Does modeling a structural break improve forecast accuracy?

Transcription

1 Does modeling a structural break improve forecast accuracy? Tom Boot University of Groningen Andreas Pick Erasmus University Rotterdam Tinbergen Institute De Nederlandsche Bank 10th ECB Workshop on Forecasting Techniques 2018 Does modeling a structural break improve forecast accuracy? 1 of 17

2 INSTABILITY IN MACROECONOMIC SERIES Stock and Watson (1996): strong evidence for structural breaks Does modeling a structural break improve forecast accuracy? 2 of 17

3 INSTABILITY IN MACROECONOMIC SERIES Stock and Watson (1996): strong evidence for structural breaks Figure: Fraction of series where a break is found macroeconomic and financial time series (FRED-MD), 1960M M10 AR(1) using a moving window of 120 observations Andrews (1993) (heteroskedasticity robust) sup-f test Does modeling a structural break improve forecast accuracy? 2 of 17

4 FORECASTING UNDER MODEL INSTABILITY Clements and Hendry (1998) view structural breaks as a key reason for forecasting failure Can explain lack of predictability in: Stock returns Ang and Bekaert (2004). Rapach and Wohar (2006). Paye and Timmermann (2006) Interest rates and inflation Rapach and Wohar (2005) Exchange rates Rossi (2006) Does modeling a structural break improve forecast accuracy? 3 of 17

5 FORECASTING UNDER MODEL INSTABILITY Clements and Hendry (1998) view structural breaks as a key reason for forecasting failure Can explain lack of predictability in: Stock returns Ang and Bekaert (2004). Rapach and Wohar (2006). Paye and Timmermann (2006) Interest rates and inflation Rapach and Wohar (2005) Exchange rates Rossi (2006) What should one do? Unbiasedness : post-break data Use some pre-break observations Pesaran & Timmermann (2005, 2007) Optimally weight all observations Pesaran, Pick & Pranovich (2013) We need to know (1) whether there is a break and (2) the break date. Does modeling a structural break improve forecast accuracy? 3 of 17

6 ARE BREAKS REALLY THAT BAD? Figure: Fraction of series where a break is found Breaks in the parameters breaks in forecasts If breaks are small, break-models are not always better Elliott and Müller (2007,2014): Large uncertainty around break date Does modeling a structural break improve forecast accuracy? 4 of 17

7 TESTING FOR BREAKS FROM A FORECASTING PERSPECTIVE We develop a break point test when forecasting under MSFE loss 1. Test for the break in the forecast, not in the parameters 2. Taking into account the full bias-variance trade-off: (a) shorter window, but no/smaller bias vs. longer window, but larger bias (b) break date uncertainty in break model Does modeling a structural break improve forecast accuracy? 5 of 17

8 TESTING FOR BREAKS FROM A FORECASTING PERSPECTIVE We develop a break point test when forecasting under MSFE loss 1. Test for the break in the forecast, not in the parameters 2. Taking into account the full bias-variance trade-off: (a) shorter window, but no/smaller bias vs. longer window, but larger bias (b) break date uncertainty in break model Main empirical finding Breaks that are relevant for forecasting occur much less frequent than existing tests indicate Does modeling a structural break improve forecast accuracy? 5 of 17

9 CASE 1: KNOWN BREAK DATE Consider two forecasts: 1) Forecast conditional on a structural break at T b = T τ b y t = x tβ 1 I[t < T b ] + x tβ 2 I[t T b ] + ε t 2) Forecast that ignores breaks y t = x tβ F + ε t Does modeling a structural break improve forecast accuracy? 6 of 17

