From prices to business cycles

From Prices to Business Cycles

João V. Issler (FGV) and Claudia Rodrigues (VALE)

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

Outline

Two talks for the price of one :-))

Our theme in both talks is: “from prices to business cycles.”

First paper covers asset prices and consumption business cycles (intertemporal substitution).

We exploit theoretical commonality among prices to identify price-quantity relationships.

Second paper covers metal-commodity prices and cycles in Global Industrial Production (and Global GDP).

We exploit short-run restrictions in the equilibrium condition for input markets to identify price-quantity relationships.

In both papers we employ common-feature techniques in estimation and testing.

A Stochastic Discount Factor Approach to Asset Pricing

using Panel Data Asymptotics

Fabio Araujo (Central Bank of Brazil) and João Victor Issler (FGV)

Motivation – Why Consumption Cycles?

Research is done mostly using GDP/GNP cycles. But:

-.05 -.04 -.03 -.02 -.01 .00 .01 .02 .03 .04 55 60 65 70 75 80 85 90 95 00 05 10 HP CYCLE GDP USA HP CYCLE CONS REAL

Synchronized cycles for HP …ltered consumption and GDP. Similar picture for Beveridge-Nelson cycles; Issler and Vahid (JME, 2001).

This Paper

We impose no-arbitrage to show that the SDF is a common feature of all asset returns.

Our estimator is based on the Asset-Pricing Equation: it is a novel consistent estimator of the SDF based on return (price) data alone; Harrison and Kreps (1979), Hansen and Richard (1987), and Hansen and Jagannathan (1991, 1997).

This allows consistent estimation of the stochastic process of the SDF

fMtgt∞=1 without resorting to any parametric assumptions on preferences.

This also allows the development of no-arbitrage estimators of the SDF and of asset-return volatility and conditional volatility. They could be used as theory-based estimators (quasi-structural) to be compared to unrestricted reduced-form estimators.

This Paper

We bridge the gap between a large literature on common features in macro [Vahid and Engle (1993, 1997), Engle and Issler (1995), Issler and Vahid (2001, 2006), Hecq, Palm and Urbain (2006), Issler and Lima (2009), and Athanasopoulos et al. (2011)] and the …nance literature using common components in mean and variance [Aït-Sahalia and Lo (1998, 2000), Lettau and Ludvigson (2001), Rosenberg and Engle (2002), Garcia, Renault, and Semenov (2006), Sentana and Peñaranda (2008), Hansen and Scheinkman (2009), and Hansen and Renault (2009), Peñaranda and Sentana (2010)].

We also propose an alternative way of imposing no-arbitrage in constructing important …nancial econometrics estimators, which became popular through the work of Diebold and Li (2006), Christensen, Diebold, and Rudebusch (2007, 2009, 2011), and Diebold and Rudebusch (2013).

The estimation technique is based on standard panel-data asymptotics.

Assumptions

Assumption 1: We assume the absence of arbitrage opportunities in asset pricing, c.f., Ross (1976). This must hold for all

t=1, 2, ..., T , and for all lapses of time, however small. Basically, Assumption 1 is a necessary and su¢ cient condition for (1) to hold; Cochrane (2002). In Hansen and Renault (2009), it implies (1): pi ,t = Et βU 0_(ct+1) U0_(ct) (pi ,t+1+di ,t+1) , 8i 1 = Et βU 0_(ct+1) U0_(ct) xi ,t+1 pi ,t , i =1, 2, ..., N 1 = EtfMt+1Ri ,t+1g, i =1, 2, ..., N. (1)

Assumptions

The absence of arbitrage opportunities has two important implications:

1 _{There exists at least one stochastic discount factor Mt, for which}

Mt >0 for all t. This is due to the fact that, when we consider the existence derivatives on traded assets, arbitrage opportunities will

arise if Mt 0; Hansen and Jagannathan (1997).

2 A weak law-of-large numbers (WLLN) holds in the cross-sectional

dimension for the level of gross returns Ri ,t; Ross (1976).

The Pricing Equation (1) is present in virtually all studies dealing with intertemporal substitution; e.g., Hansen and Singleton (1982, 1983, 1984), Mehra and Prescott (1985), Epstein and Zin (1991), Fama and French (1992, 1993), Attanasio and Browning (1995), Lettau and Ludvigson (2001), Garcia, Renault, and Semenov (2006), and Hansen and Scheinkman (2009).

Assumptions

Assumption 2: Let Rt = (R1,t, R2,t, ... RN ,t)0 be an N 1 vector stacking all asset returns in the economy and consider the vector processnln(MtR0t)0

o

. In the time (t)dimension, we

assume thatnln(MtR0t)0

o∞

t=1 is covariance-stationary and ergodic with …nite …rst and second moments uniformly across i .

