Dynamic Hierarchical Factor Models

Transcription

1 Dynamic Hierarchical Factor Models Emanuel Moench 2 Serena Ng 1 Simon Potter 2 1 Columbia University 2 New York Fed December 2009

2 Motivation A multi-level (hierarchical) factor model: A large panel of data organized by B blocks, e.g. Production, Employment, Demand, Housing,... each block b has N b series, N b large N = B b=1 N b

3 Motivation A multi-level (hierarchical) factor model: A large panel of data organized by B blocks, e.g. Production, Employment, Demand, Housing,... each block b has N b series, N b large N = B b=1 N b Each block can be divided into sub-blocks e.g. sub-blocks of Demand: Retail Sales, Auto Sales, Wholesale Trade

4 Motivation A multi-level (hierarchical) factor model: A large panel of data organized by B blocks, e.g. Production, Employment, Demand, Housing,... each block b has N b series, N b large N = B b=1 N b Each block can be divided into sub-blocks e.g. sub-blocks of Demand: Retail Sales, Auto Sales, Wholesale Trade Within block variations due to block-level factors Between block variations due to common factors Idiosyncratic noise

5 A Three Level Model Level 1: For b = 1,..., B, i = 1,..., N b, X bit = Λ G.bi (L)G bt + e Xbit, ψ X.bi (L)e Xbit = ɛ Xbit,

6 A Three Level Model Level 1: For b = 1,..., B, i = 1,..., N b, X bit = Λ G.bi (L)G bt + e Xbit, ψ X.bi (L)e Xbit = ɛ Xbit, Level 2: For j = 1,... k Gb G bjt = Λ F.bj (L)F t + e Gbjt, ψ G.bj (L)e Gbjt = ɛ Gbjt.

7 A Three Level Model Level 1: For b = 1,..., B, i = 1,..., N b, X bit = Λ G.bi (L)G bt + e Xbit, ψ X.bi (L)e Xbit = ɛ Xbit, Level 2: For j = 1,... k Gb G bjt = Λ F.bj (L)F t + e Gbjt, ψ G.bj (L)e Gbjt = ɛ Gbjt. Level 3: For r = 1,... k F ψ F.r (L)F rt = ɛ Frt, ɛ Xbit, ɛ Gbjt, ɛ Fkt iid(0, σ 2 Xbi, σ2 G b j, σ2 F r ).

8 A Four Level Model For s = 1,..., S b, b = 1,..., B, i = 1,..., N b : Z bsit = Λ H.bsi (L)H bst + e Zbsit, Individual Series H bst = Λ G.bs (L)G bt + e Hbst, Subblock Factors G bt = Λ F.b (L)F t + e Gbt, Block factors ψ F.r (L)F rt = ɛ Frt Aggregate factors with dyanmics ψ Z.bsi (L)e Zbsit = ɛ Zbsit Ψ H.bs (L)e Hbst = ɛ Hbst Ψ G.b (L)e Gbt = ɛ Gbt

9 Why another factor model? 1) Block structure arises naturally in many economic and financial analyses: real activity: production, employment, demand, housing Global, regional, and country level variations Country, regions, state level variations Aggregate, industry, firm level variations in stock returns

10 2) Can put structure on the model through Ordering of the variables within block Ordering of the blocks where X bt = Λ Gb.0 G bt Λ Gb.sGb G b,t sgb + e Xbt Λ G.b0 = Easy interpretation of factors x... 0 x x 1 λ G.b01 λ G.b0kb

11 3) Addresses a limitation of level two models: e.g. k G = k F = 1: x ibt = λ G.ib (λ F.b F t + e G.bt ) + e X.ibt = λ ib F t + v ibt v ibt = λ G.ib e G.bt + e X.ibt. Ignoring block level variations gives a level 2 model: x it = λ i F t + v it with E(v it v jt ) 0 if i, j both belong to b. A multi-level model controls for these quasi-common variations.

12 4) State Space Framework Data sampled at mixed frequencies Missing values Internally coherent (no auxiliary forecasting model necessary)

13 5) Advantages and Uses Allows block and aggregate level analysis but still achieves dimension reduction Jointly estimates block level and aggregate factors Can be used for monitoring, counterfactuals, assess relative importance of shocks etc.

