Finquiz - Item-set Answers, Study Session 3, Reading 9.pdf

Reading 9

Correlation and Regression

FinQuiz.com

FinQuiz.com CFA Level II Item-set – Solution Study Session 3 June 2017

Copyright © 2010-2017. FinQuiz.com. All rights reserved. Copying, reproduction or redistribution of this material is strictly prohibited. [email protected].

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

FinQuiz Level II 2017 – Item-sets Solution Reading 9: Correlation and Regression 1. Question ID: 11438 Correct Answer: B Statement 1 is incorrect. A correlation coefficient of zero indicates that there is absolutely no linear relation between the two variables. Variables with a correlation of 0 can have a strong non-linear relationship. If there is no linear relationship, the value of one variable tells us nothing about the value of the other variable. Statement 2 is correct. Correlation coefficients can be computed validly if the means and variances of the two variables, and the covariance between them, are finite and constant. When these assumptions are not true, correlations between two different variables can depend greatly on the sample that is used. 2. Question ID: 11439 Correct Answer: B Correlation that is induced by a calculation that mixes each of the two variables under consideration with a third is termed as a spurious correlation, since it does not measure the direct relationship between the variables. Spurious correlations can suggest results that may not be true. 3. Question ID: 11440 Correct Answer: A Exhibit 1 shows that the correlation between large-cap and small-cap stocks is positive but very low (0.13) which means the two indices represent distinct styles of investing. However, the correlation of the micro-cap index with the large-cap index (and the small-cap index) is very high (almost 1.0), which shows that there is very little difference between the two return series, and therefore, we may not be able to justify distinguishing between large-cap value and micro-cap value, or small-cap value and micro-cap value, as distinct investment styles. Therefore, Option A is most appropriate as it also provides the most opportunity of diversification. 4. Question ID: 11441 Correct Answer: B The t-statistic equals: −0.642√10 − 2 = −2.368 1 − (−0.642) Since the t-value is not less than –2.7854, the correlation coefficient is not statistically significant (we cannot reject the null hypothesis).

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

5. Question ID: 11442 Correct Answer: C The slope coefficient equals: 0.000865 = 1.248 = 0.000693 Since in a linear regression, the regression line fits through the point corresponding to the means of the dependent and independent variables, solving for the intercept using this point, we have: = 0.1156 − 1.248(0.0785) = 0.0176 6. Question ID: 11443 Correct Answer: A ! "# "#" ! "# 0.2187 = = 0.6547 0.2187 + 0.1153

=

The F-statistic equals: 0.2187% 1 = 157.434 0.1153% (85 − 2) 7. Question ID: 15545 Correct Answer: C Kitsis is incorrect with respect to Statement 1. If the correlation is positive one (perfect positive correlation), all the points on the scatter plot lie on a straight line with a positive slope. However, the slope depends on the relationship between the two variables; it could be that when one variable increases by one unit the other increases by half a unit (with an increase in one unit in one variable associated with exactly the same half-unit increase in the other variable). The slope of the line can be different (but positive), but as long as the points lie on a straight line the correlation between them will be 1. Statement 2 is incorrect. Sample correlation can be an unreliable measure when outliers are present. However, removing the outliers is not always the right thing to do, if the outliers provide important information about the variables during the period under analysis. One must use judgment to determine whether the outliers contain information about the variables’ relationship (should be included in the analysis) or contain no information (and should be excluded). 8. Question ID: 15546 Correct Answer: A Even though the correlation between NI and FCFF for the restaurant chain is much higher than the correlation for the sports-wear manufacturer, however, only the correlation coefficient for the sports-wear manufacturer is significant. This is most likely because the sample size used for the analysis of the sports-wear manufacturer is much larger than the sample size used for the restaurant chain. The larger the sample, the smaller the evidence in terms of the magnitude of the sample correlation needed to reject the null hypothesis of zero correlation. Recall that the t-statistic equals r√(n-2)/√1-r2. This shows that as n increases, the

