Approximating Fixed-Horizon Forecasts Using Fixed-Event Forecasts

Transcription

1 Approximating Fixed-Horizon Forecasts Using Fixed-Event Forecasts Malte Knüppel and Andreea L. Vladu Deutsche Bundesbank 9th ECB Workshop on Forecasting Techniques 4 June 216 This work represents the authors personal opinions and does not necessarily reflect the views of the Bundesbank 1 / 27

2 What we do Construct approximations for fixed-horizon forecasts from fixed-event forecasts; Approximation uses optimal weighting of fixed-event forecasts derived by minimizing the mean-squared approximation error; Explore gains in approximating one-year-ahead mean inflation and GDP growth for the 13 countries from the Consensus Economics (CE) survey... and the corresponding cross-sectional forecast disagreement. 2 / 27

3 Example: Fixed-event forecasts from Consensus Economics 3 / 27

4 Inflation forecasts from Consensus Economics (CE) Question: What is the forecast of 11-month-ahead year-on-year inflation rate (fixed-horizon), standing at beginning of month t? g t+11,t 1 = p t+11 p t 1 p t 1 g t,t 1 + g t+1,t + g t+11,t+1. Information at time t: Two CE fixed-event forecasts: Annual growth rate of price index for current calendar year: g (12) 1, = p(12) 1 p (12) 1, where p (12) 1 = 1 12 (p p 12 ), p (12) = 1 12 (p p ). Annual growth rate of price index for next calendar year: g (12) 2,1 = p(12) 2 p (12) 1 1, where p (12) 2 = 1 12 (p p 24 ). Drawbacks of fixed-event forecasts 4 / 27

5 Annual growth rates and monthly growth rates CE fixed-event forecasts are well approximated by linear function of monthly growth rates (Patton and Timmermann, 211): g (12) 1, ω 1 g 12,11 + ω 2 g 11,1 + + ω 24 g 11, 12 g (12) 2,1 ω 1 g 24,23 + ω 2 g 23, ω 24 g 1, where w k = 1 k PT weights current year PT weights next year Jan Apr Jul Oct Jan Apr Jul Oct Jan Apr Jul Oct previous year current year next year 5 / 27

6 Ad-hoc approximation of fixed-horizon forecasts at time t Approximate g t+11,t 1 by weighting the two CE fixed-horizon forecasts according to: ( ) ĝ t+11,t 1 wt adhoc g (12) 1, + 1 wt adhoc g (12) 2,1, with w adhoc t = 13 t 12. Example: In beginning of January (t = 1) one wants to forecast inflation Dec(current year)-to-dec(previous year): ĝ 12, g (12) 1, since w adhoc t = 1. Note: Information contained in the second fixed-event forecast g (12) 2,1 is ignored in this case. 6 / 27

7 Optimal approximation of fixed-horizon forecasts at time t We propose to determine the weight w t in: g t+11,t 1 = w t g (12) 1, + (1 w t) g (12) 2,1, by minimizing the expected squared approximation error [ min E (g t+11,t 1 g t+11,t 1 ) 2]. w Novel approximation accounts correctly for monthly growth rates embedded in g t+11,t 1 and g t+11,t 1 = w t g (12) 1, + (1 w t) g (12) 2,1. 7 / 27

8 Derive optimal approximation of fixed-horizon forecasts By introducing the vectors G = g 24,23 g 23,22. g 11, 12, A t,12 = 13 t t, B 1 = 12 ω 1 ω 2. ω 24, B 2 = we write the approximation error as linear expression of weight w t : g t+11,t 1 g t+11,t 1 = A t,12g ( w t B 1G + (1 w t ) B 2G ) E = A t,12 B 2 } {{ } :=M t ( + w t B 2 B }{{ 1) G. } :=N [ (g t+11,t 1 g t+11,t 1 ) 2] is a quadratic expression in w t. ω 1 ω 2. ω / 27

9 Optimal weights formula The optimal weight on the current year fixed-event forecast is: w t = M t ΩN / ( NΩN ). Ω is the covariance matrix of vector G of monthly growth rates Ω = E [ (G E [G]) ( G E [ G ])]. Ω needs to account for known and forecasted monthly growth rates in vector G. 9 / 27

10 Characteristics of the optimal weighting approach Optimal weighting approach can account for: type of growth rates embedded in the fixed-event forecasts (annual growth rates, quarterly growth rates,...); (assumptions about) the data generating process; the (assumed) covariance matrix of the data generating process (realized and forecast); discretionary forecast horizons (3-month, 6-month, 24-month); information from additional fixed-event forecasts, but does not consider information contained in previous or later forecasts or from other variables. 1 / 27

11 Properties of the optimal weighting approach Example: Assume data-generating process for the monthly growth rate of the variable: with ε t+1 iid N (, σ 2 ε). g t+1,t µ = ρ (g t,t 1 µ) + ε t+1, Assume that prior s month growth rate is observed, but not the one for the current month. The forecaster makes optimal forecasts. 11 / 27

12 Weight on current-year forecast from two methods w, ρ = w, ρ =.5 w, ρ =.8 w, ρ =.99 w adhoc Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Optimal weights w depending on ρ and ad-hoc weights w adhoc for current-year forecast g (12) 1,. 12 / 27

13 Relative approximation errors ρ = ρ =.5 ρ =.8 ρ = Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Ratio of expected squared approximation error with optimal weights to expected squared approximation error with ad-hoc weights for different values of ρ. 13 / 27