14 Application: Monitoring Real Activity in the US Data are released in various blocks throughout the month Releases broadly correspond to economic categories Block level factors are of independent interest Real time monitoring of F t and G bt More manageable than monitoring hundreds of series

15 Related Work 1 Diebold, Li, Yue (JOE 2008): Three level model for international bond yields Two step: estimate country level factors using OLS (with loadings fixed), then estimate global factors via MCMC. Limitations: single factor at block and global levels Information from global factors not taken into account when sampling country (block-level) factors. Global factors do not account for sampling uncertainty of block factors. Our one-step estimation solves both issues.

16 2 Kose-Otrok-Whiteman (AER 2003): three level model For each unit i in country b: x bit = c i F t + d bi e Gbt + e bit Top down vs. bottom up (G b vs e Gb ) single factors static loadings (N k F + N k G ) vs (k G k F + N k G ) parameters. Hierarchical structure: k G << N for each i, need to invert a T T matrix at each draw.

17 3 Hierarchical loadings vs. hierarchical factors Common factors across blocks, loadings differ by blocks Spatial factor models: loadings vary by distance Not a natural way of analyzing macroeconomic data All shocks are global, sensitivity to global shocks differ by regions

18 4 Principal components i One step estimation of F t (level two) ii One step estimation of F t and e Gbt (but no G bt ) iii Sequential estimation: F t ( G t ): For each b, first estimate G bt, then estimate F t from G t Ignore dynamic dependence of G bt on F t. Needs auxiliary equations for forecasting iv Block level dynamic principal components? Require N and T large.

19 Many other seemingly related state space models... Unique features of our dynamic hierarchical model: Coherent treatment of factors at different levels Produce factor estimates at both the block-level and aggregate levels Multiple factors at each level Hierarchical structure of factors, not loadings Analyze up to 4 levels, each with possibly multiple factors Can handle (but does not require) large N or T.

20 The State Space Form Block-Level Factors: G bt = Λ F.b0 F t + Λ F.b1 F t Λ F.blF F t lf + e Gbt, Ψ G.b (L)e Gbt = ɛ Gbt. implies (pseudo) measurement equation Ψ G.b (L)G bt = Ψ G.b (L)Λ F.b (L)F t + ɛ Gbt. Transition equation: Ψ F (L)F t = ɛ Ft,

21 Observed data: X bt = Λ Gb.0 G bt Λ Gb.sGb G b,t sgb + e Xbt ψ X.bi (L)e Xbit = ɛ Xbit, implies the individual level measurement equation Ψ X.b (L)X bt = Ψ X.b (L)Λ G.b (L)G bt + ɛ Xbt. Measurement equation from level 2 becomes the transition equation for state variable in level 1 G bt = α F.bt + Ψ G.b1 G bt Ψ G.bqGb G bt qgb + ɛ Gbt.

22 Observed data: X bt = Λ Gb.0 G bt Λ Gb.sGb G b,t sgb + e Xbt ψ X.bi (L)e Xbit = ɛ Xbit, implies the individual level measurement equation Ψ X.b (L)X bt = Ψ X.b (L)Λ G.b (L)G bt + ɛ Xbt. Measurement equation from level 2 becomes the transition equation for state variable in level 1 G bt = α F.bt + Ψ G.b1 G bt Ψ G.bqGb G bt qgb + ɛ Gbt. with time-varying intercept α F.bt = Ψ G.b (L)Λ F.b (L)F t.

23 Identification var(ɛ Gb ) and var(ɛ F ) are diagonal Block factor loadings Λ G0 = x... 0 x x 1 λ Gb01 λ Gb0kb Common factor loadings Λ F 0 = x.. 0 x x 1 λ F.01 λ F.0KF

24 MCMC Let Σ = (Σ F, Σ G, Σ X ), Ψ = (Ψ F, Ψ G, Ψ X ), Λ = (Λ G, Λ F ) 1 Use PCs as initial estimates of {G t } and {F t }, get initial values for Λ, Ψ, Σ 2 Conditional on Λ, Ψ, Σ and {F t }: draw {G bt } block by block using Carter-Kohn algorithm modified to allow for time varying intercept 3 Conditional on Λ, Ψ, Σ and {G t }: draw {F t } 4 Conditional on {F t } and {G t }: draw Λ, Ψ, and Σ assuming conjugate priors