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

t-statistic increases, so that even for a small value of r, the null hypothesis can be rejected. A null hypothesis is more likely to be rejected as we increase the sample size, all else equal. 9. Question ID: 15547 Correct Answer: C Assumptions 1 and 4 are incorrect. The relationship between the dependent variable and the independent variable should be linear in the parameters b0 and b1. This requirement does not exclude the independent or dependent variable from being raised to a power other than one. Assumption 4 is not an assumption of linear regression. 10. Question ID: 15548 Correct Answer: A If Tannis uses a lower level of significance, which is the same as using a higher level of confidence, the critical t-value will increase. This choice leads to wider confidence intervals and to a decreased likelihood of rejecting the null hypothesis. Decreasing the level of significance from 0.05 to 0.01 decreases the probability of the Type 1 error, but it increases the probability of the Type 2 error (failing to reject the null hypothesis when, in fact, it is false). 11. Question ID: 15549 Correct Answer: A Bergren is correct. If the standard error is reduced to half of its current value, the confidence interval will be half as large and the t-statistic twice as large (with a small standard error, the t-value increases). When this happens, the probability of rejecting the null hypothesis increases. 12. Question ID: 15550 Correct Answer: A If forecasts are unbiased, the value of b0 should be 0 and the value of b1 should be 1. At a 0.05 significance level, with 72-2 = 70 degrees of freedom, the critical t-value is 1.994. Given the information in Exhibit 2, the 95% confidence interval for b0 is: 0.0239 ± 1.994(0.4531) –0.8796 to 0.9274 The value of 0 falls within this interval so we cannot reject the null hypothesis that b0 = 0. The 95% confidence interval for b1 is as follows: 0.9248 ± 1.994(0.0948) 0.7358 to 1.1138 The value of 1 falls within this confidence interval, so we cannot reject the null hypothesis that b1=1 at the 0.05 significance level. Because we cannot reject either of the null hypotheses, we cannot reject the hypothesis that the forecasts are unbiased (which means they are unbiased).

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

13. Question ID: 15580 Correct Answer: A The type of data used by the research team at West Newman for conducting their regression analysis is cross-sectional data. This is because the analysts are comparing the effects of earnings announcements on price changes across corporations. If they had used earnings announcements for a single corporation and compared the effects of announcements on that corporation’s share price change across time, the research team would have been using time series data. Factor models are not relevant here. 14. Question ID: 15581 Correct Answer: A The first assumption is consistent with the underlying assumptions. Linear regression assumes that the relationship between the dependent variable and independent variable is linear in the parameters, b0 and b1. This requires the latter two parameters to be raised to the first power only. This requirement does not exclude, the independent variable (in this case E), from being raised to a power other than 1. The second assumption, outlined by the two senior analysts, is inconsistent with the assumptions normally underlying linear regression. Linear regression assumes that the expected value of the error term is 0 and not 1. The third assumption is inconsistent with the underlying assumptions. Linear regression assumes that the error term is normally distributed. Although linear regression may be modified and still be used in the event error term is not normally distributed, the analysts’ assumptions are inconsistent with the assumptions underlying linear regression. 15. Question ID: 15582 Correct Answer: B In order to determine the confidence interval, the following steps need to be followed: 1. Make the prediction: Using the regression model, an EPS of $2.50 indicates change in per share value of $39.056. P1 – P0 = 1.2458 + 15.1242($2.50) = $39.0563 2. Compute the variance of the prediction error: The variance is calculated using the following formula:

s

2

f

(

)

2  1 X−X  = s 1 + +  2  n (n − 1)s x  2  1 (2.50 − 1.6344 )  2 = 0.7542 1 + +   62 (62 − 1)0.4248  2

= 0.59444

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

The standard deviation of the forecast error is sf = (0.59444)0.5 = 0.7710 3. Determine the critical value of the t-statistic. Given a 95% confidence interval and 62 – 2 = 60 degrees of freedom, the critical value of the t-statistic, tc, 2.00. 4. Compute the value of the t-statistic. The 95% confidence interval for P1 – P0 extends from 39.0563 – 2.00(0.7710) to 39.0563 + 2.00(0.7710), or 37.51430 to 40.59830. 16. Question ID: 15583 Correct Answer: B The hypothesized value of the slope coefficient does not fall within the confidence interval. Therefore the null hypothesis that the slope coefficient is 3.50 is rejected. Since the null hypothesis is rejected, this indicates the slope coefficient will not contribute in producing an unbiased estimate of price change. In order to test whether the regression model is able to produce unbiased forecasts, in terms of the slope coefficient only, a confidence interval needs to be constructed for the slope coefficient. Based on the hypothesized value of b1, 3.50, the confidence interval is constructed as follows:

bˆ1 ± t c sbˆ

1

15.1242 ± 2.00*(2.6556) 9.81300 to 20.43540 *The degrees of freedom are 62 – 2 = 60. Using a 95% confidence interval, the t-statistic is 2.00. 17. Question ID: 15584 Correct Answer: A The most appropriate response question 1 is a no and to question 2 is a Type I error. The magnitude of r, the correlation coefficient, needed to reject the null hypothesis decreases as sample size n decreases, for two reasons: 1) due to the degrees of freedom and the absolute value of the critical tc value decreasing and 2) due to the absolute value of the numerator increasing with larger n, resulting in larger magnitude t-values. Selecting a 90% confidence interval, as opposed to 95%, will decrease the width of the confidence interval and increase the likelihood of rejecting the null hypothesis when it is true, i.e. increases the probability of a Type I error. The tc value increases with higher levels of confidence. This will lead to a wider confidence interval and a decreased likelihood of rejecting the null hypothesis. The converse can be said to be true for lower levels of confidence, which will lead to narrower confidence intervals and an increased chance of rejecting the null hypothesis.

