Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting 10-12 May 2006 Comparing Seasonal Forecasts of Industrial Production Pedro M.D.C.B. Gouveia Denise R Osborn Paulo M.M. Rodrigues
Comparing Seasonal Forecasts of Industrial Production Pedro M.D.C.B. Gouveia University of Algarve Denise R Osborn University of Manchester Paulo M.M. Rodrigues University of Algarve Invited presentation by Denise Osborn at Eurostat conference “Seasonality, Seasonal Adjustment and their Implications for Short-Term Analysis and Forecasting”, Luxembourg 10-12 May 2006.
1. Background � Short run properties of many economic time series dominated by seasonality; � Particularly true of industrial production in some European countries; � Various methods are available for modelling seasonality; � Forecasting now embodied in X-12-ARIMA seasonal adjustment; � But little investigation of forecast accuracy for seasonal series. 2
Recent interest in forecasting using combination methods. Two advantages: 1. Improved accuracy compared to individual models; 2. Less risky than selecting a specific model. Indeed, simple mean of individual forecasts often performs well. This paper studies forecast performance for monthly industrial production: � Individual models and combinations � Combinations may exploit different approaches to seasonal modelling 3
We consider: � 17 individual forecasting models � 18 forecast combination methods Issues of interest: 1. Do some classes of methods perform better than others? 2. Do combinations improve forecast accuracy? 3. Are some combination methods better than others? 4
2. Data � Seasonally unadjusted industrial production indices for 17 countries: Austria, Canada, Denmark, Finland, France, Germany, Greece, Hungary, Italy, Japan, Luxembourg, Netherlands, Portugal, Spain, Sweden, UK, USA Plus Euro Area � Data: January 1980 to December 2005 Estimation & modelling: January 1980 to December 2002 Forecast period: January 2003 to December 2005 � Outliers (estimation period) removed using Tramo/Seats 5
Descriptive Statistics (Selected Series) Annual Growth (%) Monthly Growth (%) Country Mean St. Dev. Mean St. Dev. France 1.14 3.15 0.08 15.13 Germany 1.37 3.37 0.10 6.71 Italy 0.90 4.36 0.06 31.85 UK 1.00 3.52 0.06 7.14 USA 2.52 3.67 0.21 2.01 Euro Area 1.55 3.09 0.12 11.12 Annual/monthly standard deviations reflect different strengths of seasonality. 6
3. Models Observation for "season" (month) s of year n : y Sn+s , s = 1,..., S ; n = 0, 1, ... with S = 12 for monthly data. Models for seasonal y Sn+s can be classified in various ways. We consider: Linear models of ARMA class; Nonlinear threshold models; Periodic (seasonally varying coefficient) models. 7
Linear Models Linear models are based on: y Sn+s = µ Sn+s + x Sn+s φ ( L ) x Sn+s = u Sn+s where: µ Sn+s = E [ y Sn+s ] is trend & seasonals, φ ( L ) embodies any nonstationarity, u Sn+s is stationary, β ( L ) u Sn+s = θ ( L ) ε Sn+s Different linear models make different assumptions about φ ( L ). 8
Framework: y Sn+s = µ Sn+s + x Sn+s φ ( L ) x Sn+s = u Sn+s Seasonally integrated model Assumes seasonal nonstationarity: φ ( L ) = 1 – L S = ∆ S = (1 – L )(1 + L + … + L S -1 ) S unit roots; seasons not cointegrated. Roots of 1 + L + … + L S -1 are seasonal unit roots. 9
Deterministic seasonal model Stationary seasonality (no seasonal unit roots): φ ( L ) = 1, or = 1 – L SARIMA model Seasonal and first differences: φ ( L ) = ∆ S ∆ 1 = (1 – L ) 2 (1 + L + … + L S -1 ) Two zero frequency unit roots plus seasonal unit roots. 10
Evidence of nonstationarity May be able to discriminate between linear models through (seasonal) unit root tests. HEGY-type test based on ∆ S = (1 – L ) (1 + L + … + L S -1 ) t 0 tests conventional unit root (first differencing) F 1…6 tests seasonal unit roots (1 + L + … + L S -1 ) F 0…6 tests first and seasonal unit roots (annual differencing) 11
Seasonal Unit Root Tests (* Significant at 5%) t 0 F 1…6 F 0…6 Implication ∆ 1 France * * ∆ 12 Germany ∆ 12 Italy ∆ 1 UK * * ∆ 1 USA * * ∆ 12 Euro Area Overall: • Reject seasonal unit roots for about half of series; • Support seasonal integration for a group of European countries (Finland, Germany, Greece, Italy, Portugal, Spain, and Euro Area); • Strongest evidence against seasonal integration for UK & USA; • Conventional unit root supported for all series. 12
