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

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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

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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.

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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.

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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

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

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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.

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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 .

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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– ).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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).

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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

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