10 CASE 1: KNOWN BREAK DATE Consider two forecasts: 1) Forecast conditional on a structural break at T b = T τ b y t = x tβ 1 I[t < T b ] + x tβ 2 I[t T b ] + ε t 2) Forecast that ignores breaks y t = x tβ F + ε t Expected mean squared forecast error [ ( 1) E x ˆβ ) ] 2 T +1 2 x T +1 β 2 ε T +1 = 2) E 1 T T b x T +1 V x T +1 + σ 2 [ ( x T +1 ˆβ F x T +1 β 2 ε T +1 ) 2 ] = 1 T x T +1 V x T +1 + σ 2 [ ] 2 Tb + T x T +1 (β 1 β 2 ) with V = plim T (T T b )Var(ˆβ 2 ) = plim T T Var(ˆβ F ) Does modeling a structural break improve forecast accuracy? 6 of 17

11 CASE 1: KNOWN BREAK DATE Consider two forecasts: 1) Forecast conditional on a structural break at T b = T τ b y t = x tβ 1 I[t < T b ] + x tβ 2 I[t T b ] + ε t 2) Forecast that ignores breaks y t = x tβ F + ε t Expected mean squared forecast error [ ( 1) E x ˆβ ) ] 2 T +1 2 x T +1 β 1 2 ε T +1 = x T +1 T T V x T +1 + σ 2 b [ ( 2) E x ˆβ ) ] 2 T +1 F x T +1 β 2 ε T +1 = 1 T x T +1 V x T +1 + σ 2 with V = plim T Var(ˆβ i ) [ ] 2 Tb + T x T +1 (β 1 β 2 ) Does modeling a structural break improve forecast accuracy? 7 of 17

12 WHEN IS THE FULL SAMPLE FORECAST BETTER? [ ] 1 E MSFE(ˆβ 2 ) = x T +1 V x T +1 + σ 2 T T b [ ] E MSFE(ˆβ F ) = 1 [ ] 2 T x T +1 V x T +1 + σ 2 Tb + T x T +1 (β 1 β 2 ) Full sample forecast is more accurate if ζ = T 1 [ x T +1 (β 1 β 2 ) ] 2 x T +1 ( V 1 τ b + V 2 1 τ b ) x T +1 Does modeling a structural break improve forecast accuracy? 8 of 17

13 TEST STATISTIC FOR FORECASTING Test for H 0 : ζ 1: ˆζ = T (ˆβ 2 ˆβ 1 ) x T +1 x T +1 (ˆβ 2 ˆβ 1 ) ( ) x ˆV 1 T +1 τ b + ˆV 2 1 τ b x T +1 H 0 χ 2 (1, 1) Does modeling a structural break improve forecast accuracy? 9 of 17

14 TEST STATISTIC FOR FORECASTING Test for H 0 : ζ 1: ˆζ = T (ˆβ 2 ˆβ 1 ) x T +1 x T +1 (ˆβ 2 ˆβ 1 ) ( ) x ˆV 1 T +1 τ b + ˆV 2 1 τ b x T +1 Differs from the standard Wald statistic for β 1 = β 2 H 0 χ 2 (1, 1) We get [ ˆV Ŵ = T (ˆβ 2 ˆβ 1 ) 1 + ˆV ] 1 2 (ˆβ τ b 1 τ 2 ˆβ 1 ) H 0 χ 2 (dim β) b 1. A test statistic weighted by x T A non-central χ 2 distribution Does modeling a structural break improve forecast accuracy? 9 of 17

15 CASE II: UNKNOWN BREAK DATE To account for uncertainty in the break date, consider local breaks of O(T 1/2 ). sup τ ( x ˆβ T +1 1 (τ) x ˆβ T +1 2 (τ) ˆζ(τ) = T ( ) x ˆV 1 T +1 τ + ˆV 2 1 τ x T +1 ) 2 Does modeling a structural break improve forecast accuracy? 10 of 17

16 CASE II: UNKNOWN BREAK DATE To account for uncertainty in the break date, consider local breaks of O(T 1/2 ). sup τ ( x ˆβ T +1 1 (τ) x ˆβ T +1 2 (τ) ˆζ(τ) = T ( ) x ˆV 1 T +1 τ + ˆV 2 1 τ x T +1 ˆζ(τ) converges to a Gaussian process indexed by τ ) 2 Does modeling a structural break improve forecast accuracy? 10 of 17

