Improving the Composite Indices of Business Cycles by Minimum Distance Factor Analysis

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1 Improving the Composite Indices of Business Cycles by Minimum Distance Factor Analysis Yasutomo Murasawa Institute of Economic Research, Kyoto University Kyoto , Japan First draft: April 999 This version: February 2000 Abstract Minimum distance factor analysis (MD-FA is applicable to stationary ergodic sequences The estimated common factors, or factor scores, are weighted averages of the current observable variables, ie, more informative variables receive larger weights MD-FA of the US coincident business cycle indicators (BCI gives a new US coincident composite index (CI of business cycles The weights on the BCIs for the new CI are similar to those for the Stock Watson Experimental Coincident Index (XCI Keywords: Covariance structure; Time series; Generalized method of moments; Stock Watson index

2 Introduction The traditional composite indices (CI of business cycles, currently published by the Conference Board in the US, are descriptive statistics No statistical model exists, at least explicitly, behind the method So the statistical meaning of the resulting indices is unclear Assuming a linear one-factor structure for the coincident business cycle indicators (BCI, Stock and Watson (989, 99 obtain their Experimental Coincident Index (XCI The XCI is an estimate of the realization of the common business cycle factor Their assumption that the factors are linear autoregressive (AR processes with normal errors, however, is inconsistent with the observed asymmetry of business cycles between expansions and recessions Diebold and Rudebusch (996 suggest to combine the factor model of Stock and Watson (989, 99 and the regime-switching model of Hamilton (989 They assume that the mean of the common factor is a two-state first-order Markov chain Kim and Yoo (995, Chauvet (998, and Kim and Nelson (998 estimate this type of models in different ways and propose alternative coincident indices Although their models are consistent with the asymmetry of business cycles, they may still be misspecified In addition, estimation of the associated nonlinear state-space models is cumbersome In this paper, we assume a linear one-factor structure for the coincident BCIs, and apply minimum distance factor analysis (MD-FA We do not assume a parametric model for the dynamics of the factors, but only require the observable variables to be stationary ergodic and square integrable The advantages of this approach are (i the obtained index is an estimate of the realization of the common business cycle factor, (ii the model is consistent with the asymmetry of business cycles, (iii we do not have to worry about misspecification of the dynamics of the factors, and (iv estimation is easy 2

3 The obtained index is essentially a weighted average of the current standardized growth rates of the BCIs, where more informative BCIs receive larger weights, while the traditional CI is essentially the simple average For the US coincident BCIs, our estimation results show that employees on nonagricultural payrolls (EMP and index of industrial production (IIP are more informative, ie, highly correlated with the common business cycle factor, than personal income less transfer payments (INC and manufacturing and trade sales (SLS, implying that it is more efficient to weight them accordingly The weights for the new CI are similar to those for the XCI In a sense, the new CI improves the traditional CI towards the XCI, although the new CI imposes less assumptions on the factors The plan of the paper is as follows Section 2 defines a factor model and derives the implied autocovariance structure Section 3 discusses identification of the model parameters Section 4 discusses estimation of the model parameters and the realization of the common factors Section 5 applies MD-FA to the US coincident BCIs and proposes a new CI Section 6 concludes 2 Factor Model Let {x t } t= L 2 be an N stationary sequence with mean µ x and autocovariance matrix function Γ xx ( We observe {x t } T t= Assume a K-factor structure for {x t } t=, where K < N, such that t Z, x t = µ x + Bf t + u t, ( where B R N K is an unknown factor loading matrix, {f t } t= is an unobservable K stationary sequence of common factor vectors with mean zero and autocovariance matrix function Γ ff (, and {u t } t= is an unobservable N stationary sequence of specific factor vectors with mean zero and autocovariance matrix function Γ uu ( Consider estimation of B and the realization of {f t } T t= In FA, for identification of B, we usually assume that (i rank(b = K, (ii 3

4 Γ ff (0 is positive definite (pd, and (ii u,t,, u N,t are uncorrelated with each other and with f t at all leads and lags Then we have s Z, Γ xx (s = BΓ ff (sb + Γ uu (s, (2 where Γ uu (s is diagonal Assume that we use Γ xx (0,, Γ xx (S, where S < T, for estimation of B Let x T := ˆΓ xx,t (s := T t=s+ T t=s+ x t, (x t x T (x t s x T, s = 0,, S Under certain conditions which we clarify later, s {0,, S}, ˆΓ xx,t (s is consistent for Γ xx (s as T Given this, the first issue is identification of B from Γ xx (0,, Γ xx (S Since (2 still has rotational indeterminacy, we need further restrictions on the model 3 Identification 3 Principal Factor Model As additional identification restrictions, we often assume that (i Γ ff (0 = I K, (ii Γ uu (0 is given and pd, and (iii B Γ uu (0 B is diagonal, ie, the columns of Γ uu (0 /2 B are orthogonal These restrictions relate FA to principal component analysis (PCA Consider identification of B Since Γ uu (0 is given and pd, rearranging (2, Γ uu (0 /2 Γ xx (0Γ uu (0 /2 I N = Γ uu (0 /2 BB Γ uu (0 /2 (3 Since rank(γ uu (0 = N and rank(b = K < N, rank (Γ uu (0 /2 Γ xx (0Γ uu (0 /2 I N = rank (Γ uu (0 /2 BB Γ uu (0 /2 = K Let λ 0 Λ := 0 λ K, W := [ w w K ], 4