Weak stationarity of returns can coexist with conditional

heteroskedasticity; see Engle (1982) and Bollerslev (1986). This …ts well established properties of asset returns.

Exact Taylor Expansion

Consider a second-order Taylor Expansion of the exponential function around x, with increment h, as follows:

ex+h =ex+hex +h

2_ex+λ(h)h

2 , (2)

with λ(h):R_{! (}0, 1). (3)

For the expansion of a generic function, λ( ) would depend on x and h.

However, dividing (2) by ex:

eh =1+h+h

2_eλ(h)h

2 , (4)

Exact Taylor Expansion

De…ne z = h2ex+₂λ(h)h, and let h=ln(MtRi ,t)in (2):

MtRi ,t =1+ln(MtRi ,t) +zi ,t. (5)

Computing now Et 1( ) of (5) and rearranging terms yields:

Et 1(MtRi ,t) = 1+Et 1[ln(MtRi ,t)] +Et 1(zi ,t), or,

Quasi-Structural Factor Model (in logs)

Let γ2_t γ2_1,t γ2_2,t ... γ_{N ,t}2 0 and εt (ε1,t ε2,t ... εN ,t)0: ln(MtRt) =Et 1fln(MtRt)g +εt = γ2_t +εt. Denoting by rt =ln(Rt)and by mt =ln(Mt): ri ,t = mt γ2i ,t+εi ,t, i =1, 2, . . . , N. (7)

mt is a common feature of all (logged) asset returns; Engle and

Kozicki (1993). Also, Hansen and Singleton (JPE, 1983). εi ,t =ln(MtRi ,t) Et 1fln(MtRi ,t)g =

Quasi-Structural Factor Model (in logs)

Wold representation from Assumption 2: ln(MtRi ,t) =µ_i+

∞

∑

j=0

bi ,jεi ,t j (8)

where, for all i , bi ,0 =1,jµ_ij <∞, ∑jb_{i ,j}2 <∞, and εi ,t is white noise. Now: γ2i ,t Et 1(zi ,t) = Et 1fln(MtRi ,t)g Then: ri ,t = mt γ2i +εi ,t+ ∞

∑

j=1 bi ,jεi ,t j, i =1, 2, . . . , N, (9) where γ2_i µ_i.

Quasi-Structural Factor Model (in logs)

Special case: assume ln(MtRt)jIt 1 N [Et 1(rt +ιmt),Σ], i.e., MtRt has a multivariate log-Normal homoskedastic distribution. Then,

Et 1(MtRi ,t) = exp Et 1(ri ,t +mt) + σ2_i 2 =1, or, Et 1(ri ,t+mt) + σ2_i 2 = 0, or, using ri ,t+mt = Et 1(ri ,t+mt) +εi ,t, yields, ri ,t = mt σ2_i 2 +εi ,t i =1, 2, . . . , N. (10)

Law-of-Large Numbers for (11)

Consider now a cross-sectional average of (9): 1 N N

∑

i=1 ri ,t = mt 1 N N

∑

i=1 γ2_i + 1 N N

∑

i=1 εi ,t + 1 N N

∑

i=1 ∞

∑

j=1 bi ,jεi ,t j. (11)

The stochastic terms in the cross-sectional dimension have the following variance: VAR 1 N N

∑

i=1 εi ,t ! +VAR 1 N N

∑

i=1 bi ,1εi ,t 1 ! +VAR 1 N N

∑

i=1 bi ,2εi ,t 2 ! +

since there is no time-series correlation in εt (ε1,t ε2,t ... εN ,t)0 due to assumption of weak stationarity.

Skepticism on Law-of-Large Numbers for (11)

We can always decompose εi ,t as:

εi ,t = ln(MtRi ,t) Et 1fln(MtRi ,t)g

= [mt Et 1(mt)] + [ri ,t Et 1(ri ,t)]

= qt +vi ,t = _|{z}αi 1+COV_VAR(vi,t ,qt₍ )

qt) =1+δi qt +ξ_{i ,t} plim N_!∞ 1 N N

∑

i=1 εi ,t = 0=) lim N_!∞ 1 N N

∑

i=1 δi = 1.

This an identi…cation issue.

Alternative (Ergodic Theorem): no-arbitrage requires _N1 ∑N_i=1Ri ,t _!p , implying _N1 _∑N_i=1ri ,t

p

!

Here: with quasi-structural system no-arbitrage implies plim N_!∞ 1 N ∑ N i=1εi ,t =0.

Main Result

Theorem

Under Assumptions 1 and 2, as N, T _!∞, with N diverging at a rate at

least as large as T , the realization of the SDF at time t, denoted by Mt,

can be consistently estimated using: b Mt = RG_t 1 T T ∑ j=1 RG_j RA_j , where RG_t =_∏Ni=1R 1 N i ,t and R A t = N1 N ∑ i=1

Ri ,t are respectively the

geometric average of the reciprocal of all asset returns and the arithmetic average of all asset returns.

