Financial Econometrics Econ 40357 ARIMA Part 2: Autoregressive - PowerPoint PPT Presentation

Financial Econometrics Econ 40357 ARIMA Part 2: Autoregressive Models N.C. Mark University of Notre Dame and NBER August 30, 2020 1 / 22

Autoregressive (AR) models. These are models with more durable, persistent dependence over time. iid ∼ ( 0 , σ 2 Let ǫ t ǫ ) , and | ρ | < 1 . Then the AR(1) model is y t = a + ρ y t − 1 + ǫ t where σ 2 a ( 1 − ρ 2 ) , ρ ( y t , y t − k ) = ρ k ǫ E ( y t ) = µ = ( 1 − ρ ) , Var ( y t ) = Note: a = µ ( 1 − ρ ) , which means we can also write it as y t = µ ( 1 − ρ ) + ρ y t − 1 + ǫ t 2 / 22

The MA representation of the AR(1) The AR(1) can also be represented as an MA ( ∞ ) . = a + ρ ( a + ρ y t − 2 + ǫ t − 1 ) + ǫ t y t � �� � y t − 1 a + ρ a + ρ 2 ( a + ρ y t − 3 + ǫ t − 2 ) = + ρǫ t − 1 + ǫ t � �� � y t − 2 a + ρ a + ρ 2 a + ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 2 y t − 3 = . . . � � 1 + ρ + ρ 2 + ρ 3 + · · · + ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 3 ǫ t − 3 + · · · = a � �� � a / ( 1 − ρ ) a 1 − ρ + ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 3 ǫ t − 3 + · · · = 3 / 22

What is the mean E ( y t ) ? � � a 1 − ρ + ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 3 ǫ t − 3 + · · · E ( y t ) = E � � a 1 − ρ + E ǫ t + ρ E ǫ t − 1 + ρ 2 E ǫ t − 2 + ρ 3 E ǫ t − 3 + · · · = a = 1 − ρ 4 / 22

What is the Variance Var ( y t ) ? � � 2 σ 2 ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 3 ǫ t − 3 + · · · = Var ( y t ) = E y � � ǫ 2 t + ρ 2 ǫ 2 t − 1 + ρ 4 ǫ 2 t − 2 + · · · + 2 ρǫ t ǫ t − 1 + 2 ρ 2 ǫ t ǫ t − 2 + · · · = E    E ǫ 2 t + ρ 2 E ǫ 2 t − 1 + ρ 4 E ǫ 2 t − 2 + · · · + 2 ρ E ǫ t ǫ t − 1 + 2 ρ 2 E ǫ t ǫ t − 2 + · · · =  � �� � 0 � � 1 + ρ 2 + ρ 4 + · · · σ 2 = ǫ σ 2 ǫ = 1 − ρ 2 5 / 22

What is the autocorrelation function? First, write the AR(1) in deviations from the mean form, = µ ( 1 − ρ ) + ρ y t − 1 + ǫ t y t y t − µ = ρ ( y t − 1 − µ ) + ǫ t Then, = Cov ( y t , y t − 1 ) = E ( y t − µ ) ( y t − 1 − µ ) γ 1 = E ( ρ ( y t − 1 − µ ) + ǫ t ) ( y t − 1 − µ ) ρ E ( y t − 1 − µ ) 2 = + E ( ǫ t ( y t − 1 − µ )) � �� � � �� � Var ( y t − 1 ) 0 = ρ Var ( y t ) = ρσ y σ y Hence, ρ ( y t , y t − 1 ) = ρ 6 / 22

γ 2 = Cov ( y t , y t − 2 ) = E ( y t − µ ) ( y t − 2 − µ ) = E ( ρ ( y t − 1 − µ ) + ǫ t ) ( y t − 2 − µ ) = ρ E ( y t − 1 − µ ) ( y t − 2 − µ ) + E ( ǫ t ( y t − 2 − µ )) � �� � � �� � γ 1 0 = ργ 1 ρ ( y t , y t − 2 ) = ργ 1 = ρρσ y σ y = ρ 2 σ y σ y σ y σ y We can infer that ρ ( y t , y t − k ) = ρ k 7 / 22

AR(1) forecasts E t ( ˜ y t + 1 ) = ρ ˜ y t y t + 1 ) = ρ 2 ˜ E t ( ˜ y t + 2 ) = ρ E t ( ˜ y t Hence, y t + k ) = ρ k ˜ E t ( ˜ y t Try it out on daily stock returns. 8 / 22

