Forecasting Lecture 2: Forecast Combination, Multi-Step Forecasts

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1 Forecasting Lecture 2: Forecast Combination, Multi-Step Forecasts Bruce E. Hansen Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

2 Today s Schedule Combination Forecasts Multi-Step Forecasting Fan Charts Iterated Forecasts If time (optional): Threshold models or Nonlinear/Nonparametric Time Series Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

3 Review Optimal point forecast of y n+1 given information I n is the conditional mean E (y n+1 I n ) Linear model E (y n+1 I n ) β x n is an approximation Estimate linear projections by least-squares Model selection should focus on performance, not truth Best forecast has smallest MSFE Unknown, but MSFE can be estimated CV is a good estimator of MSFE Good forecasts rely on selection of leading indicators Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

4 Combination Forecasts Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

5 Diversity of Forecasts Model choice is critical Issues: Classic approach: Selection Modern approach: Combination How to select from a wide set of models/forecasts? Model selection criteria How to combine a wide set of models/forecasts? Weight selection criteria Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

6 Foundation The ideal point forecast minimizes the MSFE The goal of a good combination forecast is to minimize the MSFE Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

7 Forecast Selection M forecasts: f = {f (1), f (2),..., f (M)} Selection picks ˆm to determine the forecast f = f ( ˆm) M weights: w = {w(1), w(2),..., w(m)} A combination forecast is the weighted average Combination generalizes selection f (w) = M w(m)f (m) m=1 = w f Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

8 Possible restrictions on the weight vector M m=1 w(m) = 1 Unbiasedness Typically improves performance w(m) 0 nonnegativity regularization Often critical for good performance w(m) {0, 1} Equivalent to forecast selection f (w) = f (m) Selection is a special case of combination Strong restriction Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

9 OOS Forecast Combination Sequence of true out-of-sample forecasts f t for y t+1 Combination forecast is f (w) = w f OOS empirical MSFE ˆσ 2 (w) = 1 P n ( yt+1 w ) 2 f t t=n P PLS selected the model with the smallest OOS MSFE Granger-Ramanathan combination: select w to minimize the OOS MSFE Minimization over w is equivalent to the least-squares regression of y t on the forecasts y t+1 = w f t + ε t+1 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

10 Granger-Ramanathan (1984) Unrestricted least-squares ŵ = ( n t=n P ) 1 f t f t n f t y t+1 t=n P This can produce weights far outside [0, 1] and don t sum to one Granger-Ramanathan s intuition was that this flexibility is good But they provided no theory to support conjecture Unrestricted weights are not regularized This results in poor sampling performance Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

11 Alternative Representation Take y t+1 = w f t + ε t+1, subtract y t+1 from each side 0 = w f t y t+1 + ε t+1 Impose restriction that weights to sum to one. 0 = w (f t y t+1 ) + ε t+1 Define e t+1 = w (f t y t+1 ), the (negative) forecast errors. Then 0 = w e t+1 + ε t+1 This is the regression of 0 on the forecast errors But it is still better to also impose non-negativity w(m) 0 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

12 Constrained Granger-Ramanathan The constrained GR weights solve the problem where min w w Aw subject to M w(m) = 1 m=1 0 w(m) 1 A = e t+1 e t+1 t is the M M matrix of forecast error empirical variances/covariances Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

13 Quadratic Programming (QP) The weights lie on the unit simplex The constrained GR weights minimize a quadratic over the unit simplex QP algorithms easily solve this problem Gauss (qprog) Matlab (quadprog) R (quadprog) Solution solution typical Many forecasts will receive zero weight Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

14 Bates-Granger (1969) Assume A = t e t+1 e t+1 is diagonal. Then the regression with the coeffi cients constrained to sum to one has solution 0 = w e t+1 + ε t+1 w(m) = ˆσ 2 (m) M j=1 ˆσ 2 (j) This are the Bates-Granger weights. In many cases, they are close to equality, since OOS forecast variances can be quite similar Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

15 Bayesian Model Averaging (BMA) Put priors on individual models, and priors on the probability that model m is the true model Compute posterior probabilites w(m) that m is the true model Forecast combination using w(m) Advantages Conceptually simple no theoretical analysis required applies in broad contexts Disadvantages Not designed to minimize forecast risk Similar to BIC: asymptotically picks true finite models does not distinguish between 1-step and multi-step forecast horizons Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

