Proofs

Under the conditions (A6) and (A7) we can apply the Theorem 2.2 for (d_t(θ₀))_t∈N to obtain

P Ã

Tlim→∞

d^(j)_T (θ₀) q

2V(d^(j)_T (θ₀)) ln lnV(d^(j)_T (θ₀))

= 1

!

= 1. (B.40)

where d^(j)_T (θ₀) is thej-th element of d_T(θ).

Using (A6) and let vjj =¡V¯¢

jj, then we have 1 = lim

T→∞

d^(j)_T (θ0)/√ q T

2^V(d(j)T_T^(θ0))ln ln^V(d(j)T_T^(θ0))T

= lim

T→∞

d^(j)_T (θ)/√ p T

2v_jjln lnT a.s. , which means for some ², for P-a.s. ω

d^(j)_T (θ₀(ω))

√T ≤(1 +²)p

2v_jjln lnT , (B.41) for T great enough (depending on ω).

Now using step 1 and (B.41) we have

¯¯¯

¯√

T(˜θ_T −θ₀)¯

¯¯

¯≤M¡¯¯¯

¯−D(θ¯ ₀)⁻¹¯

¯¯

¯+²¢

(1 +²)q 2 max

j v_jjln lnT . So we proved the first statement of the lemma. With the same argument we can prove ¯¯¯¯η_T −η₀¯¯¯¯

pln lnT /T ≤˜c.

Using

θˆT −θ0 =h(ˆηT)−h(η0) = dh

dη⁰(η_t^∗)(ˆηT −η0),

and also the strong consistency of ˆη_T (h is invertible following the implicit function theorem) and continuity of dd^hη in neighborhoods of η₀, the second statement is also proved.

2

Proof of Lemma 2.6:

Using the triangle inequality for the norm we have

¯¯¯

¯¹

TD_Tb(˜θ_T, θ₀)−D¯₀¯

¯¯ p ¯

ln lnT /T

≤

¯¯¯¯¹

TD_T(b(˜θ_T, θ₀))−D(b(˜¯ θ_T, θ₀))¯¯¯¯ pln lnT /T

| {z }

≤K , because of the condition (A10) +

≤

¯¯¯¯d^D¯ d^θ(θ∗)

¯¯¯¯¯¯¯¯_{b(˜θT,θ0)−θ0}¯¯¯¯

z }| {

¯¯¯¯D(b(˜¯ θ_T, θ₀))−D¯₀¯¯¯¯

√ln lnT

≤K,˜ because ¯

¯¯

¯dD^¯

dθ(θ^∗)¯

¯¯

¯ ≤ K1 for T large enough and

¯¯¯¯_{b(˜θT,θ0)−θ0}¯¯¯¯

√ln lnT ≤ c2 from Lemma 2.5.

2

Proof of Lemma 2.9:

For any B, we choose Γ = B(B₀⁻¹Γ₀), then ∆Π = (−B⁻¹Γ +B⁻¹₀ Γ₀) = 0, also A = 0. That means, for any B, we can choose Γ such that the third term on the right hand side of the equation (B.30) equals zero.⁶⁰ Since the Ω is unconstrained, for the maximum of ¯l in (B.30) we need only to decide Ω.

LetQbe the probability measure ofN(0,Ω) andq :R^G →R₊be the density function of Q. Let Q0 be the P-measure of N(0,Ω0). Then

E_Q0[lnq(·)] = 1

2ln det Ω− 1

2 tr (ΩΩ⁻¹₀ ).

Now using Jensens inequality, E_Q0[lnq(·)] has it’s unique maximum at Ω₀. 2

In order to prove Lemma 2.10, We quote at first the definition of mixingale in Mcleish (1975).

Definition 2.16 A sequence of one-dimensional r.v. is called mixingale, if there exist sequences of positive constants(c_n)_n∈N and(ϕ_m)_m≥0 withlimϕ_m = 0, such that

E£

En−m[Xn]²¤

≤ϕmcn

E£

(X_n−E_n+m[X_n])²¤

≤ϕ_m+1c_n.

60At this point the problem of observational differentiability can be seen clearly: the MLE is uniquely identified only by Π0. From Π0to (B,Γ) is the problem we have discussed in Chapter 3.

