Problems and Solutions

ARMA)

3.5 Problems and Solutions

68 3 Autoregressive Moving Average Processes (ARMA)

Solutions

3.1 The traditional way to solve the problem is curve sketching. We consider the first order autocorrelation to be a function ofb:

f.b/D.1/D b 1Cb²: Then the quotient rule for the first order derivative yields

f⁰.b/D 1Cb² 2b²

.1Cb²/² D .1 b/.1Cb/

.1Cb²/² :

The roots of the derivative are given byjbj D 1. Inb D 1 there is a change of sign off⁰.b/, namely from a negative to a positive slope. Hence, in b D 1 there is a relative (and also an absolute) minimum. Because off.b/being an odd function (symmetric about the origin), there is a maximum inbD1. Therefore, the maximum possible correlation in absolute value is

jf. 1/j Df.1/D 1 2:

One may also tackle the problem by more elementary means. Note that forb¤0 1

jbj 2C jbj D 1 pjbj

pjbj

!2

0 ; which is equivalent to

1

2 1

1

jbjC jbj D jbj

1Cb² D jf.b/j;

withf.b/D.1/defined above. Sincejf. 1/j Df.1/D ¹₂, this solves the problem.

3.2 BySnwe denote the following sum for finiten:

S_nD Xn

iD0

gⁱD1CgC: : :Cg^{n 1}Cgⁿ: Multiplication bygyields

g SnDgCg²C: : :CgⁿCg^nC1:

70 3 Autoregressive Moving Average Processes (ARMA)

Therefore, it holds that

Sn g SnD1 g^nC1: By ordinary factorization the formula

SnD 1 g^nC1

1 g

and therefore the claim is verified.

3.3 The absolute summability of.h/follows from the absolute summability of the linear coefficientsfcjgallowing for a change of the order of summation. In order to do so, we first apply the triangle inequality:

1 ²

X1 hD0

j.h/j D X1 hD0

ˇˇ ˇˇ ˇˇ

X1 jD0

cjcjCh

ˇˇ ˇˇ ˇˇ

X1 hD0

X1 jD0

ˇˇc_jcjCh

ˇˇD X1 hD0

X1 jD0

ˇˇc_jˇˇˇˇc_jChˇˇ

D X1

jD0

ˇˇc_jˇˇ X¹

hD0

ˇˇc_jChˇˇ

!

;

where at the end round brackets were placed for reasons of clarity. The final term is further bounded by enlarging the expression in brackets:

X1 jD0

ˇˇc_jˇˇ X¹

hD0

ˇˇc_jChˇˇ

!

X1 jD0

ˇˇc_jˇˇ X¹

hD0

jchj

! :

Therefore, the claim follows indeed from the absolute summability offcjg. 3.4 For the proof we denote.1 aL/ ¹asP₁

jD0˛jL^j, 1

1 aLD

X1 jD0

˛jL^j;

and determine the coefficients˛j. By multiplying this equation with1 aL, we obtain

1D.1 aL/

X1 jD0

˛jL^j D˛0C˛1L¹C˛2L²C: : :

a˛0L¹ a˛1L² a˛2L³ : : : :

Now, we compare the coefficients associated withL^jon the left- and on the right- hand side:

1D˛0; 0D˛1 a˛0; 0D˛₂ a˛₁;

:::

0D˛j a˛j 1; j1 :

As claimed, the solution of the difference equation obtained in this way, (˛j D a˛j 1), is obviously˛jDa^j.

3.5We factorizeP.z/D1Cb₁zC: : :Cbpz^pwith rootsz₁; : : : ;zpof this polynomial (fundamental theorem of algebra):

P.z/Dbp.z z₁/ : : : .z zp/ : From each bracket we factorize zjout such that

P.z/Dbp. 1/^pz₁ : : :zp

1 z

z₁

: : :

1 z

zp

:

Because ofP.0/D 1, we obtainb_p. 1/^pz₁ : : :z_pD1. Therefore the factorization simplifies to

P.z/D

1 z

z₁

: : :

1 z

zp

DP1.z/ Pp.z/ ; with

Pk.z/D1 z

zk D1 kz; kD1; : : : ;p;

72 3 Autoregressive Moving Average Processes (ARMA)

wherekD1=zk. From part a) we know that 1

Pk.L/ D X1

jD0

_k^jL^j with X1

jD0

j_k^jj<1

if and only if

jzkj D 1 jkj > 1 :

Now, consider the convolution (sometimes called Cauchy product) fork¤`:

1 Pk.L/

1 P`.L/ D

X1 jD0

cjL^j

with

cjWD Xj

iD0

_kⁱ_`^{j i}:

We have P₁

jD0jcjj < 1 if and only if both P_k¹.L/ and P_`¹.L/ are absolutely summable, which holds true if and only if

jz_kj> 1 and jz_`j> 1 : Repeating this argument we obtain that

1

P.L/D 1

P₁.L/ 1 Pp.L/ D

X1 jD0

cjL^j with X1

jD0

jcjj<1

if and only if (3.10) holds. Quod erat demonstrandum.

