Macroeconometric Modelling

Transcription

1 Macroeconometric Modelling 4 General modelling and some examples Gunnar Bårdsen CREATES November 2009

2 From system... I Standard procedure see (?), Hendry (1995), Johansen (1995), Johansen (2006), Juselius (2007), Garratt, Lee, Pesaran, and Shin (2006) and Lütkepohl (2006) for detailed expositions. Here we follow the exposition in Lütkepohl (2005). We are interested in analyzing the k-dimensional VAR(p) process of either I (1) or I (0) variables y t y t = ν + p i=1 A i y t i + u t, typically written in Equilibrium correction form (EqC) as p 1 y t = ν + Πy t 1 + i=1 Γ i y t i + u t, (1)

3 From system... II with Γ i = (A i A p ), i = 1,..., p 1 The system is stable if Π = det (I k A 1 z A p z p ) = 0 for z 1. If Π = (I k A 1 z A p z p ) has a zero determinant with (k r) unit roots, the system still contains r steady-state relationships contained in the (k r) matrix β defined by Π = αβ.

4 ...to model I From a discretized and linearized cointegrated VAR representation to a dynamic Simultaneous Model (SEM) in three steps: 1. Linearized and discretized approximation as a data-coherent statistical system representation in the form of a cointegrated VAR p 1 y t = ν + Πy t 1 + i=1 Γ i y t i + u t, (2)

5 ...to model II 2. Identify the steady state, by testing and imposing overidentifying restrictions on the cointegration space: y t = ν + α p 1 β y t 1 + i=1 Γ i y t i + u t, 3. Identify the dynamics, by testing and imposing overidentifying restrictions on the dynamics: A 0 y t = A 0 ν + A 0 α p 1 β y t 1 + i=1 A 0 Γ i y t i + A 0 u t,

6 ...to model III Comment: The steady-state parameters are invariant to the dating of the steady-state solution. However, the estimated short-run parameters, and therefore the interpretation of the model dynamics, need not be see Bårdsen (1992), Bårdsen and Fisher (1999).

7 ...to model IV This is easily seen by noting that the model can equivalently be written as with A 0 y t = A 0 ν + A 0 α p 1 β y t p + i=1 A 0 D i y t i + A 0 u t, D i = (I k A 1 A i ), i = 1,..., p 1.

8 An application: Norwegian Aggregate Model-NAM Properties Short-run growth framework Good fit Small and transparent Good forecasts?

9 An application: NAM I Overview

10 An application: NAM I Steady-state relationships (v + p p) t = 0.12 [(R π) (R π )] + µ v (3) (pi v pi ) t 0.55 (p v p ) t = µ pi (4) p t 0.7(w z) t (1 0.7) pi t = µ p (5) (w p z) t = 0.1u + µ w (6) z t 0.47 (w p) t Trend t = 0.03u + µ z (7) 0 = u 7.7 (w p) 4.5 [0.01 (RL π) 4 y] µ u (8) 0 = RL 0.41RB 0.76R µ RL (9) 0 = RB 0.43R 0.57RB µ RB (10) y t 0.9g t 0.16(v + p p) t = 0.06 (RL π) + µ y (11) (l p) t 2.65y t (RL RB) t = µ l p (12)

11 An application: NAM Stylized dynamic version v t = 0.04 (R R ) t 0.04 {(v + p p) 0.12 [(R π) (R π )]} t 1 (pi pi v) t = 0.1 v t 0.43 [(pi pi v) 0.55 (p p v)] t 1 p t = 0.09 z t pi t pe t y t 0.07 [p 0.7(w z) 0.3pi] t 1 (w p) t = 0.04 u t T 1 t 0.07 [(w p z) + 0.1u] t 1 z t = 0.09 (w p) t 0.24 [z 0.47 (w p) 0.003Trend 0.03u] t 1 u t = 0.23 {u 7.65 (w p) 4.46 [0.01 (R L π) 4 y]} t 1 R L,t = 0.58 R t 0.33 (R L 0.41R B 0.76R) t 1 (R B RB ) t = 0.43 R t 0.17 (R B 0.43R 0.57RB ) t 1 y t = 0.16 g t (l p) t 0.12 [y 0.9g t (v + p p) (R L π)] t 1 (l p) t = 0.3 y t 0.09 [(l p) 2.65y (R L, R B )] t 1 R t = 0.27 [R (π C π C ) + (U Ū) 0.86 (R R )] t 1

