Proof of theorem 2 (Correct Model Specification) -Define ηt(β)≡ ft(β)− µkqyt, so that εkqt = ut+ ηt(β∗). Then:
Et[ψkq(ut)] = Et[−2ut+ kqI(ut< 0)|ut|q−1]
= Et[−2(εkqt − ηt(β∗)) + kqI(εkqt < ηt(β∗))|ηt(β∗)− εkqt |q−1]
= Et[−2εkqt + kqI(εkqt < 0)|εkqt |q−1+
+2ηt(β∗)+ kq I(εkqt < ηt(β∗))|ηt(β∗)− εkqt |q−1− kqI(εkqt < 0)|εkqt |q−1]
Note that the term in the first row in the last equality is zero by (7). The term in the second row will be always greater than 0 when ηt(β∗) > 0 and less than 0 when ηt(β∗) < 0 (this follows by the properties of the integral). Therefore a necessary and sufficient condition for Et[ψkq(ut)] = 0 is ηt(β∗) = 0, which is equivalent to assuming that the conditional expectile model is correctly specified. ¥
Proof of theorem 3 (Consistency) - We verify that the conditions of theorem 3.5 of [22] White (1994) are satisfied.
Assumption 2.1 in [22] White (1994) is assumption C0. Assumption 2.3 is also satisfied, given that ft(β) is continuous by assumption C1 and
ρkq(yt− ft(β)) is also continuous in β.
Assumption 3.1(a) in [22] White (1994) requires to show that E[ρkq(yt− ft(β))] exists and is finite.
E[ρkq(yt− ft(β))] = E[ρkq(ut− δt(β))]
= E[(ut− δt(β))2+ kI(ut< δt(β))|δt(β)− ut|q]
≤ E[|ut− δt(β)|2+ k|ut− δt(β)|q]
≤ 2[E|ut|2+ E|δt(β)|2] + k2q−1[E|ut|q+ E|δt(β)|q]
because of the inequality E|X +Y |r≤ cr[E|X|r+E|Y |r], where cr = 2r−1, r ≥ 1 (see, e.g., Zellner, p.111, Handbook of Econometrics, Vol.1). The result follows from assumptions C2 and C3.
Assumption 3.1(b) in [22] White (1994) (the continuity of E[ρkq(yt− ft(β))] in B) follows from the continuity of ρkq(λ)in λ and the continuity of ft(β)in β (assumption C1). Assumption 3.1(c) in [22] White (1994) is simply C4.
It remains to verify Assumption 3.2 of theorem 3.5 in [22] White (1994), that is that E[T−1PT
t=1ρkq(ut − δt(β))] has identifiably unique minimizers β∗T, that is we need to show that E[T−1PT
t=1ρkq(ut−δt(β))]− E[T−1PT
t=1ρkq(ut)] > 0if min
||β−β∗||>ξT−1PT
t=1E[δt(β)2] > υ. Consider each
element at time t:
E[ρkq(ut− δt(β))− ρkq(ut)] = E[(ut− δt(β))2+ kI(ut < δt(β))|δt(β)− ut|q−
−u2t − kI(ut< 0)|ut|q]
= E[u2t + δt(β)2− 2δt(β)ut+ kI(ut < δt(β))|δt(β)− ut|q−
−u2t − kI(ut < 0)|ut|q+ kq I(ut< 0)|ut|q−1δt(β)− kqI(ut< 0)|ut|q−1δt(β)]
Now note that by assumption M, E[{−2ut+ kqI(ut < 0)|ut|q−1}δt(β)] = E[ψkq(ut)(ft(β)− ft(β∗))] = 0. Therefore:
E[ρkq(ut− δt(β))− ρkq(ut)] = E[δt(β)2+ k{I(ut< δt(β))|δt(β)− ut|q−
−I(ut< 0)|ut|q− qI (ut< 0)|ut|q−1δt(β)}]
≡ E[δt(β)2+ kAt]
If we show that At ≥ 0 ∀t, the result follows. We need to consider two cases.
i) δt(β) > 0
At = [I(ut < 0) + I(0≤ ut≤ δt(β))]|δt(β)− ut|q− I(ut< 0)[|ut|q+ q|ut|q−1δt(β)]
≥ I(ut < 0)[|δt(β)− ut|q− |ut|q− q|ut|q−1δt(β)]
≡ I(ut < 0)Bt
ii) δt(β) < 0
Showing that At ≥ 0 is therefore equivalent to showing that Bt ≥ 0 whenever ut< 0:
if min
||β−β∗||>ξT−1PT
t=1E[δt(β)2] > υ, by assumption C5. ¥
Proof of theorem 4 (Asymptotic Normality) - Since the PLS objec-tive function is not twice differentiable, we need to resort to asymptotic normality results for nonsmooth objective function. We show that the conditions of theorems 7.2 and 7.3 of [16] Newey and McFadden (1994) hold. Define the following:
To apply theorem 7.2 of [16] Newey and McFadden (1994), we need to check the following conditions:
6. √
T ˆgT(β∗)→ N(0, V )d
If these conditions are satisfied, then theorems 7.2 and 7.3 of [16] Newey and McFadden (1994) imply that √
T (ˆβT − β∗)→ N(0, Zd −1V Z−1).