5 where λ k is the kth largest eigenvalue of the left-hand side of (3, and w k is the associated normalized eigenvector By the eigenvalue decomposition, Γ uu (0 /2 Γ xx (0Γ uu (0 /2 I N = W ΛW (4 Since the columns of Γ uu (0 /2 B are orthogonal, comparing (3 and (4, Γ uu (0 /2 B = W Λ /2, or B = Γ uu (0 /2 W Λ /2 Next, consider identification of the realization of {f t } T t= Notice that, given B, ( is a generalized linear regression model for each t Given µ x, the GLS estimator of f t is, ˆf t = ( B Γ uu (0 B B Γ uu (0 (x t µ x = ( Λ /2 W W Λ /2 Λ /2 W Γ uu (0 /2 (x t µ x = Λ /2 W Γ uu (0 /2 (x t µ x Notice that if we normalize x t by Γ uu (0 /2, then FA and PCA are equivalent up to scale Let x t := Γ uu (0 /2 (x t µ x, t =,, T, and Γ x x (0 := var( x Let λ k be the kth largest eigenvalue of Γ x x (0 I N and w k be the associated normalized eigenvector, k =,, K The kth principal component of x t is w k x t The GLS estimator, which is now equivalent to the OLS estimator, of the realization of the kth common factor of x t is w k x t/ λk Unfortunately, this approach has some problems First, Γ uu (0 is usually unknown in practice Second, since the restriction on B is not explicit, it is difficult to derive the asymptotic distribution of an estimator of B Third, it does not use Γ xx (,, Γ xx (S, which may contain some information 5

6 32 Multivariate Errors-in-Variables Model Alternatively, we can simply assume that B = [I K, B 2] This restriction relates FA to multivariate errors-in-variables models (EVM Let t Z, f t := Then we can write ( as x,t µ x, x K,t µ x,k, v t := u,t u K,t f t = f t + v t, x K+,t µ x,k+ = β K+f t + u K+,t, x N,t µ x,n = β Nf t + u N,t, where β i is the ith row of B Eliminating f t, we have t Z, x i,t µ x,i = β i f t β iv t + u i,t, i = K +,, N Consider the equation for x K+,t Suppose that s {,, S}, Γ ff (s 0 Then i {K + 2,, N}, s {0,, S}, x i,t s is correlated with f t through f t and f t s, but uncorrelated with β K+ v t + u K+,t by our assumption So we can use them as instrumental variables for estimation of β K+ By the order condition, we need (N K (S + K, or K (N (S + /(S + 2, to identify β K+ The same argument applies to the other equations Notice that serial correlation in {f t } t= helps identification of B Without serial correlation (S = 0, the order condition requires that K (N /2 With sufficient serial correlation (S, the order condition requires only that K < N 6

7 4 Estimation 4 Autocovariance Matrices 4 Consistency Let We can write γ 0 := vech(γ xx (0 vec(γ xx ( vec(γ xx (S, ˆγ T := vech (ˆΓxx,T (0 vec (ˆΓxx,T ( vec (ˆΓxx,T (S ˆΓ xx,t (s := = = T t=s+ T t=s+ T t=s+ ( x T µ x (x t x T (x t s x T [(x t µ x ( x T µ x ][(x t s µ x ( x T µ x ] (x t µ x (x t s µ x T t=s+ (x t s µ x, s = 0,, S Let t Z, vech((x t µ x (x t µ x vec((x t µ x (x t µ x z t := vec((x t µ x (x t S µ x If {x t } t= is stationary ergodic, then {z t } t= is stationary ergodic; see Durrett (996, pp 336, 340 Note that E(z = γ 0 Theorem Suppose that {x t } t= is stationary ergodic, 2 x L 2 Then lim T ˆγ T = γ 0 as Proof Applying the ergodic theorem to {x t } t=, we can write ˆγ T = T t=s+ z t + o as ( 7

8 The second term is asymptotically irrelevant The result follows by applying the ergodic theorem to {z t } t= 42 Asymptotic Distribution Let (Ω, F, P ( be the probability space under consideration Let {F t } t= be a filtration on (Ω, F We use the notion of mixingale to simplify our discussion; see Davidson (994, ch 6 Definition We say that {X t } t= is an L p -mixingale with respect to {F t } t= if {c t } t=, {ψ s } s=0 R +, where lim s ψ s = 0, such that t Z, s 0, E(X t F t s Lp c t ψ s, (5 X t E(X t F t+s Lp c t ψ s+ (6 The second inequality holds trivially if {F t } t= is adapted to {X t } t= We say that a vector or matrix random sequence is an L p -mixingale if each element is an L p -mixingale Note that if {X t } t= is a mixingale, then t Z, E(X t = 0 The rate of convergence of the coefficient measures the degree of serial dependence We say that {a s } s=0 R + is of size q if s=0 a/q s < If a s = O(s r, where r > q, then it is of size q; see Davidson (994, p 20 If a sequence is of size q, then q < q, it is also of size q We say that an L p -mixingale is of size q if {ψ s } s=0 is of size q With the notion of mixingale, we can state Gordin s central limit theorem (CLT for stationary sequences in Durrett (996, pp as follows Theorem 2 (Gordin s CLT Suppose that {X t } t= is a stationary ergodic L 2 -mixingale with respect to {F t } t= of size, 2 {F t } t= is adapted to {X t } t= 8

