Empirical Macroeconomics

Transcription

1 Empirical Macroeconomics Francesco Franco Nova SBE April 18, 2018 Francesco Franco Empirical Macroeconomics 1/23

2 Invertible Moving Average A difference equation interpretation Consider an invertible MA1) Y t = µ + ɛ t + θɛ t 1 you are solving for ɛ t ɛ t = θɛ t 1 + Y t µ) that you can solve backwards provided θ < 1 ɛ t = 1 + θl) 1 Y t µ) Francesco Franco Empirical Macroeconomics 2/23

3 Non-Invertible Moving Average A difference equation interpretation Consider a non invertible MA1) ɛ t = θɛ t 1 + Y t µ) provided θ 1.Forward one period ɛ t+1 = θɛ t + Y t+1 µ) and solve for ɛ t ɛ t = 1 θ ɛ t θ Y t+1 µ) ɛ t = θ L 1 ) 1 1 θ ) L 1 Y t µ where L 1 x t = x t+1. You need future values of Y t to recover ɛ t. Francesco Franco Empirical Macroeconomics 3/23

4 Invertible Moving Average MA and VAR Take the invertible MA1) Y t = µ + ɛ t + θɛ t 1 The moments are E Y t ) = µ, E Y t µ) 2 = 1 + θ 2) σ 2 and E Y t µ) Y t 1 µ) = θσ 2. We can recover the shocks with a VAR Y t µ = 1 + θl) ɛ t provided θ < 1, you can multiply both sides by 1 + θl) 1 and get 1 + θl) 1 Y t µ) = ɛ t which is AR with infinite lags. Francesco Franco Empirical Macroeconomics 4/23

5 Non-Invertible Moving Average MA and VAR Consider a seemingly different MA1) We know that the moments are E 2 E Ỹt µ) = 1 + θ 2) σ 2 and ) E Ỹt µ Ỹt 1 µ) = θ σ 2. Suppose θ = θ 1 σ 2 = θ 2 σ 2 Ỹt ) = µ, then Y t and Ỹ t have the same moments. Notice that for any non-invertible MA1) we have found a non-invertible MA1) and vice-versa. Francesco Franco Empirical Macroeconomics 5/23

6 Non-Invertible Vector Moving Average VMA The VMA Y t = CL)ɛ t is invertible if and only if the roots of the polynomial Cz) = 0 lie outside the unit circle Example ) ) ) Y1t 1 L ɛ1t = 0 θ L Y 2t the det Cz)) = θ z is zero for θ = z. Therefore for invertibility z > 1 which is satisfied if θ > 1. ɛ 2t Francesco Franco Empirical Macroeconomics 6/23

7 Invertible and Non-invertible Recovering the shocks When we estimate a VAR we always recover the innovations of the invertible MA, or the Wold decomposition innovations. What if our model/theory implies non-fundamental shocks? Here ɛ t is non fundamental for Y t is θ 1. But suppose it is fundamental for another variable S t? This is actually what many models imply. Fundamentalness is therefore an information issue. Francesco Franco Empirical Macroeconomics 7/23

8 An example Non-fundamentalness and News Shocks Forni et al.) Consider a simple asset price model. We assume the total factor productivity a t follows a t = a t 1 + ɛ t 2 + η t Where ɛ t is a news shock and η t is a shock affecting TFP on impact. The agents observe the shock ɛ t at time t and react to it immediately while the shock will affect TFP only at time t + 2. Francesco Franco Empirical Macroeconomics 8/23

9 An example Non-fundamentalness and News Shocks Forni et al.) The RA maximizes s.t. E t β t c t t=0 c t + p t n t+1 = p t + a t ) n t FOC λ t = 1 λ t p t = βe t λ t+1 [p t+1 + a t+1 ] Combining we get p t = β j E t [a t+j ] j=1 Francesco Franco Empirical Macroeconomics 9/23

10 An example Non-fundamentalness and News Shocks Forni et al.) Now p t = j=1 β j E t [a t+j ] is using our DGP on productivity a t = a t 1 + ɛ t 2 + η t E t a t+1 = a t + ɛ t 1 E t a t+2 = E t a t+1 + ɛ t = a t + ɛ t 1 + ɛ t E t a t+3 = E t a t+2 = a t + ɛ t 1 + ɛ t and so on. Using it in the price p t = β j=1 β j 1 E t [a t+j ] we get p t = β 1 β a t + β 1 β βɛ t + ɛ t 1 ) Francesco Franco Empirical Macroeconomics 10/23

11 An example Non-fundamentalness and News Shocks Forni et al.) Now assume we observe p t and a t. Take the first difference and obtain the VMA: ) at L 2 ) ) 1 ɛt = β p 2 t 1 β + βl β η 1 β t the polynomial to be solved is determinant is β 1 β z2 β2 1 β + βz = 0 and the roots are z = 1 and z = β. Given β < 1 the shocks are non-fundamental for the variables a t and p t. Francesco Franco Empirical Macroeconomics 11/23

