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David Harper CFA FRM

David Harper CFA FRM
Staff member
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Learning objectives: Describe linear and nonlinear time trends. Explain how to use regression analysis to model seasonality. Describe a random walk and a unit root.

Questions:

21.1.1. The following seasonal dummy model estimates the quarterly growth rate (in percentage terms) of housing starts:



The model's intercept (δ) equals +1.20 and the gamma coefficients are the following: γ(1) = -2.60, γ(2) = +7.90, and γ(3) = +1.80. According to this model, when does the growth rate peak?

a. The peak depends on the year
b. The growth rate peaks at 9.10 in the second quarter
c. The growth rate peaks at 8.30 in the fourth quarter
d. It never peaks because the growth rate has a deterministic, upward trend


21.1.2. Peter wants to model the following AR(2) time series: Y(t) = 0.750*Y(t-1) - 0.1250*T(t-2) + e(t). He wonders if this AR(2) is stationary. He realizes that he can write this as a lag polynomial:
  • Y(t) = 0.750*L - 0.1250*L^2 + ε(t); first he expresses the model with lag operators
  • (1 - 0.750*L + 0.1250*L^2)*Y(t) =ε(t); then he isolates the lag polynomial
  • (1 - 0.50*L)(1 - 0.250*L)*Y(t) = ε(t); finally, he factors the polynomial
Which of the following statements about this AR(2) time series is TRUE?

a. This AR(2) is non-stationary and Peter incorrectly factored the polynomial
b. This AR(2) is non-stationary because the sum of the coefficients is less than one
c. This AR(2) is stationary because the sum of the coefficients is less than one
d. This AR(2) is stationary because both roots of the characteristic equation are greater than one


21.1.3. Sally considers two series for her model: a linear trend model (aka, deterministic trend), and a random walk with drift. Each is simulated below (n = 100 steps):



Her two models are given by the following specifications:
  • Y(t) = δ(0) + δ(1)*t + ε(t), and
  • Y(t) = δ(0) + Y(t-1) + ε(t).
Sally realizes that she prefers to transform both series so that they are stationary. Given her goal, what is her BEST approach?

a. Sally does not need to do anything because both series are already stationary
b. Sally can detrend the random walk but she should difference the linear trend model if she seeks white noise processes
c. Sally can difference the random walk but she should detrend the linear trend model if she seeks white noise processes
d. Sally wants to estimate the slope coefficients because she can observe that both series already generalize (i.e., are versions of) an MA(1) model where the parameter is the trend's slope

Answers:
 
Last edited:

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
An additional note about this question set: although they are simple applications, I wrote R code snippets for each of these questions. See link below at my data science blog (or my github). The seasonal series (#1) is a trivial simulation. For Peter's AR(2) series (#2), I found the polynom package (at https://cran.r-project.org/web/packages/polynom) but it is not necessary, we can factor and find the zeros by hand. In fact, for factoring a polynomial, I'd probably first use WolframAlpha at https://www.wolframalpha.com/. For Sally's series (#3), you'll notice that I used the arima.sim() function which is very popular and versatile for time series simulation.
 
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