Realization of an AR(1) with ρ = 0 . 96 9 / 22

How to generate in Eviews ’ Generate white noise process series e = nrnd ’Generate persistent AR(1) smpl @first @first series sto = 0 ’ Initial conditions smpl @first+1 @last series sto = .96*sto(-1)+.5*e ’ Recursion series y = sto delete sto (To get impulse response: Quick, estimate VAR) (arima models.wf1 and pgm) 10 / 22

Impulse Response Function The impulse response function traces the effect of a one time, one-standard deviation shock today ǫ t = σ ǫ , on the current and all future values y t , y t + 1 , y t + 2 , .... Stationary processes will revert to their mean values. Let’s analyze as deviations from the mean (set µ = 0). AR(1): y t = ρ y t − 1 + ǫ t , 0 < ρ < 1 . y t = ǫ t = ρ y t = ρǫ t y t + 1 ρ y t + 1 = ρ 2 ǫ t y t + 2 = ρ k ǫ t = y t + k 11 / 22

Another representation of impulse response. MA representation (mean suppressed µ = 0), y t = ǫ t + ρǫ t − 1 + ρ 2 ǫ t − 2 + ρ 3 ǫ t − 3 + · · · One time shock ǫ t , with all other shocks shut down, ǫ k = 0 , k � = t = y t ǫ t y t + 1 = ρǫ t ρ 2 ǫ t = y t + 2 and so on. Later, I will show you how to generate implulse responses in Eviews. 12 / 22

Impulse response of AR(1) 13 / 22

AR(1) with negative ρ 14 / 22

Unit Root Nonstationarity Why | ρ | < 1 is necessary for stationarity? It is usually the case that 0 < ρ < 1 in economics and finance (persistence). What happens to the mean and the variance of y t when ρ = 1? What happens to the impulse response function when ρ = 1? (permanent effect). 15 / 22

Realization of a driftless Random Walk 16 / 22

Random walk with drift 17 / 22

The AR(2) model. Back to Stationary Models. iid � � 0 , σ 2 Let ǫ t ∼ . The second-order autoregressive model (AR(2)) is ǫ y t = a + ρ 1 y t − 1 + ρ 2 y t − 2 + ǫ t and is stationary if | ρ 1 + ρ 2 | < 1 . Assume stationarity, take expectations µ y = a + ρ 1 µ y + ρ 2 µ y µ y a = 1 − ρ 1 − ρ 2 Computing variance and autocovariances by hand is too complicated. It involves taking variance and first-order covariance σ 2 ρ 2 1 σ 2 y + ρ 2 2 σ 2 y + 2 ρ 1 ρ 2 γ 1 + σ 2 = y ǫ ρ 1 σ 2 y ρ 1 σ 2 γ 1 = y + ρ 2 γ 1 → γ 1 = 1 − ρ 2 Then you must to solve these two equations for σ 2 y and γ 1 . 18 / 22

AR(2) Impulse Response Function AR(2) with µ = 0 (or in deviation from mean form). y t = ρ 1 y t − 1 + ρ 2 y t − 2 + ǫ t Let y 0 = y − 1 = 0 , One-time shock at time 1, ǫ 1 , with all other shocks shut down. Trace effect recursively = y 1 ǫ 1 = ρ 1 y 1 = ρ 1 ǫ 1 y 2 � � ρ 2 = ρ 1 y 2 + ρ 2 y 1 = 1 + ρ 2 y 3 ǫ t � � ρ 2 y 4 = ρ 1 y 3 + ρ 2 y 2 = ρ 1 1 + 2 ρ 2 ǫ 1 and so on Is possible to get cyclical impulse responses. 19 / 22

Realization and Impulse Response AR(2) ρ 1 = 0 . 8 , ρ 2 = − 0 . 8 20 / 22

AR(2) forecasts Form the forecasts and input recursively. E t ( ˜ y t + 1 ) = ρ 1 ˜ y t + ρ 2 ˜ y t − 1 E t ( ˜ y t + 2 ) = ρ 1 ( E t ( ˜ y t + 1 )) + ρ 2 ˜ y t E t ( ˜ = ρ 1 ( E t ( ˜ y t + 2 )) + ρ 2 E t ( ˜ y t + 3 ) y t + 1 ) 21 / 22

Extensions No need to stop at AR(2). Can add more and more lags. 1 In MA model, can add more and more lagged shocks. 2 Difference between MA and AR. 3 AR is dependence across time of observations. MA is dependence across time of shocks. MA memory is finite 4 AR memory is infinite (but diminishes exponentially) 5 Can combine MA and AR. Here’s ARMA(1,1) 6 y t = a + ρ y t − 1 + ǫ t + θ 1 ǫ t − 1 22 / 22