Theorem 2.17 (Strong law of large number for mixingales) ( (1.6) The-orem, p.831, Mcleish(1975) and (1.9) Corollary p.832 )

Let X_t, t ∈ N be a mixingale with constants (c_n)_n∈N and (ϕ_m)_m≥0 defined above. If (i) P_∞

i=1c²_i/i² < ∞ (ii) ∃ak > 0, k∈N such that P_∞

k=1ak < ∞ and (iii) P_∞

k=1ϕ²_k(a⁻¹_k −a⁻¹_k−1)<∞, then

Tlim→∞

P_T

t=1Xt

T = 0 a.s..

2

Proof of Lemma 2.10

Recall that there are two kinds of variables collected inXt = (Yt−1,· · · , Yt−p, ξt).

Without loss of generality we consider here only one-dimensional processes.

Consider the convergence of _T¹ P_T

t=1ξ_tV_t. This is a case of the strong law of large number (SLLN) for independent r.v’s. Using Theorem 5.4.1 p.124 and Corollary p.125 in Chung (1974) we know for independent r.v’s X_t

X∞

t=1

E[X_t²]/t² <∞ ⇒ lim

T→∞

1 T

XT

t=1

X_t = 0 a.s.. (B.42)

We know that ξtVt are independent under t and E[(ξtVt)²] = ξ_t²σ². We will use the following lemma.

Lemma 2.18 Let q_t, t∈N be a sequence.

Tlim→∞

1 T

XT

t=1

q²_t <∞ ⇒ X∞

t=1

q_t²

t² <∞ . (Without proof).

Because limT→∞ 1 T

P_T

t=1ξ_t² = Ξ², then lim_T→∞P_T

t=1

E[(ξtVt)²]

t² = lim_T→∞P_T

t=1σ^2ξ_t^2t2 < ∞. Using the theorem of Chung (B.42) it follows _T¹ξ_tV_t→0 a.s..

Consider now the convergence of _T¹ P_T

t=1Y_t−kV_t, k = 1,· · · , p. We apply at first the linear decomposition of Yt in (B.26) and consider ^PY˜t−k_T ^Vt and

Pξ˜t−kVt

T separately, where Y˜_t=

X∞

i=0

ψ_iV_t−i, ξ˜_t = X∞

i=0

ψ_iξ_t−i

For the convergence of ^Pξ˜t−k_T ^Vt, use the following lemma.

Lemma 2.19 Assuming that the dynamic of Y˜t is stationary. Then

Tlim→∞

1 T

XT

t=1

ξ_t² = Ξ² ⇒ lim

T→∞

1 T

XT

t=1

ξ˜_t² <∞.

(Without proof.)

The condition lim_{T→∞ T}¹ P_T

t=1ξ˜_t² <∞leads to lim_{T→∞ T}¹ P_T

t=1ξ_tV_t = 0 a.s.. The reason is the same as in the discussion about the convergence of _T¹ P_T

t=1ξ_tV_t above.

We consider now PY˜_t−kV_t. We note at first that ˜Y_t is mixingale with c²_n = (EV_t²)²/(1−λ²) and ϕ²_m = λ^2m, where λ maximal absolute eigenvalue of dynamics ⁶¹ is a value 0 < λ < 1. We choose a_k = k⁻², then it can be checked that thesec_n, ϕ_m, a_ksatisfy the conditions in the theorem. Therefore

1 T

P_T

t=1Y˜t−kVt= 0 a.s. . 2

Proof of Lemma 2.11

Following the linear decomposition for Y_t, we rewrite the terms in the r.h.s.

of (B.34) as the following 1

T XT

t=1

Y_t−k1Y_t−k2 = 1 T

XT

t=1

Y˜_t−k1Y˜_t−k2 + 1 T

XT

t=1

Y˜_t−k1ξ˜_t−k2

+ 1 T

XT

t=1

Y˜_t−k2ξ˜_t−k1 + 1 T

XT

t=1

ξ˜_t−k1ξ˜_t−k2 , where k₁ = 0,· · · , p, k₂ = 0,· · ·p.

1 T

XT

t=1

Yt−kξt = 1 T

XT

t=1

Y˜t−kξt+ 1 T

XT

t=1

ξ˜t−kξt, and

1 T

XT

t=1

ξ_tξ_t.

Therefore we should discuss three kinds of convergences (i) lim_{T→∞ T}¹ P_T

t=1Y˜_t−k1Y˜_t−k2 =c, a.s. because ( ˜Y_t−k1Y˜_t−k2)_t∈N is strictly stationary⁶²and is therefore ergodic by the ergodic theorem (Theorem 6.21) on p.113 in Breimen(1992).

