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(229-2)-1 Econ 229 – Handout on time series econometrics • Objective: - Primer on time series econometrics relevant for empirical macroeconomics - Distinction between unit root and trend-stationary series; notion of cointegration. • Background reading (not required): Hamilton, Time Series Econometrics, ch.15-17. Stochastic Processes Example #1: Stationary AR(1) y t = " # y t $1 + % + & t with AR coefficient | " | < 1 and constant " . • Assume " t is white noise = mean zero, finite variance, uncorrelated over time. • Projection n steps ahead: y t+ n = " # y t+ n $1 + % + & t+ n = " # [ " # y t+ n $ 2 + % + & t+ n $1 ] + % + & t+ n ... => y t+ n = " n # y t + " j j = 0 n $1 % (& + t+ n $ j ) = " n # y t + 1$ " n 1$ " & + " j j = 0 n $1 % t+ n $ j • Forecast mean: E t y t+ n = " n # y t + 1$ " n 1$ " % converges to the unconditional mean E[ y t+ n ] = E[ y t ] = " 1# $ • Forecast variance is Var t [ y t+ n ] = E t [( y t+ n " E t y t+ n ) 2 ] = E t [( # j j = 0 n "1 $ % t+ n " j ) 2 ] = # 2 j j = 0 n "1 $ & % 2 = 1" # 2 n 1" # 2 & % 2 converges to the unconditional variance Var[ y t ] = 1 1" # 2 $ % 2 • Time-t disturbance has declining effect over time: E t y t+ n " E t "1 y t+ n = # n $ t (“Impulse response” = graph against n) • Moving Average (MA) representation: y t = " n # y t $ n + 1$ " n 1$ " % + " j j = 0 n $1 & t $ j ( % 1$ " + " j j = 0 ) & t $ j

229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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Page 1: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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Econ 229 – Handout on time series econometrics • Objective: - Primer on time series econometrics relevant for empirical macroeconomics - Distinction between unit root and trend-stationary series; notion of cointegration. • Background reading (not required): Hamilton, Time Series Econometrics, ch.15-17. Stochastic Processes Example #1: Stationary AR(1)

!

yt = " # yt$1 +% +&t with AR coefficient

!

|" |<1 and constant

!

" .

• Assume

!

"t is white noise = mean zero, finite variance, uncorrelated over time.

• Projection n steps ahead:

!

yt+n = " # yt+n$1 +% +&t+n = " #[" # yt+n$2 +% +&t+n$1]+% +&t+n ...

=>

!

yt+n = "n # yt + " jj=0n$1

% (& +'t+n$ j ) = "n # yt +1$" n

1$"& + " j

j=0n$1

% 't+n$ j

• Forecast mean:

!

Etyt+n = "n # yt +1$" n

1$"% converges to the unconditional mean

!

E[yt+n ]= E[yt ]="1#$

• Forecast variance is

!

Vart[yt+n ]= Et[(yt+n " Etyt+n )2]= Et[( # j

j=0n"1

$ %t+n" j )2]= #2 j

j=0n"1

$ &%2

=1"# 2n

1"# 2&%2

converges to the unconditional variance

!

Var[yt ]=1

1"# 2$%2

• Time-t disturbance has declining effect over time:

!

Etyt+n " Et"1yt+n = #n$t (“Impulse response” = graph against n) • Moving Average (MA) representation:

!

yt = "n # yt$n +1$" n

1$"% + " j

j=0n$1

& 't$ j (%1$"

+ " jj=0)

& 't$ j

Page 2: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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Example #2: Random walk

!

yt = yt"1 +# +$t with drift coefficient

!

" .

- Interpret as limiting case of AR(1) with

!

"#1. (Sums in the AR don’t converge for

!

" =1.)

- Projection n steps ahead:

!

yt+n = yt +n" + #t+n$ jj=0n$1

%

- Conditional expectation “drifts” with current value:

!

Etyt+n = yt +n "# => No unconditional mean.

- Time-t disturbance has a permanent effect:

!

Etyt+n " Et"1yt+n = #t

- Forecast error grows over time:

!

Vart[yt+n ]= Et[( "t+n# jj=0n#1

$ )2]= n%"

2

- Taking differences:

!

"yt = yt # yt#1 = $ +%t is stationary => Random Walk is a difference-stationary process. Example #3: Trend-stationary AR(1)

!

yt = " # yt$1 +%t +& +'t with

!

|" |<1 and deterministic drift

!