16 BMA Approximation BIC weights ( ) BIC (m) w(m) exp 2 Simple approximation to full BMA method Smoothed version of BIC selection Works better than BIC selection in simulations Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

17 AIC Weights Smooted AIC ( ) AIC (m) w(m) exp 2 Proposed by Buckland, Burnhamm and Augustin (1997) Not theoretically motivated, but works better than AIC selection in simulations Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

18 Comments Combination methods typically work better (lower MSFE) than comparable selection methods BIC and BMA not optimal for MSFE Granger-Ramanathan has similar senstive as PLS to choice of P Bates-Granger and weighted AIC have no theoretical grounding Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

19 Forecast Combination ŷ n+1 (w) = = M w(m)ŷ n+1 (m) m=1 M w(m)x n (m) β(m) m=1 = x n β(w) where β(w) = M w(m) β(m) m=1 In Iinear models, the combination forecast is the same as the forecast based on the weighted average of the parameter estimates across the different models Computationally, it is easiest to calculate the M individual forecast ŷ n+1 (m), then take the weighted average to obtain ŷ n+1 (w) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

20 Combination Residuals ê t+1 (w) = y t+1 x t β(w) = = M w(m) m=1 M w(m)ê t+1 (m) m=1 ( y t+1 x t β(m) In linear models, the residual from the combination model is the same as the weighted average of the model residuals. ) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

21 Mallows Averaging Criterion C n (w) = σ 2 (w) + 2 n σ2 with σ 2 an estimate from a large model M w(m)k(m) m=1 C n (w) is an estimate of the MSFE (assuming homoskedasticity) Hansen (2007, Econometrica) Mallows Model Averaging (MMA) Hansen (Journal of Econometrics, 2008) Forecast Model Averaging (FMA) Combination weights found by constrained minimization ŵ = argmin C n (w) w subject to M w(m) = 1 m=1 0 w(m) 1 Solution by Quadratic Programming (QP) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

22 Theory of Optimal Weights Hansen (2007, Econometrica) Mallows weight selection is asymptotically optimal under homoskedasticity [In large samples, equivalent to using MSFE-minimizing weights Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

23 Comparison of Granger-Ramanathan and FMA Both are solved by Quadratic Programming (QP) Both typically yield corner solutions many forecasts will receive zero weight GR uses empirical (OOS) forecast errors, FMA uses sample residuals GR uses no penalty, FMA uses average # of parameters penalty FMA is an estimate of MSFE for homoskedastic one-step forecasts, GR has no optimality Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

24 Cross-Validation Leave-one-out estimator β t (w) = Leave-one-out prediction residual ẽ t+1 (m) = y t+1 = M w(m) β t (m) m=1 M m=1 M w(m)ẽ t+1 (m) m=1 w(m) β t (w) x t (m) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

25 CV n (w) = 1 n n 1 t=0 ẽt+1(w) 2 is an estimate of MSFE n (m) Cross-validation (CV) criterion for regression combination/averaging Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

26 Cross-validation Weights Combination weights found by constrained minimization of CV n (w) min CV n(w) = w Sw w subject to M w(m) = 1 m=1 0 w(m) 1 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

27 Cross-validation for combination forecasts (theory) Theorem: ECV n (w) C n (w) For heteroskedastic forecasts, CV is a valid estimate of the one-step MSFE from a combination forecast Hansen and Racine (Journal of Econometrica, 2012) show that the CV weights are asymptotically optimal for cross-section data under heteroskedasticity Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

28 Summary: Forecast Combination Methods Granger-Ramanathan (GR), forecast model averaging (FMA) and cross-validation (CV) all pick weight vectors by quadratic minimization GR only needs actual forecasts, the method can be unknown or a black box CV can be computed for a wide variety of estimation methods optimality theory for linear estimation FMA limited to homoskedastic one-step-ahead models Smoothed AIC (SAIC) and BMA have no forecast optimality, and are designed for homoskedastic one-step-ahead forecasts. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

29 Example: AR models for GDP Growth Fit AR(1) and AR(2) only Leave-one-out residuals ẽ 1t and ẽ 2t Covariance matrix S = [ The best-fitting single model is AR(2) The best combination is w = (.22,.78) CV = ] Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

