kasinkei

The model xt = b0 + b1 xt − 1 + b2 xt − 2  + εt is:

A) an autoregressive conditional heteroskedastic model, ARCH.

B) a moving average model, MA(2).

C) an autoregressive model, AR(2).

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This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p = 2 (that is, 2 lags).

kasinkei

The model xt = b0 + b1 xt-1 + b2 xt-2 + b3 xt-3 + b4 xt-4 + εt is:

A) an autoregressive model, AR(4).

B) an autoregressive conditional heteroskedastic model, ARCH.

C) a moving average model, MA(4).

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This is an autoregressive model (i.e., lagged dependent variable as independent variables) of order p=4 (that is, 4 lags).

kasinkei

Which of the following statements regarding covariance stationarity is CORRECT?

A) A time series that is covariance stationary may have residuals whose mean changes over time.

B) The estimation results of a time series that is not covariance stationary are meaningless.

C) A time series may be both covariance stationary and have heteroskedastic residuals.

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Covariance stationarity requires that the expected value and the variance of the time series be constant over time.

kasinkei

Which of the following statements regarding covariance stationarity is CORRECT?

A) A time series that is covariance stationary may have residuals whose mean changes over time.

B) The estimation results of a time series that is not covariance stationary are meaningless.

C) A time series may be both covariance stationary and have heteroskedastic residuals.

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Covariance stationarity requires that the expected value and the variance of the time series be constant over time.

kasinkei

To qualify as a covariance stationary process, which of the following does not have to be true?

A) Covariance(xt, xt-1) = Covariance(xt, xt-2).

B) E[xt] = E[xt+1].

C) Covariance(xt, xt-2) = Covariance(xt, xt+2).

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If a series is covariance stationary then the unconditional mean is constant across periods. The unconditional mean or expected value is the same from period to period: E[xt] = E[xt+1]. The covariance between any two observations equal distance apart will be equal, e.g., the t and t-2 observations with the t and t+2 observations. The one relationship that does not have to be true is the covariance between the t and t-1 observations equaling that of the t and t-2 observations

kasinkei

To qualify as a covariance stationary process, which of the following does not have to be true?

A) Covariance(xt, xt-1) = Covariance(xt, xt-2).

B) E[xt] = E[xt+1].

C) Covariance(xt, xt-2) = Covariance(xt, xt+2).

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If a series is covariance stationary then the unconditional mean is constant across periods. The unconditional mean or expected value is the same from period to period: E[xt] = E[xt+1]. The covariance between any two observations equal distance apart will be equal, e.g., the t and t-2 observations with the t and t+2 observations. The one relationship that does not have to be true is the covariance between the t and t-1 observations equaling that of the t and t-2 observations

kasinkei

Rhonda Wilson, CFA, is analyzing sales data for the TUV Corp, a current equity holding in her portfolio. She observes that sales for TUV Corp. have grown at a steadily increasing rate over the past ten years due to the successful introduction of some new products. Wilson anticipates that TUV will continue this pattern of success. Which of the following models is most appropriate in her analysis of sales for TUV Corp.?

A) A linear tend model, because the data series is equally distributed above and below the line and the mean is constant.

B) A log-linear trend model, because the data series can be graphed using a straight, upward-sloping line.

C) A log-linear trend model, because the data series exhibits a predictable, exponential growth trend.

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The log-linear trend model is the preferred method for a data series that exhibits a trend or for which the residuals are predictable. In this example, sales grew at an exponential, or increasing rate, rather than a steady rate.

kasinkei

Trend models can be useful tools in the evaluation of a time series of data. However, there are limitations to their usage. Trend models are not appropriate when which of the following violations of the linear regression assumptions is present?

A) Heteroskedasticity.

B) Model misspecification.

C) Serial correlation.





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One of the primary assumptions of linear regression is that the residual terms are not correlated with each other. If serial correlation, also called autocorrelation, is present, then trend models are not an appropriate analysis tool.

kasinkei

Dianne Hart, CFA, is considering the purchase of an equity position in Book World, Inc, a leading seller of books in the United States. Hart has obtained monthly sales data for the past seven years, and has plotted the data points on a graph. Which of the following statements regarding Hart’s analysis of the data time series of Book World’s sales is most accurate? Hart should utilize a:

A) log-linear model to analyze the data because it is likely to exhibit a compound growth trend.

B) mean-reverting model to analyze the data because the time series pattern is covariance stationary.

C) linear model to analyze the data because the mean appears to be constant.