61See Hamilton (1994), p.259

62See Hamilton (1994) p.46

(ii) limT→∞ 1 T

P_T

t=1Y˜t−kξ˜t =c, a.s. by using the theorem of Mcleish with ϕ²_m =λ^2m and c²_n =ξ_n²(EV_n²)²/(1−λ²).

The convergence of the series of the third kind is the assumption.

2

Proof of Lemma 2.12

at first we want to show that the linear combination of the first differentials α⁰d_t(θ₀) satisfies (W2).

Recall

∂l(θ)

∂β_ij

¯¯

θ0 =−Y¯_jtW_it(θ₀) +¡

β^ji−V_jtW_it(θ₀)¢ where the second term on the r.h.s has an expectation of zero.

We observe that the term α⁰d_t(θ₀) has the structure XI

i=1

X_it²_it (B.43)

whereX_tisF_t−1-measurable, (²_it)_i=1,···,I are i.i.d withE²_it = 0 andE|²_it|^2+δ<

∞. Let Zt =P_I

i=1Xit²it and σ²_t =Et−1Z_t², then sup

t

E_t−1|Z_t|^2+δ σ_t^2+δ <∞.

First we “orthogonize” Z_t, i.e. we can find ˜X_it, ˜²_it such that P_I

i=1X_it²_it = P_I

i=1X˜_it˜²_it and ˜X_it are still F_t−1 measurable, and ˜²_it are uncorrelated under i and also E˜²it= 0, E|˜²it|^2+δ <∞. Then

E_t−1|Z_t|^2+δ

σ^2+δ_t = E_t−1£

|P_I

i=1X˜_it˜²_it|^2+δ¤

¡E_t−1£ (P_I

i=1X˜_it˜²_it)²¤¢^2+δ

2

≤ I^2+δP_I

i=1E_t−1[|X˜_it²˜_it|^2+δ]

¡ P_I

i=1E_t−1[( ˜X_it²˜_it)²]¢^2+δ

2

≤I^2+δ P_I

i=1|X˜_it|^2+δE|˜²_it|^2+δ P_I

i=1|X˜_it|^2+δ¡

E[˜²²_it]¢^2+δ

2

≤I^2+δmax

i

E|˜²_it|^2+δ

¡E[˜²²_it]¢^2+δ

2

< ∞.

2

Proof of Lemma 2.13

We already showed in (B.35) that

Tlim→∞

1 T

XT

t=1

E_t−1£∂l(θ₀;Y_t, X_t)

∂θi

∂l(θ₀;Y_t, X_t)

∂θj

¤ = ( ¯V_d)_ij.

Because the expectation of the l.h.s. converges to the same limit, it follows that

( ¯V_d)_ij = lim

T→∞

1 T

XT

t=1

E£∂l(θ₀;Y_t, X_t)

∂θi

∂l(θ₀;Y_t, X_t)

∂θj

¤

=− lim

T→∞

1 T

XT

t=1

E[∂²l(θ₀;Y_t, X_t)

∂θ_i∂θ_j ] =−D(θ₀)_ij 2

Proof of Lemma 2.14

Analogous to the proof of Lemma 2.11 we simply need to consider the con-vergence speed for the three types of series PY˜_t−k1Y˜_t−k2/T,PY˜_t−kξ_t/T and Pξ_tξ_t/T. Since the convergence speed of the third type is assumed, we only need to consider the first and second types.

Let Z_t represent ˜Y_t−k1Y˜_t−k2 or ˜Y_t−kξ_t. The basic idea of this proof is to represent Z_t−E[Z_t] as

Z_t−E[Z_t] = lim

N→∞Z_t−E_t−N[Z_t] = lim

N→∞

XN

n=0

¡E_t−n[Z_t]−E_t−n−1[Z_t]¢ , where E[Zt] = limn→∞Et−N[Zt]. For any fixedn,¡

Et−n[Zt]−Et−n−1[Zt]¢ is a martingale difference process. Then we can apply Theorem 2.2 to controlt∈N

the convergence speed.