30 Example: AR models for GDP Growth Fit AR(0) through AR(12) AR(0) is constant only Models with positive weight are AR(0), AR(1), AR(2) w = (.06,.16,.78) S = CV = (essentially unchanged) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

31 Example: Leading Indicator Forecasts Fit AR(1), AR(2) with leading indicators Models with positive weight CV = 9.81 w AR(1), Spread, Housing 0.13 AR(1), Spread, High-Yield, Housing 0.16 AR(1), Spread, High-Yield, Housing, Building 0.52 AR(2) 0.18 AR(2), Spread 0.01 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

32 Summary: Forecast Combination by CV M forecasts f n+1 (m) from n observations For each estimate m Define the leave-one-out prediction error Store the n 1 vector ẽ(m) Construct the M M matrix ẽ t+1 (m) = y t+1 β ( t) (m)x t (m) = ê t+1 (m) 1 h tt (m) S = 1 n ẽ ẽ Find the M 1 weight vector w which minimizes w Sw Use quadratic programming (quadprog) to find solution The combination forecast is f n+1 = M m=1 w(m) f n+1 (m) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

33 Forecast Combination Criticisms There has been considerable skepticism about formal forecast combination method in the forecast literature Many researchers have found that equal weighting: (w m = 1/M) works as well as formal methods However, the formal methods which investigated are Bates-Granger simple weights Not expected by theory to work well Unconstrained Granger-Ramanathan Without imposing [0, 1] weights, work terribly! Furthermore, most investigations examine pseudo out-of-sample performance Identical to comparing models by PLS criterion This is NOT an investigation of performance Just a ranking by PLS Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

34 Another Example - 10-Year Bond Rate Estimated AR(1) through AR(24) models CV Selection picked AR(2) CV weight Selection: Models with positive weight AR(0): w = 0.04 AR(1): w = 0.04 AR(2): w = 0.47 AR(6): w = 0.23 AR(22): w = 0.22 MInimizing CV = (slightly lower than from AR(2)) Point forecast 1.96 (same as from AR(2)) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

35 Multi-Step Forecasts Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

36 Multi-Step Forecasts Forecast horizon: h We say the forecast is multi-step if h > 1 Forecasting y n+h given I n e.g., forecasting GDP growth for 2012:3, 2012:4, 2013:1, 2013:2 The forecast distribution is y n+h I n F h (y n+h I n ) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

37 Point Forecast f n+h h minimizes expected squared loss f n+h h = ) argmin E ((y n+h f ) 2 I n f = E (y n+h I n ) Optimal point forecasts are h-step conditional means Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

38 Relationship Between Forecast Horizons Take an AR(1) model y t+1 = αy t + u t+1 Iterate y t+1 = α (αy t 1 + u t ) + u t+1 = α 2 y t 1 + αu t + u t+1 or y t+2 = α 2 y t + e t+2 u t+2 = αu t+1 + u t+2 Repeat h times y t+h = α h y t + e t+h e t+h = u t+h + αu t+h 1 + α 2 u t+h α h 1 u t+1 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

39 AR(1) h-step forecast y t+h = α h y t + e t+h e t+h = u t+h + αu t+h 1 + α 2 u t+h α h 1 u t+1 E (y n+h I n ) = α h y n h step point forecast is linear in y n h-step forecast error e n+h is a MA(h 1) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

40 AR(2) Model 1-step AR(2) model 2-steps ahead Taking conditional expectations y t+1 = α 0 + α 1 y t + α 2 y t 1 + u t+1 y t+2 = α 0 + α 1 y t+1 + α 2 y t + u t+2 E (y t+2 I t ) = α 0 + α 1 E (y t+1 I t ) + α 2 E (y t I t ) + E (e t+2 I t ) which is linear in (y t, y t 1 ) = α 0 + α 1 (α 0 + α 1 y t + α 2 y t 1 ) + α 2 y t = α 0 + α 1 α 0 + ( α α 2 ) yt + α 1 α 2 y t 1 In general, a 1-step linear model implies an h-step approximate linear model in the same variables Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

41 AR(k) h-step forecasts If y t+1 = α 0 + α 1 y t + α 2 y t α k y t k+1 + u t+1 then y t+h = β 0 + β 1 y t + β 2 y t β k y t k+1 + e t+h where e t+h is a MA(h-1) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

