Craig Standish, CFA, is investigating the validity of claims associated with a fund that his company offers. The company advertises the fund as having low turnover and, hence, low management fees. The fund was created two years ago with only a few uncorrelated assets. Standish randomly draws two stocks from the fund, Grey Corporation and Jars Inc., and measures the variances and covariance of their monthly returns over the past two years. The resulting variance covariance matrix is shown below. Standish will test whether it is reasonable to believe that the returns of Grey and Jars are uncorrelated. In doing the analysis, he plans to address the issue of spurious correlation and outliers.
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|
Grey |
Jars |
|
Grey |
42.2 |
20.8 |
|
Jars |
20.8 |
36.5 |
Standish wants to learn more about the performance of the fund. He performs a linear regression of the fund’s monthly returns over the past two years on a large capitalization index. The results are below:
|
ANOVA |
|
|
|
|
|
|
df |
SS |
MS |
F |
|
Regression |
1 |
92.53009 |
92.53009 |
28.09117 |
|
Residual |
22 |
72.46625 |
3.293921 |
|
|
Total |
23 |
164.9963 |
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|
|
|
|
|
|
|
|
|
Coefficients |
Standard Error |
t-statistic |
P-value |
|
Intercept |
0.148923 |
0.391669 |
0.380225 |
0.707424 |
|
Large Cap Index |
1.205602 |
0.227467 |
5.30011 |
2.56E-05 |
Standish forecasts the fund’s return, based upon the prediction that the return to the large capitalization index used in the regression will be 10%. He also wants to quantify the degree of the prediction error, as well as the minimum and maximum sensitivity that the fund actually has with respect to the index.
He plans to summarize his results in a report. In the report, he will also include caveats concerning the limitations of regression analysis. He lists four limitations of regression analysis that he feels are important: relationships between variables can change over time, the decision to use a t-statistic or F-statistic for a forecast confidence interval is arbitrary, if the error terms are heteroskedastic the test statistics for the equation may not be reliable, and if the error terms are correlated with each other over time the test statistics may not be reliable.
Given the variance/covariance matrix for Grey and Jars, in a one-sided hypothesis test that the returns are positively correlated H0: ρ = 0 vs. H1: ρ > 0, Standish would:
|
A) |
reject the null at the 5% but not the 1% level of significance. | |
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B) |
reject the null at the 1% level of significance. | |
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C) |
need to gather more information before being able to reach a conclusion concerning significance. | |
First, we must compute the correlation coefficient, which is 0.53 = 20.8 / (42.2 × 36.5)0.5.
The t-statistic is: 2.93 = 0.53 × [(24 - 2) / (1 ? 0.53 × 0.53)]0.5, and for df = 22 = 24 ? 2, the t-statistics for the 5 and 1% level are 1.717 and 2.508 respectively. (Study Session 3, LOS 11.g)
In performing the correlation test on Grey and Jars, Standish would most appropriately address the issue of:
|
A) |
spurious correlation but not the issue of outliers. | |
|
B) |
neither outliers nor correlation. | |
|
C) |
spurious correlation and the issue of outliers. | |
Both these issues are important in performing correlation analysis. A single outlier observation can change the correlation coefficient from significant to not significant and even from negative (positive) to positive (negative). Even if the correlation coefficient is significant, the researcher would want to make sure there is a reason for a relationship and that the correlation is not caused by chance. (Study Session 3, LOS 11.b)
If the large capitalization index has a 10% return, then the forecast of the fund’s return will be:
The forecast is 12.209 = 0.149 + 1.206 × 10, so the answer is 12.2. (Study Session 3, LOS 11.h)
The standard error of the estimate is:
SEE equals the square root of the MSE, which on the ANOVA table is 72.466 / 22 = 3.294. The SEE is 1.81 = (3.294)(0.5). (Study Session 3, LOS 11.i)
A 95% confidence interval for the slope coefficient is:
The 95% confidence interval is 1.2056 ± (2.074 × 0.2275). (Study Session 3, LOS 11.f)
Of the four caveats of regression analysis listed by Standish, the least accurate is:
|
A) |
the choice to use a t-statistic or F-statistic for a forecast confidence interval is arbitrary. | |
|
B) |
if the error terms are heteroskedastic the test statistics for the equation may not be reliable. | |
|
C) |
the relationships of variables change over time. | |
The t-statistic is used for constructing the confidence interval for the forecast. The F-statistic is not used for this purpose. The other possible shortfalls listed are valid. (Study Session 3, LOS 11.i)
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发表于 2011-3-3 11:31
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