Sketch of the Proof

Stacked quasi-structural form imposing with Log-Normality: 0 B B B @ r1t r2t .. . rNt 1 C C C A= mt 0 B B B @ 1 1 .. . 1 1 C C C A 0 B B B @ σ2₁ σ2₂ .. . σ2_N 1 C C C A+ 0 B B B @ ε1t ε2t .. . εNt 1 C C C A.

An arbitrage portfolio with weights wi, all of order N 1 in absolute value,

stacked in a vector WN = (w1, w2, ..., wN)0, obeys:

W_N0 0 B B B @ 1 1 .. . 1 1 C C C A=0, limN_!∞W 0 N 0 B B B @ 1 1 .. . 1 1 C C C A=0, limN_!∞VAR 2 6 6 6 4W 0 N 0 B B B @ r1,t r2,t .. . rN ,t 1 C C C A 3 7 7 7 5=0. (12)

Sketch of the Proof

No-arbitrage imposes restrictions on the limit return of arbitrage portfolios:

plim N_!∞ W_N0 0 B B B @ r1,t r2,t .. . rN ,t 1 C C C A=0. (13)

A weak inequality ( 0)does not work, because portfolio W_N0 would still

Sketch of the Proof

No arbitrage (13) requires: 0 = mt lim N_!∞W 0 N 0 B @ 1 .. . 1 1 C A lim N_!∞W 0 N 0 B B @ σ21 2 .. . σ2N 2 1 C C A+_Nplim_!∞WN0 0 B @ ε1,t .. . εN ,t 1 C A = lim N_!∞W 0 N 0 B B @ σ21 2 .. . σ2N 2 1 C C A+_Nplim_!∞WN0 2 6 4 0 B @ 1 .. . 1 1 C A+ 0 B @ δ1 .. . δN 1 C A 3 7 5 qt +plim N_!∞ W_N0 0 B @ ξ_1,t .. . ξN ,t 1 C A .

Sketch of the Proof

W_N0 must lie in the left-null space of 0 B @ 1 .. . 1 1 C A, to prevent plim N!∞W 0 N 0 B @ r1,t .. . rN ,t 1 C A to depend on qt: 0 B @ δ1 .. . δN 1 C A= 0 B @ δ .. . δ 1 C A ,jδj <∞,

Otherwise, for some realizations of qt we could not prevent that

plim N_!∞ W_N0 0 B @ r1,t .. . rN ,t 1 C A >0, or plim N_!∞ W_N0 0 B @ r1,t .. . rN ,t 1 C A<0, which violate no

Sketch of the Proof

Thus, we establish that:

plim N!∞W 0 N 0 B @ ε1,t .. . εN ,t 1 C A=0. (14)

In…nite possible δs leading to (14): identi…cation issue, since

qt = [mt Et 1(mt)]is the innovation of mt (latent variable) and vi ,t = [ri ,t Et 1(ri ,t)] is the innovation of ri ,t (data).

Sketch of the Proof – Identi…cation of SDF

Identi…cation of qt and thus of mt, requires a choice of δ.

plim N_!∞ 1 N N

∑

i=1 εi ,t = (1+δ)qt. (15)

Identi…cation of qt and thus of mt, requires a choice of δ. The choice

δ = 1, for all i , yields plim N_!∞ 1

N ∑

N

i=1εi ,t =0.

Sketch of the Proof – Meaning of Identi…cation

Assumption

Take (10), average across i and take the probability limit to obtain:

plim N!∞ 1 N N

∑

i=1 ri ,t = mt 1 2 Nlim_!∞ 1 N N

∑

i=1 σ2i ! + plim N!∞ 1 N N

∑

i=1 εi ,t = mt 1 2 Nlim_!∞ 1 N N

∑

i=1 σ2_i ! , which yields: VAR plim N!∞ 1 N N

∑

i=1 ri ,t ! =VAR(mt), (16)

Sketch of the Proof

We have: 1 N N

∑

i=1 ri ,t = mt 1 2 1 N N

∑

i=1 σ2i + 1 N N

∑

i=1 εi ,t. (17)

Take the probability limit as N _!∞:

1 N N

∑

i=1 e( 12σ2i) !_eγ2_{, say.}

Using Slutsky’s Theorem, we can then propose a consistent estimator for a tilted version of Mt (eγ

2 Mt =Met): be Mt = N

∏

i=1 R 1 N i ,t . (18)

Sketch of the Proof

Multiply the Pricing Equation for every asset by e( 12σ 2 i) _{to get:} e( 12σ 2 i) =E t 1 n e( 12σ 2 i)_{MtRi ,t} o =Et 1 n e MtRi ,to. Take now the unconditional expectation, use the law-of-iterated

expectations, and average across i =1, 2, ..., N to get, for large enough N:

eγ2 ₌ 1 N N

∑

i=1 EnMte Ri ,to.

