Reading 13: Time-Series Analysis-LOS a 习题精选 center, either, 2011

土豆妮

Session 3: Quantitative Methods for Valuation
Reading 13: Time-Series Analysis

LOS a: Calculate and evaluate the predicted trend value for a time series, modeled as either a linear trend or a log-linear trend, given the estimated trend coefficients.

 

 

David Wellington, CFA, has estimated the following log-linear trend model: LN(x_t) = b₀ + b₁t + ε_t. Using six years of quarterly observations, 2001:I to 2006:IV, Wellington gets the following estimated equation: LN(x_t) = 1.4 + 0.02t. The first out-of-sample forecast of x_t for 2007:I is closest to:

A) 1.88.

B) 4.14.

C) 6.69.

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Wellington’s out-of-sample forecast of LN(x_t) is 1.9 = 1.4 + 0.02 × 25, and e^1.9 = 6.69.

土豆妮

Modeling the trend in a time series of a variable that grows at a constant rate with continuous compounding is best done with:

A) a moving average model.

B) simple linear regression.

C) a log-linear transformation of the time series.

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The log-linear transformation of a series that grows at a constant rate with continuous compounding (exponential growth) will cause the transformed series to be linear.

土豆妮

In the time series model: y_t=b₀ + b₁ t + ε_t, 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.

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The slope is the change in the dependent variable per unit of time. The intercept is the estimate of the value of the dependent variable before the time series begins. The disturbance term should be independent and identically distributed. There is no reason to expect the disturbance term to be mean-reverting, and if the residuals are autocorrelated, the research should correct for that problem.

土豆妮

Clara Holmes, CFA, is attempting to model the importation of an herbal tea into the United States. She gathers 24 years of annual data, which is in millions of inflation-adjusted dollars. The real dollar value of the tea imports has grown steadily from $30 million in the first year of the sample to $54 million in the most recent year.

She computes the following equation:

(Tea Imports)_t = 3.8836 + 0.9288 × (Tea Imports)_{t ? 1} + e_t

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 model’s specification.

B) the sample size.

C) the value of the Durbin-Watson statistic.

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

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

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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) $54.108 million.

B) $54.543 million.

C) $47.151 million.

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Briars’ forecasts for he 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, and his conclusion is probably accurate.

B) Briars computes is correct, but the 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. However, the evidence suggests that the data is not covariance stationary. The imports have grown steadily from $30 million to $54 million. (Study Session 3, LOS 13.a)

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Given the output, the most obvious potential problem that Briars and Holmes need to investigate is:

A) conditional heteroskedasticity.

B) a unit root.

C) multicollinearity.

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Multicollinearity cannot be a problem because there is only one independent variable. Although heteroskedasticity may be a problem, nothing in the output provides information in this regard. A unit root is a likely problem because the slope coefficient is so close to one. In fact, if Holmes and Briars divide the t-statistic of the slope coefficient by the value of the coefficient, they could determine the standard error: 0.1032 = 0.9288 / 9.0025. They could then test the null hypothesis:

H₀ : slope coefficient = 1

H₀ : slope coefficient ≠ 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 this t-statistic value of -0.6899 is not significant so the hypothesis of the slope equaling one cannot be rejected. Given that serial 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 should provide a clue that the data may have a unit root. Note that this suggests that the true value of the slope also equals one, since with a unit root the dependent variable will grow by approximately the amount of the intercept each year. (Study Session 3, LOS 13.k)

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