Consider the following time-series process given by

𝑦
𝑡
=
(
1
+
0.5
𝐿
+
0.6
𝐿
2
)
𝜀
𝑡

where
𝐿
denotes the lag operator and
𝜀
𝑡
∼
𝑊
𝑁
(
0
,
3
)
. What is the lag 3 autocorrelation of
𝑦
𝑡
,
𝜌
(
3
)
?

Question

Consider the following time-series process given by

𝑦
𝑡
=
(
1
+
0.5
𝐿
+
0.6
𝐿
2
)
𝜀
𝑡

where 
𝐿
 denotes the lag operator and 
𝜀
𝑡
∼
𝑊
𝑁
(
0
,
3
)
. What is the lag 3 autocorrelation of 
𝑦
𝑡
, 
𝜌
(
3
)
?

BlackTom AI · Accepted Answer

We are given a time-series process y_t = (1 + 0.5 L + 0.6 L^2) ε_t, where L is the lag operator and ε_t ~ WN(0, 3).

First, recognize that y_t is an MA(2) process in terms of ε_t, since y_t = ε_t + 0.5 ε_{t-1} + 0.6 ε_{t-2}.

To compute the lag-3 autocorrelation ρ(3) of y_t, we consider the covariance between y_t and y_{t-3} and then standardize by the variance of y_t.

The term y_t depends only on the current and two past innovations: ε_t, ε_{t-1}, and ε_{t-2}.
Similarly, y_{t-3} depends on ε_{t-3}, ε_{t-4}, and ε_{t-5}.

Because the white-noise innovations ε_t are uncorrelated across time (and have finite variance), there is no overlap between the set {ε_t, ε_{t-1}, ε_{t-2}} and the set {ε_{t-3}, ε_{t-4}, ε_{t-5}}.

As a result, the joint product terms that would appear in Cov(y_t, y_{t-3}) involve products like ε_t ε_{t-3}, ε_{t-1} ε_{t-4}, etc., each with zero expectation due to independence and zero mean of ε_t, so the cross-covariance is zero.

Consequently, the cross-covariance at lag 3 is zero. In the context of an MA(q) process, autocovariances vanish for lags beyond q, and the autocorrelation at lag 3 is determined by Cov(y_t, y_{t-3}) divided by Var(y_t). Since Cov(y_t, y_{t-3}) = 0, the autocorrelation at lag 3 must be zero.

Note that while Var(y_t) is nonzero (it equals the variance contributed by ε_t, ε_{t-1}, and ε_{t-2} scaled by the filter), it does not affect the conclusion that the correlation at lag 3 is zero because the numerator is zero.

In summary, because y_t is an MA(2) process in ε_t, there is no overlap in the underlying shocks at lag 3, leading to a zero autocovariance, and hence a zero autocorrelation at lag 3.

econ_475_120251_244434 Problem Set 3

Consider the following time-series process given by 𝑦 𝑡 = ( 1 + 0.5 𝐿 + 0.6 𝐿 2 ) 𝜀 𝑡 where 𝐿 denotes the lag operator and 𝜀 𝑡 ∼ 𝑊 𝑁 ( 0 , 3 ) . What is the lag 3 autocorrelation of 𝑦 𝑡 , 𝜌 ( 3 ) ?

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