Sketch of the Proof

Thus, for large enough N, it is straightforward to obtain a consistent

estimator for eγ2 _{= lim}

N_!∞ 1 N ∑ N i=1e( 1 2σ 2 i) _{using (18):} c eγ2 ₌ 1 N N

∑

i=1 1 T T

∑

t=1 be MtRi ,t ! = 1 T T

∑

t=1 N

∏

i=1 R 1 N i ,t 1 N N

∑

i=1 Ri ,t ! (19) = 1 T T

∑

t=1 RG_t RA_t, (20)

where, in this last step, N must diverge at a rate at least as fast as T , otherwise we would not be able to exchange eMt by bMet in (19).

Sketch of the Proof

Then, a consistent estimator for Mt is:

b Mt = Mbet c eγ2 = R G t 1 T ∑ T τ=1R G τR A τ .

Remarks

Hansen and Jagannathan (1991):

M_t+1 = ι0 Et Rt+1Rt0+1 1 Rt+1, or, M_t+1 = ι0 E Rt+1Rt0+1 1 Rt+1,

Theoretical problems for Et Rt+1Rt0+1 and E Rt+1Rt0+1 which are

of in…nite order when N _!∞. Theoretical problems inverting them

when N _!∞ faster than T _!∞.

Need a model to compute Et Rt+1Rt0+1 .

Numerical problems invertingEt Rt+1Rt0+1 andE Rt+1Rt0+1 for large N.

Remarks

Identi…cation: b Mt

p

!Mt =Mt >0, unique SDF, if markets are complete,

In complete markets we identify the unique SDF which is equal to the mimicking portfolio.

b

Mt p_!Mt >0, SDF>0, if markets are incomplete.

In incomplete markets we can only identify SDFs up to addition of an error term, irrelevant for pricing assets. Our estimator identi…es positive SDFs or combinations of them. They all have identical pricing properties.

Testing Preference Speci…cations

Popular Preference representations:

CRRA Mt+1 = β ct_c+1 t γ External Habit Mt+1 = β ct ct 1 κ(γ 1) γct+1 ct γ Kreps-Porteus Mt+1= β ct_c+1 t γ α ρ h 1 Bt+1 i1 α ρ

Testing Preference Speci…cations

Our sample period starts in 1972:1 and ends in 2000:4 in quarterly frequency.

We averaged the real U.S.$ returns on 18 portfolios or individual assets, taking into account the returns or more than thousands of assets.

Real returns were computed using the consumer price index in the U.S. Our database covers U.S.$ real returns on G7-country stock indices and short-term government bonds. In addition, we also use U.S.$ real returns on Gold, real estate, bonds on AAA U.S. corporations, and on the SP 500. The U.S. government bond is chosen to be the 90-day T-Bill.

Consumption data and income data: seasonally adjusted real total private consumption and GNP per-capita.

Testing Preference Speci…cations

The sample mean of cMt is 0.9927, implying an annual discount factor of

0.9711, or an annual discount rate of 2.97%.

0.85 0.90 0.95 1.00 1.05 1.10 1.15 1975 1980 1985 1990 1995 2000

Testing Preference Speci…cations

Table 1

Power Utility Function Estimates

[

mt+1=ln β γ∆ ln ct+1 ηCRRA_t+1

Instrument Set (plus C ) β γ J Test

(SE) (SE) (P-Val.)

OLS Estimate 1.002 1.979 – (0.006) (0.884) ri ,t 1, ri ,t 2,8i =1, 2, N 0.999 1.125 (0.995) (0.003) (0.517) ri ,t 1, ri ,t 2,8i =1, 2, N 1.001 1.370 (0.996) ∆ ln ct 1,∆ ln ct 2, (0.003) (0.511) ri ,t 1, ri ,t 2,8i =1, 2, N 1.000 1.189 (0.995) ∆ ln yt 1,∆ ln yt 2 (0.003) (0.523) ri ,t 1, ri ,t 2,₈i ,∆ ln ct 1,∆ ln ct 2 0.999 1.204 (0.998) ∆ ln yt 1,∆ ln yt 2, lnct 1 yt 1 (0.003) (0.514)

Testing Preference Speci…cations

Table 2

External Habit Utility Function Estimates

[

mt+1 =ln β γ∆ ln ct+1+κ(γ 1)∆ ln ct ηEH_t+1

Instrument Set (plus C ) β γ κ J-Test

(SE) (SE) (SE) (P-Val)