At first let Z_t= ˜Y_t−k1Y˜_t−k2 and ϕ⁽ⁿ⁾_t =¡

E_t−n[Z_t]−E_t−n−1[Z_t]¢

. We consider the case n≥k₁∨k₂. After some calculation we get

ϕ⁽ⁿ⁾_t = ψn−k1ψn−k2(V_t−n² −E[V_t−n² ]) +¡

ψn−k2Et−n−1Y˜n−k1 +ψn−k1Et−n−1Y˜n−k2

¢Vt−n. LetEV_t² =σ². Because of stationarity there existsc₁⁶³such thatP_∞

i=0ψ_i² ≤ c₁. Calculating the variation process V_T⁽ⁿ⁾ =V¡ P_T

t=1ϕ⁽ⁿ⁾_t ¢ V_T⁽ⁿ⁾

T = 1 T

XT

t=1

E_t−n−1(ϕ⁽ⁿ⁾_t )²

= 2ψ_n−k² ₁ψ²_n−k2σ⁴ +σ²ψ_n−k² ₂ 1 T

XT

t=1

E_t−n−1[ ˜Y_t−k1]²

| {z }

→σ²P_∞

i=1+n−k1ψ_i² a.s.

+ 2σ²ψ_n−k2ψ_n−k1 1 T

XT

t=1

E_t−n−1[ ˜Y_t−k1]E_t−n−1[ ˜Y_t−k2]

| {z }

→σ²P_∞

i=1+n−k2ψi+k2−k1ψi a.s.

+σ²ψ²_n−k1 1 T

XT

t=1

E_t−n−1[ ˜Y_t−k2]²

| {z }

P_∞

i=1+n−k2ψ²_i a.s.

.

63see Hamilton (1994)

The a.s. convergences above are obtained by applying the ergodic theorem.

Therefore

Tlim→∞

V_T⁽ⁿ⁾

T =V⁽ⁿ⁾ ≤σ⁴(1 +c1)(|ψn−k1|+|ψn−k2|)².

Because ϕ⁽ⁿ⁾_t also has the product form (B.43), (W2) is satisfied, so we can apply Theorem 2.2 to get

1 = lim

T→∞

P_T

t=1ϕ⁽ⁿ⁾_t q

2V_T⁽ⁿ⁾ln ln 2V_T⁽ⁿ⁾

= lim

T→∞

P_T

t=1ϕ⁽ⁿ⁾_t q

T2V_T⁽ⁿ⁾/Tln ln(T2V_T⁽ⁿ⁾/T)

= lim

T→∞

P_T

t=1ϕ⁽ⁿ⁾_t

√T2V⁽ⁿ⁾ln lnT a.s.

Therefore

Tlim→∞

P_T

t=1ϕ⁽ⁿ⁾_t

√2Tln lnT =√

V⁽ⁿ⁾ ≤σ²√

1 +c₁(|ψ_n−k1|+|ψ_n−k2|).

Because XT

t=1

(Z_t−E[Z_t]) = XT

t=1 N→∞lim

XN

n=0

ϕ⁽ⁿ⁾_t = lim

N→∞

XN

n=0

XT

t=1

ϕ⁽ⁿ⁾_t , then

Tlim→∞ lim

N→∞

XN

n=0

P_T

t=1ϕ⁽ⁿ⁾_t

√T ln lnT ≤ lim

N→∞

XN

n=0 Tlim→∞

P_T

t=1ϕ⁽ⁿ⁾_t

√T ln lnT, therefore

Tlim→∞

P_T

t=1(Zt−E[Zt])

√T ln lnT ≤σ²√ 1 +c₁

X∞

n=0

(|ψ_n−k1|+|ψ_n−k2|) = c <∞.

Now for Z_t = ˜Y_t−kξ_t we can easily have ϕ⁽ⁿ⁾_t = ξ_tψ_n−kV_t−n. It also has the product form (B.43) and we can have V_T⁽ⁿ⁾/T =σ²ψ_n−k² P

tξ²_t/T. Then

Tlim→∞

P_T

t=1(Z_t−E[Z_t])

√T ln lnT ≤ p

Ξ²σ X∞

n=0

ψ_n−k² <∞.

2

Remark to the proof of Theorem 2.15

We saw that in Theorem 2.3 the important properties for proving the LR barrier are (i) the P-a.s. convergences of l_T/T and its differentials, (ii) the local uniformity w.r.t parameters of these convergences, and (iii) the control of the convergence speed. After checking the conditions for the structural model we summarize here which properties of the structural model lead to the properties above. It is important that we decompose Y_t into two parts:

the exogeneous part and the autoregressive part. The convergences of lT/T and its differentials are only possible when the autoregressive part is station-ary and the exogeneous part satisfies the moment condition (B.36). We used three theorems of strong law of large number: for strict stationary processes, for independent processes, and for mixingales. The local uniformity of the convergences is due to the normal distributions of U_t. Thanks this all series considered for convergences have a product form of parameters and the av-erages which are P-a.s. convergent. For the control of convergence speed, we use LIL Theorem for martingales. The conditions for this theorem can be satisfied is because of the normal distribution of Ut and the stationarity of (the autoregressive part of) the process.