Let ∆(xt) ≡ £
ψkq(ut)∇ββft(β∗) + wt(k, q)∇βft(β∗)∇0βft(β∗)¤
. Condi-tions 4, 5 and 6 are automatically satisfied by assumpCondi-tions AN3, AN0 and AN4, respectively.
For condition 1, we first check that E|ψkq(ut)∇βft(β)| < ∞ and then apply the Lebesgue Dominated Convergence Theorem (LDCT).
E|ψkq(ut)∇βft(β)| = E|[−2ut+ kqI(ut< 0)|ut|q−1]∇βft(β)|
≤ E[(2|ut| + kq|ut|q−1)||∇βft(β)||]
≤ E[max[2, kq]|ut|q−1||∇βft(β)||]
≤ E[max[2, kq]U1(Xt)F1(Wt)||]
< ∞
Having found a dominating function for ψkq(ut)∇βft(β) with finite ex-pected value, we can apply the LDCT:
E|ψkq(ut)∇βft(β∗)| = ∂
∂βE[ψkq(ut)ft(β)]|β=β∗
= 0
by assumption M.
To establish condition 2, consider the components of r(xt; β):
g(xt; β)− g(xt; β∗) ≡ ψkq(ut− δt(β))∇βft(β)− ψkq(ut)∇βft(β∗)
= −2(ut− δt(β))∇βft(β) + kqI(ut< δt(β))|δt(β)− ut|q−1 +2ut∇βft(β∗)− kqI(ut < 0)|ut|q−1∇βft(β∗)
=−2ut(∇βft(β)− ∇βft(β∗)) + 2δt(β)∇βft(β)+
+ kq{I(ut< δt(β))|δt(β)− ut|q−1(∇βft(β)− ∇βft(β∗))+
+ [I(ut< δt(β))|δt(β)− ut|q−1− I(ut< 0)|ut|q−1]∇βft(β∗)}
= [−2ut+ kqI(ut < δt(β))|δt(β)− ut|q−1]∇ββft(˜β)(β− β∗)+
+ 2∇βft(β)∇0βft(˜β)(β− β∗)+
+ (q− 1)kqI(ut < δt(β))|δt(β)− ut|q−2∇βft(β∗)∇0βft(˜β)(β− β∗)
where ˜β comes from the mean value theorem.
Substituting everything into r(xt, β), we get:
r(xt, β)≤ ||[−2ut+kqI(ut< δt(β))|δt(β)−ut|q−1]∇ββft(˜β)−ψkq(ut)∇ββft(β∗)||+
+||2∇βft(β)∇0βft(˜β)− 2∇βft(β∗)∇0βft(β∗)||+
+||(q − 1)kqI(ut < δt(β))|δt(β)− ut|q−2∇βft(β∗)∇0βft(˜β)−
− (q − 1)kqI(ut< 0)|ut|q−2∇βft(β∗)∇0βft(β∗)||
≡ r1(˜β) + r2(˜β) + r3(˜β)
These functions are continuous in β and obviously converge to zero almost surely as β → β∗.
We next prove condition 3. From the previous step we have:
sup
whose expectation is finite by assumptions AN1 and AN2.
sup
whose expectation is finite by assumptions AN1 and AN2.
To conclude the proof, note that
T−1
by the LDCT and assumption M. ¥
Proof of theorem 5 (Variance-Covariance Matrix Estimation) -Define ˜VT ≡ T−1PT
Consistency of ˆβ guarantees that the expressions in the last three lines converge to zero in probability. The proof for ˆZT is similar.
Proof of proposition 6 - The (k, q)-projectile of yt is
ft = −0.5kq Z ft
−∞|ft− y|q−1dF (y)
= −0.5σq−1t kq Z ft/σt
−∞ |ft/σt− ε|q−1dF (ε) Rearranging:
ft/σt=−0.5σq−2t kq Z ft/σt
−∞
|ft/σt− ε|q−1dF (ε)
Note that ft/σt is the (˜kt, q)-projectile of ε, where ˜kt ≡ σq−2t k. The result follows from noting that ˜kt is constant if q = 2 and/or if σt is constant, and from the fact that ε is i.i.d. ¥
Proof of theorem 7 (Convexity) - Let y, y1 and y2 denote the returns associated to portfolios a, a1 and a2, respectively, where a = λa1+(1−λ)a2 and 0 < λ < 1. Define also µ ≡ E[y − 0.5kqI(µ > y)|µ − y|q−1}] and µi ≡ E[yi− 0.5kqI(µi > yi)|µi− yi|q−1], i = 1, 2. Finally, define x ≡ µ − y and xi ≡ µi− yi, i = 1, 2.
