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TIME SERIES

Solutions to Exercise Sheet 2

1. (a) First note that, sinced

k=1 sk = 0, the period average may be written as

mj =1d

dk=1

Xjk =1d

dk=1

(mj + Yjk).

Thus, we have

E(mj) =1

d

dk=1

mj = mj ,

which shows that mj is an unbiased estimator ofmj.(b) To show that the seasonal component estimator satisfies the model assumptions,we need to show that

dk=1 sk = 0. Now, we may write

dk=1

sk =1b

dk=1

bj=1

(Xjk mj) =1b

bj=1

(dmj dmj) = 0,

as required. Further, we have

E(sk) =1

b

bj=1

E(Xjk mj) =1

b

bj=1

sk = sk,

which shows that sk is also an unbiased estimator of sk.

2. (a) For the additive modelX

t =m

t +st +

Yt, we assume that

st =

st12

and thatmt = 0 + 1t. So the differenced series is

12Xt = Xt Xt12

= mt + st + Yt (mt12 + st12 + Yt12)

= mt mt12 + 12Yt

= 0 + 1t {0 + 1(t 12)} + 12Yt

= 121 + 12Yt.

For stationarity, we check that the expectation and variance are constant, and that the

covariances do not depend on t. Clearly, E(12Xt) = 121, which is a constant. Wealso have

cov(12Xt,12Xt+) = cov(12Yt,12Yt+)

= cov(Yt Yt12, Yt+ Yt+12)

= cov(Yt, Yt+) cov(Yt12, Yt+) cov(Yt, Yt+12)

+cov(Yt12, Yt+12)

= 2Y() Y( 12) Y(+ 12).

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Since this does not depend on t, the differenced series is stationary.(b) For the mixed model Xt = mtst + Yt, we have

12Xt = mtst + Yt (mt12st12 + Yt12)

= (0 + 1t)st {0 + 1(t 12)}st12 + 12Yt

= 121st12 + 12Yt,

which givesE(12Xt) = 121st12.

Since this still depends on t, it is not a constant. However, as st = st12 = st24, wecan eliminate the seasonal effect from 12Xt by applying the same operator again:

212Xt = 12(12Xt) = 121st12 + 12Yt (121st24 + 12Yt12)

= Yt 2Yt12 + Yt24

= 2

12Yt.

We then have E(212Xt) = 0. It can also easily be shown that cov(212Xt,

212Xt+) is

a constant. Hence, 212Xt is a stationary process.

3. Since {Xt} WN(0, 2), we know that E(Xt) = 0, cov(Xt, Xt+) = 0 for = 0 andvar(Xt) =

2. This means that both the expectation ofXt and the covariance ofXtand Xt+ do not depend on t, and so white noise is a weakly stationary process. Thus,the autocovariance function is given by

() = 2 if = 0,

0 if = 0.

It follows that the autocorrelation function is

() =

1 if = 0,0 if = 0.

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