Econometrics I. Professor William Greene Stern School of Business Department of Economics 25-1/25. Part 25: Time Series

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1 Econometrics I Professor William Greene Stern School of Business Department of Economics 25-1/25

2 Econometrics I Part 25 Time Series 25-2/25

3 Modeling an Economic Time Series Observed y 0, y 1,, y t, What is the sample Random sampling? The observation window 25-3/25

4 Estimators Functions of sums of observations Law of large numbers? Nonindependent observations What does increasing sample size mean? Asymptotic properties? (There are no finite sample properties.) 25-4/25

5 Interpreting a Time Series Time domain: A process y(t) = ax(t) + by(t-1) + Regression like approach/interpretation Frequency domain: A sum of terms y(t) = βjcos( α jt) + ε( t) j Contribution of different frequencies to the observed series. ( High frequency data and financial econometrics frequency is used slightly differently here.) 25-5/25

6 For example, 25-6/25

7 In parts 25-7/25

8 Studying the Frequency Domain Cannot identify the number of terms Cannot identify frequencies from the time series Deconstructing the variance, autocovariances and autocorrelations Contributions at different frequencies Apparent large weights at different frequencies Using Fourier transforms of the data Does this provide new information about the series? 25-8/25

9 Autocorrelation in Regression Y t = b x t + ε t Cov(ε t, ε t-1 ) 0 Ex. RealCons t = a + brealincome + ε t U.S. Data, quarterly, /25

10 Autocorrelation How does it arise? What does it mean? Modeling approaches Classical direct: corrective Estimation that accounts for autocorrelation Inference in the presence of autocorrelation Contemporary structural Model the source Incorporate the time series aspect in the model 25-10/25

11 z t = b 1 y t-1 + b 2 y t b P y t-p + e t Autocovariance: γ k = Cov[y t,y t-k ] Stationary Time Series Autocorrelation: ρ k = γ k / γ 0 Stationary series: γ k depends only on k, not on t Weak stationarity: E[y t ] is not a function of t, E[y t * y t-s ] is not a function of t or s, only of t-s Strong stationarity: The joint distribution of [y t,y t-1,,y t-s ] for any window of length s periods, is not a function of t or s. A condition for weak stationarity: The smallest root of the characteristic polynomial: 1 - b 1 z 1 - b 2 z b P z P = 0, is greater than one. The unit circle Complex roots Example: y t = ρy t-1 + e e, 1 - ρz = 0 has root z = 1/ ρ, z > 1 => ρ < /25

12 Stationary vs. Nonstationary Series 25-12/25

13 The Lag Operator Lx t = x t-1 L 2 x t = x t-2 L P x t + L Q x t = x t-p + x t-q Polynomials in L: y t = B(L)y t + e t A(L) y t = e t Invertibility: y t = [A(L)] -1 e t 25-13/25

14 Inverting a Stationary Series y t = ρy t-1 + e t (1- ρl)y t = e t y t = [1- ρl] -1 e t = e t + ρe t-1 + ρ 2 e t ρl ( ) ( ) ( ). = +ρ L +ρ L +ρ L + Stationary series can be inverted Autoregressive vs. moving average form of series 25-14/25

15 Regression with Autocorrelation y t = x t b + e t, e t = ρe t-1 + u t (1- ρl)e t = u t e t = (1- ρl) -1 u t E[e t ] = E[ (1- ρl) -1 u t ] = (1- ρl) -1 E[u t ] = 0 Var[e t ] = (1- ρl) -2 Var[u t ] = 1+ ρ 2 σ u 2 + = σ u2 /(1- ρ 2 ) Cov[e t,e t-1 ] = Cov[ρe t-1 + u t, e t-1 ] = =ρcov[e t-1,e t-1 ]+Cov[u t,e t-1 ] = ρ σ u2 /(1- ρ 2 ) 25-15/25

16 OLS Unbiased? OLS vs. GLS Consistent: (Except in the presence of a lagged dependent variable) Inefficient GLS Consistent and efficient 25-16/25

17 Ordinary least squares regression LHS=REALCONS Mean = Autocorrel Durbin-Watson Stat. = Rho = cor[e,e(-1)] = Variable Coefficient Standard Error t-ratio P[ T >t] Mean of X Constant REALDPI Robust VC Newey-West, Periods = 10 Constant REALDPI AR(1) Model: e(t) = rho * e(t-1) + u(t) Final value of Rho = Iter= 6, SS= , Log-L= Durbin-Watson: e(t) = Std. Deviation: e(t) = Std. Deviation: u(t) = Durbin-Watson: u(t) = Autocorrelation: u(t) = N[0,1] used for significance levels Variable Coefficient Standard Error b/st.er. P[ Z >z] Mean of X Constant REALDPI RHO /25

18 Use residuals Durbin-Watson d= Detecting Autocorrelation ( e e ) T t= 2 t t 1 T 2 Σt= 1et Assumes normally distributed disturbances strictly exogenous regressors Variable addition (Godfrey) y t = β x t + ρε t-1 + u t Use regression residuals e t and test ρ = 0 Assumes consistency of b. Σ 2 2(1 r) 25-18/25

19 How to test for ρ = 1? A Unit Root? By construction: ε t ε t-1 = (ρ - 1)ε t-1 + u t Test for γ = (ρ - 1) = 0 using regression? Variance goes to 0 faster than 1/T. Need a new table; can t use standard t tables. Dickey Fuller tests Unit roots in economic data. (Are there?) Nonstationary series Implications for conventional analysis 25-19/25

20 Reinterpreting Autocorrelation Regression form y = β ' x + ε, ε = ρε + u t t t t t 1 t Error Correction Form y y =β'( x x ) +α( y β ' x ) + u, ( α=ρ 1) t t 1 t t 1 t 1 t 1 t β ' x = the equilibrium The model describes adjustment of y to equilibrium when x changes. t t t 25-20/25

21 Integrated Processes Integration of order (P) when the P th differenced series is stationary Stationary series are I(0) Trending series are often I(1). Then y t y t-1 = y t is I(0). [Most macroeconomic data series.] Accelerating series might be I(2). Then (y t y t-1 )- (y t y t-1 ) = 2 y t is I(0) [Money stock in hyperinflationary economies. Difficult to find many applications in economics] 25-21/25

22 Cointegration: Real DPI and Real Consumption 25-22/25

23 Cointegration Divergent Series? 25-23/25

24 Cointegration X(t) and y(t) are obviously I(1) Looks like any linear combination of x(t) and y(t) will also be I(1) Does a model y(t) = bx(t) + u(u) where u(t) is I(0) make any sense? How can u(t) be I(0)? In fact, there is a linear combination, [1,-β] that is I(0). y(t) =.1*t + noise, x(t) =.2*t + noise y(t) and x(t) have a common trend y(t) and x(t) are cointegrated /25

25 Cointegration and I(0) Residuals 25-25/25