# Friday AM Quant

Winston Collier, CFA, has been asked by his supervisor to develop a model for predicting the warranty expense incurred by Premier Snowplow Manufacturing Company in servicing its plows. Three years ago, major design changes were made on newly manufactured plows in an effort to reduce warranty expense. Premier warrants its snowplows for 4 years or 18,000 miles, whichever comes first. Warranty expense is higher in winter months, but some of Premier’s customers defer maintenance issues that are not essential to keeping the machines functioning to spring or summer seasons. The data that Collier will analyze is in the following table (in \$ millions): WarrantyExpense (Column 2) Change inWarranty Expense (Column 3) Lagged Change inWarranty Expense yt-1 (COlumn 4) Seasonal LaggedChange inWarranty Expense yt-4 (Column 5) t 2 3 4 5 2002.1 103 2002.2 52 -51 2002.3 32 -20 -51 2002.4 68 +36 -20 2003.1 91 +23 +36 2003.2 44 -47 +23 -51 2003.3 30 -14 -47 -20 2003.4 60 +30 -14 +36 2004.1 77 +17 +30 +23 2004.2 38 -39 +17 -47 2004.3 29 -9 -39 -14 2004.4 53 +24 -9 +30 Winston submits the following results to his supervisor. The first is the estimation of a trend model for the period 2002:1 to 2004:4. The model is below. The standard errors are in parentheses. (Warranty expense)t = 74.1 - 2.7* t + et R-squared = 16.2% (14.37) (1.97) Winston also submits the following results for an autoregressive model on the differences in the expense over the period 2004:2 to 2004:4. The model is below where “y” represents the change in expense as defined in the table above. The standard errors are in parentheses. yt = -0.7 - 0.07* yt-1 + 0.83* yt-4 + et R-squared = 99.98% (0.643) (0.0222) (0.0186) After receiving the output, Collier’s supervisor asks him to compute moving averages of the sales data. Collier’s supervisors would probably not want to use the results from the trend model for all of the following reasons EXCEPT: A) the model is a linear trend model and log-linear models are always superior. B) it does not give insights into the underlying dynamics of the movement of the dependent variable. C) the slope coefficient is not significant. ________________________________________ The mean reverting level for the first equation is closest to: A) 20.0. B) -0.8. C) 43.6. ________________________________________ Based upon the output provided by Collier to his supervisor and without any further calculations, in a comparison of the two equations’ explanatory power of warranty expense it can be concluded that: A) the provided results are not sufficient to reach a conclusion. B) the autoregressive model on the first differenced data has more explanatory power for warranty expense. C) the two equations are equally useful in explaining warranty expense. ________________________________________ Based on the autoregressive model, expected warranty expense in the first quarter of 2005 will be closest to: A) \$65 million. B) \$78 million. C) \$60 million. Based upon the results, is there a seasonality component in the data? A) Yes, because the coefficient on yt-4 is large compared to its standard error. B) No, because the slope coefficients in the autoregressive model have opposite signs. C) Yes, because the coefficient on yt is small compared to its standard error. Collier most likely chose to use first-differenced data in the autoregressive model: A) in order to avoid problems associated with unit roots. B) to increase the explanatory power. C) because the time trend was significant.

Collier’s supervisors would probably not want to use the results from the trend model for all of the following reasons EXCEPT: C) the slope coefficient is not significant. ________________________________________ The mean reverting level for the first equation is closest to: A) 20.0. ________________________________________ Based upon the output provided by Collier to his supervisor and without any further calculations, in a comparison of the two equations’ explanatory power of warranty expense it can be concluded that: A) the provided results are not sufficient to reach a conclusion. ________________________________________ Based on the autoregressive model, expected warranty expense in the first quarter of 2005 will be closest to: B) \$78 million. Based upon the results, is there a seasonality component in the data? A) Yes, because the coefficient on yt-4 is large compared to its standard error. Collier most likely chose to use first-differenced data in the autoregressive model: B) to increase the explanatory power.

