Empirical Macroeconomics

Transcription

1 Empirical Macroeconomics Francesco Franco Nova SBE May 9, 2013 Francesco Franco Empirical Macroeconomics 1/18

2 Growth and Fluctuations Supply and Demand Figure : US dynamics Francesco Franco Empirical Macroeconomics 2/18

3 Unit root Non stationarity In the OLS estimation of an AR(1) y t = fly t 1 + t with t iid N! 0, 2" and y 0 = 0 the OLS estimate of fl is given by ˆfl = q nt=1 y t 1 y t q nt=1 y 2 t Francesco Franco Empirical Macroeconomics 3/18

4 Unit root Non stationarity If fl < 1, then Ô n (ˆfln fl) æ N 10, 1 fl 22 But if this result was valid wjen fl = 1 then the distribution would have variance zero. (Theory in Hamilton chap 17). We need to find a suitable non degenrate distribution to test hypotesis H 0 : fl = 1 Francesco Franco Empirical Macroeconomics 4/18

5 Unit root Augmented Dickey-Fuller we fit y t = + y t 1 + t + kÿ j y t j + e t j=1 via OLS. the test statistic for H 0 : = 0isZ t = ˆ /ˆ.Critical values. Francesco Franco Empirical Macroeconomics 5/18

6 Unit root Augmented Dickey-Fuller Francesco Franco Empirical Macroeconomics 6/18

7 Autocorrelation Test Definition Á t is not iid since it is correlated with some Á t s. Does not change the consistency result but now the OLS estimator is ine cient and you should use GLS. You can use a test of autocorrelation H 0 : no autocorrelation H 1 : autocorrelation 1 Estimate fl = corr(ˆá t, ˆÁ t 1 ) and use a t-test on fl 2 Durbin-Watson: DW 2 2fl. Reject H 0 if DW Æ 1.5. Francesco Franco Empirical Macroeconomics 7/18

8 Heteroskedasticity Phillips Perron PP correct both autocorrelation and heteroskedasticity but fits: y t = + fly t 1 + t + t Phillips-Perron test for unit root Number of obs = 259 Newey-West lags = 1 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(rho) Z(t) MacKinnon approximate p-value for Z(t) = lrgnp Coef. Std. Err. t P> t [95% Conf. Interval] lrgnp L _cons Francesco Franco Empirical Macroeconomics 8/18

9 Stationary Check code for tests on unemployment Augmented Dickey-Fuller test for unit root Number of obs = 257 Interpolated Dickey-Fuller Test 1% Critical 5% Critical 10% Critical Statistic Value Value Value Z(t) MacKinnon approximate p-value for Z(t) = Francesco Franco Empirical Macroeconomics 9/18

10 Explaining the dynamics Shocks There are two types of disturbances a ecting unemployment and output: 1 The first has no long-run e ect on either unemployment or output 2 The second has no long-run e ect on unemployment, but may have a long-run e ect on output 3 Finally, these two disturbances are uncorrelated at all leads and lags Francesco Franco Empirical Macroeconomics 10/18

11 The Model MA(Œ) Let Y and U denote the logarithm of GNP and the level of the unemployment rate, e d and e S be the two disturbances. X =( Y, U), andlet =(e d, e s ). The vector moving average (VMA) representation of the postulated process: X t = A(0)Á t + A(1)Á t Œÿ = A(j) t j j=0 with q Œ j=1 a 11 (j) =0andVar( ) =I Francesco Franco Empirical Macroeconomics 11/18

12 Wold representation Fundamental Since X is stationary, it has a Wold-moving average representation with Var(v) = X t = v t + Cv t C k 1 v t k Francesco Franco Empirical Macroeconomics 12/18

13 From Wold to Model Identification v = A(0) =A(0)A(0) Õ three restrictions Francesco Franco Empirical Macroeconomics 13/18

14 Identification In levels, we have: Y t = v 1t +(I 1 + C 1 )v 1t (I 1 + C C k 1 )v t k +... LR =(I + C + C )A(0) =(I C) 1 A(0) LR has the following interpretation: C LR11 : long run e ect of e LR = d on Y LR 21 : long run e ect of e d on U LR 12 : long run e ect of e d on Y LR 22 : long run e ect of u s on U D Francesco Franco Empirical Macroeconomics 14/18

15 IRF X t+s = i = s 1 ÿ i=0 C i A(0) t+s i Ë È (C) i A(0) you want to study ˆX t+s ˆ t = s Francesco Franco Empirical Macroeconomics 15/18

16 Variance Decomposition FEV D = E t0 5 1 X t0 +s X t0 +s 21 X t0 +s X 2 6 Õ t0 +s = s 1 ÿ t=0 C t C Õ t =AA Õ = n vars ÿ j=1 a j a Õ j Francesco Franco Empirical Macroeconomics 16/18

17 Estimation by VAR Invertibility Take a MA (for simplicity MA(1)) y t = v t + c 1 v t 1 y t =(1 + c 1 L) v t provided c 1 < 1, you can multiply both sides by (1 + c 1 L) 1 and get (1 + c 1 L) 1 y t = v t which is VAR with infinite lags. This generalize to matrices. Francesco Franco Empirical Macroeconomics 17/18

18 Estimation by VAR Fundamentalness You can only recover shocks with c 1 < 1. Consider the MA y t = u t + 1 c 1 u t with v 2 = 1 c1 2 u, then the two MA (invertible and not invertible have the same moments). The problem is that if your model is an MA non invertible, by estimating the VAR you are going to recover the shocks of the invertible MA, you get v and not u. Theu are not fundamental. Francesco Franco Empirical Macroeconomics 18/18