17 CASE II: UNKNOWN BREAK DATE To account for uncertainty in the break date, consider local breaks of O(T 1/2 ). sup τ ( x ˆβ T +1 1 (τ) x ˆβ T +1 2 (τ) ˆζ(τ) = T ( ) x ˆV 1 T +1 τ + ˆV 2 1 τ x T +1 ˆζ(τ) converges to a Gaussian process indexed by τ For each break date τ b, there is a unique break size ζ(τ b ) such that [ ] [ ] E MSFE Asy (ˆβ F ) = E MSFE Asy (ˆβ 2 (ˆτ)), ˆτ = arg max ˆζ(τ) τ When the break date is known ζ(τ) = 1. ) 2 Does modeling a structural break improve forecast accuracy? 10 of 17

18 EQUAL MSFE UNDER LOCAL BREAKS OF UNKNOWN TIMING ζ 1/ τ b Does modeling a structural break improve forecast accuracy? 11 of 17

19 WEAK OPTIMALITY The test statistic, and therefore critical values, depend on the unknown break date What if we substitute ˆτ τ? Does modeling a structural break improve forecast accuracy? 12 of 17

20 WEAK OPTIMALITY The test statistic, and therefore critical values, depend on the unknown break date What if we substitute ˆτ τ? We show that the test is weakly optimal: In the limit where the nominal size of the test α 0 ) ( ) P Ha (sup ζ(τ) > b(ˆτ) P Ha ζ(τ b ) > v(τ b ) = 0 τ where ˆτ = arg max τ ζ(τ) and P H0 (sup τ ζ(τ) > b(τ b )) = α and P H0 (ζ(τ b ) > v(τ b )) = α Does modeling a structural break improve forecast accuracy? 12 of 17

21 ASYMPTOTIC POWER (α = 0.05) 1 τ b = τ b = ζ 1/ ζ 1/2 1 τ b = τ b = ζ 1/2 ζ 1/2 Does modeling a structural break improve forecast accuracy? 13 of 17

22 BACK TO THE DATA 130 macroeconomic/financial time series between 1960:1-2015:10 1 Estimate AR(1) model on a moving window of 120 observations y t = µ 1 I[t < T b ] + µ 2 I[t T b ] + ρy t 1 + ε t Forecast period 1970:7-2015: FRED-MD - Does modeling a structural break improve forecast accuracy? 14 of 17

23 BACK TO THE DATA 130 macroeconomic/financial time series between 1960:1-2015:10 2 Estimate AR(1) model on a moving window of 120 observations y t = µ 1 I[t < T b ] + µ 2 I[t T b ] + ρy t 1 + ε t Forecast period 1970:7-2015: FRED-MD - Does modeling a structural break improve forecast accuracy? 15 of 17

24 RELATIVE MSFE y t = µ 1 I[t < T b ] + µ 2 I[t T b ] + ρy t 1 + ε t y t = µ + ρy t 1 + ε t Decide between the post-break forecast and full-sample forecast using (1) test derived here / (2) standard sup-f test Series Relative MSFE AR(1) AR(6) All series OI LM CO OrdInv MC IRER P S Excluding forecasts where both tests do not indicate a break Does modeling a structural break improve forecast accuracy? 16 of 17

25 CONCLUSIONS Existing structural break tests are inappropriate when forecasting We develop a nearly optimal test to find breaks that are important when forecasting The test has good finite sample performance Far fewer breaks that are important for forecasting Shrinkage estimators can be treated along the same lines: Optimal weights of Pesaran, Pick, and Pranovich (2013) can be writting in terms of our forecast Wald test statistic Does modeling a structural break improve forecast accuracy? 17 of 17