42 Leading Indicator Models If then y t+1 = x t β + u t E (y t+h I t ) = E (x t+h 1 I t ) β If E (x t+h 1 I t ) is itself (approximately) a linear function of x t, then E (y t+h I t ) = x t γ y t+h = x t γ + e t+h Common Structure: h-step conditional mean is similar to 1-step structure, but error is a MA. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

43 Forecast Variable We should think carefully about the variable we want to report in our forecast The choice will depend on the context What do we want to forecast? Future level: y n+h interest rates, unemployment rates Future differences: y t+h Cummulative Change: y t+h Cummulative GDP growth Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

44 Forecast Transformation f n+h n = E (y n+h I n ) = expected future level Level specification y t+h = x t β + e t+h f n+h n = x t β Difference specification y t+h = x t β h + e t+h f n+h n = y n + x t β x t β h Multi-Step difference specification y t+h y t = x t β + e t+h f n+h n = y n + x t β Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

45 Direct and Iterated There are two methods of multistep (h > 1) forecasts Direct Forecast Model and estimate E (y n+h I n ) directly Iterated Forecast Model and estimate one-step E (y n+1 I n ) Iterate forward h steps Requires full model for all variables Both have advantages and disadvantages For now, we will forcus on direct method. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

46 Direct Multi-Step Forecasting Markov approximation E (y n+h I n ) = E (y n+h x n, x n 1,...) E (y n+h x n,..., x n p ) Linear approximation E (y n+h x n,..., x n p ) β x n Projection Definition β = (E (x t x t )) 1 (E (x t y t+h )) Forecast error e t+h = y t+h β x t Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

47 Multi-Step Forecast Model y t+h = β x t + e t+h β = ( E ( x t x t )) 1 (E (xt y t+h )) E (x t e t+h ) = 0 σ 2 = E ( et+h 2 ) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

48 Least Squares Estimation β = ( ) 1 ( ) n 1 n 1 x t x t x t y t+h t=0 t=0 ŷ n+h n = f n+h n = β x n Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

49 Residuals Least-squares residuals ê t+h = y t+h β x t Standard, but overfit Leave-one-out residuals ẽ t+h = y t+h β t x t Does not correct for MA errors Leave h out residuals ẽ t+h = y t+h β t,h x t ( ) 1 ( ) β t,h = x j x j x j y j+h j+h t h j+h t h The summation is over all observations outside h 1 periods of t + h. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

50 Example: GDP Forecast y t = 400 log(gdp t ) Forecast Variable: GDP growth over next h quarters, at annual rate y t+h y t h = β 0 + β 1 y t + β 1 y t 1 + Spread t + HighYield t + β 2 HS t + e t+h HS t =Housing Starts t h = 1 h = 2 h = 3 h = 4 β (1.0) 0.38 (1.3) 0.01 (1.6) 0.47 (1.8) y t 0.16 (.10) 0.18 (.09) 0.13 (.08) 0.13 (.09) y t (.10) 0.04 (.05) 0.05 (.07) 0.02 (.06) Spread t 0.61 (.23) 0.65(.19) 0.65 (.22) 0.65 (.25) HighYield t 1.10 (.75) 0.68 (.70) 0.48 (.90) 0.41 (1.01) HS t 1.86 (.65) 1.64 (.70) 1.31 (.80) 1.01 (.94) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

51 Example: GDP Forecast Cummulative Annualized Growth Forecast Actual 2012: : : : : : : :1 3.2 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

52 Selection and Combination for h step forecasts AIC routinely used for model selection PLS (OOS MSFE) routinely used for model evaluation Neither well justified Not well studied problem I recommend leave h out cross-validation. Topic of afternoon seminar Minimize sum of squared leave h out residuals, separately for each forecast horizon. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

53 Example: GDP Forecast Weights by Horizon h = 1 h = 2 h = 3 h = 4 h = 5 h = 6 h = 7 AR(1) AR(2).30 AR(1)+HS AR(1)+HS+BP AR(2)+HS.04 ŷ n+h n Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

54 h-step Variance Forecasting Not well developed using direct methods Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

55 h-step Interval Forecasts Similar to 1-step interval forecasts But calculated from h step residuals Use constant variance specification Let q e (α) and q e (1 α) be the α th and (1 α) th percentiles of residuals ẽ t+h Forecast Interval: [ µ n + q ε (α), µ n + q e (1 α)] Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