OLS Estimate 1.002 1.975 -0.008 – (0.006) (0.97) (0.99) ri ,t 1, ri ,t 2,8i =1, 2, N 1.005 1.263 -2.847 (0.991) (0.003) (0.61) (8.33) ri ,t 1, ri ,t 2,8i =1, 2, N, 0.9954 1.308 1.997 (0.995) ∆ ln ct 1,∆ ln ct 2 (0.003) (0.56) (3.27) ri ,t 1, ri ,t 2,8i =1, 2, N, 0.987 1.592 3.588 (0.995) ∆ ln yt 1,∆ ln yt 2 (0.003) (0.68) (3.74) ri ,t 1, ri ,t 2,₈i ,∆ ln ct 1,∆ ln ct 2 0.987 1.161 8.834 (0.998) ∆ ln yt 1,∆ ln yt 2, lnct 1 yt 1 (0.002) (0.62) (32.76)

Testing Preference Speci…cations

Table 3

Kreps–Porteus Utility Function Estimates

[

mt+1= θ ln β θγ∆ ln ct+1 (1 θ)ln Bt+1 ηKP_t+1

Instrument Set (plus C ) β γ θ J-Test

(SE) (SE) (SE) (P-Val.)

OLS Estimate 1.007 3.141 0.831 – (0.006) (0.88) (0.02) ri ,t 1, ri ,t 2,8i =1, 2, N 1.001 1.343 0.933 (0.996) (0.004) (0.72) (0.01) ri ,t 1, ri ,t 2,8i =1, 2, N, 1.003 1.360 0.922 (0.998) ∆ ln ct 1,∆ ln ct 2 (0.004) (0.76) (0.01) ri ,t 1, ri ,t 2,8i =1, 2, N, 1.000 0.926 0.927 (0.996) ∆ ln yt 1,∆ ln yt 2 (0.004) (0.75) (0.01) ri ,t 1, ri ,t 2,₈i ,∆ ln ct 1,∆ ln ct 2 0.997 0.362 0.901 (0.999) ∆ ln yt 1,∆ ln yt 2, lnct 1 yt 1 (0.004) (0.76) (0.01)

Testing Preference Speci…cations – Consumption Factor

Alone

-3 -2 -1 0 1 2 3 4 -.020 -.015 -.010 -.005 .000 .005 .010 .015 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 Cons. Growth (X-1) SDF Estimate ρ=0.21

Testing Preference Speci…cations - Two-Factor

Kreps-Porteus Model

-4 -3 -2 -1 0 1 2 3 4 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 SDF K-P Model (scaled) SDF Estimate (scaled) ρ=0.61

Conclusions and Further Results

1 Assuming no arbitrage, and that the we can consistently estimate the

…rst two moments of asset-return and SDF data, we can use asset return (price) data to model intertemporal substitution without specifying preferences.

2 All our estimators can be labelled no-arbitrage estimators.

3 We generate a template to evaluate preferences and intertemporal

substitution models (similarly to Hansen and Jagannathan, 1991, 1997).

4 Our results could be use a mixed-frequency to model and predict

consumption cycles, as well as GDP cycles, as long as we can tie up these two together in a structural-equation setup.

Common Features for Metal-Commodity Prices and

Global Business Cycles

João V. Issler (FGV), Claudia Rodrigues (VALE), Rafael Burjack (FGV)

Global GDP and IP growth rates

-6 -5 -4 -3 -2 -1 0 1 2 3 92 94 96 98 00 02 04 06 08 10 GDP Growth Rates (std)

Copper prices and global IP growth

-5 -4 -3 -2 -1 0 1 2 3 92 94 96 98 00 02 04 06 08 10 Copper Prices Growth Rates (std)

Global Ind. Prod. Growth Rates (std)

Outline of the Talk

1 Basic concepts: Features and Common Features

2 Econometric models that incorporate common features

3 _{Common Cycles between Metal Prices and Industrial Production}

4 Forecastability performance of econometric models with and without

Outline of the Talk

1 Basic concepts: Features and Common Features

2 Econometric models that incorporate common features

3 _{Common Cycles between Metal Prices and Industrial Production}

4 Forecastability performance of econometric models with and without

Outline of the Talk

1 Basic concepts: Features and Common Features

2 Econometric models that incorporate common features

3 _{Common Cycles between Metal Prices and Industrial Production}

4 Forecastability performance of econometric models with and without

Outline of the Talk

1 Basic concepts: Features and Common Features

2 Econometric models that incorporate common features

3 _{Common Cycles between Metal Prices and Industrial Production}

4 Forecastability performance of econometric models with and without

Common Features – Basic Ideas

1 Series y_1t has property A.

2 Series y_2t has property A.

3 There exists a linear combination of them, y_1t eαy2t, that does not

have property A.