Nonlinear SETAR Models SETAR is AR model with regime-dependent coefficients y Sn+s = ∑ α s,j D s,Sn+s + φ 1, j y Sn+s -1 + … + φ p , j y Sn+s - p + ε Sn+s , j , j = 1, 2 D s,Sn+s are seasonal dummy variables and regime j is determined by j = 1 if ∆ S y Sn+s-d ≤ γ j = 2 if ∆ S y Sn+s-d > γ with γ and d unknown. Test for nonlinearity, with max p of 12 and d ≤ p , via bootstrap. Null hypothesis is linearity, or same coefficients for j = 1, 2. 13
Linearity Tests γ d p -value France .019 8 .167 Germany .037 5 .007* Italy .078 12 .112 UK .045 1 .129 USA -.051 2 .067 Euro Area .014 10 .001* Overall: Reject linearity (5%) for about half of series. 14
Periodic Models Periodic models allow coefficients to change with the seasons. PAR ( p ): y Sn+s = α s D s,Sn+s + φ 1, s y Sn+s -1 + … + φ p , s y Sn+s - p + ε Sn+s , s = 1, …, 12 with not all φ i ,1 = … = φ i , S = φ i , i = 1, …, p Can test for periodic variation either directly on the coefficients, or using residuals from a non-periodic model. S = 12 seasons may be excessive; also group months to construct S = 3 “seasons” (negative, low positive, high positive average monthly growth). 15
Periodicity Tests (* Significant at 5%) Coefficient Tests Residual Tests PAR (3) PAR (12) PAR (3) PAR (12) France * * * * Germany * * * Italy * * * UK * * * USA * * * * Euro Area * * * Overall: evidence of periodic coefficients for all series. 16
Summary: � Mixed evidence about appropriate differences (first or annual) � Mixed evidence about presence of nonlinearity � Strong evidence for seasonally-varying autoregressive coefficients Nevertheless, apply all models to all series. 17
4. Forecast Accuracy Use m post-sample observations to evaluate h -step ahead forecasts from models fitted to the first T observations. Accuracy measured by Root Mean Squared Predic- tion Error ( RMSPE ) v u T + m X u ¡ ¢ 2 1 t b RMSPE ( h ) = y T + j | T + j − h − y T + j m − h + 1 j = h Nonlinear model forecasts computed using bootstrap. Forecast Combination Methods Can be simple measures (mean, median), or more sophisticated methods based on perfor- mance.
Historical RMSPE y i Combine k separate h -step forecasts, b Sn + s + h | Sn + s ( i = 1 , ..., k ) as X k y c w h y i b i b Sn + s + h | Sn + s = Sn + s + h | Sn + s i =1 with weight w h i X k w h i = [1 /RMSPE ( h ) i ] λ / [1 /RMSPE ( h ) i ] λ j =1 for λ ∈ { 0 , 1 , 1 . 25 , 1 . 5 , 2 } . RMSPE ( h ) i computed using estimation period y i y i b T − 35 | T − 35 − h to b T | T − h . Also use discounted RMSPE weights X k ¡ ¢ − 1 / ¡ ¢ − 1 w h m h m h i = i j j =1 where v u δ j − 1 ³ ´ 2 T − h X u t m h y i b i = T − j +1 | T − j +1 − h − y T − j +1 j =1 for discount factor δ ∈ { 1 , 0 . 95 , 0 . 90 } .
Regression Methods Weights determined from k X y j β j b y Sn + s + h = β 0 + Sn + s + h | Sn + s + ε Sn + s + h j =1 estimated using observations y T − 35 to y T . In practice, only best 5 forecasts used.
5. Forecast Performance Methods Employed Forecast Models Code Individual Models Filter Deterministic terms ARMA (1 , 1) ∆ 12 M1 Intercept ARMA (2 , 2) ∆ 12 M2 Intercept ARMA (3 , 3) ∆ 12 M3 Intercept AR ( p ) ∆ 12 M4 Intercept SETAR ( p 1 , p 2 ) ∆ 12 M5 Intercept AR ( p ) levels Seas. intercepts + trend M6 PAR (12 , 3) levels Seas. intercepts + trend M7 PAR (3 , 3) levels Seas. intercepts & trends M8 M9 SETAR ( p 1 , p 2 ) levels Seasonal intercepts M10 AR ( p ) ∆ 1 Seasonal intercepts M11 PAR (3 , 3) ∆ 1 Seas. intercepts + trend M12 PAR (12 , 3) ∆ 1 Seas. intercepts & trends M13 SETAR ( p 1 , p 2 ) ∆ 1 Seasonal intercepts M14 ARMA (1 , 1) ∆ 1 ∆ 12 None M15 ARMA (2 , 2) ∆ 1 ∆ 12 None M16 ARMA (3 , 3) ∆ 1 ∆ 12 None M17 AR ( p ) ∆ 1 ∆ 12 None
Combination Methods Code Combination method parameters C1 mean of M1 to M17 C2 median of M1 to M17 λ = 0 C3 RMSPE weight C4 RMSPE weight λ = 1 λ = 1 . 25 C5 RMSPE weight λ = 1 . 5 C6 RMSPE weight λ = 2 C7 RMSPE weight C8 mean of best 5 models C9 mean of best 10 models C10 mean of best 15 models C11 median of best 5 models C12 median of best 10 models C13 median of best 15 models δ = 1 C14 discounted RMSPE weight C15 discounted RMSPE weight δ = 0 . 95 δ = 0 . 9 C16 discounted RMSPE weight C17 regression weight C18 mean of combinations
Recommend
More recommend
