)"),
          custom.coef.names = c("Alpha", "Market Excess Return", "HML", "SMB"),
          robust = TRUE, type = "HC1"
)
#> task

Take a look at Alpha, the idiosyncratic return of closed-end funds compared to open-end funds. We see that Alpha is significantly smaller than zero, which means that closed-end funds yielded lower returns than there open-end competitors.

As a consequence, we can reject the argument that closed-end funds charged higher fees because they offered higher returns.

#< task_notest
"End"
# Enter a space somewhere and run RTutor to finish this exercise
#> task
"End"

Exercise 5 -- Recapitulation

Congratulations! You have finished the problem set! I hope you gained a lot of new insights into the impact of law changes on funds' and investors' decision making.

One last excercise! Recapitulate the problem set. Assign the boolean operator TRUE to an argument, if it is true, and FALSE, if it is false. Example: a0 = "This problem set was fun" --> a0 = TRUE (Note: This time, do NOT use quotation marks.)

#< task_notest
# a1 = # "In Regime 2, Closed-end funds charged amortized fees that were lower than the up-front fees charged by open-end funds."

# a2 = # "In Regime 2, a significantly higher proportion of closed-end equity funds started compared to Regime 1. During Regime 3, this proportion declined to levels of Regime 1."

# a3 = # "Allowing heteroskedasticity changes the estimates' values"

# a4 = # "During Regime 2, open-end funds offered lower returns than closed-end funds"
#> task
a1 = FALSE
#< hint
display("Ok, this is the last excercise. I will give you the solution as a gift. a1 = FALSE; a2 = TRUE; a3 = FALSE; a4 = FALSE. Thank you very much for your interest. :-)")
#>
a2 = TRUE
#< hint
display("Ok, this is the last excercise. I will give you the solution as a gift. a1 = FALSE; a2 = TRUE; a3 = FALSE; a4 = FALSE. Thank you very much for your interest. :-)")
#>
a3 = FALSE
#< hint
display("Ok, this is the last excercise. I will give you the solution as a gift. a1 = FALSE; a2 = TRUE; a3 = FALSE; a4 = FALSE. Thank you very much for your interest. :-)")
#>
a4 = FALSE
#< hint
display("Ok, this is the last excercise. I will give you the solution as a gift. a1 = FALSE; a2 = TRUE; a3 = FALSE; a4 = FALSE. Thank you very much for your interest. :-)")
#>

References

Author: Tobias Markus Krabel
License: This problem set is licensed under CC-BY 4.0 International.



skranz/RTutorShroudedFees documentation built on May 30, 2019, 2:02 a.m.

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