4 Cointegration is the most well-known example of common features.

5 Serial correlation-common features (SCCF) or common cycles are also

well-known: stationary series y1t and y2t both have serial correlation

(are predictable), but there exists y1t eαy2t which is white noise

Common Features – Basic Ideas

1 Series y_1t has property A.

2 Series y_2t has property A.

3 There exists a linear combination of them, y_1t eαy2t, that does not

have property A.

4 Cointegration is the most well-known example of common features.

5 Serial correlation-common features (SCCF) or common cycles are also

well-known: stationary series y1t and y2t both have serial correlation

(are predictable), but there exists y1t eαy2t which is white noise

Common Features – Basic Ideas

1 Series y_1t has property A.

2 Series y_2t has property A.

3 There exists a linear combination of them, y_1t eαy2t, that does not

have property A.

4 Cointegration is the most well-known example of common features.

5 Serial correlation-common features (SCCF) or common cycles are also

well-known: stationary series y1t and y2t both have serial correlation

(are predictable), but there exists y1t eαy2t which is white noise

Common Features – Basic Ideas

1 Series y_1t has property A.

2 Series y_2t has property A.

3 There exists a linear combination of them, y_1t eαy2t, that does not

have property A.

4 Cointegration is the most well-known example of common features.

5 Serial correlation-common features (SCCF) or common cycles are also

well-known: stationary series y1t and y2t both have serial correlation

(are predictable), but there exists y1t eαy2t which is white noise

Common Features – Basic Ideas

1 Series y_1t has property A.

2 Series y_2t has property A.

3 There exists a linear combination of them, y_1t eαy2t, that does not

have property A.

4 Cointegration is the most well-known example of common features.

5 Serial correlation-common features (SCCF) or common cycles are also

well-known: stationary series y1t and y2t both have serial correlation

(are predictable), but there exists y1t eαy2t which is white noise

Common Features – Basic Ideas

1 Engle and Kozicki (1993) main example: No cointegration for

log-levels of GDP for the U.S. and Canada. Instantaneous growth rates of GDP for the U.S. and Canada have serial correlation and there is a linear combination of growth rates that is white noise. Cycles in U.S. and Canadian GDP growth are synchronized.

2 Our main …nding: common serial correlation between growth rates of

industrial production and metal commodity prices (e.g., Copper):

∆ ln PCOPPER t ∆ ln IPt = λ1 ft+ εCOPPERt εIPt , or, ∆ ln PCOPPER t λ∆ ln IPt = εCOPPERt λεIPt ,

Common Features – Basic Ideas

1 Engle and Kozicki (1993) main example: No cointegration for

log-levels of GDP for the U.S. and Canada. Instantaneous growth rates of GDP for the U.S. and Canada have serial correlation and there is a linear combination of growth rates that is white noise. Cycles in U.S. and Canadian GDP growth are synchronized.

2 Our main …nding: common serial correlation between growth rates of

industrial production and metal commodity prices (e.g., Copper):

∆ ln PCOPPER t ∆ ln IPt = λ1 ft+ εCOPPERt εIPt , or, ∆ ln PCOPPER t λ∆ ln IPt = εCOPPERt λεIPt ,

Common Features – Useful Dynamic Representations

Vahid and Engle (1993): VAR for yt, an n-vector of I(1) variables:

yt =Γ1yt 1+. . .+Γpyt p+et. (1)

VECM:

∆ yt =Γ1∆ yt 1+ . . . +Γp 1∆ yt p+1+γα0yt 1+et. (2)

Normalized cofeature vectors:

˜α= _˜α Is

(n s) s Quasi-structural model (restricted VECM):

" Is ˜α 0 0 (n s) s In s # ∆ yt = " 0 s (np+r) Γ1 . . . Γp 1 γ # 2 6 6 6 4 ∆ yt 1 .. . ∆ yt p+1 α0yt 1 3 7 7 7 5+vt. (3)

Common Cycles not considered in the model

VECM:₈ > > > > > > > > > > > > > > < > > > > > > > > > > > > > > :

∆ y1t= Γ₁₁∆ y1t 1+Γ₁₂∆ y2t 1+. . .+Γ_1n∆ ynt 1+...

+Γp 1,1∆ y_{1,t p}+1+. . .+Γp 1,n∆ yn,t (p 1) +γα0yt 1+e1t .. . .. . ∆ ynt= Φ11∆ y1t 1+Φ12∆ y2t 1+. . .+Φ1n∆ ynt 1+... +Φp 1,1∆ y1,t p+1+. . .+Φp 1,n∆ yn,t (p 1) +ηα0yt 1+ent

Common Cycles incorporated to the model

Quasi-structural model:₈ > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : ∆ y1t = λ1s∆ y(s+1)t +...+λ1n∆ ynt +e1t .. . ∆ yst = λss∆ y(s+1)t +...+λsn∆ ynt+e1t ∆ y(s+1)t = Γ11∆ y1t 1+Γ12∆ y2t 1+. . .+Γ1n∆ ynt 1+... +Γp 1,1∆ y1,t p+1+. . .+Γp 1,n∆ yn,t (p 1) +γα0yt 1+e1t .. .

