Long-Run Covariability
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1 Long-Run Covariability Ulrich K. Müller and Mark W. Watson Princeton University October 2016
2 Motivation Study the long-run covariability/relationship between economic variables great ratios, long-run Phillips curve, nominal exchange rates and relative price levels, etc. Challenge: many economic time series are persistent spurious regression effects Cointegration framework is highly constraining Very specific model of persistence, rigid relationship between persistence and long-run covariability 1
3 This Paper Defines population measures about long-run covariability of general bivariate process Derives confidence intervals about these measures that are valid in flexible parametric model of long-run properties Descriptive statistics with confidence intervals Statistical framework reflects sparsity of long-run information (Müller and Watson (2008, 2016a, 2016b)) No consistent estimation of long-run properties Inference in presence of nuisance parameters (Elliott, Müller and Watson (2015), Müller and Norets (2016)) 2
4 Long-Run Correlation Matrix TFP ˆ % CI [0 71; 0 97] [0 02; 0 81] [0 95; 0 99] [0 45; 0 95] ˆ % CI [0 03; 0 81] [0 68; 0 97] [0 28; 0 91] ˆ % CI [0 02; 0 80] [ 0 08; 0 71] ˆ % CI [0 38; 0 93] 3
5 Outline of Talk 1. Introduction 2. Statistical framework, running examples 3. Measuring long-run covariability 4. A flexible parametric model of long-run properties 5. Construction of confidence intervals 6. Additional applications 4
6 5 GDP and Consumption Growth
7 6 GDP and Consumption Growth
8 Low-Pass Moving Averages vs Long-Run Projections 7
9 Cosine Transforms and Long-Run Projections Let Ψ ( ) = 2cos( ) For a scalar sequence { } =1,define = 1 X =1 Ψ ( ) = 2 X =1 cos( ), =1 8
10 9 =12Cosine Weights
11 10 GDP Growth
12 11 =12Cosine Transforms for GDP
13 Long-Run Projections Define ˆ as the predicted values from a regression of on {Ψ ( )} =1 Equivalently ˆ = 1 X =1 Ψ ( )= 2 because the weights Ψ ( ) are orthonormal X =1 cos( ) ˆ usefully thought of as low-pass filter for frequencies lower than 2 periods (66 years of data, =12 lowerthan11yearcycles) ˆ fully characterized by regression coefficients 12
14 13 GDP and Consumption Growth
15 14 Short and Long-Term Interest Rates
16 Asymptotic Behavior of Cosine Transforms Let ( 0 0 )0 be 2 1 cosine transforms of {( )} =1 Parameters of interest and suggested confidence intervals are defined in terms of ( 0 0 )0 Consider asymptotics where is fixed, as in Müller (2004, 2007, 2014), Phillips (2006), Müller and Watson (2008, 2013, 2016a, 2016b), Sun (2013, 2016) Defines notion of long-run : Periods of interest are 2 and longer Reflects that in samples of interest, reasonable are small Ignoring data information beyond ( 0 0 )0 avoids modelling higher frequency properties 15
17 Asymptotic Behavior of Cosine Transforms Suppose ( ) is stationary with a spectral density such that 2 stabilizes in 1 vicinity of 0 Under linear process assumption and additional weak regularity conditions, by CLT in Müller and Watson (2016) Ã! Ã! Ã! Σ Σ N (0 Σ), Σ = Σ Σ and Σ =VarÃ! Σ where Σ is a function of limiting behavior of 2 16
18 CLT for 1 2Z 1 0 ( )0 b c+1 Let = P =,where are and is 1. Supposethat (i) { F } is a martingale difference sequence with [ 0 ] = Σ, Σ invertible, sup [ 2+ ] for some 0, and for some sequence 0. [ 0 Σ F ] (ii) For every 0 the exists and integer 0 such that lim sup 1 P = +1 sup 2. (iii) P = (but not necessarily uniformly in ); denote spectral density of by :[ ] 7 H. (iii.a) Assume that there exists a function : R 7 H such that R 1 ( ), R 0 ( ) 2 and for all fixed, 1 Z 0 ( ) ( ) 0. 17
