TSS module 9 Time series differencing

Cryer and Chan show how differencing converts a non-stationary ARIMA process to a stationary time series.

** Question 1.2: Linear over three time points

Suppose that Y_t = M_t + e_t     with      M_t = M_t-1 + å_t

The time series M(t) is linear over three consecutive points, so the least squares estimator of M(t) is ⅓ (Y_t-1 + Y_t + Y_t+1)  

Which of the following is stationary?

A.      Y(t)

B.       Y(t)

C.       ²Y(t)

D.        Y(t)

E.      Y(t) – Y(t-1) + Y(t-2)

Answer 1.2: C

If M_t is linear over three points, the best estimate for M_t is the centered moving average of three Y_t values:

If we remove the trend, the time series is stationary.

Simplify the right hand side of this equation:

= –⅓ (Y_t+1 –2 Y_t + Y_t-1)

= –⅓ ∇(∇Y_t+1)

= –⅓ ∇²(Y_t+1)

See Cryer and Chan, chapter 5, page 91

**Question 1.3: First difference

Suppose Y_t = M_t + e_t and M_t = M_t-1 + å_t

What is ∇Y_t (the first difference of Y_t)?

A.      å_t

B.      e_t

C.      å_t – e_t – e_t-1

D.     å_t + e_t – e_t-1

E.      å_t + e_t + e_t-1

Answer 1.3: D

(See Cryer and Chan, chapter 5, page 90, Equation 5.1.9)

∇Y_t = Y_{t –} Y_t-1 = M_t + e_t – (M_t-1 + e_t-1) = å_t + e_t – e_t-1

Final exam problems give Y_t = k × M_t + e_t (where k is a scalar) and the variances of e_i and å_t. You must work out variances, auto-covariances, and autocorrelations.

**Question 1.4: ARIMA(p,1,q) process

An ARIMA(p,1,q) process Y_t has first differences ∇ Y_t that are an ARMA(p,q) process with parameters ö_i and è_j.

The ARIMA(p,1,q) process is written as a non-stationary moving average process ARMA(p+1, q).

What is the coefficient of Y_t-2 in the non-stationary ARMA(p+1, q) process?

A.      1 – ö₂

B.      ö₁ – ö₂

C.      ö₂ – ö₁

D.     ö₂ – 1

E.      1 – ö₁ – ö₂

Answer 1.4: C

See equation 5.2.2 on page 92.

Intuition: The ARIMA(p,q) process is

Y_t – Y_t-1 = ö₁ (Y_t-1 – Y_t-2) + ö₂ (Y_t-2 – Y_t-3) + … + ö_p (Y_t-p – Y_t-p-1) + å_t – è₁ å_t-1 – è₂ å_t=2 – … – è_q å_t-q

Rewrite this as

Y_t = (1 + ö₁) Y_t-1 + (ö₂ – ö₁) Y_t-2 + (ö₃ – ö₂) Y_t-3 + … + (ö_p – ö_p-1) Y_t-p + + å_t – è₁ å_t-1 – è₂ å_t=2 – … – è_q å_t-q

**Question 1.5: ARIMA(p,1,q) process

An ARIMA(p,1,q) process Y_t has first differences ∇ Y_t that are an ARMA(p,q) process with parameters ö_i and è_j.

The ARIMA(p,1,q) process is written as a non-stationary moving average process ARMA(p+1, q).

What is the coefficient of å_t-2 in the non-stationary ARMA(p+1, q) process?