56 Fan Charts Plots of a set of interval forecasts for multiple horizons Pick a set of horizons, h = 1,..., H Pick a set of quantiles, e.g. α =.10,.25,.75,.90 Recall the quantiles of the conditional distribution are q n (α, h) = µ n (h) + σ n (h)q ε (α, h) Plot q n (.1, h), q n (.25, h), µ n (h), q n (.75, h), q n (.9, h) against h Graphs easier to interpret than tables Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

57 Illustration I ve been making monthly forecasts of the Wisconsin unemployment rate Forecast horizon h = 1,..., 12 (one year) Quantiles: α =.1,.25,.75,.90 This corresponds to plotting 50% and 80% forecast intervals 50% intervals show likely region (equal odds) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

58 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

59 Comments Showing the recent history gives perspective Some published fan charts use colors to indicate regions, but do not label the colors Labels important to infer probabilities I like clean plots, not cluttered Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

60 Illustration: GDP Growth Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

61 Figure: GDP Average Growth Fan Chart Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

62 It doesn t fan because we are plotting average growth Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

63 Figure: Fan Chart with Actuals Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

64 Iterated Forecasts Estimate one-step forecast Iterate to obtain multi-step forecasts Only works in complete systems Autoregressions Vector autoregressions Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

65 Vector Autoregresive Models y t is an p vector x t are other variables (including lags) Ideal point forecast E (y n+1 I n ) Linear approximation E (y n+1 I n ) A 1 y t + A 2 y t A k y t k+1 + Bx t Vector Autoregression (VAR) y t+1 = A 1 y t + A 2 y t A k y t k+1 + Bx t + e t+1 Estimation: Least squares y t+1 =  1 y t +  2 y t 1 + +A k y t k+1 + Bx t + e t+1 One-Step-Ahead Point forecast ŷ n+1 =  1 y n +  2 y n  k y n k+1 + Bx n Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

66 Vector Autoregresive versus Univariate Models Let x t = (y t, y t 1,..., x t ) Then a VAR is a set of p regression models y 1t+1 = β 1 x t + e 1t y pt+1 = β p x t + e pt All variables x t enter symmetrically in each equation Sims (1980) argued that there is no a priori reason to include or exclude an individual variable from an individual equation.. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

67 Model Selection Do not view selection as identification of truth Rather, inclusion/exclusion is to improve finite sample performance minimize MSFE Use selection methods, equation-by-equation Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

68 Example: VAR with 2 variables y 1t+1 = β 11 y 1t + β 12 y 1t 1 + β 13 y 2t + ê 1t. y 2t+1 = β 21 y 1t + β 22 y 2t + β 23 y 2t 1 + ê 2t Selection picks y 1t, y 1t 1, y 2t for equation for y 1t+1 Selection picks y 1t, y 2t, y 2t 1 for equation for y 2t+1 The two equations have different variables Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

69 Same as system y t+1 = A 1 y t + A 2 y t 1 + e t+1 with A 1 = A 2 = [ β11 β 13 β 21 β 22 [ β β 23 ] ] The VAR system notation is still quite useful for many purposes (including multi-step forecasting) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

70 Iterative Forecast Relationships in Linear VAR vector y t y t+1 = A 0 + A 1 y t + A 2 y t A k y t k+1 + u t+1 1-step conditional mean E (y t+1 I t ) = A 0 + A 1 E (y t I t ) + + A k E (y t k+1 I t ) 2-step conditional mean = A 0 + A 1 y t + A 2 y t A k y t k+1 E (y t+1 I t 1 ) = E (E (y t+1 I t ) I t 1 ) = A 0 + A 1 E (y t I t 1 ) + + A k E (y t k+1 I t 1 ) = A 0 + A 1 E (y t I t 1 ) + A 2 y t A k y t k+1 h step conditional mean E (y t+1 I t h+1 ) = E ( E (y t+1 I t ) It h+1 ) = A 0 + A 1 E (y t I t h+1 ) + + A k E (y t k+1 I t h+1 Linear in lower-order (up to h 1 step) conditional means Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

