Changes in persistence, spurious regressions and the Fisher hypothesis

A Proofs

Proof of Lemmas 1 and 2.

Results 2(a), 2(c), 3(b), and 3(d), can be found in Phillips’ (1986, Lemma 1, 314–315). As for result 2(b), note that, for 0<λ_y <1:

∑t=1Tyt+1=∑t=1[λyT]ξy,t︷Op(T3/2)+∑t=1[λyT]εy,t+1︷Op(T1/2)+T(1−λy)∑t=1[λyT]εx,t+1︷Op(T1/2)+∑t=[λyT]+1Tεy,t+1︷Op(T1/2),=∑t=1[λyT]ξy,t+T(1−λy)∑t=1[λyT]εy,t+1+Op(T1/2).

Following Phillips’ (1986, Lemma 1, 314–315), is it easy to see that:

T−3/2∑t=1[λyT]ξy,t→Dσy∫0λyWy, and T−1/2(1−λy)∑t=1[λyT]εy,t+1→D(1−λy)σyWy(λy). Result 2(b) then ensues. Results 3(a), 4(a), 4(b), 5(a), 5(b), 8(a), and 9(a) are the same. As for result 2(d), note that, for 0<λ_y <1:

∑t=1Tyt+12=∑t=1[λyT]ξy,t2︷Op(T2)+∑t=1[λyT]εy,t+12︷Op(T)+2∑t=1[λyT]ξy,tεy,t+1︷Op(T)+T(1−λy)(∑t=1[λyT]εx,t+1)2︷Op(T2)+∑t=λyT+1Tεy,t+12︷Op(T)+2(∑t=1[λyT]εx,t+1)∑t=λyT+1Tεy,t+1︷Op(T),=∑t=1[λyT]ξy,t2+T(1−λy)(∑t=1[λyT]εy,t+1)2+Op(T).

Again, is it easy to see that T−2∑t=1[λyT]ξy,t2→Dσy2∫0λyWy2, and, using the continuous mapping theorem, that T−1(1−λy)∑t=1[λyT]εy,t+12→D(1−λy)σy2[Wy(λy)]2. Result 2(d) then ensues. Results 3(c), 4(c), 4(d) 5(c), 5(d), 8(c), and 9(c) are the same. As for result 2(e), note that, for 0<λ_y <1:

∑t=1Tyt+1xt=∑t=1[λyT]ξy,tξx,t−1+(∑t=1[λyT]εx,t+1)∑t=[λyT]+1Tξx,t−1︷Op(T2)+Op(T).

Following Phillips (1986, Lemma 1, 314–315), it is easy to see that the first O_p (T²) term, T−2∑t=1[λyT]ξy,tξx,t−1→Dσyσx∫0λyWxWy(r). The second O_p (T²) term was proved previously. Result 2(e) then ensues. Result 3(e) follows in a similar vein. As for result 4(e), note that, for 0<λ_y =λ_x <1:

∑t=1Tyt+1xt=∑t=1[λyT]ξy,tξx,t−1+T(1−λy)(∑t=1[λyT]εx,t+1)(∑t=1[λxT]εx,t)︷Op(T2)+Op(T).

The asymptotics of all the O_p (T²) terms were proved previously. Result 4(e) ensues. As for result 5(e), note that, for 0<λ_y =λ_x <1:

∑t=1Tyt+1xt=∑t=1[λyT]ξy,tξx,t−1+(∑t=1[λyT]εy,t+1)∑t=[λyT]+1[λxT]ξx,t−1+ T(1−λx)(∑t=1[λxT]εx,t)(∑t=1[λyT]εy,t+1)︷Op(T2) + Op(T).