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

18. Question ID: 15585 Correct Answer: A Limitation 1 fails to correspond with the limitations generally underlying linear regression. A limitation of regression analysis is that public knowledge of the regression relationships may negate their future usefulness. Public knowledge of stocks categorized as ‘High’, i.e. with strong price reactions, may lead to analysts acting upon the relationship and bidding the price of the stock up, rather than down. Thus stocks in this category may exhibit substantial price increases. This may decrease the usefulness of the model to predict future price changes. The standard estimate of the model may increase as the regression model produces inaccurate forecasts of price changes. Limitation 2 accurately captures a limitation of linear regression. Parameter instability can be a particular problem when comparing the price changes of stocks across the three different categories, in a cross-sectional context, as the characteristics of each category will certainly differ. Limitation 3 accurately captures a limitation of linear regression. If any of the regression assumptions are violated, hypothesis tests and predictions based on linear regression will not be valid. 19. Question ID: 18459 Correct Answer: B The regression model tries to explain the dependent variable through the use of the independent variable. 20. Question ID: 18460 Correct Answer: A The coefficient of determination measures the fraction of the total variation in the dependent variable that is caused by the independent variable. 21. Question ID: 18461 Correct Answer: B The critical value at 80% significance level with 10 degrees of freedom is 1.372. 22. Question ID: 18462 Correct Answer: B R2 = (Multiple R)2 = 0.40692 = 0.1656 23. Question ID: 18463 Correct Answer: B s2f = 0.002391×[1+(1/12) + {(0.0518 – 0.0412)2 / (11 × 0.0846)}] = 0.002391×[1 + 0.0833 + (0.00011236/0.9306)] = 0.002391×[1 + 0.0833 + 0.00012074] = 0.002391×(1.08342) = 0.0026

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

24. Question ID: 18464 Correct Answer: C Sf = 0.00261/2 = 0.0510 Predicted value of interest rate +/- tcsf = 0.02546 +/– (1.372 × 0.0510) = 0.02546 +/– (0.0700) = –0.0445 to 0.0954 25. Question ID: 18466 Correct Answer: A Regression residual is the difference between the actual and the predicted values of the dependent variable. Standard error of estimate measures the degree of variability between the actual and the predicted values of the dependent variable. 26. Question ID: 18467 Correct Answer: A The covariance factor cannot tell the magnitude of the relation between two variables. It can only point out the direction of the relationship. 27. Question ID: 18468 Correct Answer: B It is not possible to measure slope of a scatter graph with zero correlation as there is no linear relation between the observations. 28. Question ID: 18469 Correct Answer: A The standard error of a correlation coefficient is used to determine the confidence intervals around a true correlation of zero. If your correlation coefficient falls outside of this range, then it is significantly different than zero. 29. Question ID: 18470 Correct Answer: C SSE = 0.00872 = 0.000076 SEE = [SSE/ (n-k-1)]1/2 = [0.000076/ 43]½ = 0.001329 30. Question ID: 18471 Correct Answer: A SST = 0.00922 = 0.000085 RSS = SST – SSE = 0.000085 – 0.000076 = 0.000009

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

31. Question ID: 18474 Correct Answer: C Uncertainty in error term and uncertainty in the estimated parameters are the two sources of uncertainty while carrying out regression analysis. 32. Question ID: 18475 Correct Answer: A Parameter instability states that the regression relations like correlations can change over time. Regardless of whether the analyst is carrying out cross- section regression or time-series regression, collecting samples from more than one population would increase parameter instability. 33. Question ID: 18476 Correct Answer: A ANOVA is the procedure through which total variability is divided into components that are matched to their respective sources. This analysis of variance is carried out through the F-test, which is used to test whether all the slope coefficients are equal to zero. 34. Question ID: 18477 Correct Answer: A Confidence Range = B1+/– tc (sb1) = 0.8 +/– 1.734 (0.48) = –0.03 to 1.63 35. Question ID: 18478 Correct Answer: B Since the null hypothesis lies within the critical range, Brick would fail to reject the null hypothesis. 36. Question ID: 18479 Correct Answer: A t = (B1 – b1)/sb1 = (0.8 – 1.2)/ 0.48 = –0.833 37. Question ID: 18481 Correct Answer: B Acceptance of the null hypothesis is a binary decision. It will lead to an imminent rejection of the alternative hypothesis. 38. Question ID: 18482 Correct Answer: A We cannot assume the independent variable is caused by the dependent variable, simply because of the existence of correlation. A third variable could be the reason for the change in both of the previous variables.

