Macroeconomic Analysis: VAR, DSGE and Factor Models

Transcription

1 Macroeconomic Analysis: VAR, DSGE and Factor Models Marco Lippi, Einaudi Institute for Economics and Finance Scuola di Dottorato in Economia 24 Febbraio 2016

2 Outline Motivation The simplest model Singularity of RBC and DSGE

3 Outline Motivation The simplest model Singularity of RBC and DSGE

4 Outline Motivation The simplest model Singularity of RBC and DSGE

5 Outline Motivation The simplest model Singularity of RBC and DSGE

6 Outline Motivation The simplest model Singularity of RBC and DSGE

7 The simplest model Singularity of RBC and DSGE RBC max E 0 [ β t u(c t, c t 1, N t )] t (1) c t + K t+1 f (K t, N t, u t ) + (1 δ)k t where 0 < N t < 1, and u t is a stochastic disturbance term to the production function. A solution is a sequence fulfilling (1). (c t, K t, N t ), t = 0, 1, 2,...

8 The simplest model Singularity of RBC and DSGE RBC Solution Suppose we can find a solution in the form c t K t = H c t 1 + α c u t N t K t 1 N t 1 α K α N You see that this system is singular, i.e. three variables are driven by just one shock.

9 The simplest model Singularity of RBC and DSGE Singularity as a general feature of DSGE This simple fact generalizes to. The number of shocks driving the system is independent of the number of variables. Shocks to technology, to labour supply, to money, etc. are motivated by theory, not from the number of variables to be modeled.

10 The simplest model Singularity of RBC and DSGE DSGE s and the data Kydland and Prescott 1. Solution of the model, linearization 2. Calibration-Estimation 3. Comparison with the covariance structure of actual macroeconomic data, the business cycle, etc.

11 Measurement errors and singularity However, the calibration-estimation approach can be questioned and replaced by estimation-estimation. How do we reconcile the theory, i.e. singularity, with the obvious fact that macroeconomic vectors are not singular? Adding measurement errors: c t c t 1 α c e ct K t = H K t 1 + α K u t + e Kt N t N t 1 α N e Nt

12 Unobserved Components Back to the model with measurement errors: c t K t N t = H c t 1 K t 1 N t 1 + α c α K α N u t + e ct e Kt e Nt Given this structure one may think of estimating the deep parameters: H, the parameters α, the variances etc. This requires identification, but in principle...

13 Direct estimation of a VAR Replace with c t c t 1 K t = H K t 1 + α K u t + e Kt N t N t 1 α N e Nt c t K t N t = H c t 1 K t 1 N t 1 α c + ɛ 1t ɛ 2t ɛ 3t e ct If the measurement errors are small then H and H are close etc.

14 But where do VAR models come from: Prediction VAR models are often introduced as a relatively theory-free approach. This is not completely true. Where do they come from. Not from macroeconomics. Rather, Engineering and Mathematics. Prediction problems: Given a process x t, the best linear prediction of x t, given x t 1, x t 2,..., is the projection x t = a 1 x t 1 + a 2 x t e t, where e t is orthogonal to x t k, k 0. It is shown that e t has no autocorrelation.

15 Where VAR models come from: Prediction This generalizes to stochastic vectors: A VAR is x 1t x 2t. = X t = A 1 X t 1 + A 2 X t E t. x nt X t = A 1 X t 1 + A 2 X t A p X t p + E t, which may result from truncation or assumption.

16 The example of an MA: fundamentalness Suppose that x t = u t + bu t 1, that is an MA(1). The prediction autoregression is x t = bx t 1 b 2 x t 2 + b 3 x t 3 + u t. But of course we need the assumption b < 1. This is called invertibility. Note the invertibility is necessary for prediction. Now suppose that y t is the rate of change of productiivity and that x t = 0.2v t + 0.8v t 1 = w t + 4w t 1. is the structural equation. This cannot be estimated by a VAR. This is known as the fundamentalness problem within VAR theory.

17 Fundamentalness Motivation Thus VARs rely on the assumption that the structural shocks are also fundamental. There is a vast literature trying to do without this implicit assumption. The point can be restated by saying that what is good for prediction is not necessarily good for structural analysis. The fundamentalness problem arises both with VARs and DSGE models.

18 The simplest example of a factor model Some of the problems arising with VARs and DSGE models can be solved within factor models. An example: c t = x 1t = a 1 u t + ξ 1t y t = x 2t = a 2 u t + ξ 2t x 3t = a 3 u t + ξ 3t x 4t = a 4 u t + ξ 4t. The ξ s are called idiosyncratic components. They are orthogonal to one another. The terms a i u t are the common components. This is a huge number of macroeconomic variables, but we are interested in the first two. Precisely we want to estimate u t and the coefficients a 1 and a 2.

19 Estimation of the factor model Take the average of the xs: [ ] x n = 1 n 1 n x it = a i u t + 1 n n n i=1 The variance is i=1 n ξ it = ā n u t + 1 n i=1 n ξ it. i=1 var( x n ) ā 2 nσ 2 u + 1 n 2 n max i var(ξ it ) = ānσ 2 u n max var(ξ it ) i This is a weak law of large numbers and we get rid of the idiosyncratic components.

20 The general form of the factor model x it = χ it + ξ it = a i1 (L)u 1t + + a iq (L)u qt + ξ it. We want to estimate the common shocks u jt and the functions a j (L). The model has been used for prediction and structural analysis.

21 Solving the fundamentalness problem in the singular case Consider a singular MA vector: χ 1t = u t + au t 1 χ 2t = u t + bu t 1 (1) Each of the equation is not necessarily invertible. But the system is. Multiply the first equation by b, the second by a and subtract: Then, from (1), u t = bχ 1t aχ 2t b a χ 1t = abχ 1,t 1 a 2 χ 2,t 1 b a χ 2t = b2 χ 1,t 1 abχ 2,t 1 b a + u t + u t