1 MF-852 Financial Econometrics Lecture 11 Distributed Lags and Unit Roots Roy J. Epstein Fall 2003

2 Topics Dynamic models Serial correlation and lagged dependent variables Short run vs. long-run dynamic effects Random walks and trends Unit roots Stationarity Dickey-Fuller test

3 Dynamic models Suppose the regression is Y t =  0 +  1 Y t–1 + e t Lagged dependent variable (Y t–1 ) on right-hand side —> dynamic model. Can also include other X variables on right-hand side. Estimate as usual by least squares.

4 Dynamic models and serially correlated residuals Serially correlated residuals in dynamic model yield biased least squares coefficient estimates. Unlike last week, where serial correlation just affected standard errors and t-tests. Cannot use Durbin-Watson statistic to test for serial correlation in dynamic model. Use Durbin h-test instead (assuming 1 st order autoregressive process). See RR, p. 447.

5 Dynamics Y t evolves based on Y t–1, so each observation depends on the entire history of the process. Y t =  Y t–1 +  0 + e t Y t–1 =  Y t–2 +  0 + e t–1 Y t–2 =  Y t–3 +  0 + e t–2 … By substitution: Y t =  (  j )(  0 + e t–j ) +  0 + e t

6 Note: Calculation of Geometric Series S = 1 +  (  j ) = 1 +  +  2 +  3 + …  S=  +  2 +  3 + … S(1–  ) = 1 S = 1/(1–  )

7 “Steady-State” Y t =  (  j )(  0 + e t–j ) +  0 + e t Steady-state Y: eventual value of Y, assuming no additional shocks. Equal to expected value of process Y SS =  (  j )  0 +  0 =  0 [1 +  (  j )] =  0 /(1 –  ) =  Y SS +  0 Y remains constant at steady-state value.

8 Short Run vs. Long Run Suppose Y in steady-state: Y t–1 = Y SS. What is effect of one-period unit shock (e t = 1)? Y t =  Y SS +  E(Y t+1 ) =  (Y SS +  0 + 1) +  0 E(Y t+2 ) =  2 (Y SS +  0 + 1) +  0 (1 +  ) E(Y t+3 ) =  3 (Y SS +  0 + 1) +  0 (1 +  +  2 ) E(Y t+  ) =  0 (1 +  +  2 ) =  0 /(1 –  ) = Y SS In long-run, Y reverts to Y SS (mean-reverting process). Short-run effect, duration depends on .

9 Permanent Shock (A) Suppose Y in steady-state: Y t–1 = Y SS. What is effect of unit increase in  0 ? Y t =  Y SS + (  0 + 1) E(Y t+1 ) =  (Y SS +  0 + 1) + (  0 + 1) E(Y t+2 ) =  2 (Y SS +  0 + 1) + (  0 +1)(1 +  ) E(Y t+3 ) =  3 (Y SS +  0 + 1) + (  0 +1)(1 +  +  2 ) E(Y t+  ) = (  0 +1)(1 +  +  2 + …) = Y SS + 1/(1 –  ) New steady-state Process mean increases by 1/(1–  ).

10 Permanent Shock (B) Suppose Y in steady-state: Y t–1 = Y SS. Assume one-period unit shock (e t = 1) and  = With   1, one-period shock has nearly permanent effect.

11 Random Walk Suppose Y t = Y t–1 + e t e t is serially independent mean zero error. Then Y is a random walk process. All shocks are permanent. Current period Y is best prediction of next period.

12 Random Walk with Drift Suppose Y t =  + Y t–1 + e t e t is serially independent mean zero error.  is drift (i.e., trend component) Then Y is a random walk process. All shocks are permanent. Current period Y +  is best prediction of next period.

13 Unit Roots Suppose the model for Y is: Y t =  Y t–1 +  0 +  1 X t + e t If  = 1, then Y t is said to have a unit root. Unit root means Y t is determined by a random walk along with other variables.

14 Stationarity Stationary time-series: finite variances that do not change over time Unit root process for Y t : Y t = Y t–1 +  0 +  1 X t + e t =  e t-j + e t + terms in  0 and X t Var(Y t ) =  ! Y t is non-stationary process.

15 Stationarity and Model Specification Suppose model is Y t =  0 +  1 X t + e t If e t non-stationary (i.e., has unit root) then misspecified model. Error term must have constant finite variance.

16 Dynamic Models and Stationarity Suppose model is Y t =  Y t–1 +  0 +  1 X t + e t If  = 1 (unit root for Y t ) then: Least squares estimate of  is biased toward zero; T-tests overstate statistical significance, potentially by a lot.

17 Dickey-Fuller Test Notation:  Y t = Y t – Y t–1 Dickey-Fuller test for non-stationarity.  Y t =  Y t–1 +  0 + e t  Y t =  Y t–1 +  0 +  1  Y t–1 + e t (augmented Dickey-Fuller) Get correct critical value for t-statistic on  from Dickey-Fuller table (RR, p. 565).

18 Correction for Non-Stationarity Suppose the model is non- stationary (e t has unit root): Y t =  0 +  1 X t + e t First-difference the data to remove unit-root. Estimate model by least squares as:  Y t =  0 +  1  X t + u t