71 Iterative Least Squares Forecasts Estimate 1-step VAR(k) by least-squares y t+1 =  0 +  1 y t +  2 y t  k y t k+1 + û t+1 Gives 1-step point forecast ŷ n+1 n =  0 +  1 y n +  2 y n  k y n k+1 2-step iterative forecast ŷ n+2 n =  0 +  1 ŷ n+1 n +  2 y n + +  k y n k+2 h step iterative forecast ŷ n+h n =  0 +  1 ŷ n+h 1 n +  2 ŷ n+h 2 n + +  k ŷ n+h k n This is (numerically) different than the direct LS forecast Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

72 Illustration 1: GDP Growth AR(2) Model y t+1 = y t +.16y t 1 y n = 1.8, y n 1 = 2.9 ŷ n+1 = = 2.6 ŷ n+2 = = 2.7 ŷ n+3 = = 2.9 ŷ n+4 = = 3.0 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

73 Point Forecasts 2012: : : : : : : : Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

74 Illustration 2: GDP Growth+Housing Starts VAR(2) Model y 1t = GDP Growth, y 2t =Housing Starts x t = (GDP Growth t, Housing Starts t, GDP Growth t 1, Housing Starts t 1 y t+1 = Â 0 + Â 1 y t + Â 2 y t 1 + û t+1 y 1t+1 = y 1t y 2t y 1t y 2t 1 y 2t+1 = y 1t + 1.2y 2t 0.001y 1t y 2t 1 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

75 Illustration 2: GDP Growth+Housing Starts y 1n = 1.8, y 2n = 0.71, y 1n 1 = 2.9, y 2n 1 = 0.68 y 1n+1 = = 2.3 y 2t+1 = = 0.76 y 1n+2 = = 2.4 y 2t+1 = = 0.80 Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

76 Point Forecasts GDP Housing 2012: : : : : : : : Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

77 Model Selection It is typical to select the 1-step model and use this to make all h-step forecasts However, there theory to support this is incomplete (It is not obvious that the best 1-step estimate produces the best h-step estimate) For now, I recommend selecting based on the 1-step estimates Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

78 Model Combination There is no theory about how to apply model combination to h-step iterated forecasts Can select model weights based on 1-step, and use these for all forecast horizons Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

79 Variance, Distribution, Interval Forecast While point forecasts can be simply iterated, the other features cannot Multi-step forecast distributions are convolutions of the 1-step forecast distribution. Explicit calculation computationally costly beyond 2 steps Instead, simple simulation methods work well The method is to use the estimated condition distribution to simulate each step, and iterate forward. Then repeat the simulation many times. Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

80 Multi-Step Forecast Simulation Let µ (x) and σ (x) denote the models for the conditional one-step mean and standard deviation as a function of the conditional variables x Let µ (x) and σ (x) denote the estimates of these functions, and let { ε 1,..., ε n } be the normalized residuals x n = (y n, y n 1,..., y n p ) is known. Set x n = x n To create one h-step realization: Draw ε n+1 iid from normalized residuals { ε 1,..., ε n } Set yn+1 = µ (x n) + σ (xn) εt+1 Set xn+1 = (y n+1, y n,..., y n p+1 ) Draw εn+2 iid from normalized residuals { ε 1,..., ε n } Set yn+2 = µ ( xn+1 ) ( ) + σ x n+1 ε t+2 Set xn+2 = (y n+2, y n+1,..., y n p+2) Repeat until you obtain yn+h yn+h is a draw from the h step ahead distribution Repeat this B times, and let yn+h (b), b = 1,..., B denote the B repetitions Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

81 Multi-Step Forecast Simulation The simulation has produced yn+h (b), b = 1,..., B For forecast intervals, calculate the empirical quantiles of yn+h (b) For an 80% interval, calculate the 10% and 90% For a fan chart Calculate a set of empirical quantiles (10%, 25%, 75%, 90%) For each horizon h = 1,..., H As the calculations are linear they are numerically quick Set B large For a quick application, B = 1000 For a paper, B = 10, 000 (minimum)) Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82

82 VARs and Variance Simulation The simulation method requires a method to simulate the conditional variances In a VAR setting, you can: Treat the errors as iid (homoskedastic) Easiest Treat the errors as independent GARCH errors Also easy Treat the errors as multivariate GARCH Allows volatility to transmit across variables Probably not necessary with aggregate data Bruce Hansen (University of Wisconsin) Forecast Combination and Multi-Step Forecasts October 29-31, / 82