We need to show that µ ≥ λµ1+ (1− λ)µ2. Suppose, by contradiction, that µ ≤ λµ1+ (1− λ)µ2. Then, noticing that −I(x > 0)|x|q−1 is convex
in x:
µ ≥ E[y − 0.5kqI(λµ1+ (1− λ)µ2 > y)|λµ1+ (1− λ)µ2− y|q−1]
= E[y− 0.5kqI(λx1+ (1− λ)x2 > 0)|λx1+ (1− λ)x2|q−1]
≥ E[y − 0.5kq{λI(x1 > 0)|x1|q−1+ (1− λ)I(x2 > 0)|x2|q−1}]
= λE[y1− 0.5kqλI(µ1 > y1)|µ1− y1|q−1] + (1− λ)E[y2− 0.5kqλI(µ2 > y2)|µ2− y2|q−1]
= λµ1+ (1− λ)µ2
which contradicts the initial assumption. ¥
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European Central Bank Working Paper Series
For a complete list of Working Papers published by the ECB, please visit the ECB’s website (http://www.ecb.int)
691 “The yield curve as a predictor and emerging economies” by A. Mehl, November 2006.
692 “Bayesian inference in cointegrated VAR models: with applications to the demand for euro area M3” by A. Warne, November 2006.
693 “Evaluating China’s integration in world trade with a gravity model based benchmark”
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694 “Optimal currency shares in international reserves: the impact of the euro and the prospects for the dollar”
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695 “Geography or skills: What explains Fed watchers’ forecast accuracy of US monetary policy?” by H. Berger, M. Ehrmann and M. Fratzscher, November 2006.
696 “What is global excess liquidity, and does it matter?” by R. Rüffer and L. Stracca, November 2006.
697 “How wages change: micro evidence from the International Wage Flexibility Project”
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698 “Optimal monetary policy rules with labor market frictions” by E. Faia, November 2006.
699 “The behaviour of producer prices: some evidence from the French PPI micro data”
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700 “Forecasting using a large number of predictors: Is Bayesian regression a valid alternative to principal components?” by C. De Mol, D. Giannone and L. Reichlin, December 2006.
701 “Is there a single frontier in a single European banking market?” by J. W. B. Bos and H. Schmiedel, December 2006.
702 “Comparing financial systems: a structural analysis” by S. Champonnois, December 2006.
703 “Comovements in volatility in the euro money market” by N. Cassola and C. Morana, December 2006.
704 “Are money and consumption additively separable in the euro area? A non-parametric approach”
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705 “What does a technology shock do? A VAR analysis with model-based sign restrictions”
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710 “Pricing of settlement link services and mergers of central securities depositories” by J. Tapking, January 2007.
711 “What “hides” behind sovereign debt ratings?” by A. Afonso, P. Gomes and P. Rother, January 2007.
712 “Opening the black box: structural factor models with large cross-sections” by M. Forni, D. Giannone, M. Lippi and L. Reichlin, January 2007.
713 “Balance of payment crises in emerging markets: How early were the “early” warning signals?” by M.
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714 “The dynamics of bank spreads and financial structure” by R. Gropp, C. Kok Sørensen and J.-D. Lichtenberger, January 2007.
715 “Emerging Asia’s growth and integration: How autonomous are business cycles?” by R. Rüffer, M. Sánchez and J.-G. Shen, January 2007.
716 “Adjusting to the euro” by G. Fagan and V. Gaspar, January 2007.
717 “Discretion rather than rules? When is discretionary policy-making better than the timeless perspective?”
by S. Sauer, January 2007.
718 “Drift and breaks in labor productivity” by L. Benati, January 2007.
719 “US imbalances: the role of technology and policy” by R. Bems, L. Dedola and F. Smets, January 2007.
720 “Real price wage rigidities in a model with matching frictions” by K. Kuester, February 2007.
721 “Are survey-based inflation expectations in the euro area informative?” by R. Mestre, February 2007.
722 “Shocks and frictions in US business cycles: a Bayesian DSGE approach” by F. Smets and R. Wouters, February 2007.
723 “Asset allocation by penalized least squares” by S. Manganelli, February 2007.