9 Then and s 2 := lim T var T ( T X t t= T T t= X t <, d N ( 0, s 2 If E(X 0, then the CLT applies to {X t E(X t } t= even if E(X is unknown, because {F t } t= is adapted to {X t E(X t } t= The following result is now immediate Theorem 3 Suppose that {x t } t= is stationary ergodic, 2 {F t } t= is adapted to {x t } t=, 3 {x t µ x } t= and {z t γ 0 } t= are L 2 -mixingales with respect to {F t } t= of size Then and Σ := lim T var ( T t=s+ z t <, (ˆγT γ 0 d N(0, Σ Proof Applying Gordin s CLT to {x t µ x } t=, we can write ˆγT = T t=s+ z t + o p ( By asymptotic equivalence, the second term is asymptotically irrelevant Notice that a R N(N+/2+SN 2, (i {a z t } t= is stationary ergodic, (ii {F t } t= is adapted to {a (z t γ 0 } t=, and (iii {a (z t γ 0 } t= is an L 2 -mixingale with respect to {F t } t= of size The result follows by Gordin s CLT and the Crámer Wold device 9

10 Since a transformation of a mixingale is not a mixingale in general, we need separate mixingale conditions on {x t µ x } t= and {z t γ 0 } t= Using the notion of mixing, we can give sufficient conditions on {x t } t= for the mixingale conditions to hold Definition 2 The sth-order α-mixing and φ-mixing coefficients of {X t } t= are α s := sup sup t Z A F t,b F t+s P (A B P (AP (B, φ s := sup sup P (B A P (B, t Z A F t,b F t+s ;P (A>0 where t Z, s 0, F t+s t := σ(x t,, X t+s Definition 3 We say that {X t } t= is α-mixing [φ-mixing] if lim s α s [φ s ] = 0 Since P (A B = P (B AP (A, φ-mixing implies α-mixing As before, we say that a mixing sequence is of size q if the mixing coefficient is of size q Mixing inequalities give the following relations between mixing sequences and mixingales; see Davidson (994, sec 42 Theorem 4 Suppose that {X t } t= is stationary, 2 X L p, where p >, 3 E(X = 0, 4 {F t } t= is adapted to {X t } t= Then if {X t } t= is α-mixing of size q, where q > 0, then p [, p, {X t } t= is an L p -mixingale with respect to {F t } t= of size q(/p /p, 2 if {X t } t= is φ-mixing of size q, where q > 0, then p [, p], {X t } t= is an L p -mixingale with respect to {F t } t= of size q( /p 0

11 Proof Since {F t } t= is adapted to {X t } t=, (6 holds trivially It remains to show that (5 also holds By Theorem 42 in Davidson (994, p [, p, t Z, s 0, ( E(X t F t s Lp 2 2 /p + X t+s Lp α /p /p s ( where c := 2 2 /p + X Lp = cψ s, q, {ψ s } s=0 is of size q(/p /p and ψ s := α /p /p s Since {α s } s=0 is of size 2 By Theorem 44 in Davidson (994, p [, p], t Z, s 0, E(X t F t s Lp 2 X t+s Lp φ /p s = cψ s, where c := 2 X Lp and ψ s := φ /p s Since {φ s } s=0 is of size q, {ψ s } s=0 is of size q( /p Next corollary gives sufficient conditions on {x t } t= for {z t γ 0 } t= to be an L 2 -mixingale with respect to {F t } t= of size Corollary Suppose that {x t } t= is stationary, 2 x L p, where p > 2, 3 {F t } t= is adapted to {x t } t= Then if {x t } t= is α-mixing of size q, where q > 0, then p [, p/2, {z t γ 0 } t= is an L p -mixingale with respect to {F t } t= of size q(/p 2/p,

12 2 if {x t } t= is φ-mixing of size q, where q > 0, then p [, p/2], {z t γ 0 } t= is an L p -mixingale with respect to {F t } t= of size q( 2/p Proof By Theorem 4 in Davidson (994, if {x t } t= is α-mixing [φ-mixing] of size q, then {z t γ 0 } t= is α-mixing [φ-mixing] of size q We have z γ 0 L p/2, where p > 2, and E(z γ 0 = 0 The result follows by applying the previous theorem to {z t γ 0 } t= So if (i x L p, where p > 4, and {x t } t= is α-mixing of size 2p/(p 4, or (ii x L p, where p 4, and {x t } t= is φ-mixing of size p/(p 2, then {x t µ x } t= and {z t γ 0 } t= are L 2 -mixingales with respect to {F t } t= of size 43 Covariance Matrix Estimation Consider estimation of Σ If we observe {z t } T t=s+, then we can apply various heteroskedasticity and autocorrelation consistent (HAC covariance matrix estimators Let Γ zz ( be the autocovariance matrix function for {z t } t= Then Let Σ := lim T var ( ( T = lim T var = lim T = s= = Γ zz (0 + Γ zz (s T t=s+ t=s+ T S s= (T S z t z t (Γ zz (s + Γ zz (s s= ( s Γ zz (s z T := ˆΓ zz,t (s := T t=s+ T z t, t=s+s+ (z t z T (z t s z T, s = 0,, 2