∆ ynt = Φ11∆ y1t 1+Φ12∆ y2t 1+. . .+Φ1n∆ ynt 1+...

+Φp 1,1∆ y1,t p+1+. . .+Φp 1,n∆ yn,t (p 1)+

Common Features: A Test for Common Cycles

GMM approach: exploits the following moment restriction and test

H0 :existence of s linearly independent SCCF:

0=E 2 6 6 6 6 6 6 6 6 4 0 B B B B B B B B @ " Is ˜α 0 0 (n s) s In s # ∆ yt " 0 s (np+r) Γ1 . . . Γp 1 γ # 2 6 6 6 4 ∆ yt 1 .. . ∆ yt p+1 α0yt 1 3 7 7 7 5 1 C C C C C C C C A Zt 1 3 7 7 7 7 7 7 7 7 5 ,

where the elements of Zt 1 are the instruments comprising past series:

α0yt 1,∆ yt 1,∆ yt 2, ,∆ yt p+1. The test for common cycles is an over-identifying restriction test – the J test proposed by Hansen (1982). This test is robust to HSK of unknown form if it uses a White-correction in its several forms.

Common Trends and Cycles: Estimation

Over-identi…ed Quasi-structural model (restricted VECM) has

contemporaneous restrictions among the elements of ∆ yt:

" Is ˜α 0 0 (n s) s In s # ∆ yt = " 0 s (np+r) Γ1 . . . Γp 1 γ # 2 6 6 6 4 ∆ yt 1 .. . ∆ yt p+1 α0yt 1 3 7 7 7 5+vt.

FIML estimation imposing common-cycle (contemporaneous) restrictions.

GMM estimation imposing common-cycle (contemporaneous) restrictions.

Common Cycles: Copper prices and Global IP growth

-5 -4 -3 -2 -1 0 1 2 3 92 94 96 98 00 02 04 06 08 10 Copper Prices Growth Rates (std)

Global Ind. Prod. Growth Rates (std)

Sources: LME and J.P.Morgan

Deaton and Laroque (JPE, 1996) stress the importance of demand factors for commodity prices.

Common Cycles: From Metal Prices to IP Quantum

Firm’s cost minimization problem when producing y0 using a given metal

as an input in production: min

x C(w , x) =w x s.t. f(x) y0.

Derived demand for inputs. From Shepard’s Lemma: ∂C(w , x )

∂wi

=x_i (w , y0).

Suppose that, in the short-run, metal supply cannot be increased, therefore we treat it as …xed (xi). In equilibrium:

Common Cycles: From Metal Prices to IP Quantum

Short-run relationship between changes in metal prices (wi) (ceteris

paribus) and in industrial production (y0)for a representative …rm: 0 = ∂xi (w , y0) ∂wi dwi +∂xi (w , y0) ∂y0 dy0, or, dwi dy0 = ∂xi(w ,y0) ∂y0 ∂xi(w ,y0) ∂wi >0, since ∂xi(w ,y0) ∂y0 >0 and ∂xi(w ,y0) ∂wi <0.

As the equilibrium horizon becomes larger, supply cannot be treated as …xed, which reduces the importance of demand factors.

Econometric model

Quasi-structural model:₈ < : ∆ pCOPPER t = λ∆ipt +e1t

∆ ipt = α1∆ipt 1+α2∆ipt 2

Bi-variate Global Common-Cycle Tests (Monthly)

Sample from 1992 to 2012, comprises 252 observations

Table: Global Industrial Production

∆y1,t ∆y2,t ˜α J-statistic

(1, ˜α ) (SD) [p-value]

Aluminum Global Industrial Production -5.316 0.0427 (0.969) [0.036]

Lead Global Industrial Production -4.052 0.0331

(2.101) [0.047]

Copper Global Industrial Production -7.523 0.0310 (1.504) [0.189]

Tin Global Industrial Production -5.23 0.0096

(1.603) [0.512]

Nickel Global Industrial Production -6.034 0.0292 (1.728) [0.219]

Zinc Global Industrial Production -5.827 0.0337

Bi-variate Global Common-Cycle Tests (Quarterly)

Sample from 1992 to 2012, comprises 84 observations

Table: Global Industrial Production

∆y1,t ∆y2,t ˜α J-statistic

(1, ˜α ) (SD) [p-value]

Aluminum Global Industrial Production -4.523 0.0411 (0.410) [0.667]

