JoeyDVivre

Barry Phillips, CFA, has the following time series observations from earliest to latest: (5, 6, 5, 7, 6, 6, 8, 8, 9, 11). Phillips transforms the series so that he will estimate an autoregressive process on the following data (1, -1, 2, -1, 0, 2, 0, 1, 2). The transformation Phillips employed is called:

A) beta drift.

B) first differencing.

C) moving average.

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Phillips obviously first differenced the data because the 1=6-5, -1=5-6, .... 1 = 9 - 9, 2 = 11 - 9.

JoeyDVivre

Barry Phillips, CFA, has estimated an AR(1) relationship (xt = b0 + b1 × xt-1 + et) and got the following result: xt+1 = 0.5 + 1.0xt + et. Phillips should:

A) first difference the data because b1 = 1.

B) not first difference the data because b0 = 0.5 < 1.

C) not first difference the data because b1 − b0 = 1.0 − 0.5 = 0.5 < 1.

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The condition b1 = 1 means that the series has a unit root and is not stationary. The correct way to transform the data in such an instance is to first difference the data.

JoeyDVivre

A time series that has a unit root can be transformed into a time series without a unit root through:

A) calculating moving average of the residuals.

B) first differencing.

C) mean reversion.

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First differencing a series that has a unit root creates a time series that does not have a unit root

JoeyDVivre

Suppose that the following time-series model is found to have a unit root:

Salest = b0 + b1 Sales t-1+ εt

What is the specification of the model if first differences are used?

A) Salest = b0 + b1 Sales t-1 + b2 Sales t-2 + εt.

B) (Salest - Salest-1)= b0 + b1 (Sales t-1 - Sales t-2) + εt.

C) Salest = b1 Sales t-1+ εt.

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Estimation with first differences requires calculating the change in the variable from period to period.

JoeyDVivre

Which of the following statements regarding unit roots in a time series is least accurate?

A) A time series that is a random walk has a unit root.

B) Even if a time series has a unit root, the predictions from the estimated model are valid.

C) A time series with a unit root is not covariance stationary.

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The presence of a unit root means that the least squares regression procedure that we have been using to estimate an AR(1) model cannot be used without transforming the data first.
A time series with a unit root will follow a random walk process. Since a time series that follows a random walk is not covariance stationary, modeling such a time series in an AR model can lead to incorrect statistical conclusions, and decisions made on the basis of these conclusions may be wrong. Unit roots are most likely to occur in time series that trend over time or have a seasonal element.

JoeyDVivre

Marvin Greene is interested in modeling the sales of the retail industry. He collected data on aggregate sales and found the following:

Salest = 0.345 + 1.0 Salest-1

The standard error of the slope coefficient is 0.15, and the number of observations is 60. Given a level of significance of 5%, which of the following can we NOT conclude about this model?

A) The model has a unit root.

B) The slope on lagged sales is not significantly different from one.

C) The model is covariance stationary.

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The test of whether the slope is different from one indicates failure to reject the null H0: b1=1 (t-critical with df = 58 is approximately 2.000, t-calculated = (1.0 - 1.0)/0.15 = 0.0).  This is a 2-tailed test and we cannot reject the null since 0.0 is not greater than 2.000. This model is nonstationary because the 1.0 coefficient on Salest-1 is a unit root. Any time series that has a unit root is not covariance stationary which can be corrected through the first-differencing process.

JoeyDVivre

An AR(1) autoregressive time series model:

A) can be used to test for a unit root, which exists if the slope coefficient equals one.

B) cannot be used to test for a unit root.

C) can be used to test for a unit root, which exists if the slope coefficient is less than one.

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If you estimate the following model xt = b0 + b1 × xt-1 + et and get b1 = 1, then the process has a unit root and is nonstationary.

JoeyDVivre

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):

Quarter	Warranty Expense	Change in Warranty Expense yt	Lagged Change in Warranty Expense yt-1	Seasonal Lagged Change in Warranty Expense yt-4
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) it does not give insights into the underlying dynamics of the movement of the dependent variable.

B) the model is a linear trend model and log-linear models are always superior.

C) the slope coefficient is not significant.

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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)

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The mean reverting level for the first equation is closest to:

A) -0.8.

B) 43.6.

C) 20.0.

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The mean reverting level is X1 = bo/(1-b1)
X1 = 74.1/[1-(-2.7)] = 20.03
(Study Session 3, LOS 13.f)

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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 autoregressive model on the first differenced data has more explanatory power for warranty expense.

B) the provided results are not sufficient to reach a conclusion.

C) the two equations are equally useful in explaining warranty expense.

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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.f)

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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.

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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)

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Based upon the results, is there a seasonality component in the data?

A) No, because the slope coefficients in the autoregressive model have opposite signs.

B) Yes, because the coefficient on yt-4 is large compared to its standard error.

C) Yes, because the coefficient on yt is small compared to its standard error.

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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.l)

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Collier most likely chose to use first-differenced data in the autoregressive model:

A) to increase the explanatory power.

B) because the time trend was significant.

C) in order to avoid problems associated with unit roots.

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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.k)

JoeyDVivre

Barry Phillips, CFA, is analyzing quarterly data. He has estimated an AR(1) relationship (xt = b0 + b1 × xt-1 + et) and wants to test for seasonality. To do this he would want to see if which of the following statistics is significantly different from zero?

A) Correlation(et, et-1).

B) Correlation(et, et-5).

C) Correlation(et, et-4).

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Although seasonality can make the other correlations significant, the focus should be on correlation(et, et-4) because the 4th lag is the value that corresponds to the same season as the predicted variable in the analysis of quarterly data.

JoeyDVivre

The table below shows the autocorrelations of the lagged residuals for quarterly theater ticket sales that were estimated using the AR(1) model: ln(salest) = b0 + b1(ln salest − 1) + et. Assuming the critical t-statistic at 5% significance is 2.0, which of the following is the most likely conclusion about the appropriateness of the model? The time series:

Lagged Autocorrelations of the Log of Quarterly Theater Ticket Sales
Lag	Autocorrelation	Standard Error	t-Statistic
1	−0.0738	0.1667	−0.44271
2	−0.1047	0.1667	−0.62807
3	−0.0252	0.1667	−0.15117
4	0.5528	0.1667	3.31614

A) contains ARCH (1) errors.

B) contains seasonality.

C) would be more appropriately described with an MA(4) model.

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The time series contains seasonality as indicated by the strong and significant autocorrelation of the lag-4 residual.