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

39. Question ID: 18483 Correct Answer: A Outliers are small numbers of observations at either extreme of a sample. Analysts need to use judgment to determine whether the outliers contain information pertaining to the relationship between the two variables or not. 40. Question ID: 18484 Correct Answer: B A Type II error = failure to reject the null hypothesis although it is wrong. A decrease in the sample size will lead to an increase in the magnitude of the correlation coefficient needed to reject the null hypothesis. This will also lead to an increase in the critical value of the t-statistic, because of which the t-statistic will most likely fall within the critical range. 41. Question ID: 18485 Correct Answer: C R2 = (Total Variation – Unexplained Variation)/Total Variation R2 = (0.004348 – 0.000532)/ 0.004348 = 0.877645 42. Question ID: 18486 Correct Answer: C SEE = [SSE/ (n – k – 1)]1/2 SEE = [0.000532/ (40 – 1 – 1)]½ SEE = 0.003742 = 0.37% or use SEE = MSE1/2 SEE = 0.0000141/2 SEE = 0.003742 43. Question ID: 18488 Correct Answer: A Since the data uses many observations from across time periods for the same company, it can be classified as time-series data. The expected value of the error term is assumed to be zero in the linear regression model. 44. Question ID: 18489 Correct Answer: B t = r √(n-2)/ (1-r2) t = 0.154 √123/ √(1-0.1542) = 1.708/0.988 = 1.729 The calculated t-statistic is 1.729. Since this value falls outside the critical range, Cooper can reject the null hypothesis and conclude that the relation between Revco and the automotive index is statistically significant.

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

45. Question ID: 18490 Correct Answer: C SEE = √MSE = √0.0064 = 0.08 46. Question ID: 18491 Correct Answer: B The calculated t-statistics for the intercept and the slope coefficients are –9.269 and 3.576 respectively. Since both these value fall outside the critical range, Cooper can reject the null hypothesis and conclude that the relation between Revco and the automotive index is statistically significant. Intercept t-statistic = –0.0482/0.0052 = –9.269 Slope Coefficient t-statistic = 0.5893/ 0.1648 = 3.576 47. Question ID: 18492 Correct Answer: B Individual independent variables and their respective residuals are assumed to be uncorrelated. 48. Question ID: 18493 Correct Answer: A Linear regression is also known as linear least squares since the selected values of the intercept and the slope coefficient minimize the sum of squared vertical distances between the observations and the regression line. 49. Question ID: 18495 Correct Answer: A The error of regression coefficient is the standard error of the slope. 50. Question ID: 18496 Correct Answer: A A higher level of significance will lead to a narrower confidence interval. In this case, probability of rejecting the null hypothesis increases. Thus, the probability of Type I error increases. 51. Question ID: 18497 Correct Answer: A A lower level of significance results in a higher critical value of the t-statistic. This would decrease the probability of rejecting the null hypothesis. Thus, the probability of Type I error would decrease. 52. Question ID: 18498 Correct Answer: C The p-value is the smallest level of significance at which the null hypothesis can be rejected. When p<significance level, H0 can be rejected. If p>significance level, H0 cannot be rejected.

FinQuiz.com © 2017 - All rights reserved.

Reading 9

Correlation and Regression

FinQuiz.com

53. Question ID: 18499 Correct Answer: B The sum of squared regression residuals is the accumulated square of the difference between the actual values of the dependent variable and the predicted value of the dependent variable. The estimated parameters help in reducing this difference. 54. Question ID: 18500 Correct Answer: B Mean Square Error = Sum of Squared Errors/(n-2) MSE = 125000/ (76) = 1645 55. Question ID: 18502 Correct Answer: A The correlation coefficient measures both the direction and the magnitude of a linear relationship. 56. Question ID: 18503 Correct Answer: A The significance level of the correlation coefficient needs to be tested regardless of its value. The t-test used to check the significance of the coefficient requires the number of observations recorded. 57. Question ID: 18504 Correct Answer: C Correlation = Covariance/ (Std Devx × Std Devy) = 79.37/ {(50)1/2 (350)1/2} = 0.600 58. Question ID: 18505 Correct Answer: A t = r (n – 2)1/2/ (1 – r2)1/2 = 0.6 √(43)/ √(1 – 0.36) = 3.934/0.8 = 4.918 59. Question ID: 18506 Correct Answer: A The critical value of the t-statistic is 1.302 with 43 degrees of freedom and a p-value of 0.10. Since the t-statistic falls out of the critical rang, Crater can reject the null hypothesis. 60. Question ID: 18507 Correct Answer: C Option C indicates a reliable correlation between two variables. However, Option A is spurious as the two variables indicated can only have a viable correlation through dividing them by another variable i.e. the number of shares. Option B is also spurious, as many other factors affect a manager’s performance other than their gender.

FinQuiz.com © 2017 - All rights reserved.