The asymptotics of all the O_p (T²) terms were proved previously. The first part of Result 5(e) ensues. When 0<λ_x =λ_y <1 (the second part of Result 5(e)), the result is analogous. Finally, for Results 8(b), 8(d), 8(e), 9(b), 9(d), and 9(e) it is straightforward to see that

∑t=1Tyt+1=αT+β∑t=1Txt︷Op(T3/2)+∑t=1Tεy,t+1︷Op(T1/2),∑t=1Tyt+12=α2T+β2∑t=1Txt2︷Op(T2)+∑t=1Tεy,t+12︷Op(T)+2αβ∑t=1Txt+2β∑t=1Txtεy,t+1︷Op(T)+Op(T1/2),∑t=1Tyt+1xt=α∑t=1Txt+β∑t=1Txt2+∑t=1Txtεy,t+1,

and,

∑t=1Tyt+1=αT+β∑t=1Txt+∑t=1Tξy,t︷Op(T3/2)+∑t=1Tεy,t+1,∑t=1Tyt+12=α2T+β2∑t=1Txt2+∑t=1Tξy,t2︷Op(T2)+∑t=1Tεy,t+12+2αβ∑t=1Txt+2α∑t=1Tξy,t+2β∑t=1Txtξy,t︷Op(T2)+Op(T),∑t=1Tyt+1xt=α∑t=1Txt+β∑t=1Txt2+∑t=1Txtξy,t+∑t=1Txtεy,t+1,

Again, the asymptotics of the higher order terms have been already presented previously.

Proof of Theorem 1 and Corollary 2.

To obtain the asymptotic expression of the R² using OLS in cases, M2–M4′ and M10, we employ the classical formulae, Θ^=(α^, β^)′. All sums run from t=1 to T unless stated otherwise:

where,

X′X=[T∑xt∑xt∑xt2],   X′Y=[∑yt+1∑xtyt+1],

and,

The estimated parameters α^,β^,σ^2 and the R² are functions of the sums ∑xt,∑yt+1,∑xt2,∑yt+12, and ∑xtyt+1. These expressions, conveniently normalized by T^−3/2, T^−3/2, T⁻², T⁻², and T⁻² (respectively), converge in distribution to 𝒮_x , 𝒮_y , 𝒮_xx , 𝒮_yy , and 𝒮_xy , again, respectively:

Cases M2–M4′. Let x_t and y_t+1 be generated by eqs. (2) and (1). The formulae (10), (11) and (12) are then:

α^=∑xt2∑yt+1−∑xt∑xtyt+1T∑xt2−(∑xt)2,T−1/2α^→D𝒮xx𝒮y−𝒮x𝒮xy𝒮xx−(𝒮x)2≡A,β^=T∑xtyt+1−∑xt∑yt+1T∑xt2−(∑xt)2,→D𝒮xy−𝒮y𝒮x𝒮xx−(𝒮x)2≡B,σ^2=(T−2)−1[∑yt2+α^2T+β^2∑xt2−2α^∑yt−2β^∑xtyt+2α^β^∑xt],T−1σ^2→DSyy+A2+B2Sxx−2A𝒮y−2BT2𝒮xy+2AB𝒮x≡S2.

Finally, for the R²,

R2=1−(T−2)σ^2∑yt+12−(∑yt+1/T)∑yt+1,→D1−S2𝒮yy−(𝒮y)2.

An algebraic simplification is needed to present this result as

1−2𝒮y(𝒮x𝒮xy−12𝒮y𝒮xx)−𝒮xy2−𝒮yyϒxϒxϒy

as in Theorem 1 (ii). The reader may also verify that all long-term variances cancel out in the limit expression of R².

Cases M9 and M10. Case M9 is a special case of M10 with λ_x =1, so we focus on the second one. Let x_t and y_t+1 be generated by eqs. (2) and (3). The formulae (10), (11) and (12) are:

α^=∑xt2∑yt+1−∑xt∑xtyt+1T∑xt2−(∑xt)2.

It is easy to see by simple substitution (of y_t+1) that all the terms but α cancel each other, and therefore:

α^→Pα,β^=T∑xtyt+1−∑xt∑yt+1T∑xt2−(∑xt)2.

Again, it is easy to see that all the terms but β cancel each other, and therefore:

β^→Pβ+Op(T−1/2),σ^2=(T−2)−1[∑yt2+α^2T+β^2∑xt2−2α^∑yt−2β^∑xtyt+2α^β^∑xt].