19 CLT for 1 2Z 1 0 ( )0 b c+1, ctd. (iii.b) For every diverging sequence 1 Z ( ) 2 = Z ( ) 2 0 (iii.c) 1 2 Z 1 ( ) = 1 2 Z 1 ( ) (iv) Each component of the function :[0 1] 7 R is of bounded variation. Then Z Ã ( ) 0 b c+1 N 0 0 and Z 1 Var[ 1 2 ( ) 0 b c+1 ] 0 Z µz 1 0 µz 1! ( ) i ( ) i ( ) 0 Z µz 1 0 µz 1 ( ) i ( ) i ( )
20 Corollary Let =,with as in the Theorem. In addition to the conditions there, assume R 1 ( ) =0.Then 0 Z Ã 1 Z µz 1 0 µz ( ) 1! 1 2 ( ) 0 b c+1 N 0 i ( ) i ( ) 0 and Z 1 Var[ 1 2 ( ) 0 b c+1 ] 0 Z µz µz ( ) 1 i ( ) 2 i ( ) 0 In combination, Theorem and Corollary cover wide range shapes of (pseudo-)spectral density close to zero. 19
21 Measuring Long-Run Covariability Ω = 1 X = = =1 Ã tr Σ tr Σ Ã Ω Ω Ã ˆ ˆ Ω Ω!Ã ˆ tr Σ tr Σ! = Ω ˆ!! 0 = X =1 Ã Ã tr Σ tr Σ tr Σ tr Σ!Ã!! 0 Scalar parameters derived from Ω = Ω Ω Ω = Ω Ω = q Ω Ω 2 Ω 20
22 21 GDP and Consumption Growth
23 Interpretation of and Define = argmin 1 X (ˆ ˆ ) 2 =argmin = v u tmin =1 1 X (ˆ ˆ ) 2 =1 = v u tmin X =1 X =1 ( ) 2 ( ) 2 Then and, since 1 X (ˆ ˆ ) 2 = 1 =1 X =1 ˆ 2 2 X =1 ˆ ˆ + 2 X =1 ˆ 2 = Ω 2 Ω + 2 Ω Population 2 is Ω 2 (Ω Ω ) 2 22
24 Parametrizing Σ Let ( ) = ( ) 1 i 2 be the pseudo-spectrum of ( ),and suppose ( ) ( ) in a suitable sense. Then Σ is a function of the local-to-zero spectrum In I(0) model, ( ) Λ, ( ) 0 N (0 Λ) Ω Λ In I(1) model, ( ) Λ 2, ( ) 0 N (0 Λ 2 ) Ω Λ 23
25 24 Short and Long-Term Interest Rates
26 I(0) and I(1) Inference Follows from well-known small sample results (cf. Anderson (1984)) and I-Rates I(0) I(1) I(0) I(1) Estimates % CI [0.81;0.97] [0.82;0.97] [0.93;0.99] [0.82;0.97] Estimates % CI [0.60;0.92] [0.67;1.01] [0.84;1.08] [0.68;1.03] Estimates % CI [0.26;0.55] [0.26;0.54] [0.47;0.97] [0.36;0.74] 25
27 AFlexibleModelfor ( ) model: ( ) = Ã ( ) ( ) 2 with [ 0 4 1], [0 ] and lower triangular! parameter model that embeds bivariate fractional, local-to-unity and local-level models as special cases Allows for various long-run phenomena such as (stochastic) breaks in means, slow mean reversion, overdifferencing, etc. Ω can be computed from implied Σ = Σ( ), =( ) 26
28 Construction of Confidence Sets Under asymptotic approximation, a parametric small sample problem: Observe = ( 0 0 ) 0 N (0 Σ), where Σ = Σ( ), = ( ) Θ Seek confidence interval ( ) for parameter of interest = ( ), for known Impose appropriate invariance on -weighted average expected length minimizing program Z min [lgth( ( ))] ( ) s.t. ( ( ) ( )) 1 Θ Optimal depends on unknown Lagrange multipliers (least favorable distribution Λ on Θ) 27
29 Form of Optimal Confidence Set For simplicity, ignore invariance. 0 : ( ) = 0 of the form 0 ( ) =1[ Z Then optimal inverts tests 0 of ( ) ( ) cv Z ( ) Λ 0 ( )] for some probability distribution Λ 0 on Θ with support on a subset of { : ( ) = 0 }, Λ 0 ({ : ( ( ) ( )) 1 }) =0and [ 0 ( )] for all ( ) = 0 Similar form under invariance, but not a family of distributions Λ if parameter of interest is affected by invariance see Müller and Norets(2016)for details. Effective dimension of parameterspaceafterinvarianceis8forallthreeproblems. 28
30 Computation of ALFD Λ Power upper bounds in Elliott, Müller and Watson (2015) generate lower bounds on length criterion for arbitrary Λ Basic algorithm 1. Let Θ = { 1 } be candidate set for support of Λ 2. Compute ALFD weights and cv such that size of test is on Θ. 3. Check size control on Θ: (a) If size is controlled, we are done (compare length to bound generated from cv 0 that induces Λ-weighted size equal to ). (b) If size violated, add violating to Θ andgotostep2. 29