25 Step 2: allows lower level factors to depend on the factors at the next level. Level two transition equation: α F.bt = Ψ Gb (L)Λ F (L)F t G bt = α F.bt + Ψ Gb.1 G bt Ψ Gb.qGb G bt qgb + ɛ Gbt The time-varying intercept α F.bt is known given Ψ, Λ, {F t }: Modify standard updating and smoothing equations for G bt to take this into account. Any filtering/sampling method for linear state space models can be adapted to hierarchical models this way.

26 Data Release Calendar June 2009 June 2 ISM Manufacturing Survey (ISM) Auto Sales (AUTO) June 6 Wholesale Trade (WT) Establishment Survey (ES) Household Survey (HS) June 17 Housing Starts (H-Starts) Industrial Production (IP) Capacity Utilization (CU) June 25 New Home Sales (H-NewSales) June 1 June 15 June 30 June 3 Manufacturing Shipments, Inventories, and Orders (DG) June 12 Retail Sales (RS) June 19 Philadelphia Business Outlook Survey (PhilaFed) June 26 Existing Home Sales (H-ExistSales) Chicago Midwest Mfg. Survey (ChicagoFed)

27 A Three Level Model: 315 series Block N Variable Ordered First Variable Ordered Second 1 CU 25 Machinery Motor Vehicles and Parts 2 IP 38 Durable Consumer Goods Nondurable Consumer Goods 3 ES 82 All Employ: Wholesale Trade Avg Wkly Earnings: Construction 4 HS 92 Civ. Labor Force: Men: Unemp. Rate Full-Time Men Worker 5 MS 35 PMI Composite Index Phila FRB General Activity Index 6 DG 60 Inventories: Machinery Mfrs Unfilled Orders: Machinery Parameters: k F = 1, k G = 2 s F = 2, s G = 2 q X = q G = q F = 1

28 Table 4: Level 3, 6 Block Model Block j ΨG.bj σ ɛbj 2 S.E CU: CU: IP: IP: ES: ES: HS: HS: MS: MS: DG: DG: Factor Ψ F.k σ F 2.k S.E

29 Decomposition of Variance Estimates S.E. block share F share G share X share F share G share X 1 CU: IP: ES: HS: MS: DG:

30 Figure: 6 Block Model of Real Activity: 3 Three Level Six Block Model for Output ˆF F

31 Is F t picking up block-level common variations? Bai and Ng (2002): IC 2 finds 2 factors common to G t for r = 1,... k F, regress F rt on F t. Let ẽ rt be the residuals regress ẽ rt on Ĝt R 2 measures variations in F kt that are not genuinely common: orthogonal to our F t but correlated with G bt.

32 Table 5: Correlation Between Ĝbkt and ẽ rt F b k b j ρ

33 Four Level Model: 447 Series Block subblock N K Gb K Hbs Variable Ordered First Production CU Capacity Utilization IP IP: Durables DG Manufacturers Inventorie Employment ES All Employees: Wholesale HS Civilian Labor Force: Men Demand RS Retail Sales: General Mer WS Merchant Wholesalers: S AUTO Domestic Car Retail Sale Housing H-STARTS Housing Starts: 1-Unit: W H-NEWSALES New 1-Family Houses So H-EXISTSALES NAR Total Existing Hom Mfg Surveys ISM ISM Mfg: PMI Composit PHILAFED Phila FRB Bus Outlook: CHICFED Chicago FRB: Midwest M

34 3 Four Level Model for Real Activity with 5 Blocks and 14 Subblocks: ˆF and all Ĝ Ĝ1 Ĝ2 5 Ĝ3 Ĝ4 Ĝ5 ˆF

35 Decomposition of Variance block sub-block share F share G share H share X Output IP Output CU Output DG Employment HS Employment ES Demand RS Demand WT Demand AUTO Housing H-starts Housing H-newsales Housing H-existsales Mfg Surveys ISM Mfg Surveys PhilaFed Mfg Surveys ChicagoFed