12 An application: NAM Dynamic simulation

13 An application: NAM Policy analysis 100 basis points permanent reduction in the short rate from 2007

14 An application: NAM Forecasts from march 2007: evaluated 02 June, 2009

15 An application: NAM Forecast comparison: NAM and Norges Bank

16 An application: NAM Forecasts: 2 february 2009

17 An example: How to make a textbook model Let us start with the stylized representation of the full-scale model v t = 0.04 (R R ) t 0.04 {(v + p p) 0.12 [(R π) (R π )]} t 1 (pi pi v) t = 0.1 v t 0.43 [(pi pi v) 0.55 (p p v)] t 1 p t = 0.09 z t pi t pe t y t 0.07 [p 0.7(w z) 0.3pi] t 1 (w p) t = 0.04 u t T 1 t 0.07 [(w p z) + 0.1u] t 1 z t = 0.09 (w p) t 0.24 [z 0.47 (w p) 0.003Trend 0.03u] t 1 u t = 0.23 {u 7.65 (w p) 4.46 [0.01 (R L π) 4 y]} t 1 R L,t = 0.58 R t 0.33 (R L 0.41R B 0.76R) t 1 (R B RB ) t = 0.43 R t 0.17 (R B 0.43R 0.57RB ) t 1 y t = 0.16 g t (l p) t 0.12 [y 0.9g t (v + p p) (R L π)] t 1 (l p) t = 0.3 y t 0.09 [(l p) 2.65y (R L, R B )] t 1 R t = 0.27 [R (π C π C ) + (U Ū) 0.86 (R R )] t 1

18 An example: How to make a textbook model I If we make some simplifying assumptions, like: 1. closed economy 2. no public sector 3. one interest rate 4. no debt 5. disregard energy 6. disregard unemployment 7. and make productivity a stochastic trend

19 An example: How to make a textbook model II...we have a textbook model still a bit too rich but close p t = a 12 y t c 11 [p (w z) µ 1 ] t 1 y t = c 22 [y + β 23 (R p) z µ 2 ] t 1 R t = c 33 [ Rt 1 a 31 ( pt p ) a 32 ( yt y ) µ 3 ] (w p z) t = c 44 (w p z + µ 1 ) t 1 z t = µ 5 NB: note that R is I (0) and endogenous. What problems will this cause?

20 Calibration for simulation I p t =.1 y t.2 [p (w z).2] t 1 y t =.1 [y +.5 (R p) z 1] t 1 R t =.1 [R t ( p t 0.02) 0.5 ( y t 0.04) 0.05] (w p z) t =.2 (w p z +.2) t 1 z t =.04

21 Simultaneous equations form of model p t a 12 y t = c 11 µ 1 c 11 [p (w z)] t 1 y t = c 22 µ 2 c 22 (y + β 23 R z) t 1 + c 22 β 23 p t 1 R t c 33 a 31 p t c 33 a 32 y t = c 33 ( a31 p + c 33 y µ 3 ) c 33 R t 1 (w p z) t = c 44 µ 1 c 44 (w p z) t 1 z t = µ 5

22 Calibrated model on matrix form Ready for simulation p y R =.1 ( ) w z.04 }{{} t }{{}}{{} A 0 y t p p y y R R w w }{{} z z }{{} β t 1 }{{}}{{} A 0α y t 1 A 0` 1 A 0ν t 1

23 References I Bårdsen, G. and P. G. Fisher (1999). Economic theory and econometric dynamics in modelling wages and prices in the united kingdom. Empirical Economics 24(3), Bårdsen, G. (1992). Dynamic modeling of the demand for narrow money in norway. Journal of Policy Modeling 14(3), Garratt, A., K. Lee, M. H. Pesaran, and Y. Shin (2006). Global and National Macroeconometric Modelling: A Long-Run Structural Approach. Oxford University Press. Hendry, D. F. (1995). Dynamic Econometrics. Oxford: Oxford University Press.

24 References II Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press. Johansen, S. (2006). Cointegration: An overview. In T. C. Mills and K. Patterson (Eds.), Palgrave Handbook of Econometrics, Volume 1 of Econometric Theory. Palgrave-MacMillan. Juselius, K. (2007). The Cointegrated VAR Model: Methodology and Applications. Oxford University Press. Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer.

25 References III Lütkepohl, H. (2006). Vector autoregressive models. In T. C. Mills and K. Patterson (Eds.), Econometric Theory, Volume 1 of Palgrave Handbook of Econometrics, pp Palgrave MacMillan.