All the O_p (T²), O_p (T^3/2) terms cancel each other. Only the O_p (T^1/2) terms and one O_p (T) term remain:

σ^2=(T−2)−1[∑uy,t+12+(α^2T+2β∑xtuy,t+1+α^2T−2α^2T−2β^∑xtuy,t+1)︸=0+Op(T1/2)],σ^2→Pσy2.

As for the R²,

R2=1−(T−2)σ^2∑yt+12−(∑yt+1/T)∑yt+1,=1−Op(T)Op(T2),=1−Op(T−1).

Cases M9′ and M10′. Case M9′ is a special case of M10′ with λ_x =1, so we focus on the second one. Let x_t and y_t+1 be generated by eqs. (2) and (4). The formulae (10), (11) and (12) are:

β^=T∑xtyt+1−∑xt∑yt+1T∑xt2−(∑xt)2,→Dβ+𝒮xey−𝒮x𝒮ey𝒮xx−(𝒮x)2+Op(T−1/2),

where T−2∑t=1Txtξy,t→D𝒮xey, and T−3/2∑t=1Tξy,t→D𝒮ey. To simplify notation, we define 𝒮xey−𝒮x𝒮ey𝒮xx−(𝒮x)2≡βerror:

β^→Dβ+βerror,α^=y¯−β^x¯,=T−1[∑yt+1−β^∑xt],T−1/2α^→D𝒮ey−βerror𝒮x.

Again, to simplify notation, we define 𝒮_ey −β_error𝒮_x ≡α_error:

T−1/2α^⟶Dαerror,σ^2=(T−2)−1∑yt2+α^2T+β^2∑xt2−2α^∑yt−2β^∑xtyt+2α^β^∑xt.

Contrary to cases M9 and M10, the O_p (T²) do not cancel each other (because the estimator β^ does not converge to the true β). Therefore,

T−1σ^2→D𝒮yy+αerror+(β+βerror)𝒮xx−2αerror𝒮y−2(β+βerror)𝒮yx+2αerror𝒮x.

In other words, σ^2=Op(T). As for the R²,

R2=1−(T−2)σ^2∑yt+12−(∑yt+1/T)∑yt+1,=1−Op(T2)Op(T2),=1−Op(1).

Proof of Theorem 3.

To prove Theorem 3, we need to extend Lemma 1. Results 2, 5, 6, 8, 11, and 12, in Lemma 3 are simple sums whose orders in convergence depend on results 1, 3, 4, 7, 9, and 10 (also in Lemma 3), which were demonstrated by TC2000, 159–161, and are reproduced here for convenience.

Lemma 3Let Assumption 1 in TC2000 (158) hold. Then, as T→∞, we have the following results.

Case M11 with 0<d_y , d_x <1/2 and d_y <d_x :

1. T−(1/2+dx)∑t=1Txt=Op(1),

2. ∑t=1Tyt+1=αT+β∑t=1Txt+∑t=1Tεy,t+1,

3. T−1∑t=1Txt2=Op(1),

4. ∑t=1Txtεy,t+1={Op(Tdy+dx)if dy+dx>1Op(Tdy+dx)O((lnT)1/2)if dy+dx=1Op(T−1/2)otherwise

5. ∑t=1Tyt+12=α2T+β2∑t=1Txt2+∑t=1Tεy,t+12+2αβ∑t=1Txt+2α∑t=1Tεy,t+1+2β∑t=1Txtεy,t+1, and

6. ∑t=1Tyt+1xt=α∑t=1Txt+β∑t=1Txt2+∑t=1Txtut+1.

   Case M11 with 0<d_y <1/2 and −1/2<d_x <0:

7. T−(3/2+dx)∑t=1Txt=Op(1),

8. ∑t=1Tyt+1=αT+β∑t=1Txt+∑t=1Tεy,t+1,

9. T−2(1+dx)∑t=1Txt2=Op(1),

10. T−(1+dy+dx)∑t=1Txtεy,t+1=Op(1),

11. ∑t=1Tyt+12=α2T+β2∑t=1Txt2+∑t=1Tεy,t+12+2αβ∑t=1Txt+2α∑t=1Tεy,t+1+2β∑t=1Txtεy,t+1, and

12. ∑t=1Tyt+1xt=α∑t=1Txt+β∑t=1Txt2+∑t=1Txtεy,t+1.

