Empirical Macroeconomics Francesco Franco Nova SBE April 21, 2015 Francesco Franco Empirical Macroeconomics 1/33
Growth and Fluctuations Supply and Demand Figure : US dynamics Francesco Franco Empirical Macroeconomics 2/33
ARMA Processes Expectations, stationarity and ergodicity Suppose we observe a sample of size T of some random variables, say i.i.d t { 1, 2,..., T }, with t N(0, 2 ) which is only one possible outcome of the underlying stochastic process. Same for { } Πt= Π: one outcome. Francesco Franco Empirical Macroeconomics 3/33
ARMA Processes Expectations, stationarity and ergodicity Now consider I realizations ) * y 1 Œ t t= Œ, ) yt 2 * Ó Ô Œ Œ t= Œ,..., yt I and select one observation t= Œ associated with the I realizations: Ó Ô yt 1, yt 2,...,yt I This is a sample of I realizations of the random variable Y t.this random variable has some density, f Yt (y t ) which is called unconditional denisity, for example for the Gaussian white noise. 1 2 f Yt (y t )= Ô 1 y 2 t 2 e 2 2fi Francesco Franco Empirical Macroeconomics 4/33
ARMA Processes Expectations The expectation ot the tth observation of a time series refers to the mean of this probability distribution E (Y t )= Œ Œ y t f Yt (y t )dy t which you might see as the probability limit of the ensemble average Iÿ E (Y t )=plim IæŒ (1/I) i=1 Y (i) t which is sometimes called the unconditional mean of Y t : E(Y t )=µ t (examples: Y t = + t, Y t = t + t ). Francesco Franco Empirical Macroeconomics 5/33
ARMA Processes Expectations The variance same examples. 0,t = E (Y t µ t ) 2 = Œ Œ (y t µ t ) 2 f Yt (y t )dy t Francesco Franco Empirical Macroeconomics 6/33
ARMA Processes Autocovariance Given a particular realization such as ) yt 1 * Œ t= Œ on a time series process consider xt 1 consisting of the [j + 1] most recent observations on y as of date tfor that realization: S T x 1 t = W U yt 1 yt 1 1... y 1 t j and think of each realization of y t as generating one particular value of the vector x t.thejthautocovariance of Y t is X V jt = E (Y t µ t )(Y t j µ t j )= Œ Œ Œ Œ... Œ Œ (y t µ t )(y t j µ t j ) f Yt,Y t 1,...,Y t j (y t, y t 1,...,y t j )dy t dy t 1...dy t j Francesco Franco Empirical Macroeconomics 7/33
ARMA Processes Autocovariance Notice it has the form of a covariance on lagged values. The Variance is the zero lag autocovariance. Again again examples. jt = plim IæŒ (1/I) Iÿ i=1 Ë Y (i) t ÈË µ t Y (i) È t j µ t j Francesco Franco Empirical Macroeconomics 8/33
ARMA Processes Stationarity If neither the mean µ t nor the autocovariances jt depend on the date t then the process for Y t is said to be covariance-stationary or weakly stationary. Consider the two examples. It follows that for a covariance statonary process j = j Strict sationarity is related to the joint distribution of Y t, Y t+j1,... depending only on the intervals and not on t (higher moments) Francesco Franco Empirical Macroeconomics 9/33
ARMA Processes Ergodicity Consider the sample mean which in this case in not an ensemble average but rather a time average ȳ =(1/T ) whether time averages eventually converge to ensemble averages has to do with ergodicity. A Gaussian covariance-stationary process is ergodic for the mean if the autocovariances j goes to zero su ciently quickly for as j becomes large: q Œ j=0 j < Œ Tÿ t=1 y 1 t Francesco Franco Empirical Macroeconomics 10/33
ARMA Processes Ergodicity Usually stationarity and ergodicity coincide but not always Y i t = µ i + t with µ generated from a N(0, 2 ) distribution. Which is covariance-stationary but not ergodic. Francesco Franco Empirical Macroeconomics 11/33
ARMA Processes MA consider a MA(1) Y t = µ + t + t 1 Mean and Variance, autocovariances -> covariance-stationary and ergodicity condition is satisfied. E (Y t )=µ E (Y t µ) 2 = 1 1 + 22 2 Francesco Franco Empirical Macroeconomics 12/33
ARMA Processes MA Autocovariance E (Y t µ)(y t 1 µ) = 2 Ergodicity Œÿ j = j=0 E (Y t )=µ 1 1 + 22 2 + - - 2- - - Francesco Franco Empirical Macroeconomics 13/33
ARMA Processes MA The autocorrelation of a covariance-stationary process (denoted fl j )isdefinedas fl j = j 0 first order autocorrelation 1 + 2 max,min,same if is replaced by 1/ Francesco Franco Empirical Macroeconomics 14/33
ARMA Processes MA(Œ) Y t = µ + Œÿ  j t j j=0 condition for covariance-stationary is Œÿ Âj 2 < Œ j=0 with q Œ j=0  j < Œ we also have ergodicity. Francesco Franco Empirical Macroeconomics 15/33
ARMA Processes AR(1) Consider Y t = c + fly t 1 + t Œÿ Y t = fl j (c + t j ) j=0 if fl Ø1 there is no covariance stationary process, if fl < 1there is a covariance stationary process. Francesco Franco Empirical Macroeconomics 16/33
ARMA Processes AR(1) The unconditional mean is µ = c 1 fl and the variance is while the autocovariance j = 2 1 fl 2 Ë È fl j /(1 fl 2 ) 2 Francesco Franco Empirical Macroeconomics 17/33
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 1 Francesco Franco Empirical Macroeconomics 18/33
Unit root Non stationarity If fl < 1, then Ô n (ˆfln fl) æ N 10, 1 fl 22 But if this result was valid when 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 19/33
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 20/33
Unit root Augmented Dickey-Fuller Francesco Franco Empirical Macroeconomics 21/33
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 22/33
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) -0.703-20.307-14.000-11.200 Z(t) -2.300-3.459-2.880-2.570 MacKinnon approximate p-value for Z(t) = 0.1720 lrgnp Coef. Std. Err. t P> t [95% Conf. Interval] lrgnp L1..9973338.0010155 982.14 0.000.9953341.9993335 _cons.030857.0087953 3.51 0.001.013537.0481769 Francesco Franco Empirical Macroeconomics 23/33
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) -3.669-3.459-2.880-2.570 MacKinnon approximate p-value for Z(t) = 0.0046 Francesco Franco Empirical Macroeconomics 24/33
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 25/33
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 1 +... Œÿ = A(j) t j j=0 with q Œ j=1 a 11 (j) =0andVar( ) =I Francesco Franco Empirical Macroeconomics 26/33
Wold representation Fundamental Since X is stationary, it has a Wold-moving average representation with Var(v) = X t = v t + Cv t 1 +.. + C k 1 v t k+1 +... Francesco Franco Empirical Macroeconomics 27/33
From Wold to Model Identification v = A(0) =A(0)A(0) Õ three restrictions Francesco Franco Empirical Macroeconomics 28/33
Identification In levels, we have: Y t = v 1t +(I 1 + C 1 )v 1t 1 +.. +(I 1 + C 1 +.. + C k 1 )v t k +... LR =(I + C + C 2 +...)A(0) =(I C) 1 A(0) LR has the following interpretation: C LR11 :LRe LR = d on Y LR 21 :LR e d on U LR 12 :LR e d on Y LR 22 :LR u s on U D Francesco Franco Empirical Macroeconomics 29/33
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 30/33
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 31/33
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 32/33
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 1 1 2 2 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 33/33