3.6At first we reformulate the autoregressive polynomialA.z/D1 a₁z : : : apz^p in its factorized form with rootsz₁; : : : ;zp (again by the fundamental theorem of algebra):

A.z/D ap.z z₁/ : : : .z zp/ : ForzD1this amounts to

A.1/D ap.1 z1/ : : : .1 zp/ : (3.15)

Because ofA.0/D1we obtain as well:

1D ap. 1/^pz₁ : : :zp: (3.16) Now we proceed in two steps, treating the cases of complex and real roots separately.

(A) Complex roots: Note that for a rootz₁2Cit holds that the complex conjugate, z₂ D z₁;is a root as well. Then calculating with complex numbers yields for the product

.1 z₁/.1 z₂/D.1 z₁/.1 z₁/ D.1 z₁/.1 z₁/ D j1 z₁j²> 0 :

Hence, for p > 2, complex roots contribute positively to A.1/in (3.15). If pD2, the roots are only complex ifa2< 0, since the discriminant isa²₁C4a2; hence,A.1/ > 0by (3.15).

(B) Since the effect of complex roots is positive, we now concentrate on real roots zi, for which it holds thatjzij > 1by assumption. So, we assume without loss of generality that the polynomial has no complex roots, or that all complex roots have been factored out. Two sub-cases have to be distinguished. (1) Even degree: For an evenpwe again distinguish between two cases. Case 1,ap> 0:

Because of (3.16) there has to be an odd number of negative roots and therefore there has to be an odd number of positive roots as well. For the latter it holds that.1 zi/ < 0while the first naturally fulfill.1 zi/ > 0. Hence, as claimed, it follows from (3.15) thatA.1/is positive. Case 2,ap < 0: In this case one argues quite analogously. Because of (3.16) there is an even number of positive and negative roots such that the requested claim follows from (3.15) as well. (2) Odd degree: For an oddpone obtains the requested result as well by distinction of the two cases for the sign ofap. We omit details.

Hence, the proof is complete.

3.7 The normality off"tgimplies a multivariate Gaussian distribution of 0

B@

"tC1

:::

"tCs

1 CAiiN_s

0 B@ 0 B@ 0

::: 0

1 CA; ²I_s

1 CA

with the identity matrixI_sof dimensions. Thes-fold substitution yields xtCsDa^s₁xtCa^s₁ ¹"tC1C: : :Ca₁"tCs 1C"tCs

Da^s₁x_tC

s 1

X

iD0

aⁱ₁"tCs i:

74 3 Autoregressive Moving Average Processes (ARMA)

The sum over the white noise process has the moments

E

s 1

X

iD0

aⁱ₁"tCs i

!

D0; Var

s 1

X

iD0

aⁱ₁"tCs i

! D²

s 1

X

iD0

a²ⁱ₁ ; and, furthermore, it is normally distributed:

s 1

X

iD0

aⁱ₁"tCs i N 0; ²

s 1

X

iD0

a²ⁱ₁

! :

Hence, xtCs given xt follows a Gaussian distribution with the corresponding moments:

xtCsjxt N a^s₁xt; ²

s 1

X

iD0

a²ⁱ₁

! :

AsxtCscan be expressed as a function ofxtand"tC1; : : : ; "tCsalone, the further past of the process does not matter for the conditional distribution ofxtCs. Therefore, for the entire informationI_tup to timetit holds that:

xtCsjI_t N a^s₁xt; ²

s 1

X

iD0

a²ⁱ₁

! :

Hence, the Markov property (2.9) has been shown. It holds independently of the concrete value ofa₁.

3.8 For

x_tDa₂x_{t 2}C"t;

we obtain forsD1the conditional expectations E.xtC1jI_t/Da₂xt 1;and E.xtC1jxt/DE.a₂xt 1C"tC1jxt/Da₂E.xt 1jxt/ ;

withI_tD.xt;xt 1; : : : ;x₁/:As the conditional expectations are not equivalent, the conditional distributions are not the same. Hence, it generally holds that

P.xtC1xjxt/¤P.xtC1xjI_t/ ; which proves thatfxtgis not a Markov process.

References

Andrews, D. W. K., & Chen, H.-Y. (1994). Approximately median-unbiased estimation of autoregressive models.Journal of Business & Economic Statistics, 12, 187–204.

Brockwell, P. J., & Davis, R. A. (1991).Time series: Theory and methods(2nd ed.). New York:

Springer.

Campbell, J. Y., & Mankiw, N. G. (1987). Are output fluctuations transitory?Quarterly Journal of Economics, 102, 857–880.

Fuller, W. A. (1996).Introduction to statistical time series(2nd ed.). New York: Wiley.

Sydsæter, K., Strøm, A., & Berck, P. (1999). Economists’ mathematical manual (3rd ed.).

Berlin/New York: Springer.

Wold, H. O. A. (1938).A study in the analysis of stationary time series. Stockholm: Almquist &

Wiksell.

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