Forecasting with large-scale macroeconometric models

Transcription

1 Forecasting with large-scale macroeconometric models Econometric Forecasting January 8th 2008

2 Agenda Introduction 1 Introduction

3 Agenda Introduction 1 Introduction 2 Taxonomy General formulation

4 Agenda Introduction 1 Introduction 2 Taxonomy General formulation 3

5 Agenda Introduction 1 Introduction 2 Taxonomy General formulation 3 4 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation

6 Introduction They often consist of hundreds of equations. They typically share a focus on economic aggregates. They are also subject of critique.... existing Keynesian macroeconometric models are incapable of providing reliable guidance in formulating monetary, fiscal, or other types of policy. This conclusion is based in part on the spectacular recent failures of these models, and in part on their lack of a sound theoretical or econometric basis ROBERT E. LUCAS Jr. and THOMAS J. SARGENT (1978)

7 Macroeconometric model building choosing the variables to be included in the model; separating variables: endogenous-exogenous; sketching a priori causal relationships among the variables, in the style of a flow chart; specification of estimable equations; estimation; forecasting.

8 Taxonomy General formulation Problems with large-scale ME forecasting Parameter non-constancy Parameters are unknown Forecast origin may be subject to error and revision, or may not yet be available. Non-modelled variables Error process distribution may change over time Extraneous information - not included in the model Cost of asymmetric loss - over and under prediction

9 Taxonomy General formulation Taxonomy x t = τ + Υx t 1 + ν t (1) ν t N[0, Ω], with expectation E[ν t ] = 0, and variance matrix V [ν t ] = Ω (1) satisifies r < n cointegration relation, such that: Υ = I n + αβ (2) where α and β are n r full rank matrices.

10 Taxonomy General formulation x t = τ + αβ x t 1 + ν t (3) τ = γ αµ (4) We define γ as the expected growth rate of the system: γ E[ x t ], Taking expectation through (3): µ is the equilibrium mean µ E[β x t ] γ = αe[β x t 1 ] + τ (5)

11 Taxonomy General formulation x t γ = α(β x t 1 µ) + ν t (6) In (4) τ is n 1 and from: β γ = E[β x t ] = E[ β x t ] = 0, where β is n r, then γ consists of only n r free parameters. γ = β γ

12 Taxonomy General formulation General formulation as VAR(q) x t = x t x t 1. x t q+1 = τ + Υx t 1 + ν t

13 Taxonomy General formulation General formulation as VAR(q) τ = Υ 1 Υ 2 Υ q 1 Υ q I n I n I n 0 x t 1 x t 2. x t q + ν t 0. 0

14 Taxonomy General formulation General formulation as VAR(q) (ˆτ : ˆΥ : ˆΩ) τ p τ Υ p Υ x t = τ p + Υ p x t 1 + ν t (7)

15 Taxonomy General formulation Stationarity If π = E[x t ] the relationship between π and τ is given by: τ = (I n Υ)π x t = γ + ξ t, which is correctly specified when α = 0 in (6), in which case ξ t = ν t.

16 Taxonomy is developed for a structural change over the forecast period, when the model and DGP may differ over the sample period. The model: x t = τ + Υx t 1 + ν t h-step forecast is given by: ˆx T +h = ˆτ + Υˆx T +h 1 = h 1 i=1 Υ i ˆτ + Υ hˆx T h = 1,..., H,

17 Assume there is a single (one-step) break in the system, such that (τ : Υ) (τ : Υ ) and the variance, autocorrelation and distribution of the error may change to ν T +h D n (0, Ω )

18 Thus, the data generated by the process for the next H periods are given by: x T +h = τ + Υ x T +h 1 + ν T +h = h 1 h 1 (Υ ) i τ + (Υ ) h x T + (Υ ) i ν T +h i i=1 i=0

19 From above, the h-step forecast error is: ˆν T +h = h 1 (Υ ) i τ i=0 h 1 i=0 Υ i ˆτ + ( Υ ) h x T Υ hˆx h 1 T + (Υ ) i ν T +h i (1) i=0

20 Let s denote deviations between sample estimates and population parameters by δ τ = ˆτ τ p and δ Υ = Υ Υ p Then, i 1 Υ i = (Υ p + δ Υ ) i Υ i p + Υ j pδ Υ Υp i j 1 j=0 = Υ i p + C i

21 Vectoring gives the following: ) ν ( Υi ( ( ) i 1 ) Υ i ν p + Υ j p Υp i j 1 Υ i ˆτ = j=0 j=0 δ ν Υ ( ) i 1 Υ i p + Υ j pδ Υ Υ i j 1 p (τ p + δ τ ) i=1 Υ j pδ Υ Υp i j 1 τ p + Υ i pτ p + Υ i pδ τ j=0