13 A kernel estimator of Σ is l(t ˆΣ T := ˆΓ zz,t (s + k(s (ˆΓ zz,t (s + ˆΓ zz,t (s s= where k( is a kernel function and l( is a bandwidth function Since we do not observe {z t } T t=s+, this estimator is infeasible Let z T,t := z T := ˆΓ zz,t (s := vech((x t x T (x t x T vec((x t x T (x t x T, t = S +,, T, vec((x t x T (x t S x T T z T,t, t=s+ T t=s+s+ (z T,t z T (z T,t s z T, s = 0,, Then z T,t = z t + o p (, z T = T t=s+ t=s+s+ z t + o p ( = E(z t + o p (, ˆΓ zz,t (s = T (z t E(z t + o p ((z t E(z t + o p ( = Γ zz (s + o p ( So we can apply a kernel estimator to {z T,t } T t=s+ to estimate Σ 42 Parameters 42 Minimum Distance Estimator Assume that B = [I K, B 2] Then we can write (2 as s Z, Γ xx (s = [ IK B 2 ] Γ ff (s [ I K B 2 ] + Γ uu (s (7 3

14 Let vec(b 2 vech(γ ff (0 diag(γ uu (0 vec(γ ff ( θ 0 := diag(γ uu ( vec(γ ff (S diag(γ uu (S Let Θ R (N KK+K(K+/2+N+S(K2 +N be the parameter space and g : Θ R N(N+/2+SN 2 be such that θ Θ, ([ ] IK vech Γ B ff (0 [ I K B 2 ] + Γ uu (0 2 ([ ] IK vec Γ g(θ := B ff ( [ I K B 2 ] + Γ uu ( 2 ([ ] IK vec Γ B ff (S [ I K B 2 ] + Γ uu (S 2 Then we have γ 0 = g(θ 0 (8 An MD estimator of θ 0 is ˆθ T := arg min θ Θ (ˆγ T g(θ W T (ˆγ T g(θ, (9 where W T is a weighting matrix that is finite, pd, and plim T W T = W, where W is fixed, finite, and pd Different weighting matrices give different MD estimators of θ 0 If W T is the identity matrix, then we have the equally-weighted MD (EMD estimator If W T is such that W = Σ, then we have an optimal MD (OMD estimator, ie, its asymptotic variance-covariance matrix is the smallest among the MD estimators 422 Consistency Let {Q T : Ω Θ R} T =S+ be a sequence of random criterion functions such that T S +, θ Θ, Q T (θ := (ˆγ T g(θ W T (ˆγ T g(θ 4

15 By definition, T S +, Let Q : Θ R be such that θ Θ, ˆθ T = arg min θ Θ Q T (θ Q(θ := (γ 0 g(θ W (γ 0 g(θ Since γ 0 = g(θ 0 and W is pd, given the identification restrictions, θ 0 = arg min θ Θ Q(θ To prove consistency of ˆθ T, it suffices to check conditions (A (C for Theorem 4 in Amemiya (985 The only nontrivial part is in condition (C, where we have to show that Q T ( converges in probability to Q( uniformly on Θ Next lemma gives sufficient conditions for this to hold Lemma Suppose that Θ is compact, 2 plim T ˆγ T = γ 0 Then plim sup Q T (θ Q(θ = 0 T θ Θ Proof By the triangle inequality, T S +, θ Θ, Q T (θ Q(θ = (ˆγ T g(θ W T (ˆγ T g(θ (γ 0 g(θ W (γ 0 g(θ = [(ˆγ T γ 0 + (γ 0 g(θ] (W T W [(ˆγ T γ 0 + (γ 0 g(θ] +[(ˆγ T γ 0 + (γ 0 g(θ] W [(ˆγ T γ 0 + (γ 0 g(θ] (γ 0 g(θ W (γ 0 g(θ = (ˆγ T γ 0 (W T W (ˆγ T γ 0 +2(ˆγ T γ 0 (W T W (γ 0 g(θ +(γ 0 g(θ (W T W (γ 0 g(θ 5

16 +(ˆγ T γ 0 W (ˆγ T γ 0 + 2(ˆγ T γ 0 W (γ 0 g(θ ˆγ T γ 0 W T W ˆγ T γ 0 +2 ˆγ T γ 0 W T W γ 0 g(θ + γ 0 g(θ W T W γ 0 g(θ + ˆγ T γ 0 W ˆγ T γ ˆγ T γ 0 W γ 0 g(θ Since Θ is compact and g( C 0, M < such that sup θ Θ γ 0 g(θ M Taking the supremum on both sides, T S +, sup Q T (θ Q(θ ˆγ T γ 0 W T W ˆγ T γ 0 θ Θ +2 ˆγ T γ 0 W T W M + M W T W M + ˆγ T γ 0 W ˆγ T γ ˆγ T γ 0 W M The result follows by taking the probability limit on both sides and applying Slutsky s theorem Consistency of ˆθ T is now immediate Theorem 5 Suppose that Θ is compact, 2 plim T ˆγ T = γ 0 Then plim ˆθ T = θ 0 T Proof Verify the conditions for Theorem 4 in Amemiya ( Asymptotic Distribution To prove asymptotic normality of ˆθ T, redefine Q T ( and Q( as θ Θ, Q T (θ := 2 (ˆγ T g(θ W T (ˆγ T g(θ, Q(θ := 2 (γ 0 g(θ W (γ 0 g(θ 6

