9. Regression, Multivariate, and Nonlinear Methods

Question 1

1. What are the two distinguishing characteristics that make a regression a simple linear regression?

Answer 1

• One independent variable

* The relationship between the dependent and independent variable is liner

Question 2

2. In a linear regression analysis of realized fund returns based on the single factor market model, what parameters or variables of the regression would be associated with a fund’s estimated ex ante alpha, a fund’s estimated beta and a fund’s estimated idiosyncratic returns?

Answer 2

• Ex ante alpha is the intercept of the regression (relative to the riskless rate)
• A fund’s estimated beta is the slope coefficient of the regression
• A fund’s estimated idiosyncratic returns is the residuals (estimated error terms).

Question 3

3. List the three primary assumptions used in a least squares regression to justify that the estimated parameters are unbiased and most likely.

Answer 3

➢ The model’s error terms are assume to be:
• Normally distributed
• Uncorrelated
• Homoskedastic (i.e., have the same finite variance)

Question 4

4. Why is multicollinearity an issue in a multiple regression model but not a single regression model?

Answer 4

• There is only one independent variable in a single regression model, but two or more independent variables are needed to have multicollinearity. A multiple regression model is a regression model with more than one independent variable. Multicollinearity is when two or more independent variables in a regression model have high correlation to each other.

Question 5

5. The excess returns of a fund are being analyzed using a quadratic regression approach with an intercept and one independent variable: the squared value of the excess return of the overall market. What would be the likely interpretations of a result in which both the intercept and the slope coefficient are significantly positive?

Answer 5

• A positive slope coefficient indicates that a manager has been able to successfully time the market.
• A positive intercept indicates superior security selection.

Question 6

6. In the context of a dummy variable approach to dynamic risk exposures, what is a “down market beta”?

Answer 6

• The down market beta, bi,d is the responsiveness of the fund’s return to the market return when the market return is less than the riskless rate (i.e., when the market’s excess return is negative or “down”).

Question 7

7. A fund specializing in market timing of listed equities is estimated to have exhibited negative conditional correlation with the returns of a major equity market index. The fund alternates between net short positions and net long positions. What is primary interpretation of this finding?

Answer 7

• The manager is mis-timing the market by having higher risk exposure (higher betas or, more net long) when the market falls having less exposure when the market rises.

Question 8

8. Why would an analyst use a rolling window analysis of the systematic risk exposures of an investment strategy rather than a single analysis based on the entire dataset?

Answer 8

• The analyst is concerned about style drift (specifically, systematic risk exposures that change through time). By using a short-term analysis that moves through time the analyst can get estimates of the change in risk exposures through time.

Question 9

9. Consider a style analysis of fund returns based on Sharpe’s seminal approach. Based on past observations, how would you expect the goodness of fit of a regression to change based on whether the fund returns were from traditional mutual funds or from hedge funds?

Answer 9

• Traditional mutual fund returns are well explained by the returns of the asset classes that the funds hold but the same is not true for hedge funds. Empirical evidence indicates that the returns on most hedge funds are not well explained by passive return indices of their underlying assets. This is because hedge funds are more likely to have quickly and/or substantially changing risk exposures.

Question 10

10. What are two major shortcoming of an empirical study that examines performance persistence of funds by comparing the correlation of returns in an earlier period with returns in a subsequent period when returns are based on appraised values?

Answer 10

• The results could be driven by serial correlation in returns that does not reflect true performance correlations
• The returns are not risk-adjusted

Question 11

conditional correlation

Answer 11

is a correlation between two

variables under specified circumstances.

Question 12

dependent variable

Answer 12

is the variable supplied by the
researcher that is the focus of the analysis and is determined at
least in part by other (independent or explanatory) variables.

Question 13

down market beta

Answer 13

bi,d, is the responsiveness of the
fund’s return to the market return when the market return is
less than the riskless rate (i.e., when the market’s excess return
is negative, or down).

Question 14

goodness of fit

Answer 14

of a regression is the extent to which the
model appears to explain the variation in the dependent
variable.

Question 15

independent variables

Answer 15

are those explanatory variables that
are inputs to the regression and are viewed as causing the
observed values of the dependent variable.

