TS Module 2 Time series concepts practice problems

(The attached PDF file has better formatting.)

Time series practice problems variances and covariances

*Question 2.1: Random walk

The time series Y_t = Y_t-1 + e_t is a random walk with ó²_e = 0.25 and Y_t = 0 for t < 1.

What is the standard deviation of Y₉?

A.   0.15

B.   0.25

C.   0.50

D.   1.50

E.   2.25

Answer 2.1: D

        Y_t = å₁ + å₂ + … + å₉

        The error terms are independent, with a variance of 0.25 each.

        The variance of Y₉ is 9 × 0.25 = 2.25.

        The standard deviation of Y₉ is (9 × 0.25)^0.5 = 1.50.

*Question 2.2: Equally weighted moving average

Let Y_t = ½ × (å_t + å_t-1). What is ñ_t,t-1, the correlation of Y_t and Y_t-1?

A.   –1

B.   –0.5

C.   0

D.   0.5

E.   1

Answer 2.2: D

Let ó² = the variance of the error term.

        The covariance of Y_t with Y_t-1 = covariance (½ × (å_t + å_t-1), ½ × (å_t-1 + å_t-2) ) = ¼ ó².

        The variance of Y_t is ¼ × 2 ó² = ½ ó².

        The correlation of Y_t with Y_t-1 = ¼ ó² / ½ ó² = 0.500.

See Cryer and Chan page 15, equation 2.2.16:

 for |t–s| = 1

(2.2.16)

*Question 2.3: Random walk

        Let Y_t be a random walk with zero drift: Y_t = Y_t-1 + å_t, with ó²_e = 1.

        The time series starts at t = 1. (This time series is not stationary.)

What is ñ_4,25?

A.   0.00

B.   0.16

C.   0.40

D.   4.00

E.   25.0

Answer 2.3: C

(4/25)^0.5 = 0.400

(See Cryer and Chan equation 2.2.13)

*Question 2.4: Stationary time series

        X_t is a stationary time series.

        Y_t = â₀ + â₁ t + â₂ t² + X_t

Which of the following is a stationary time series?