17 By differentiation, θ Θ, Q T (θ θ i = g (θ W T (ˆγ T g(θ, θ i Q (θ θ i = g (θ W (γ 0 g(θ, θ i i =,, (N KK + K(K + /2 + N + S ( K 2 + N, and 2 Q T (θ = 2 g (θ W T (ˆγ T g(θ + g (θ g W T (θ, θ i θ j θ i θ j θ i θ j 2 Q (θ = 2 g (θ W (γ 0 g(θ + g (θ W g (θ, θ i θ j θ i θ j θ i θ j i, j =,, (N KK + K(K + /2 + N + S ( K 2 + N We need the following lemma first Lemma 2 Suppose that plim T ˆθT = θ 0 Then plim sup T θ K(θ 0 where K(θ 0 is a compact neighborhood of θ 0 2 Q T (θ 2 Q(θ = 0, Proof The proof is similar to that of Lemma Given this, the proof of asymptotic normality of ˆθ T is standard Theorem 6 Suppose that plim T ˆθT = θ 0, 2 (ˆγ T γ 0 d N(0, Σ, where Σ < Then (ˆθT θ 0 d N(0, V, where V := (GW G GW ΣW G (GW G, G := g(θ 0 7

18 ] Proof By the mean value theorem, T S +, θ T [ˆθT, θ 0 such that Q T (ˆθT = Q T (θ Q T ( θt (ˆθT θ 0 By the first-order condition, T S +, Q T (θ Q T ( θt (ˆθT θ 0 = 0, or (ˆθT θ 0 = 2 Q T ( θt QT (θ 0 = 2 Q T ( θt GWT (ˆγT γ 0 = (GW G GW (ˆγ T γ 0 [ + 2 ] Q T ( θt GWT (GW G GW O p ( By asymptotic equivalence, it suffices to show that the second term is o p ( We can write T S +, 2 Q T ( θt GW G = ( 2 Q T ( θt 2 Q ( θt + ( 2 Q ( θt GW G Since θ T is consistent for θ 0, the first term is o p ( by the previous lemma, and the second term is o p ( by Slutsky s theorem Note that if W = Σ, then V = ( GΣ G 43 Factor Scores Consider estimation of the realization of {f t } T t= Stacking ( for t =,, T, and taking B and µ x as given, we have a seemingly unrelated regressions (SUR model such that x µ x = Bf + u, x T µ x = Bf T + u T 8

19 Let x := Γ uu := x x T Let l T := (,, Then we have, f := f f T, u := Γ uu (T Γ uu (0 Γ uu (0 Γ uu (T u u T, x l T µ x = (I T Bf + u, where E(u = 0 and var(u = Γ uu The infeasible GLS estimator of f is ˆf GLS,T = [ (I T B Γ uu (I T B ] (IT B Γ uu (x l T µ x By the Gauss Markov theorem, ˆf GLS,T is the BLUE of f To make the GLS estimator feasible, we need a consistent estimator of the whole Γ uu, which requires a parametric model for the dynamics of {u t } t= Alternatively, we may give up system estimation and apply a feasible single-equation GLS estimator, ie, ˆf T,t = ( ˆB T ˆΓuu,T (0 ˆBT ˆB T ˆΓuu,T (0 (x t x T, t =,, T, (0 where ˆB T and ˆΓ uu,t (0 are consistent estimators of B and Γ uu (0 respectively Although estimation errors in ˆB T and ˆΓ uu,t (0 cause a measurement-error bias in ˆf T,t, it disappears as T Note that if {u t } t= is serially uncorrelated, then Γ uu is block-diagonal So the full-equation GLS estimator and the single-equation GLS estimator are equivalent 5 New Composite Index 5 Data 5 Business Cycle Indicators We analyze the four BCIs in Table that currently make up the US coincident CI of business cycles Our data are from CITIBASE The sample period is 959: 998:2 (480 observations We take the first difference of the log of each series and 9

20 Table : Components of the US Coincident Composite Index BCI EMP INC IIP SLS Description Employees on nonagricultural payrolls (thousands, SA Personal income less transfer payments (billions of chained $, SA, AR Index of industrial production (992 = 00, SA Manufacturing and trade sales (millions of chained $, SA Note: SA means seasonally-adjusted, and AR means annual rate Table 2: Descriptive Statistics of the Business Cycle Indicators BCI Mean SD Min Max EMP INC IIP SLS multiply it by 00, which is approximately equal to the monthly percentage growth rate series Table 2 and 3 summarize some descriptive statistics of the transformed series We see that they have significantly different means and standard deviations (sd: EMP has lower mean than the others, and EMP and INC are much smoother than the other two (Table 2 We eliminate these differences by standardizing each series so that the sample mean is 0 and the sample sd is We also see that all variables are positively correlated In particular, EMP and IIP have the highest correlation, while INC and SLS have the lowest correlation (Table 3 Figure shows the sample autocorrelation functions of the BCIs We see that they have different autocorrelation structures: EMP has persistent autocorrelation, Table 3: Sample Correlation Coefficients of the Business Cycle Indicators EMP INC IIP SLS EMP 00 INC IIP SLS

