Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting 10-12 May 2006 Revision confidence limits for recent data on trend levels, trend growth rates and seasonally adjusted levels Peter B. Kenny
Revision confidence limits for recent data on trend levels, trend growth rates and seasonally adjusted levels Peter B. Kenny (PBK Research)
Statement of problem • The latest seasonally adjusted and trend values will be revised as new data are added… • …even if the unadjusted data are not changed. • Data compilers know this, but it is not usually highlighted in statistical output. • Even the compilers seldom have a quantitative idea of the likely extent of revisions. • There are situations where knowledge of the likely extent of future revisions will aid data users in interpreting current data.
Possible solution • SEATS output includes estimates of revision standard error for trend and seasonally adjusted. • X-12-ARIMA has no corresponding output… • …so can we define a procedure to obtain one? • If X-12-ARIMA analysis includes an ARIMA model, we can formulate the requirement as: – What range of revisions is consistent with the future evolution of the ARIMA model of the series? • Look for a mathematical representation of this requirement.
Calculation Method (1) • The default forecast of the ARIMA model assumes all innovations are zero. • Actual realisations are generated by sampling a set of innovations from a Gaussian model. • We could generate a large set of realisations and see the extent of revisions (Monte Carlo). • But it would be more convenient to have a way of expressing the revisions as a function of the innovations.
Calculation Method (2) • Use the ‘black box’ method – find the response of the filter system to an impulse innovation at each forecast time point. • Consider how far ahead to estimate revisions – we must be confident that the ARIMA model will provide a reasonable representation. • Chosen forecast horizon is 36 months. • How far back to calculate revision limits? – more than 36 months back change is negligible.
Calculation Method (3) • The first step is to carry out a full modelling and estimation run, identifying an ARIMA model and producing 36 months of forecasts (with zero innovations). • Table B1 from this run is the basis for all the later variations with non-zero innovations. • We need to assume that the effects of the future innovations on past values are additive. • Experiment shows that we cannot make this assumption if extreme values are modified. – (This applies only to revision limit estimation)
Calculation Method (4) • First run to calculate base series for trend and seas. adj. – apply x-11 with no modification of extremes to saved Table B1. • Modify saved B1 by assuming an innovation of one s.e. in the first forecast period (all others remain zero). • Apply x-11 to this modified series, again with no extremes. • Repeat with the non-zero innovation in forecast period 2, 3, …, 36.
Calculation Method (5) • Calculate differences between the 36 variant results and the base case. • These are the coefficients of the linear approximation to the relationship between the actual innovations and the revision to the base trend and s.adj. series. • Since the innovations are by definition independent, the sum of squares of these coefficients gives the variance of the revision.
Illustration and Testing • Use Box-Jenkins airline passengers series. • Initial modelling with X-12-ARIMA identifies model ( 0 1 1)(0 1 1) and trading day effect. • Rerun with TD gives model (1 1 0)(0 1 1) - the TD effect is weekday – weekend contrast. • Take (1 1 0)(0 1 1) with TD as basic model for this series. • Model (0 1 1)(0 1 1) with TD is a close second and is used as a variant.
