CFA Level 2 - Quantitative Methods Session 3 - Reading 13 Time-Series Analysis - LOS a (Practice Questions, Sample Questions) 1. David Wellington, CFA, has estimated the following log-linear trend model: LN(xt) = b0 + b1t + εt. Using six years of quarterlyobservations, 2001:I to 2006:IV, Wellington gets the followingestimated equation: LN(xt) = 1.4 + 0.02t. The ﬁrst out-of-sampleforecast of xt for 2007:I is closest to: A) 1.88.B) 4.14. C) 6.69 ( Explanation : Wellington’s out-of-sample forecast of LN(xt) is 1.9 = 1.4 + 0.02 × 25, and e1.9 = 6.69) 2. Modeling the trend in a time series of a variable that grows at a constant rate with continuous compounding is best donewith: A) a moving average model.B) simple linear regression. C) a log-linear transformation of the time series. ( Explanation : The log-linear transformation of a series that grows at aconstant rate with continuous compounding (exponentialgrowth) will cause the transformed series to be linear) 3. In the time series model: yt=b0 + b1 t + εt, t=1,2,…,T, the: A) disturbance terms are autocorrelated.B) disturbance term is mean-reverting.
C) change in the dependent variable per time period is b1( Explanation : The slope is the change in the dependent variable per unit of time. The intercept is the estimate of thevalue of the dependent variable before the time seriesbegins. The disturbance term should be independent andidentically distributed. There is no reason to expect thedisturbance term to be mean-reverting, and if the residualsare autocorrelated, the research should correct for thatproblem) 4. Clara Holmes, CFA, is attempting to model the importation of an herbal tea into the United States. She gathers 24 yearsof annual data, which is in millions of inﬂation-adjusteddollars. The real dollar value of the tea imports has grownsteadily from $30 million in the ﬁrst year of the sample to $54million in the most recent year. She computes the following equation: (Tea Imports)t = 3.8836 + 0.9288 × (Tea Imports)t ? 1 + et t-statistics (0.9328) (9.0025) R2 = 0.7942Adj. R2 = 0.7844SE = 3.0892N = 23 Holmes and her colleague, John Briars, CFA, discuss theimplication of the model and how they might improve it.Holmes is fairly satisﬁed with the results because, as she says“the model explains 78.44 percent of the variation in thedependent variable.” Briars says the model actually explainsmore than that. Briars asks about the Durbin-Watson statistic. Holmes saidthat she did not compute it, so Briars reruns the model and
computes its value to be 2.1073. Briars says “now we knowserial correlation is not a problem.” Holmes counters bysaying “rerunning the model and computing theDurbin-Watson statistic was unnecessary because serialcorrelation is never a problem in this type of time-seriesmodel.” Briars and Holmes decide to ask their company’s statisticianabout the consequences of serial correlation. Based on whatBriars and Holmes tell the statistician, the statistician informsthem that serial correlation will only a ect the standarderrors and the coe cients are still unbiased. The statisticiansuggests that they employ the Hansen method, whichcorrects the standard errors for both serial correlation andheteroskedasticity. Given the information from the statistician, Briars and Holmesdecide to use the estimated coe cients to make someinferences. Holmes says the results do not look good for thefuture of tea imports because the coe cient on (Tea Import)t? 1 is less than one. This means the process is mean reverting.Using the coe cients in the output, says Holmes, “we knowthat whenever tea imports are higher than 41.810, the nextyear they will tend to fall. Whenever the tea imports are lessthan 41.810, then they will tend to rise in the following year.”Briars agrees with the general assertion that the resultssuggest that imports will not grow in the long run and tend torevert to a long-run mean, but he says the actual long-runmean is 54.545. Briars then computes the forecast of importsthree years into the future. With respect to the statements made by Holmes and Briarsconcerning serial correlation and the importance of theDurbin-Watson statistic: A) they were both incorrect. ( Explanation : Briars was incorrect because the DW statistic is not appropriate for testing serialcorrelation in an autoregressive model of this sort. Holmes
was incorrect because serial correlation can certainly be aproblem in such a model. They need to analyze the residualsand compute autocorrelation coe cients of the residuals tobetter determine if serial correlation is a problem.) B) Holmes was correct and Briars was incorrect.C) Briars was correct and Holmes was incorrect 5. With respect to the statement that the company’s statistician made concerning the consequences of serialcorrelation, assuming the company’s statistician iscompetent, we would most likely deduce that Holmes andBriars did not tell the statistician: A) the model’s speciﬁcation. ( Explanation : Serial correlation will bias the standard errors. It can also bias the coe cientestimates in an autoregressive model of this type. Thus,Briars and Holmes probably did not tell the statistician themodel is an AR(1) speciﬁcation.) B) the sample size.C) the value of the Durbin-Watson statistic. 6. The statistician’s statement concerning the beneﬁts of the Hansen method is: A) not correct, because the Hansen method only adjusts forproblems associated with serial correlation but notheteroskedasticity. B) correct, because the Hansen method adjusts for problemsassociated with both serial correlation andheteroskedasticity. ( Explanation : The statistician is correct because the Hansen method adjusts for problems associatedwith both serial correlation and heteroskedasticity.) C) not correct, because the Hansen method only adjusts forproblems associated with heteroskedasticity but not serialcorrelation.
7. Using the model’s results, Briar’s forecast for three years into the future is: A) $54.108 million. ( Explanation : Briars’ forecasts for he next three years would be:year one: 3.8836 + 0.9288 × 54 = 54.0388year two: 3.8836 + 0.9288 × (54.0388) = 54.0748year three: 3.8836 + 0.9288 × (54.0748) = 54.1083) B) $54.543 million.C) $47.151 million. 8. With respect to the comments of Holmes and Briars concerning the mean reversion of the import data, thelong-run mean value that: A) Briars computes is correct, and his conclusion is probablyaccurate. B) Briars computes is correct, but the conclusion is probablynot accurate. ( Explanation : Briars has computed a value that would be correct if the results of the model were reliable. Thelong-run mean would be 3.8836 / (1 ? 0.9288)= 54.5450. However,the evidence suggests that the data is not covariancestationary. The imports have grown steadily from $30 millionto $54 million.) C) Briars computes is not correct, but his conclusion isprobably accurate. 9. Given the output, the most obvious potential problem that Briars and Holmes need to investigate is: A) conditional heteroskedasticity. B) a unit root. ( Explanation : Multicollinearity cannot be a problem because there is only one independent variable.
Although heteroskedasticity may be a problem, nothing inthe output provides information in this regard. A unit root isa likely problem because the slope coe cient is so close toone. In fact, if Holmes and Briars divide the t-statistic of theslope coe cient by the value of the coe cient, they coulddetermine the standard error: 0.1032 = 0.9288 / 9.0025. Theycould then test the null hypothesis: H0 : slope coe cient = 1 H0 : slope coe cient ≠ 1 The t-statistic is: t = -0.6899 = (0.9288 ? 1) / 0.1032 They would not have to go to a t-table to realize that thist-statistic value of -0.6899 is not signiﬁcant so the hypothesisof the slope equaling one cannot be rejected. Given thatserial correlation generally underestimates standard errors,this statistic would become even smaller if that is the case.Finally, the fact that they know that imports have grown from$30 million to $54 million over a 24-year period shouldprovide a clue that the data may have a unit root. Note thatthis suggests that the true value of the slope also equals one,since with a unit root the dependent variable will grow byapproximately the amount of the intercept each year.) C) multicollinearity.
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CFA Level 2 - Quantitative Methods Session 3 - Reading 13 Time-Series Analysis - LOS a