21 EMP INC + IIP SLS Lag + + Figure : Sample Autocorrelation Functions of the Business Cycle Indicators while SLS has almost no serial correlation 52 Composite Indices The leading, coincident, and lagging CIs of business cycles are summary statistics of the selected leading, coincident, and lagging BCIs respectively In the US, the Conference Board calculates the coincident CI in the following five steps: Calculate the monthly symmetric growth rate series of the BCIs 2 Exclude outliers and normalize each symmetric growth rate series so that the sample standard deviation is 3 Take the simple cross-section average of the normalized symmetric growth rate series This is the monthly symmetric growth rate series of the CI 4 Calculate the level series from the symmetric growth rate series 5 Rebase the level series to average 00 in the base year See the December 996 issue of Business Cycle Indicators for more details 2

22 Table 4: Principal Component Analysis of the Business Cycle Indicators Principal Component st 2nd 3rd 4th Eigenvalue Proportion Eigenvector EMP INC IIP SLS Note that it could be a weighted average in Step 3 For example, if we apply PCA, then the first principal component is a weighted average, where the weight vector is proportional to the normalized eigenvector associated with the largest eigenvalue of the sample correlation coefficient matrix Table 4 is the result of PCA of the US coincident BCIs (in terms of the standardized first differences of their logs The weight vector for the first principal component, which accounts for 63% of the total variation, is (027, 024, 027, 022 The weights on EMP and IIP are larger than those on INC and SLS PCA is still a descriptive method that reduces the dimension of a multivariate sample without assuming a statistical model We apply FA because it is a statistical method that estimates the realization of the latent common factors underlying the observable variables 52 Estimation Results 52 Parameters We apply MD estimators to estimate a one-factor model for the standardized first differences of the logs of the US coincident BCIs Let x,t,, x 4,t be EMP, INC, IIP, and SLS respectively First, we have to set S, the highest order of the autocovariances included Unfortunately, we do not have a criterion for selecting an optimal S given T So we simply try S = 0, Second, we have to choose W T, a weighting matrix for the MD 22

23 Table 5: Results of Minimum Distance Estimation of the Factor Model Parameter EMD OMD S = 0 S = S = 0 S = β β (007 (008 (008 (02 β (008 (008 (008 (02 β (006 (005 (007 (00 γ ff ( (04 (04 (03 (007 γ uu, ( (005 (006 (005 (004 γ uu,2 ( (008 (007 (008 (005 γ uu,3 ( (006 (005 (006 (005 γ uu,4 ( (006 (007 (006 (005 Note: Numbers in parentheses are asymptotic se s estimator Although OMD estimators are asymptotically more efficient than the EMD estimator, they have small-sample bias in analysis of covariance structures; see Altonji and Segal (996 and Clark (996 So we try both EMD and OMD estimators To estimate Σ, we apply a HAC covariance matrix estimator proposed by Newey and West (994 (To implement the Newey West estimator, we use a GAUSS code written by Ka-fu Wong, which is available from the GAUSS Sourse Code Archive at American University Table 5 summarizes the estimation results Although the estimates are a little sensitive to the choice of S and W T, EMP and IIP seem to have larger factor loadings and smaller specific-factor variances than INC and SLS In other words, EMP and IIP are more informative about the common business cycle factor than INC and SLS Hence, it is more efficient to weight them accordingly to construct a CI 23

24 Table 6: Weights for the New Composite Index BCI EMD OMD PCA CI S = 0 S = S = 0 S = EMP (004 (004 (003 (004 INC (003 (002 (003 (002 IIP (004 (004 (005 (004 SLS (002 (002 (002 (002 Note: Numbers in parentheses are asymptotic se s 522 Weights for the New Composite Index In this application, ( becomes x t = βf t + u t ( Given the model parameters, the single-equation GLS estimator of f t is ˆf t = ( β Γ uu (0 β β Γ uu (0 x t 4 i= β i γ uu,i (0 x i,t, ie, ˆf t is essentially a weighted average of x,t,, x 4,t Since the weight on x i,t is β i /γ uu,i (0, more informative variables receive larger weights Table 6 summarizes estimates of the weight vector The weights are not equal but larger on EMP and IIP than on INC and SLS We call the CI associated with one of these weight vectors the new CI 53 Comparison with Other Indices 53 Stock Watson Experimental Coincident Index The Stock Watson XCI assumes that t Z, φ f (Lf t = v t, Φ u (Lu t = w t, ( ( [ ] vt σ 2 NID 0, v 0, w t 0 Σ ww 24