12 State Space Representation Economic Models as state space models Most Dynamic macro models can be casted into a State Space representation: S t+1 = AS t + Bη t+1 which we can re-write as X t = CS t 1 S t = AS t 1 + Bη t X t = CS t 1 + Dη t where S t is a vector of states, X t is a vector of observables and η t are shocks. Francesco Franco Empirical Macroeconomics 12/23

13 State Space Representation Our example The previous example S t = AS t 1 + Bη t is a t ɛ t ɛ t 1 = a t 1 ɛ t 1 ɛ t ɛt η t ) and X t = CS t 1 + Dη t is at p t ) = β 1 β β 1 β β 1 β ) a t 1 ɛ t 1 ɛ t β 2 1 β β 1 β ) ɛt η t ) Francesco Franco Empirical Macroeconomics 13/23

14 Non Fundamentnalness An information issue Consider D to be square: number of observations is equal to number of structural shocks η t = D 1 X t CS t 1 ) Francesco Franco Empirical Macroeconomics 14/23

15 Non Fundamentnalness An information issue Plugging ɛ into the state equation: S t = AS t 1 + BD 1 X t CS t 1 ) S t = ) A BD 1 C S t 1 + BD 1 X t solve backwards I ) ) A BD 1 C L S t = BD 1 X t S t = I ) 1 A BD 1 C L) BD 1 X t provided the eigenvalues ofa BD 1 C all be strictly less than one in modulus. Then the past of X t reveals S t. Francesco Franco Empirical Macroeconomics 15/23

16 Non Fundamentnalness An information issue: shocks are non-fundamental wrt to observed variables In our example A BD 1 C = β the eigenvalues are 1 β the pas X t do not reveal the S t. 1 β β ) β 1 β β 1 β β 1 β, 1) not respected as expected, therefore ) Francesco Franco Empirical Macroeconomics 16/23

17 Non Fundamentnalness An information issue Therefore the condition that A BD 1 C ) : X t = CS t 1 + Dη t X t = C I ) 1 A BD 1 C L) BD 1 X t + u t X t = Φ j X t j + u t j=1 is a sufficient condition for a VAR on observables to have innovations that map directly back into structural shocks in population. The variance of the residuals Σ u = DΣ η D. Francesco Franco Empirical Macroeconomics 17/23

18 DFM-FAVAR Solving the Information problem using factors? The idea is to have sufficient information to reveal the states. Go back to the state equation and reverse the reasoning: S t = AS t 1 + Bη t and notice that the structural shocks are always fundamental to the states. If you could observe the states you would simply run a VAR in the states and recover the shocks. Suppose B 1 exists almost always, here left inverse) η t = B 1 S t B 1 AS t 1 Francesco Franco Empirical Macroeconomics 18/23

19 DFM-FAVAR Solving the Information problem using factors? Plug into the observation the shocks you recover from the state equation X t = CS t 1 + Dη t X t = DB 1 + Write the previous equation as C DB 1 A X t = ΛF t ) ) L S t 1 where Λ = [ DB 1 C DB 1 A ] and F t = S t S t 1). Which is to underline the connection between the states and what are called factors. The idea of the FAVAR is to measure the unobserved states with factors and run a VAR with factors and observed variables. Now we turn to the exampl of our paper. Francesco Franco Empirical Macroeconomics 19/23

20 BBE Motivating the FAVAR Structure: An Example Backward-looking model as an example Supply and Demand): π t = δπ t 1 + κ y t 1 y n t 1) + st y t = φy t 1 ψ R t 1 π t 1 ) + d t y n t = ρy n t 1 + η t s t = αs t 1 + ν t Francesco Franco Empirical Macroeconomics 20/23

21 SVAR/FAVAR Motivating the FAVAR Structure: An Example The monetary policy authority follows the Taylor Rule R t = βπ t + γ y t y n t ) + ɛ r t Monetary policy has two dimensions: 1 systematic: rule 2 unexpected ɛ t Francesco Franco Empirical Macroeconomics 21/23

22 SVAR/FAVAR Motivating the FAVAR Structure: An Example The model can be casted in State Space Form: ] ] [ Ft X t = ΦL) [ Ft 1 X t 1 + ɛ t ρ α where Φ = κ α δ κ ψ φ ψ γψ βα βδ + γψ) βκ + γφ) βκ + γρ) η t v and ɛ t = t d t and ɛ γ β γ 1 r t F t X t) = y n t s t π t y t R t ) Francesco Franco Empirical Macroeconomics 22/23

23 SVAR/FAVAR Motivating the FAVAR Structure: An Example The division between what you observe and what you do not observe determine if the model has unobserved states. If yes we know there are potential information issues discuss ηshock) and we treat the unobserved states as factors. Assume the large data set is related to the variables in our model as X t = Λ y n t s t π t y t R t ) + e t and treat F t = y n t s t ) X t = π t y t R t ) Francesco Franco Empirical Macroeconomics 23/23