Lead Global Industrial Production -12.442 0.0855

(5.110) [0.080]

Copper Global Industrial Production -2.586 0.1099 (0.603) [0.251]

Tin Global Industrial Production -3.269 0.0671

(0.647) [0.387] Nickel Global Industrial Production -16.187 0.0986 (7.389) [0.050]

Zinc Global Industrial Production -2.202 0.0817

Trends and Cycles in GDP and IP

-6 -5 -4 -3 -2 -1 0 1 2 3 92 94 96 98 00 02 04 06 08 10 GDP Growth Rates (std)

Ind. Prod. Growth Rates (std)

Global (log) GDP and (log) IP cointegrate and share a common cycle on growth rates with vectors:

Multivariate Analysis: Metal Prices, GDP, and IP

2 6 6 6 6 6 6 6 6 6 4 1 0 0 0 6.87 (0.32) 0 1 0 0 8.12 (0.38) 0 0 1 0 3.92 (0.17) 0 0 0 1 0.30 (0.01) 0 0 0 0 1 3 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 4 ∆ pAl t ∆ pCo t ∆ pTin t ∆yGdp t ∆ipt 3 7 7 7 7 5 | {z } ∆Xt = 2 6 6 6 6 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 7 7 7 7 5 2 6 6 6 6 4 ∆ pAl t 1 ∆ pCo t 1 ∆ pTin t 1 ∆yGdp t 1 ∆ipt 1 3 7 7 7 7 5 + 2nd Lag + 2 6 6 4 0 .. . 0.043 (0.01) 3 7 7 5 yt 1Gdp 1.39ipt 1 J-test [P-value] = 0.2259 [0.9848]

Multivariate Analysis: IP, GDP and Metal Prices

Special Trend-Cycle Decomposition for previous (5) series Xt

5 1

with (4)

common trends and (1)common cycle. Let:

A 5 5 = 2 4 α 0 1 5 eα0 4 5 3 5 and A 1 5 5 = h α 5 1 5 4eα i . Then, A 1A Xt 5 1 = α α0Xt | {z } 5 Cycles +eα eα0Xt | {z } 5 Trends

Cycles are generated by I(0)cointegrating-vector linear combinations

α0Xt.

Trends are generated by I(1)martingale cofeature-vector linear

Multivariate Analysis: IP, GDP and Metal Prices

Trend-Cycle Decomposition for Global GDP:

-.06 -.04 -.02 .00 .02 .04 4.8 5.0 5.2 5.4 5.6 5.8 92 94 96 98 00 02 04 06 08 10 Global GDP (logs) Trend Cycle

Multivariate Analysis: IP, GDP and Metal Prices

Trend-Cycle Decomposition for Global Industrial Production:

-.16 -.12 -.08 -.04 .00 .04 .08 5.3 5.4 5.5 5.6 5.7 5.8 5.9 92 94 96 98 00 02 04 06 08 10

Ind. Production (logs) Trend

Multivariate Analysis: IP, GDP and Metal Prices

Cycles for Metal Prices and Global Industrial Production:

-.8 -.6 -.4 -.2 .0 .2 .4 92 94 96 98 00 02 04 06 08 10

log-Al. Price Cycle log-Co. Price Cycle log-Tin Price Cycle log-Ind. Prod. Cycle

Alternative Model: Only GDP and IP (no Metal Prices)

" 1 0.342 (0.022) 0 1 # ∆yt ∆ipt = 0 0 ∆yt 1 ∆ipt 1 + " 0 0.123 (0.033) # (yt 1 1.505ipt 1) J-test [P-value] = 0.0353 [0.2480]

h-quarter Ahead GDP Forecast with and without Metal

Prices

Forecast GDP RMSE( 100) w/ and w/o Metal Prices

Sys./Hor. h=1 h=2 h=3 h=4 h=5 h=6 h=7 h=8

W/ 0.53 1.15 1.67 2.05 2.30 2.41 2.48 2.06

W/O 0.56 1.29 1.91 2.38 2.66 2.74 2.71 2.70

System with metal prices uses Aluminium, Copper and Tin. System without metal prices uses only Global GDP and IP. Uses 42

out-of-sample observations with a recursively growing window. Denotes signi…cance using Clark-West test.

Here we do not take advantage of the informational gap between GDP and metal-commodity price data availability.

Forecasting GDP Growth with Metal Prices (h=1)

-.020 -.015 -.010 -.005 .000 .005 .010 .015 .020 92 94 96 98 00 02 04 06 08 10

Global GDP growth (observed) Global GDP growth (forecast)

Conclusions

We …nd widespread evidence of common cycles between

metal-commodity prices and global industrial production (and also global GDP).

These common cycles when incorporated to our econometric model improved its forecast performance of business cycles.