25 where L is the lag operator, φ f ( is a pth-order polynomial on R, and Φ u ( is a qth-order polynomial on R N N For identification, we assume that (i β = (, β 2 and (ii Φ u ( and Σ ww are diagonal To obtain the maximum likelihood (ML estimator of the model parameters, we rewrite the model into a state-space form, and apply the Kalman filter (KF to evaluate the likelihood function; see Appendix The XCI, obtained as a by-product of the KF, is the conditional expectation of f t given (y,, y t associated with the ML estimate of the model parameters To determine p and q, we use a model selection criterion such as Akaike s information criterion (AIC or Schwartz s Bayesian information criterion (BIC In our case, AIC and BIC are defined as AIC := BIC := { ln L T q T q (ˆθML [ (N KK + K 2 p + Nq ]}, { ln(t q [ ln L (ˆθML (N KK + K 2 p + Nq ]}, 2 where N = 4, K =, and T = 479 As shown in Table 7, AIC selects (p, q = (, 2 while BIC selects (p, q = (, 0 Although AIC selects too large models with positive probability as T, selecting too large models may be less harmful than selecting too small models The likelihood ratio (LR test statistic for testing H 0 : (p, q = (, 0 against H : (p, q = (, 2 is 3390 The asymptotic distribution of the LR test statistic under H 0 is χ 2 (8 Since the test strongly rejects H 0 in favor of H, we select (p, q = (, 2 Table 8 presents the ML estimates of the model parameters for (p, q = (, 2 Not surprisingly, the ML estimate of the factor loading vector is close to the MD estimates in Table 5 Table 9 shows the implicit weights on the coincident BCIs for the XCI associated with the steady state KF The weights on the current coincident BCIs for the XCI are close to those for the new CI in Table 6 The difference is that the XCI puts 25

26 Table 7: Lag-Order Selection for the Factor Model (p, q ln L (ˆθML AIC BIC (0, (0, (0, (, (, (, (, (2, (2, (2, (2, (3, (3, (3, (3, Table 8: Result of Maximum Likelihood Estimation of the Factor Model Parameter EMP INC IIP SLS β (007 (008 (006 φ f 056 (005 σf (004 φ u, (005 (005 (007 (005 φ u, (005 (005 (006 (005 σw (003 (004 (003 (004 Note: Numbers in parentheses are asymptotic se s 26

27 Table 9: Weights for the Stock Watson Index Lag EMP INC IIP SLS ( ( ( ( ( Table 0: Sample Correlation Coefficients of the Indices New CI CI XCI New CI 000 CI XCI nonzero weights on the lagged coincident BCIs 532 Comparison Table 0 shows the sample correlation coefficient matrix of the three indices Since the new CI has a higher correlation with the XCI than the traditional CI, the new CI improves the traditional CI towards the XCI A difficulty in comparing alternative indices is absence of a criterion for good indices If we agree that real GDP is the most important coincident BCI, then a possible criterion is correlation with real GDP in quarterly growth rates Table shows the sample correlations of the three indices with real GDP The XCI has the highest correlation with real GDP, while the new CI has the lowest Moreover, the new CI hardly improves the traditional CI towards the XCI in quarterly series Table : Sample Correlation Coefficients of the Indices with Real GDP New CI CI XCI Real GDP New CI 0000 CI XCI Real GDP

28 This criterion implicitly assumes that a coincident index is an estimate of unobservable monthly real GDP For this purpose, however, we can obtain a better index by combining monthly coincident BCIs and quarterly real GDP using statespace models This seems to be an interesting topic for future research 6 Discussion We applied MD-FA to the US coincident BCIs and obtained a new CI of business cycles The new CI, as well as the XCI, puts larger weights on more informative BCIs (EMP and IIP and smaller weights on others (INC and SLS So it is more efficient than the traditional CI, which simply puts equal weights One drawback of the new CI is that it is not unique for two reasons First, we do not know how to choose S, the highest order of the autocovariance matrices included, in finite samples Second, we do not know which MD estimator to use in finite samples OMD estimators are asymptotically more efficient than the EMD estimator, but feasible OMD estimators are biased in small samples Recently, Kitamura and Stutzer (997 proposed an estimator that avoids the small-sample bias of feasible optimal GMM estimators Their estimator seems to be applicable to our problem 7 Acknowledgments This paper is based on my PhD dissertation, Minimum Distance Factor Analysis of Time Series: Theory and Application, University of Pennsylvania, 999 I am grateful to my advisor Roberto Mariano for his guidance and encouragement I also thank Frank Diebold, Frank Schorfheide, Robert Stein, and Yoshiyuki Takeuchi for their valuable comments and suggestions I am solely responsible for any remaining errors 28

29 A Stock Watson Experimental Coincident Index A State-Space Representation Premultiplying both sides of ( by Φ u (L, t Z, y t = Φ u (LBf t + w t, where y t := Φ u (Lx t When p q, a state-space model for {y t } t= is t Z, f t f t p = [ Φf, Φ f,p O K K I pk O pk K ] f t ( IK + v O t, pk K f t (p+ y t = [ B Φ u, B Φ u,q B O N (p qk ] f t f t p + w t, where o n is the n zero vector and O m n is the m n zero matrix When p q, a state-space model for {y t } t= is t Z, f t f t q = [ Φf, Φ f,p O K (q p+k I qk O qk K y t = [ B Φ u, B Φ u,q B ] In either case, we can write t Z, ] f t f t f t q f t (q+ + w t + ( IK O qk K v t, s t = F s t + Gv t, (2 y t = Hs t + w t, (3 where s t, F, G, and H are defined appropriately A2 Likelihood Function Let Θ be the parameter space Let f(; θ, θ Θ, be a pdf of (y,, y T Then T f(y,, y T ; θ = f t t (y t y,, y t ; θ, t= where t 2, f t t ( y,, y t ; θ is a conditional pdf of y t given (y,, y t (for t =, it is a marginal pdf Let F 0 := {, Ω} and t, F t := σ(y,, y t 29

