CFA 2 - Reading 10: Multiple Regression and Issues in Regression Analysis

3.10.k If GDP rises 2.2% and the price of fuels falls $0.15, Baltz's model will predict Company sales to be (in $ millions) closest to: Using: Predictor Coefficient Standard Error of the Coefficient Intercept 78 13.710 Δ GDP 30.22 12.120 Δ $ Fuel −412.39 183.981

A: $206. Sales will be closest to $78 + ($30.22 × 2.2) + [(−412.39) × (−$0.15)] = $206.34 million. Conditional Heteroskedasticity: What is it? Residual variance related to level of independent variables Effect? Coefficients are consistent. Standard errors are underestimated. Too many Type I errors. Detection? Breusch-Pagan chi-square test = n × R^2 Correction? Use White-corrected standard errors Serial Correlation: What is it? Residuals are correlated Effect? Coefficients are consistent. Standard errors are underestimated. Too many Type I errors (positive correlation). Detection? Durbin-Watson test ≈ 2(1 − r) Correction? Use the Hansen method to adjust standard errors Multicollinearity: What is it? Two or more independent variables are correlated Effect? Coefficients are consistent (but unreliable). Standard errors are overestimated. Too many Type II errors. Detection? Conflicting t and F statistics; correlations among independent variables if k = 2 Correction? Drop one of the correlated variables

3.10.a Upon further analysis, Turner concludes that multicollinearity is a problem. What might have prompted this further analysis and what is intuition behind the conclusion?

A: At least one of the t-statistics was not significant, the F-statistic was significant, and a positive relationship between the number of analysts and the size of the firm would be expected. Multicollinearity occurs when there is a high correlation among independent variables and may exist if there is a significant F-statistic for the fit of the regression model, but at least one insignificant independent variable when we expect all of them to be significant. In this case the coefficient on ln(market value) was not significant at the 1% level, but the F-statistic was significant. It would make sense that the size of the firm, i.e., the market value, and the number of analysts would be positively correlated.

3.10.k Baltz proceeds to test the hypothesis that none of the independent variables has significant explanatory power. He concludes that, at a 5% level of significance: Using: Analysis of Variance Table (ANOVA) Source Degrees of Freedom Sum of Squares Regression - 291.30 Error 27 132.12 Total 29 423.42

A: MSE = SSE / [n − (k + 1)] = 132.12 ÷ 27 = 4.89. From the ANOVA table, the calculated F-statistic is (mean square regression / mean square error) = 145.65 / 4.89 = 29.7853. From the F distribution table (2 df numerator, 27 df denominator) the F-critical value may be interpolated to be 3.36. Because 29.7853 is greater than 3.36, Baltz rejects the null hypothesis and concludes that at least one of the independent variables has explanatory power. I don't know where they get 145.65 - the mean square regression.

3.10.k With regards to violation of regression assumptions, Baltz should most appropriately be concerned about: Baltz is concerned that violations of regression assumptions may affect the utility of the model for forecasting purposes. He is especially concerned about a situation where the coefficient estimate for an independent variable could take on opposite sign to that predicted. Baltz is also concerned about important variables being left out of the model. He makes the following statement: "If an omitted variable is correlated with one of the independent variables included in the model, the standard errors and coefficient estimates will be inconsistent."

A: Multicollinearity. Multicollinearity is a violation of regression assumptions that may cause estimates of the regression coefficients to become extremely imprecise and unreliable and possibly lead to estimates having the opposite sign to that expected.

3.10.g Which of the following statements about the F-statistic is least accurate?

A: Rejecting the null hypothesis means that only one of the independent variables is statistically significant - is least accurate. An F-test assesses how well the set of independent variables, as a group, explains the variation in the dependent variable. That is, the F-statistic is used to test whether at least one of the independent variables explains a significant portion of the variation of the dependent variable. The F-distributed test statistic can be used to test the significance of all (or any subset of) the independent variables (i.e., the overall fit of the model) using a one-tailed test: F = MSR/MSE = (RSS/k) / (SSE/(n - k - 1))

3.10.h Using Model ONE, what is the sales forecast for the second quarter of the next year? Coefficients Standard Error t-Statistic Intercept 31.40833 1.4866 21.12763 Q1 −3.77798 1.485952 −2.54246 Q2 −2.46310 1.476204 −1.66853 Q3 −0.14821 1.470324 −0.10080 TREND 0.851786 0.075335 11.20848 Regression Statistics Multiple R 0.941828 R2 0.887039 Adjusted R2 0.863258 Standard Error 2.543272 Observations 24 Coefficients Standard Error t-Statistic Intercept 31.40833 1.4866 21.12763 Q1 −3.77798 1.485952 −2.54246 Q2 −2.46310 1.476204 −1.66853 Q3 −0.14821 1.470324 −0.10080 TREND 0.851786 0.075335 11.20848

A: The estimate for the second quarter of the following year would be (in millions): 31.4083 + (−2.4631) + (24 + 2) × 0.851786 = 51.091666.

3.10.k Baltz then tests the individual variables, at a 5% level of significance, to determine whether sales are explained by changes in GDP and fuel prices. Baltz concludes that: Predictor Coefficient Standard Error of the Coefficient Intercept 78 13.710 Δ GDP 30.22 12.120 Δ $ Fuel −412.39 183.981

A: both GDP and fuel price changes explain changes in sales. From the ANOVA table, the calculated t-statistics are (30.22 / 12.12) = 2.49 for GDP and (−412.39 / 183.981) = −2.24 for fuel prices. These values are both beyond the critical t-value at 27 degrees of freedom of ±2.052. Therefore, Baltz is able to reject the null hypothesis that these coefficients are equal to zero, and concludes that both variables are important in explaining sales.

3.10.k Presence of conditional heteroskedasticity is least likely to affect the:

A: coefficient estimates - least likely affect. Conditional heteroskedasticity results in consistent coefficient estimates, but it biases standard errors, affecting the computed t-statistic and F-statistic

3.10.k Regarding the statement about omitted variables made by Baltz, which of the following is most accurate? The statement: "If an omitted variable is correlated with one of the independent variables included in the model, the standard errors and coefficient estimates will be inconsistent."

A: is correct. Baltz's statement is correct. If an omitted variable is correlated with one of the independent variables in the model, the coefficient estimates will be biased and inconsistent and standard errors will be inconsistent.

3.10.l A variable is regressed against three other variables, x, y, and z. Which of the following would NOT be an indication of multicollinearity? X is closely related to:

A: y^2 If x is related to y2, the relationship between x and y is not linear, so multicollinearity does not exist. If x is equal to a constant (3), it will be correlated with the intercept term.