22 This yields Υ i ˆτ Υ i pτ p + Υ i pδ τ + ( i 1 j=0 = Υ i pτ p + Υ i pδ τ + D i δ ν Υ, Υ j p τ pυ i j 1 p ) δ ν Υ and we obtain... h 1 [ (Υ ) i τ Υ ] h 1 [ i ˆτ (Υ ) i τ (Υ i p + Υ i pδ τ + D i δυ) ] ν (2) i=0 i=0

23 We decompose the term h 1 i=1 [(Υ ) i τ Υ i ˆτ] in (2) into h 1 [ (Υ ) i (τ τ) + (Υ ) i (τ τ p ) + ( (Υ ) i Υ i) ] τ p i=0 h 1 + [(Υ i Υ i p)τ p Υ i pδ τ D i δυ]. ν i=0

24 Similarly, for the terms in (1) post-multiplied by x T or ˆx T, using: ( h 1 h 1 ) C h x T = Υ j pδ Υ Υp h j 1 x T = Υ k p x T Υ h j 1 p δυ ν = F h δυ ν j=0 and letting (x T ˆx T ) = δ x : j=0 (Υ ) h x T Υ hˆx T ( (Υ ) h Υ h) x T + (Υ h Υ h p)x T F h δ ν Υ +(Υ h p + C h )δ x

25 Forecast error ˆν T +h h 1 i=0 ((Υ ) i Υ i )τ p + ((Υ ) h Υ h )x T - slope change + h 1 i=0 (Υ ) i (τ τ) - intercept change + h 1 i=1 (Υi Υ i p)τ p + (Υ h Υ h p)x T - slope misspecification + h 1 i=0 (Υ ) i (τ τ p ) - intercept misspecification

26 h 1 i=0 (D i F i )δυ ν - slope estimation h 1 i=0 Υi pδ τ - intercept estimation +(Υ h p + C h )δ x - initial estimation + h 1 i=0 (Υ ) i ν T +h i - error estimation

27 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation In general, there are five different sources of forecast errors, both in the closed system (or closed loop model) and in the open system (or open loop model), where the latter additionally comprises non-modelled (exogenous) policy variables. Let us generalize the different parameter vectors or matrices τ and Υ from the closed system (x t = τ + Υx t 1 + ν t ) or Γ and Φ from the open system (y t = Γz t + Φy t 1 + η t ) to a generic parameter vector θ. Then the error sources can be expressed as follows:

28 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation 1 Parameter change: θ θ 2 Model mis-specification: θ θ p 3 Estimation uncertainty: E[(ˆθ θ p )(ˆθ θ p ) ] 0 4 Variable uncertainty such as variable mis-measurement in the closed system (x T ˆx T ) and some other forms in the open system 5 Error accumulation because of M[ h 1 i=0 (Υ ) i ν T +h i ] 0

29 Parameter change: θ θ Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Probably the most important source of forecast uncertainty Usually: DGP is assumed to be time-invariant, which may not be the case due to structural change or regime shifts Paramter change and model mis-specification are often hard to distinguish Model mis-specification and changes in relevant parameters may result in forecasting errors without structural breaks occurring in the underlying behavioral equations.

30 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation From table 7.1 in Clements and Hendry (1998, A taxonomy of forecast errors, p. 166), the h-step ahead forecast error ˆν T +h = x T +h ˆx T +h in case of slope and intercept change reads as follows: ˆν T +h = h 1 h 1 (Υ ) i τ Υ i τ i=0 i=0 h 1 + (Υ ) h x T Υ h x T + (Υ ) i ν T +h i i=0

31 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Assuming that only the intercept vector is non-constant τ τ, the forecast error ˆν T +h is simplified to the following: h 1 h 1 ˆν T +h = Υ i (τ τ) + Υ i ν T +h i i=0 This highlights a persistent (and typically increasing) bias in case (τ τ) > 0. i=0

32 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation However, changes in the equilibrium mean π = E[x t ] are the major sources of error. As one may recall from the closed stationary system: π = (I n Υ) 1 τ Changes in π may be direct results from changes in τ or indirect results from changes in Υ.