30 Let t, µ t t (θ := E θ (y t F t, Σ t t (θ := var θ (y t F t Then t, y t y,, y t N ( µ t t (θ, Σ t t (θ So t, f t t (y t y,, y t ; θ = (2π N/2 det ( Σ t t (θ /2 ( exp ( yt µ t t (θ Σt t (θ ( y t µ t t (θ 2 The log-likelihood function for θ given (y,, y T is ln L(θ; y,, y T = NT 2 ln 2π 2 2 T ln det ( Σ t t (θ t= T ( yt µ t t (θ Σt t (θ ( y t µ t t (θ t= To evaluate the log-likelihood function, we must evaluate { µ t t (θ, Σ t t (θ } T t= Let t, s 0, ŝ t s := E θ (s t F s, P t s := var θ (s t F s From (3, t, µ t t (θ = Hŝ t t, Σ t t (θ = HP t t H + Σ ww Given θ, we can apply the KF to evaluate { ŝ t t, P t t } T t= 30

31 A3 Kalman Filter A3 Initial State First, we must specify ŝ 0 and P 0 To obtain the exact ML estimator, we set ŝ 0 = µ s, P 0 = Γ ss (0, where µ s := E(s and Γ ss (0 := var(s Since {s t } t= is stationary, taking expectations on both sides of (2, µ s = F µ s Assuming that I (max{p,q}+k F is nonsingular, µ s = 0 From (2, we also get Γ ss (0 = F Γ ss ( + GΣ uu G, Γ ss ( = F Γ ss (0 Eliminating Γ ss (, Γ ss (0 = F Γ ss (0F + GΣ uu G, or vec(γ ss (0 = vec(f Γ ss (0T + vec(gσ uu G = (F F vec(γ ss (0 + vec(gσ uu G = ( I [(max{p,q}+k] 2 F F vec(gσuu G In practice, we can simply set ŝ 0 0 = 0, P 0 0 = 0, 3

32 which implies that ŝ 0 = 0, P 0 = GΣ vv G The resulting estimator is asymptotically equivalent to the ML estimator A32 Updating Since (s 0, u is joint normal, s is normal Similarly, s 2, s 3, are also normal So t, s t y,, y t N ( ŝ t t, P t t We have t, y t ŷ t t = H ( s t ŝ t t + vt, where ŷ t t := E(y t F t So t, ( st y t y,, y t N (( ŝt t ŷ t t [ ] Pt t P, t t H HP t t HP t t H + Σ vv Let t, B t := P t t H ( HP t t H + Σ vv The updating equations for ŝ t t and P t t given ŝ t t and P t t are t, ŝ t t = ŝ t t + B t ( yt Hŝ t t, P t t = P t t B t HP t t A33 Prediction From (2, the prediction equations for ŝ t t and P t t given ŝ t t and P t t are t, ŝ t t = F ŝ t t, P t t = F P t t F + GΣ uu G Combining the updating and prediction equations, we get { ŝ t t, P t t } T t= 32

33 References Altonji, J G & Segal, L M (996, Small-sample bias in GMM estimation of covariance structures, Journal of Business and Economic Statistics 4, Amemiya, T (985, Advanced Econometrics, Harvard University Press Chauvet, M (998, An econometric characterization of business cycle dynamics with factor structure and regime switching, International Economic Review 39, Clark, T E (996, Small-sample properties of estimators of nonlinear models of covariance structure, Journal of Business and Economic Statistics 4, Davidson, J (994, Stochastic Limit Theory, Oxford University Press Diebold, F X & Rudebusch, G D (996, Measuring business cycles: A modern perspective, Review of Economics and Statistics 78, Durrett, R (996, Probability: Theory and Examples, second edn, Duxbury Press Hamilton, J D (989, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57, Kim, C-J & Nelson, C R (998, Business cycle turning points, a new coincident index, and tests of duration dependence based on a dynamic factor model with regime switching, Review of Economics and Statistics 80, Kim, M-J & Yoo, J-S (995, New index of coincident indicators: A multivariate Markov switching factor model approach, Journal of Monetary Economics 36,

34 Kitamura, Y & Stutzer, M (997, An information-theoretic alternative to generalized method of moments estimation, Econometrica 65, Newey, W K & West, K D (994, Automatic lag selection in covariance matrix estimation, Review of Economic Studies 6, Stock, J H & Watson, M W (989, New indexes of coincident and leading economic indicators, NBER Macroeconomics Annual pp Stock, J H & Watson, M W (99, A probability model of the coincident economic indicators, in K Lahiri & G H Moore, eds, Leading Economic Indicators, Cambridge University Press, chapter 4 34