100 index. He utilizes monthly data from the past five years, and assumes that the sum of the squared errors is .0039. The calculated standard error of the estimate (SEE) is closest to:
Bea Carroll, CFA, has performed a regression analysis of the relationship between 6-month LIBOR and the U.S. Consumer Price Index (CPI). Her analysis indicates a standard error of estimate (SEE) that is high relative to total variability. Which of the following conclusions regarding the relationship between 6-month LIBOR and CPI can Carroll most accurately draw from her SEE analysis? The relationship between the two variables is:
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The SEE is the standard deviation of the error terms in the regression, and is an indicator of the strength of the relationship between the dependent and independent variables. The SEE will be low if the relationship is strong and conversely will be high if the relationship is weak.
The most appropriate measure of the degree of variability of the actual Y-values relative to the estimated Y-values from a regression equation is the:
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The SEE is the standard deviation of the error terms in the regression, and is an indicator of the strength of the relationship between the dependent and independent variables. The SEE will be low if the relationship is strong, and conversely will be high if the relationship is weak.
Which of the following statements about the standard error of estimate is least accurate? The standard error of estimate:
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Note: The coefficient of determination (R2) is the square of the correlation coefficient in simple linear regression.
The standard error of estimate is closest to the:
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The standard error of the estimate measures the uncertainty in the relationship between the actual and predicted values of the dependent variable. The differences between these values are called the residuals, and the standard error of the estimate helps gauge the fit of the regression line (the smaller the standard error of the estimate, the better the fit).
The standard error of the estimate measures the variability of the:
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The standard error of the estimate (SEE) measures the uncertainty in the relationship between the independent and dependent variables and helps gauge the fit of the regression line (the smaller the standard error of the estimate, the better the fit).
Remember that the SEE is different from the sum of squared errors (SSE). SSE = the sum of (actual value - predicted value)2. SEE is the the square root of the SSE "standardized" by the degrees of freedom, or SEE = [SSE / (n - 2)]1/2
Jason Brock, CFA, is performing a regression analysis to identify and evaluate any relationship between the common stock of ABT Corp and the S& 100 index. He utilizes monthly data from the past five years, and assumes that the sum of the squared errors is .0039. The calculated standard error of the estimate (SEE) is closest to:
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The standard error of estimate of a regression equation measures the degree of variability between the actual and estimated Y-values. The SEE may also be referred to as the standard error of the residual or the standard error of the regression. The SEE is equal to the square root of the mean squared error. Expressed in a formula, SEE = √(SSE / (n-2)) = √(.0039 / (60-2)) = .0082
The standard error of the estimate in a regression is the standard deviation of the:<
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The standard error is se = √[SSE/(n-2)]. It is the standard deviation of the residuals.
Which of the following statements about the standard error of the estimate (SEE) is least accurate?
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The R2, or coefficient of determination, is the percentage of variation in the dependent variable explained by the variation in the independent variable. A higher R2 means a better fit. The SEE is smaller when the fit is better.
If X and Y are perfectly correlated, regressing Y onto X will result in which of the following:
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If X and Y are perfectly correlated, all of the points will plot on the regression line, so the standard error of the estimate will be zero. Note that the sign of the correlation coefficient will indicate which way the regression line is pointing (there can be perfect negative correlation sloping downward as well as perfect positive correlation sloping upward). Alpha is the intercept and is not directly related to the correlation.
A simple linear regression is run to quantify the relationship between the return on the common stocks of medium sized companies (Mid Caps) and the return on the S& 500 Index, using the monthly return on Mid Cap stocks as the dependent variable and the monthly return on the S& 500 as the independent variable. The results of the regression are shown below:
Coefficient
Standard Error
of coefficient
t-Value
Intercept
1.71
2.950
0.58
S&
1.52
0.130
11.69
R2= 0.599
The strength of the relationship, as measured by the correlation coefficient, between the return on Mid Cap stocks and the return on the S& 500 for the period under study was:
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You are given R2 or the coefficient of determination of 0.599 and are asked to find R or the coefficient of correlation. The square root of 0.599 = 0.774.
Assume an analyst performs two simple regressions. The first regression analysis has an R-squared of 0.90 and a slope coefficient of 0.10. The second regression analysis has an R-squared of 0.70 and a slope coefficient of 0.25. Which one of the following statements is most accurate?
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The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The larger R-squared (0.90) of the first regression means that 90% of the variability in the dependent variable is explained by variability in the independent variable, while 70% of that is explained in the second regression. This means that the first regression has more explanatory power than the second regression. Note that the Beta is the slope of the regression line and doesn’t measure explanatory power.
Assume you perform two simple regressions. The first regression analysis has an R-squared of 0.80 and a beta coefficient of 0.10. The second regression analysis has an R-squared of 0.80 and a beta coefficient of 0.25. Which one of the following statements is most accurate?
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The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The R-squared (0.80) being identical between the first and second regressions means that 80% of the variability in the dependent variable is explained by variability in the independent variable for both regressions. This means that the first regression has the same explaining power as the second regression.
An analyst performs two simple regressions. The first regression analysis has an R-squared of 0.40 and a beta coefficient of 1.2. The second regression analysis has an R-squared of 0.77 and a beta coefficient of 1.75. Which one of the following statements is most accurate?
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The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The larger R-squared (0.77) of the second regression means that 77% of the variability in the dependent variable is explained by variability in the independent variable, while only 40% of that is explained in the first regression. This means that the second regression has more explaining power than the first regression. Note that the Beta is the slope of the regression line and doesn’t measure explaining power.
Consider the following estimated regression equation:
ROEt = 0.23 - 1.50 CEtThe standard error of the coefficient is 0.40 and the number of observations is 32. The 95% confidence interval for the slope coefficient, b1, is:
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The confidence interval is -1.50 ± 2.042 (0.40), or {-2.317 < b1 < -0.683}.
Consider the following estimated regression equation:
AUTOt = 0.89 + 1.32 PItThe standard error of the coefficient is 0.42 and the number of observations is 22. The 95% confidence interval for the slope coefficient, b1, is:
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The degrees of freedom are found by n-k-1 with k being the number of independent variables or 1 in this case. DF = 22-1-1 = 20. Looking up 20 degrees of freedom on the student's t distribution for a 95% confidence level and a 2 tailed test gives us a critical value of 2.086. The confidence interval is 1.32 ± 2.086 (0.42), or {0.444 < b1 < 2.196}.
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.
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
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:
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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)
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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)
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The forecast is 12.209 = 0.149 + 1.206 × 10, so the answer is 12.2. (Study Session 3, LOS 11.h)
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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)
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The 95% confidence interval is 1.2056 ± (2.074 × 0.2275). (Study Session 3, LOS 11.f)
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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)
What does the R2 of a simple regression of two variables measure and what calculation is used to equate the correlation coefficient to the coefficient of determination?
| R2measures: | Correlation coefficient |
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R2, or the Coefficient of Determination, is the square of the coefficient of correlation (r). The coefficient of correlation describes the strength of the relationship between the X and Y variables. The standard error of the residuals is the standard deviation of the dispersion about the regression line. The t-statistic measures the statistical significance of the coefficients of the regression equation. In the response: "percent of variability of the independent variable that is explained by the variability of the dependent variable," the definitions of the variables are reversed.
The R2 of a simple regression of two factors, A and B, measures the:
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The coefficient of determination measures the percentage of variation in the dependent variable explained by the variation in the independent variable.
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