14 Empirical application based on Consensus Economics data Approximate each forecaster s unobservable forecast for CPI and GDP four-quarters-ahead year-on-year quarterly growth rate: g (3) t+4,t = p(3) t+4 p (3) t 1, where p (3) t = 1 3 (p 3(t 1)+1 + p 3(t 1)+2 + p 3(t 1)+3 ), t = 1, 2, 3, 4. using their individual forecasts for annual growth rates of CPI and GDP for current and next year, i.e. g (12) 1, and g (12) 2,1 ; in each March, June, September and December. 14 / 27

15 Assumptions of the optimal approximation method inflation data is known up to the previous month; GDP growth data is available up to previous quarter; both inflation and GDP growth are assumed to be monthly processes; growth rates of both variables are assumed to be iid, so that [ ] Ω = σε 2 I where I is the identity matrix. 15 / 27

16 Comparison of optimal and ad-hoc method weights w w adhoc inflation GDP growth March June September December.8.3. Weights for current-year forecasts In the first three quarters, the ad-hoc approach places (much) larger weight on the current-year forecasts 16 / 27

17 Optimal and ad-hoc approximations for Eurozone (EZ) 3 Euro Zone - Inflation Mean.5 Euro Zone - Inflation Disagreement Euro Zone - GDP Mean 1 Euro Zone - GDP Disagreement True forecast 2.8 Optimal appr. Adhoc appr Actual four-quarter-ahead Consensus mean forecasts and the approximations based on optimal and ad-hoc weights for inflation and GDP growth (left panels) and time series of disagreement (measured by the standard deviation) among the actual individual four-quarter-ahead Consensus forecasts and the approximations based on optimal and ad-hoc weights for inflation and GDP growth (right panels). 17 / 27

18 Relative approximation errors - Mean forecast inflation 2 Inflation Mean, Forecast horizon = 4q USA Japan Germany France UK Italy Canada Netherlands Norway Spain Sweden Switzerland Euro Zone Average Q1 Q2 Q3 Q4 Ratios of the average squared approximation errors using the optimal weights to the average squared approximation errors using the ad-hoc weights for inflation mean forecasts four quarters ahead. 18 / 27

19 Relative approximation errors - Mean forecast GDP growth 2 GDP Mean, Forecast horizon = 4q USA Japan Germany France UK Italy Canada Netherlands Norway Spain Sweden Switzerland Euro Zone Average Q1 Q2 Q3 Q4 Ratios of the average squared approximation errors using the optimal weights to the average squared approximation errors using the ad-hoc weights for GDP growth mean forecasts four quarters ahead. 19 / 27

20 Relative approximation errors - mean forecast Avg US JP GE FR UK IT CA NL NO ES SE CH EZ Inflation MSE ratio GDP growth MSE ratio Ratio of the mean-squared approximation error obtained with optimal weights to the mean-squared approximation error obtained with ad-hoc weights in all quarters. 2 / 27

21 Relative approx. errors - disagreement inflation forecasts 2 Inflation Disagreement, Forecast horizon = 4q USA Japan Germany France UK Italy Canada Netherlands Norway Spain Sweden Switzerland Euro Zone Average Q1 Q2 Q3 Q4 Ratios of the average squared approximation errors using optimal weights to the average squared approximation errors using ad-hoc weights for the standard deviations among individual inflation forecasts four quarters ahead. 21 / 27

22 Relative approx. errors - disagreement future GDP growth USA Japan Germany France UK Italy Canada Netherlands Norway Spain Sweden Switzerland Euro Zone Average GDP Disagreement, Forecast horizon = 4q Q1 Q2 Q3 Q4 Ratios of the average squared approximation errors using optimal weights to the average squared approximation errors using ad-hoc weights for the standard deviations among individual GDP growth forecasts four quarters ahead. Nr. Forecasters 22 / 27

23 Relative approximation errors - forecast disagreement Avg US JP GE FR UK IT CA NL NO ES SE CH EZ Inflation MSE ratio Bias optim ad-hoc Corr. true with optim with ad-hoc GDP growth MSE ratio Bias optim ad-hoc Corr. true with optim with ad-hoc Ratio of the mean-squared approximation error obtained with optimal weights to the mean-squared approximation error obtained with ad-hoc weights in all quarters, together with bias and correlation between true disagreement and disagreement based on the two approximations. 23 / 27

24 Conclusions Optimal approximation... gives easily-computable weights for constructing fixed-horizon forecasts from fixed-event forecasts; is flexible with respect to fixed-horizon forecasts of interest and availability of fixed-event forecasts; significantly decreases the mean-squared approximation error for mean forecasts of inflation and GDP growth empirically; is a first step towards a better measurement of disagreement among forecasters. 24 / 27

25 Background slides 25 / 27

26 Annual growth rates for EZ HICP - forecasts and realizations Euro Zone Consumer Prices Nowcast Realised CPI (at year end) Mean forecast for the year end event 1 Jan28 Jan29 Jan21 Jan211 Jan212 Jan213 Black line represents realized annual growth rate of Eurozone s HICP known (values known just at year end). Blue dots represent monthly individual forecasts of the current year s annual growth rate. Green line represents the mean of these fixed-event forecast. Main 26 / 27

27 Average number of respondents for fixed-event and fixed-horizon forecasts US JP GE FR UK IT CA NL NO ES SE CH EZ inflation fixed horizon fixed event GDP growth fixed horizon fixed event Average number of forecasters for Consensus Economics fixed-horizon (four-quarter-ahead) and fixed-event forecasts. The number displayed for the fixed-event forecasts is the average of the numbers for current- and next-year forecasts which tend to be virtually identical. main 27 / 27