Regression Analysis: Evaluating Fund Performance and Model Fitting

This assignment solution focuses on evaluating regression models for financial fund performance. It begins by explaining the coefficient of determination and its role in assessing model fit, comparing two initial models (ALFAX and PRDSX) using VTI as an independent variable. The solution then expands to include multi-factor models with VTI, SMB, HML, and UMD, analyzing the statistical significance of each factor and comparing the models' explanatory power. The analysis includes the interpretation of regression coefficients, p-values, and t-statistics. The assignment concludes with a critique of a portfolio manager's approach, highlighting the importance of stepwise regression analysis, adjusted R-squared, and multicollinearity checks (VIF) for constructing effective investment models. The student demonstrates an understanding of statistical concepts and their application in evaluating financial models.

3)
The coefficient of determination tells me the percentage of variation in the dependent
variable which is explained by all independent variables considered in the model. A percentage
of 75% or greater indicates that fitted model is a good fit to the data.
Model 1:
Here my dependent variable is ALFAX and independent variable is VTI. VTI is the
Monthly return to the Vanguard Total Market Index. The regression equation for this model is
given by: ALFAX = 0.001174 + 0.8832*VTI
With coefficient of determination, R square equal to 0.2600 I can say that 26.2%
variation in the dependent variable, ALFAX is explained by all independent variable VTI. This
percentage is very less and hence fitted model is not considered a good fit to the data.
Average Monthly Excess Return is 0.011255 with Standard Deviation of 0.050215. The
high value of Standard Deviation indicates that mean is not reliable.
When a unit increase in VTI there is 0.883 to units increase ALFAX.
Model 2:
Here my dependent variable is PRDSX and independent variable is VTI. PRDSX is the
Monthly return to the T Rowe Price - Small Cap Mutual Fund. VTI is the Monthly return to the
Vanguard Total Market Index. The regression equation for this model is given by: PRDSX =
0.0000010026 + 1.0667*VTI

With coefficient of determination, R square equal to 0.7361 I can say that 73.6%
variation in the dependent variable, PRDSX is explained by all independent variable VTI. This
percentage seems reasonable and hence fitted model is considered a good and better fit to the
data as compared to model 1.
Average Monthly Excess Return is 0.01217516 with Standard Deviation of 0.030184.
The high value of Standard Deviation indicates that mean is not reliable.
When a unit increase in VTI there is 1.0667 to units increase PRDSX. Hence I can say
that VTI has a stronger effect on PRDSX as compared to ALFAX.

5)
Model 1:
Here my dependent variable is ALFAX and independent variables are VTI, SMB, HML
and UMD. Regression equation is given as: ALFAX = .002906 + 0.7694*VTI + 0.5869*SMB –
0.4083*HML -0.06557*UMD
With a unit increase in VTI there is 0.7694 units increase in ALFAX. With a unit increase
in SMB there is 0.5869 units increase in ALFAX. With a unit increase in HML there is 0.4083
units decrease in ALFAX. With a unit increase in UMD there is 0.06557 units decrease in
ALFAX. These are the same signs as expected.
Consider the null hypothesis that beta_i is not significant, that is beta_i is equal to zero.
This is tested against an alternative hypothesis that beta_i is significant, that is beta_i is not equal
to zero. i =1, 2, 3, 4. The critical value is given by t(a/2, n-2) = t(.05/2, 60-2) = 2.00171.
If p-value is less than alpha or test statistic t is greater than the critical value then I reject
the null hypothesis at 5% level of significance. Else if P value is greater than Alpha or the test
statistic t is less than the critical value I fail to reject the null hypothesis at 5% level of
significance.
Here, with t = 3.826492 and p-value < .05, I reject the null hypothesis and conclude that
VTI is significant. With t = 1.3768 and p-value < .05, I reject the null hypothesis and conclude
that SMB is significant. With t = -1.4931 and p-value > .05, I fail to reject the null hypothesis
and conclude that HML is not significant. With t = -0.29772 and p-value > .05, I fail to reject the
null hypothesis and conclude that UMD is not significant.

With a greater value of coefficient of determination being 34.76% I can say that 4 factor
ALFAX model is better than initial model.
Model 2:
Here my dependent variable is PRDSX and independent variables are VTI, SMB, HML
and UMD. Regression equation is given as: ALFAX = 0.009895 + 0.3385*VTI + 0.7136*SMB –
0.1715*HML + .0759*UMD
With a unit increase in VTI there is 0.3385 units increase in ALFAX. With a unit increase
in SMB there is 0.7136 units increase in ALFAX. With a unit increase in HML there is 0.1715
units decrease in ALFAX. With a unit increase in UMD there is 0.0759 units increase in
ALFAX. These are the same signs as expected.
Consider the null hypothesis that beta_i is not significant, that is beta_i is equal to zero.
This is tested against an alternative hypothesis that beta_i is significant, that is beta_i is not equal
to zero. i =1, 2, 3, 4
If p-value is less than alpha or test statistic t is greater than the critical value then I reject
the null hypothesis at 5% level of significance. Else if P value is greater than Alpha or the test
statistic t is less than the critical value I fail to reject the null hypothesis at 5% level of
significance.
Here, with t = 0.809406 and p-value > .05, I fail to reject the null hypothesis and
conclude that VTI is not significant. With t = 1.3768 and p-value > .05, I fail to reject the null
hypothesis and conclude that SMB is not significant. With t = -0.30147 and p-value > .05, I fail

to reject the null hypothesis and conclude that HML is not significant. With t = 0.16558 and p-
value > .05, I fail to reject the null hypothesis and conclude that UMD is not significant.
With a less value of coefficient of determination being 5.59% I can say that 4 factor
PRDSX model is worse than initial model.

6)
The portfolio manager of my fund doesn’t has the skill as he hasn’t used step wise
regression analysis and adjusted R^2. Adjusted R square increases with the addition of
significant variables. step wise regression analysis helps to choose the best model on basis of
value of adjusted R square and considering the value of VIF to judge the multi-co-linearity in the
model.