Proof: TC2000 159–161.

Stationary fractional cointegration: Let y_t+1 and x_t be generated by eqs. (3) and (6), respectively. We use the results of Lemma 3 in the formulae (10), (11) and (12). For α^ and β^, the terms with the highest order in convergence is O_p (T²),

α^=αT∑xt2T∑xt2+Op(T−(1−2dx)),α^→Pα,β^=β∑xt2∑xt2+Op(T−(1−2dx)),→Pβ,

whether d_y +d_x >1/2, d_y +d_x =1/2 or otherwise. Likewise, for σ^2, all the terms with an order in converge higher than O_p (T) cancel each other. One O_p (T) term remains:

σ^2=(T−2)−1∑uy,t+12,σ^2→Pσy2.

Finally, for the R²,

R2=1−(T−2)σ^2∑yt+12−(∑yt+1/T)∑yt+1,=1−Op(T)Op(T)+Op(T2dx),=1−Op(1).

Non-stationary fractional cointegration: Let x_t and y_t+1 be generated by eqs. (2) and (3). We use expressions in the formulae (10), (11) and (12). When we use the results in Lemma 3, we find, for α^, that the terms with the highest order in convergence are:

α^=αT∑xt2−α(∑xt)2T∑xt2−(∑xt)2+Op(T−(12 − dy)),α^→Pα.

Likewise, for β^:

β^=β∑xt2−β(∑xt)2∑xt2−(∑xt)2+Op(T−(1+dx− dy)),→Pβ.

As for σ^2, all the terms with an order in converge higher than O_p (T) cancel each other. One O_p (T) term remains:

σ^2=(T−2)−1∑uy,t+12,σ^2→Pσy2.

Finally, for the R²,

R2=1−(T−2)σ^2∑yt+12−(∑yt+1/T)∑yt+1,=1−Op(T)∑yt+12−T−1(∑yt+1)2.

The term ∑yt+12−(∑yt+1/T)∑yt+1 can be further developed. After simplifying for the terms that cancel each other, we have:

T−2(1+dx)[∑yt+12−T−1(∑yt+1)2]→Dβ2Sxx−β2(Sx)2+Op(T−(1−dy+dx)),

and thus, [∑yt+12−T−1(∑yt+1)2] is Op(T2(1+dx)). Back to the R², we have:

R2=1−Op(T)Op(T2(1+dx)),=1−Op(T−(1+2dx)).

When the linear combination of the fractionally integrated variables becomesI(0). To prove this case we need to obtain the order in convergence of the sum ∑xtey,t+1, where e_y,t+1~I(0) replaces ε_y,t+1 in equation (3). It is actually easier to obtain an upper bound of this order. Note that

∑xtey,t+1≤(∑ey,t+12)1/2︸Op(T1/2)(∑xt2)1/2︸Op(T1/2) if xt∼I(dx) with−12<dx<12,∑xtey,t+1≤(∑ey,t+12)1/2︸Op(T1/2)(∑xt2)1/2︸Op(T1+dx) if xt∼I(1+dx) with −12<dx<0.

Therefore:

∑xtey,t+1=op(T) if xt∼I(dx) with−12<dx<12,∑xtey,t+1=op(T3/2+dx) if xt∼I(1+dx) with−12<dx<0.

With this result, it is straightforward to see that Theorem 3 is not affected when we replace ε_y,t+1~I(d_y ) with e_y,t+1~I(0).

B Estimated response curves for critical values

Critical values are simulated with T=1000 and 20,000 replications on a fine grid of values for λ, i.e. λ={0.10, 0.11, …, 0.89, 0.90}. Estimated response curves are reported: c_1−α (λ)=a₀+a₁λ+a₁λ²+…+u; R² is the coefficient of determination from these polynomial regressions. The nominal significance level is set equal to 10%, 5% and 1%.