Question 16

intercept

Answer 16

is the value of the dependent variable when all

independent variables are zero.

Question 17

look-back option

Answer 17

has a payoff that is based on the value of
the underlying asset over a reference period rather than
simply the value of the underlying asset at the option’s
expiration date.

Question 18

multicollinearity

Answer 18

is when two or more independent variables

in a regression model have high correlation to each other.

Question 19

multiple regression model

Answer 19

is a regression model with more

than one independent variable.

Question 20

negative conditional correlation

Answer 20

When the correlation in the down sample is higher than the

correlation in the up sample, it is termed as this.

Question 21

nonlinear exposure

Answer 21

of a position to a market factor is when
the sensitivity of the position’s value varies based on the
magnitude of the level of change in the market factor’s value.

Question 22

nonstationary

Answer 22

The return distributions of hedge funds and hedge fund
indices are this, meaning that return volatilities and
correlations vary through time.

Question 23

positive conditional correlation

Answer 23

of investment returns to
market returns is when the correlation in the up sample is
higher than the correlation in the down sample. Investors prefer
investment strategies with positive conditional correlation,
since the strategies offer higher participation in profits during
bull markets and lower participation in losses during bear
markets.

Question 24

principal components analysis

Answer 24

is a statistical technique that
groups the observations in a large data set into smaller sets of
similar types based on commonalities in the data.

Question 25

regression

Answer 25

is a statistical analysis of the relationship that
explains the values of a dependent variable as a function of
the values of one or more independent variables based on a
specified model.

Question 26

residuals

Answer 26

of the regression, eit, reflect the regression’s
estimate of the idiosyncratic portion of asset i’s realized
returns above or below its mean idiosyncratic return (i.e., the
regression’s estimates of the error term).

Question 27

rolling window analysis

Answer 27

is a relatively advanced technique
for analyzing statistical behavior over time, using overlapping
subsamples that move evenly through time.

Question 28

r-squared

Answer 28

value of the regression, which is also called the
coefficient of determination, is often used to assess goodness
of fit, especially when comparing models. In a simple linear
regression, the r-squared is simply the squared value of the
estimated correlation coefficient between the dependent
variable and the independent variable.

Question 29

serial correlation

Answer 29

is the same as autocorrelation: It is the
correlation of a variable, such as return, in one time period
(e.g., year) to the same variable in another time period.

Question 30

simple linear regression

Answer 30

is a linear regression in which the

model has only one independent variable.

Question 31

slope coefficient

Answer 31

is a measure of the change in a
dependent variable with respect to a change in an
independent variable.

Question 32

stepwise regression

Answer 32

is an iterative technique in which
variables are added or deleted from the regression equation
based on their statistical significance.

Question 33

style analysis

Answer 33

is the process of understanding an investment
strategy, especially using a statistical approach, based on
grouping funds by their investment strategies or styles.

Question 34

t-statistic

Answer 34

of a parameter is formed by taking the
estimated absolute value of the parameter and dividing by its
standard error.

Question 35

t-test

Answer 35

is a statistical test that rejects or fails to reject a

hypothesis by comparing a t-statistic to a critical value.

Question 36

up market beta

Answer 36

bi,u, is the responsiveness of the fund’s
return to the market return when the excess market return is
positive, and is estimated as the sum of bi,d and bi,diff.

Question 37

Consider a regression with an alpha estimate of
0.5% (with a standard error of 0.3) and a beta estimate of 1.1 (with a standard
error of 0.3). Are the regression parameters statistically significant?

Answer 37

Calculate the t-statistic for the alpha estimate
Step One: Press 0.5 → ÷ → 0.3
Step Two: Press =
Answer: 1.67

Calculate the t-statistic for the beta estimate
Step One: Press 1.1 → ÷ → 0.3
Step Two: Press =
Answer: 3.67

Question 38

A 50-week rolling window analysis is performed
with exactly four years of data (208 weeks). How many analyses would be
performed, and how many statistically independent analyses would there be?

Answer 38

Calculate the number of windows of analyses performed
Step One: Press 208 → - → 50
Step Two: Press =
Answer: 158

Calculate the number of independent analyses performed
Step One: Press 208 → ÷ → 50
Step Two: Press =
Answer: 4.16, round down to 4