36 Monitoring in a data rich environment Update state of the block as new information in sub-block arrives Predict state of blocks for which new data are not yet released Use updated and predicted block factors to update aggregate factors

37 Figure: Four Level Model with 5 Blocks and 14 Subblocks 0 ˆF Monthly update Continuous update 0.9 Feb07 Jun07 Sep07 Dec07 Apr08 Jul08 Oct08 Jan09 May09 Aug09

38 Figure: Demand Block 0.4 Ĝ3: Demand Monthly update Continuous update 1 Feb07 Jun07 Sep07 Dec07 Apr08 Jul08 Oct08 Jan09 May09 Aug09

39 Figure: Housing 0.1 Ĝ4: Housing Monthly update Continuous update 0.6 Feb07 Jun07 Sep07 Dec07 Apr08 Jul08 Oct08 Jan09 May09 Aug09

40 Figure: Manufacturing 0.5 Ĝ5: Mfg Surveys Monthly update Continuous update 1.5 Feb07 Jun07 Sep07 Dec07 Apr08 Jul08 Oct08 Jan09 May09 Aug09

41 A Housing Model Three features of the model: i) Distinguishes regional from national variations 4 Census regions: North-East, West, Mid-West, South. ii) Distinguishes house price from housing market variations, i.e. use data on prices and volume. iii) Combines data that are sampled at monthly and quarterly frequencies and available over different time spans. iv) Allows observed factors.

42 Regional Series Source Frequency Price data Median Sales Price of Single Family Existing Homes NAR mly Single Family Median Home Sales Price CENSUS qly Average Existing Home Prices NAR qly Average New Home Prices NAR qly Conventional Mortgage Home Price Index FHLMC qly OFHEO Purchase-only Index OFHEO mly OFHEO Home Prices OFHEO qly Volume data New 1-Family Houses Sold CENSUS mly New 1-Family Houses For Sale CENSUS mly Single-Family Housing Units Under Construction CENSUS mly Multifamily Units Under Construction CENSUS mly Homeownership Rate CENSUS qly Homeowner Vacancy Rate CENSUS qly Rental Vacancy Rate CENSUS qly

43 National Series Source Frequency Price data Median Sales Price of Single Family Existing Homes NAR mly Median Sales Price of Single Family New Homes Census mly Single Family Median Home Sales Price CENSUS qly Average Existing Home Prices NAR mly Average New Home Prices CENSUS mly S&P/Case-Shiller Home Price Index S&P qly Conventional Mortgage Home Price Index FHLMC qly OFHEO Purchase-only Index OFHEO mly OFHEO Home Prices OFHEO qly Volume data New 1-Family Houses For Sale CENSUS mly Housing Units Authorized by Permit: 1-Unit CENSUS mly Multifamily Units Under Construction CENSUS mly Multifamily Permits US CENSUS mly Multifamily Starts US CENSUS mly Multifamily Completions CENSUS mly

44 Data used in this study: Regional level 7 price series 14 price and volume series. National level 9 price series 17 price and volume series.

45 Three Level Dynamic Factor model For each series i in region b: X bit = Λ G.bi (L)G bt + e Xbit. Let G t = (G 1t G 2t... G Bt ) the set of regional factors Let Y t be observed aggregate indices: ( ) Gt = Λ F (L)F t + Y t ( egt e Yt ) Identification: Λ Gb (0) is lower triangular with 1 s on the diagonal Λ F (0) is lower triangular with 1 s on the diagonal.

46 Figure 2 4 NAR 2 CMHPI F p 1 F p 3 BLS 1.5 CMHPI OFHEO 1.5 Case Shiller F p 1.5 F p 3 OFHEO Case Shiller

47 1.5 Figure 4: House Price vs. Housing Factor F p 1.5 F pq

48 Conclusion No model can serve all needs! A multilevel state space model with unique features Hierarchy in factors of up to 4 levels Explicitly model within block correlations Multiple factors at each level Dynamic evolution of the factors modeled in an internally coherent fashion We provide algorithms for estimation and monitoring in a data rich environment Wide range of applications.

49 Thank You!