33 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation With π = (I n Υ) 1 τ: x t = τ + Υx t 1 + ν t x t π = Υ(x t 1 π) + ν t h-step ahead predictor at forecast origin T : ˆx T +h π = Υ(ˆx T +h 1 π) = Υ h (x T π)

34 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation If one assumes that (τ, Υ) changes to (τ, Υ ) in period T + h, with τ = (I n Υ )π : x T +h π = Υ (x T +h 1 π ) + ν T +h h 1 = (Υ ) h (x T π ) + (Υ ) i ν T +h i i=0

35 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation h-step ahead forecast error ˆν T +h = x T +h ˆx T +h : ˆν T +h = π π + (Υ ) h (x T π ) Υ h (x T π) h 1 + (Υ ) i ν T +h i i=0 = ( I n (Υ ) h) (π π) + ( (Υ ) h Υ h) (x T π) h 1 + (Υ ) i ν T +h i i=0

36 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Taking expectations from the last equation, with E[x T π] = 0, we get: E[ˆν T +h ] = ( I n (Υ ) h) (π π) > 0 Therefore, forecasts are only biased in case the long-run mean is not time-constant. They would be unbiased if the long-run mean were time-constant (π = π), if the observed process had zero mean (π = π = 0), or if, e.g., changes in Υ could be compensated by changes in τ.

37 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Systematic mean-shift structural changes are the major source of forecast error in case of parameter change. Changes in Υ also affect the estimates of the forecast-error variances M[ˆν T +h ], but these effects are secondary.

38 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Model mis-specification: θ θ p All models are wrong but some are useful. (G.E.P. Box) One usually does not know the true DGP. Parameterized (forecasting) models are designed to be congruent to the assumed DGP. Therefore, the included variables of the fitted model should be relatively orthogonal to wrongly excluded influences (omitted variable bias) to obtain consistent parameter estimates.

39 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation To find a good parameterized model, one should apply the so-called Hendry s method (general-to-specific approach). However, one should avoid over-fitting of the model because then secondary or irrelevant features of the data-set are likely to undermine forecasting performance. For forecasting purposes, it may also be useful to omit variables suspicious to structural breaks in case their omission has no crucial impact on the fit.

40 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Estimation uncertainty: E[(ˆθ θ p )(ˆθ θ p ) ] 0 Estimation uncertainty depends on: Size and information content of the sample Quality of the model (goodness of fit) Choice of parameterization Selection of the estimator Degree of integration and existence of unit roots

41 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Most of these problems can be improved by efficient specification, modelling and estimation strategies (Wald F tests on significance of variables, general-to-specific approach etc.).

42 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Variable mis-measurement (closed system): x T ˆx T Initial condition uncertainty (open system): y T ŷ T A badly measured forecast origin may be an important source of short-term forecast errors. This argument can be backed by the frequency with which provisional data is revised. One should only use revised and reliable data.

43 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Non-modelled variable uncertainty (open system): z T +h ž T +h for h = 0,..., H Off-line extrapolations of policy variables can also constitute a source of forecast error to the model if not deliberately conducted. Maybe some part of this flaw is compensated by intercept correction and other forms of correction such as add factors... Putting emphasis on reasonably predicting future developments of policy variables would be advisable.

44 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Feedbacks on non-modelled variables (open system) It could be the case that non-modelled variables z are Granger-caused by the modelled variables y as well (Granger feedback), although the zs are assumed to be exogenous. This would contradict the chosen parameterization of the model. However, one can test for Granger feedback using Wald F tests.

45 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Invariance to policy changes (open system) Maybe the model is invariant to changes in the policy variables. This can be the case if the policy variables are not super exogenous, but only weakly exogenous. Super exogeneity may be violated if agents change the way they form expectations (Lucas critique). Then, future policy changes modelled off-line may have little or no impact in reality, contrary to what the forecast suggests.

46 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation One can control at least for part of this error source namely possible impacts of forecasts on outcomes by incorporating expected future values of the modelled variables into the forecasting model, e.g.: y t = αŷ t+1 + βx t + ν t ŷ τ = αŷ τ+1 + βx τ with ŷ t+1 = E[y t+1 Ω t ], for τ = T + 1,..., T + H.

47 Parameter change Model mis-specification Estimation uncertainty Variable uncertainty Error accumulation Error accumulation: M[ h 1 i=0 (Υ ) i ν T +h i ] 0 In the long run, the variance of a non-stationary DGP is unbounded and therefore the forecast error ν T +h grows arbitrarily large. Fortunately, only time ranges in terms of quarters or years are usually needed in applications such as economic stabilization policy or in investment strategies. In addition, taking differences, e.g., from a I (1) process to obtain a I (0) process imposes a boundary on the variance, which may be advantageous for forecasting.