Figure 1: Box-Jenkins Airline Data Original, SA and Trend 700 600 500 400 300 200 100 0 1948 1950 1952 1954 1956 1958 1960 1962 Original SA Trend
Figure 2: Box-Jenkins Airline Data Revision Limits for X-11 SA 550 500 450 400 350 1958 1959 1960 1961 Estimate Upper bound Lower bound
Figure 3: Box-Jenkins Airline Data Revision Limits for X-11 Trend 550 500 450 400 350 1958 1959 1960 1961 Estimate Upper bound Lower bound
Figure 4: Box-Jenkins Airline Data Revision Limits for X-11 Alternative SA 550 500 450 400 350 1958 1959 1960 1961 Estimate Upper bound Lower bound
Figure 5: Box-Jenkins Airline Data Revision Limits for Seats SA 550 500 450 400 350 1958 1959 1960 1961 Estimate Upper bound Lower bound
Figure 6: Box-Jenkins Airline Data Revision Limits for Seats Trend 550 500 450 400 350 1958 1959 1960 1961 Estimate Upper bound Lower bound
Figure 7: Box-Jenkins Airline Data Comparison of Trend Revision Limits for Seats and X-11 18 16 14 12 10 8 6 4 2 0 1958 1959 1960 1961 X-11 Seats (PBK formula) Seats (Maravall formula)
Figure 8: Box-Jenkins Airline Data Comparison of Formula and Monte Carlo Trend Revision Limits (no extreme modification) 18 16 14 12 10 8 6 4 2 0 1958 1959 1960 1961 X-11 (PBK formula) X-11(Monte Carlo)
Figure 8a: Box-Jenkins Airline Data Comparison of Formula and Monte Carlo Trend Revision Limits (with extreme modification) 40 35 30 25 20 15 10 5 0 1958 1959 1960 1961 X-11 (PBK formula) X-11(Monte Carlo) X-11(MC with no extreme removal)
Figure 8b: Box-Jenkins Airline Data Comparison of Formula and Monte Carlo SA Revision Limits (with extreme modification) 40 35 30 25 20 15 10 5 0 1958 1959 1960 1961 X-11 (PBK formula) X-11(Monte Carlo) X-11(MC with no extreme removal)
Example 2 • UK Claimant Count Unemployment. • No longer the preferred measure – superseded by harmonised measure based on Labour Force Survey. • But a useful indicator – – precise (no sampling) – rapidly available – clear turning points
Figure 9: UK Claimant Count Unemployment Original, SA and Trend 3000 2500 2000 1500 1000 500 0 1994 1996 1998 2000 2002 2004 2006 2008 Original SA Trend
Figure 10: UK Claimant Count Unemployment Revision Limits for X-11 Trend 980 960 940 920 900 880 860 840 820 2003 2004 2005 2006 2007 Estimate Upper bound Lower Bound
Figure 11: UK Claimant Count Unemployment Revision Limits for X-11 Trend (data to March 2005) 980 960 940 920 900 880 860 840 820 2002 2003 2004 2005 2006 Estimate Upper bound Lower Bound
Figure 12: UK Claimant Count Unemployment Revision Limits for X-11 Trend (data to April 2005) 980 960 940 920 900 880 860 840 820 2002 2003 2004 2005 2006 Estimate Upper bound Lower Bound
Example 3 • UK Consumer Price Index • Harmonised (standard European measure) • Growth rate is official inflation target for Bank of England Monetary Policy Committee. • No official seasonal adjustment • Growth rate is 12-month change in unadjusted index level.
Figure 13: UK Consumer Price Index Original, SA and Trend 115 110 105 100 95 90 1994 1996 1998 2000 2002 2004 2006 Original SA Trend
Figure 14: UK Consumer Price Index Revision Limits for X-11 Trend (H9) 115 114 113 112 111 110 109 108 2002 2003 2004 2005 2006 Estimate Upper bound Lower Bound
Figure 15: UK Consumer Price Index Revision Limits for X-11 Trend (H23) 115 114 113 112 111 110 109 108 2002 2003 2004 2005 2006 Estimate Upper bound Lower Bound
Figure 16: UK Consumer Price Index Revision Limits for X-11 Trend Growth Rate (H23) 4 3.5 3 2.5 Growth rate (% p.a.) 2 1.5 1 0.5 0 2002 2003 2004 2005 2006 Estimate Upper bound Lower Bound
Outstanding Questions • We are looking at revisions in the X-11 process; what about revisions to regARIMA (level shifts, trading day factors, etc.)? • If the published s.adj. is based on 12 months’ forecasts, should we still use 36 to estimate the revision limits? • If there are serious doubts about the additivity of the individual innovation effects, should we use the Monte Carlo approach? • How best to present the results graphically?
UK Claimant Count Unemployment Current situation (data to April 2005) 980 960 940 920 900 880 860 840 820 2002 2003 2004 2005 2006 Trend Estimate Upper bound Lower Bound S.A.Estimate
Summary and Conclusions • The examples show that the inclusion of revision limits can give useful information to data users. • SEATS can already provide limits; the method proposed here enables X-12-ARIMA to do the same. • The method does not involve long computation; even if we use the Monte Carlo approach it is just a matter of minutes. • More research is needed!
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