References

Akaike, H. (1973). Information theory and extension of maximum likeli-hood priciple. 2nd International Symposium on Information theory, pages 267–281.

Amemiya, T. (1985). Advanced Econometrics. Basil Blackwell, 1st edition.

Amisano, G. and Giannini, C. (1997). Topics in Structural VAR Eco-noetrics. Springer, 2nd edition.

Charemza, W. (1997). New directions in econometric practice : general to specific modelling, cointegration, and vector autoregression. Elgar, 2nd edition.

Crowder, M. J. (1976). Maximum likelihood estimation for dependent observations. Journal of the Royal Statistical Society, Series B, 38:45–53.

Davidson, R. and Mackinnon, J. (1993a). Esitmation and Inference in Econometrics. Oxfort University Press, 1st edition.

— (1993b). Estimation and Inference in Econometrics. Oxford University Press, 1st edition.

Dhaene, G. (1997). Encompassing, Formulation, Properties and Testing.

Springer Verlag, 1st edition.

Dhrymes, P. J.(1993).Topics in Advanced Econometrics. Springer-Verlag, 1st edition.

Engle, R. F. and Granger, W. J. (1987). Cointegration and error correction: representation, estimation and testing. Econometrica, 55:251–

276.

Fair, R. C.(1984).Specification, estimation, and analysis of macroeconomic models. Harvard Univ. Pr, 1st edition.

Frohn, J. (1995). Grundausbildung in Okonometrie. de Gruyter, 2nd edi-tion.

Frydman, R. (1980). A proof of the consistency of maximum likelihood es-timators of nonlinear regression models with autocorrelated errors. Econo-metrica, 48:853–860.

Gourieroux, C. and Monfort, A. (1984). Pseudo-maximum likelihood methods: theory. Ecomonetrica, 52:681–700.

— (1996). Testing non-nested econometric models. Handbook of Economet-rics, 4:2583–2637.

Granger, C. W. J. and White, H. (1995). Comments on testing eco-nomic theories and the use of model selection criteria. Journal of Econo-metrics, 67:173–187.

Granger, J. (1990). Modelling Economic Series. Clarendon, 1st edition.

Hamilton, J. D. (1994). Time Series Analysis. Princeton, 1st edition.

Hargreaves, C. P.(1994).Nonstationary Time Series Analysis and Coin-tegration. Oxfort University Press, 1st edition.

Hendry, D. (1995). Dynamic Econometrics. Oxfort University Press, 1st edition.

Hendry, D. F. and Krolzig, H. M. (2001). New development in au-tomatic general to specific modelling. Econometrics and Philosophy of Economics, edited by B. P. Stigum.

Hendry, D. F. and Richard (1988). The Real Term Structure and Con-sumption Growth. Journal of Financial Economics, 22:305–333.

Johansen, S. (1991). estimation and hypothesis testing of cointegration vectors in Gausian vector autoregressive models. Econometrica, 59:1151–

1180.

— (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, 1st edition.

Judge, G.(1985). The Theory and Practice of Econometirics. John Willey and Sons, 2nd edition.

Klein, L. R.(1983). Lectures in Econometrics. North-Holland, 1st edition.

Koopmans, T. C. and Hood, W. C. (1953). The estimation of simulta-neous linear economic relations. Studies in Econometric Method edited by T. C. Koopmans and W. C. Hood.

Leamer, E. E. (1983). Let’s take the con out of econometrics. American Economic Review, 23:31–43.

Lucas, R. E. (1976). Econometric policy evaluation: a critique. Carneie-Rochester Conference Series on Public Policy, pages 19–46.

P¨otscher, B. M. and Prucha, I. R. (1997). Dynamic Econometrics.

Springer Verlag, 1st edition.

Powell, A. A. and Murphy, C. W. (1997). Inside a modern macroe-conometric model : a guide to the Murphy model. Springer Verlag, 1st edition.

Im Dokument Econometric structural models : a model selection approach (Seite 98-109)