Not done Quant since long and don’t remember this now. Ans - Q1.A Q2.A Q3.C Q4.A [64.73] Q5.A Q6.A

Collier’s supervisors would probably not want to use the results from the trend model for all of the following reasons EXCEPT: A) the model is a linear trend model and log-linear models are always superior. B) it does not give insights into the underlying dynamics of the movement of the dependent variable. C) the slope coefficient is not significant. The correct answer was A. Linear trend models are not always inferior to log-linear models. To determine which specification is better would require more analysis such as a graph of the data over time. As for the other possible answers, Collier can see that the slope coefficient is not significant because the t-statistic is 1.37=2.7/1.97. Also, regressing a variable on a simple time trend only describes the movement over time, and does not address the underlying dynamics of the dependent variable. (Study Session 3, LOS 13.a) ________________________________________ The mean reverting level for the first equation is closest to: A) 20.0. B) -0.8. C) 43.6. The correct answer was A. The mean reverting level is X1 = bo/(1-b1) X1 = 74.1/[1-(-2.7)] = 20.03 (Study Session 3, LOS 13.e) ________________________________________ Based upon the output provided by Collier to his supervisor and without any further calculations, in a comparison of the two equations’ explanatory power of warranty expense it can be concluded that: A) the provided results are not sufficient to reach a conclusion. B) the autoregressive model on the first differenced data has more explanatory power for warranty expense. C) the two equations are equally useful in explaining warranty expense. The correct answer was A. Although the R-squared values would suggest that the autoregressive model has more explanatory power, there are a few problems. First, the models have different sample periods and different numbers of explanatory variables. Second, the actual input data is different. To assess the explanatory power of warranty expense, as opposed to the first differenced values, we must transform the fitted values of the first-differenced data back to the original level data to assess the explanatory power for the warranty expense. (Study Session 3, LOS 12.e) ________________________________________ Based on the autoregressive model, expected warranty expense in the first quarter of 2005 will be closest to: A) \$65 million. B) \$78 million. C) \$60 million. The correct answer was A. Substituting the 1-period lagged data from 2004.4 and the 4-period lagged data from 2004.1 into the model formula, change in warranty expense is predicted to be higher than 2004.4. 11.73 =-0.7 - 0.07*24+ 0.83*17. The expected warranty expense is (53 + 11.73) = \$64.73 million. (Study Session 3, LOS 13.d) ________________________________________ Based upon the results, is there a seasonality component in the data? A) Yes, because the coefficient on yt-4 is large compared to its standard error. B) No, because the slope coefficients in the autoregressive model have opposite signs. C) Yes, because the coefficient on yt is small compared to its standard error. The correct answer was A. The coefficient on the 4th lag tests the seasonality component. The t-ratio is 44.6. Even using Chebychev’s inequality, this would be significant. Neither of the other answers are correct or relate to the seasonality of the data. (Study Session 3, LOS 13.k) ________________________________________ Collier most likely chose to use first-differenced data in the autoregressive model: A) in order to avoid problems associated with unit roots. B) to increase the explanatory power. C) because the time trend was significant. The correct answer was A. Time series with unit roots are very common in economic and financial models, and unit roots cause problems in assessing the model. Fortunately, a time series with a unit root may be transformed to achieve covariance stationarity using the first-differencing process. Although the explanatory power of the model was high (but note the small sample size), a model using first-differenced data often has less explanatory power. The time trend was not significant, so that was not a possible answer. (Study Session 3, LOS 13.j)

a, a, b (higher r sq?), maybe i’m reading this chart all funky b/c of the text but not .7 - 53(.07) + 77(.83) = 59.5 or c? i might have f’d this up with the chart, a, a i stink at quant

5/6 But how did they deicde to have every answer option as A?

note to self- review easy plug and play stuff (q4 i f’d)… that one should nail. the formatting didn’t do me any favors here, but not an excuse. dumb of me on q3 also. if i could go 4/6 in quant on the real thing, though, it’d be a win for me. i was under 50% last year and it wasn’t a hard quant item set. i just didn’t know quant well. trying to step this up a bit in 2009.

(Warranty expense)t = 74.1 - 2.7* t + et The mean reverting level is X1 = bo/(1-b1) X1 = 74.1/[1-(-2.7)] = 20.03 Does the reverting level formula apply for linear trend model? I thought it apply for AR(1) model only.

3/6… would have been 4 if I could have read that table. Thanks for the checkup!

5th question.
How to check if there is a seasonality component? what degrees of freedom to use?