31 Approximation to Coverage Standard Monte Carlo approximation is not continuous in, evenifsame draws of underlying pseudo-random numbers are used Importance sampling approximation: Z ( ) [ ( )] = ( ) ( ) ( ) with iid of density = [ ( ) ( )] 1 ( ) X =1 ( ) ( ) ( ) Proposal density is finite mixture of s, which are determined iteratively to minimize (empirical estimate of) variance of importance weights Var [ ( ) ( ) ] 30
32 Computational Details Size control: 500 consecutive BFGS searches with random starting values don t find violation = yields Monte Carlo standard errors of coverage of 0.1%- 0.35% for =0 1 ALFDs have about points of support CSarewithin5%oflowerboundonlengthcriterionfor =0 1 Single problem takes about 1 hour on fast PC using Fortran 31
33 Choice of Weighting Function Recall ( ) = Ã ( ) ( ) 2! Set =0, 1 = 2 =0, 1 2 [ ], = ( 1 )diag(15 0 1) ( 2 ) where [0 1] and ( ) is 2 2 rotation matrix of angle. 32
34 Credibility of Resulting CS Optimal ( ) couldbeemptyforsome, or unreasonably short Perversely, this is optimal for very uninformative draws, as coverage then costly in terms of length Generic problem in nonstandard problems: Frequentist (optimality) properties don t rule out unreasonable descriptions of uncertainty for some realizations Analyzed in detail in Müller and Norets (2016), building on old literature (Fisher (1956), Buehler (1959), Wallace (1959), Cornfield (1969), Pierce (1973), Robinson (1977), etc.) Suggested solution: constrained to be superset of equal-tailed credible set relative to prior 33
35 Spurious Regression? Coverage of weighted average length minimizing nominal 90% CIs for under =0 DGP\Assumed Model I(0) I(1) I(0) I(1)
36 Results in Running Examples I(0) I(1) and Estimates % CI [0.81;0.97] [0.82;0.97] [0.71;0.97] Estimates % CI [0.60;0.92] [0.67;1.01] [0.48;0.95] Estimates % CI [0.26;0.55] [0.26;0.54] [0.27;0.66] I-Rates Estimates % CI [0.93;0.99] [0.82;0.97] [0.89;0.99] Estimates % CI [0.84;1.08] [0.68;1.03] [0.75;1.14] Estimates % CI [0.47;0.97] [0.36;0.74] [0.53;0.92] 35
37 Real Variables: Correlation TFP ˆ % CI [0 71; 0 97] [0 02; 0 81] [0 95; 0 99] [0 45; 0 95] ˆ % CI [0 03; 0 81] [0 68; 0 97] [0 28; 0 91] ˆ % CI [0 02; 0 80] [ 0 08; 0 71] ˆ % CI [0 38; 0 93] 36
38 Real Variables: Selected Regressions ˆ 90% CI ˆ 0 76 [0 48; 0 95] [0 21; 2 21] [ ] 0 27 TFP 1.21 [0 91; 1 47]
39 38 PCE and CPI Inflation
40 PCE and CPI Inflation Estimate % CI [0.95;0.99] [0.98;1.24] 39
41 40 PCE Inflation and Unemployment
42 PCE Inflation and Unemployment Estimate % CI [-0.27;0.82] [-0.24;0.78] 41
43 42 PCE Inflation and Short I-Rates
44 PCE Inflation and Short I-Rates Estimate % CI [0.00;0.91] [-0.09;1.91] 43
45 44 TFP and Unemployment
46 TFP and Unemployment Estimate % CI [-0.75;-0.35] [-1.64;-0.27] 45
47 Conclusions Study statistical significance of long-run relationships that is robust to many DGPs Underlying derivations are involved, but not difficult or computationally intensive to apply to data In paper: Sensitivity of results to alternative and frequencies of interest Extension to more than two time series? Conceptually straightforward, but numerical determination of CIs very challenging Not too difficult to conduct Bayesian analysis based on multivariate normal approximation 46
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