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A log-linear model is more appropriate when analyzing data that is growing at a compound rate. Sales are a classic example of a type of data series that normally exhibits compound growth.

kasinkei

Clara Holmes, CFA, is attempting to model the importation of an herbal tea into the United States which last year was $ 54 million. She gathers 24 years of annual data, which is in millions of inflation-adjusted dollars.
She computes the following equation:

(Tea Imports)t = 3.8836 + 0.9288 × (Tea Imports)t − 1 + et

t-statistics

(0.9328)

(9.0025)

R2 = 0.7942
Adj. R2 = 0.7844
SE = 3.0892
N = 23

Holmes and her colleague, John Briars, CFA, discuss the implication 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 the dependent variable.” Briars says the model actually explains more than that.
Briars asks about the Durbin-Watson statistic. Holmes said that she did not compute it, so Briars reruns the model and computes its value to be 2.1073. Briars says “now we know serial correlation is not a problem.” Holmes counters by saying “rerunning the model and computing the Durbin-Watson statistic was unnecessary because serial correlation is never a problem in this type of time-series model.”
Briars and Holmes decide to ask their company’s statistician about the consequences of serial correlation. Based on what Briars and Holmes tell the statistician, the statistician informs them that serial correlation will only affect the standard errors and the coefficients are still unbiased. The statistician suggests that they employ the Hansen method, which corrects the standard errors for both serial correlation and heteroskedasticity.
Given the information from the statistician, Briars and Holmes decide to use the estimated coefficients to make some inferences. Holmes says the results do not look good for the future of tea imports because the coefficient on (Tea Import)t − 1 is less than one. This means the process is mean reverting. Using the coefficients in the output, says Holmes, “we know that whenever tea imports are higher than 41.810, the next year they will tend to fall. Whenever the tea imports are less than 41.810, then they will tend to rise in the following year.” Briars agrees with the general assertion that the results suggest that imports will not grow in the long run and tend to revert to a long-run mean, but he says the actual long-run mean is 54.545. Briars then computes the forecast of imports three years into the future.With respect to the statements made by Holmes and Briars concerning serial correlation and the importance of the Durbin-Watson statistic:

A) they were both incorrect.

B) Holmes was correct and Briars was incorrect.

C) Briars was correct and Holmes was incorrect.

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Briars was incorrect because the DW statistic is not appropriate for testing serial correlation in an autoregressive model of this sort. Holmes was incorrect because serial correlation can certainly be a problem in such a model. They need to analyze the residuals and compute autocorrelation coefficients of the residuals to better determine if serial correlation is a problem. (Study Session 3, LOS 12.i)

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With respect to the statement that the company’s statistician made concerning the consequences of serial correlation, assuming the company’s statistician is competent, we would most likely deduce that Holmes and Briars did not tell the statistician:

A) the sample size.

B) the value of the Durbin-Watson statistic.

C) the model’s specification.

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Serial correlation will bias the standard errors. It can also bias the coefficient estimates in an autoregressive model of this type. Thus, Briars and Holmes probably did not tell the statistician the model is an AR(1) specification. (Study Session 3, LOS 12.k)

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The statistician’s statement concerning the benefits of the Hansen method is:

A) not correct, because the Hansen method only adjusts for problems associated with serial correlation but not heteroskedasticity.

B) not correct, because the Hansen method only adjusts for problems associated with heteroskedasticity but not serial correlation.

C) correct, because the Hansen method adjusts for problems associated with both serial correlation and heteroskedasticity.

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The statistician is correct because the Hansen method adjusts for problems associated with both serial correlation and heteroskedasticity. (Study Session 3, LOS 12.i)

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Using the model’s results, Briar’s forecast for three years into the future is:

A) $54.543 million.

B) $47.151 million.

C) $54.108 million.

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Briars’ forecasts for the next three years would be:

year one: 3.8836 + 0.9288 × 54 = 54.0388
year two: 3.8836 + 0.9288 × (54.0388) = 54.0748
year three: 3.8836 + 0.9288 × (54.0748) = 54.1083
(Study Session 3, LOS 13.a)

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With respect to the comments of Holmes and Briars concerning the mean reversion of the import data, the long-run mean value that:

A) Briars computes is correct.

B) Briars computes is not correct, and his conclusion is probably not accurate.

C) Briars computes is not correct, but his conclusion is probably accurate.

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Briars has computed a value that would be correct if the results of the model were reliable. The long-run mean would be 3.8836 / (1 − 0.9288)= 54.5450. (Study Session 3, LOS 13.a)

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Given the nature of their analysis, the most likely potential problem that Briars and Holmes need to investigate is:

A) autocorrelation.

B) unit root.

C) multicollinearity.

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Multicollinearity cannot be a problem because there is only one independent variable. For a time series AR model, autocorrelation is a bigger worry. The model may have been misspecified leading to statistically significant autocorrelations. Unit root does not seem to be a problem given the value of b1<1. (Study Session 3, LOS 13.e)