" .

- Projection n steps ahead:

!

yt+n = "n # yt + " jj=0n$1

% (t + j)& +1$" n

1$"' + " j

j=0n$1

% (t+n$ j

- Conditional expectation:

!

Etyt+n = "n # yt +1$" n

1$"%t + j" j

j=0n$1

& % +1$" n

1$"'

converges to a deterministic trend with slope

!

" /(1#$).

[ Note:

!

" jj=0n#1

$ (t + j)% = [1#" n

1#"t +

1#n" n+(n#1)" n+1

(1#")2] &% n'(

) ' ) ) %1#"(t + 1

1#")]

- Impact of time-t disturbance vanishes as in a stationary AR(1).

!

Etyt+n " Et"1yt+n = #n$t

- Forecast error converges to

!

"#2/(1$ %2 ) as in a stationary AR(1).

- Note: Detrended process

!

xt = yt "# /(1" $) %t is stationary. • Conclude: 1. Deterministic time trends do not add uncertainty. 2. Distinction between trend-stationary and difference-stationary processes is crucial for projections.

Page 3: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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General Stationary Processes: 1. Trend-stationary stochastic process with Moving Average representation:

!

yt = µ +"t + # jj=0$

% &t' j where

!

{" j} is a sequence of coefficients.

Common notation with lag operator L defined by

!

Lxt

= xt"1.

Write:

!

yt = µ +"t + ( # jj=0$

% Lj)&t = µ +"t +#(L)&t

- Projection:

!

yt+n = µ +"t + # jj=0$

% &t+n' j

=> Expectation:

!

Etyt+n = µ +"t + # jj=n$

% &t+n' j

- Forecast error:

!

yt+n " Etyt+n = # jj=0n"1

$ %t+n" j

=> Variance:

!

E[( " jj=0n#1

$ %t+n# j )2]= " j

2j=0n#1

$ &%2

- Find: forecast error has finite variance if

!

" j2

j=0#

$ <#.

Common to impose stronger condition:

!

|" j |j=0#

$ <# .

Page 4: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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2. Trend-stationary ARMA process, and a motivation for “unit roots” - Assume

!

yt = µ +"t +ut where {u} is a mean-zero ARMA process:

!

ut = " jj=1m

# ut$ j +%t + & jj=1k

# %t$ j

- Let

!

" j denote the roots of the polynomial

!

"(L) =1# " jj=1m

$ Lj

= (1# % jL)j=1m

&

- If

!

" j <1 for all j, one can show that

!

ut =(1+ " jj=1

k# Lj )

(1$% j L)j=1

m&

't = ((L)'t is stationary => {y} is trend-stationary

- If

!

" j =1 for some j (say: j=1), write

!

(1" L)ut =(1+ # jj=1

k$ Lj )

(1"% j L)j=2

m&

't = (*(L)'t => {y} is difference-stationary

=> Long-run behavior of the process depends critically on the roots of the AR polynomial: Unit root = difference stationary.

Page 5: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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Estimation and Testing for Unit roots • Basic idea: if

!

yt = " # yt$1 +%t +& +'t

- then

!

"yt = # $ yt%1 +&t +' +(t with

!

" = # $1

- If the original process is stationary with

!

" <1, regression of

!

"yt on

!

yt"1should find a NEGATIVE slope coefficient

!

" . (sometimes called mean-reversion coefficient: High y-values are on average followed by declines.) - If the process has a unit root,

!

" should be insignificant. • Practical problems: What if the error is not white noise, but has AR or MA components? What about heteroskedasticity? - Augmented Dickey-Fuller regression (ADF): Most common standard approach: Include lags of

!

"yt. Obtain:

!

"yt = # $ yt%1 + & jj=1m

' "yt% j +(t +) +*t with

!

" = # $1

- Phillips-Perron approach (PP): Estimate without lags, then compute corrected standard errors. • Theoretical problems: If

!

yt"1 is non-stationary, usual asymptotic theory does not apply.

- Critical values for

!

" depend on inclusion/exclusion of trend and constant => use Dickey Fuller Tables. • Caution about the interpretation: - Failure to reject a unit root does not prove a unit root; question of power - KPSS test for null hypothesis of stationarity available but less commonly used.

Page 6: 229-08-L12 UnitRoots - UCSB Department of Economicsecon.ucsb.edu/~bohn/229/229-08 files/229-08-L12.pdf · Econ 229 – Handout on time series econometrics ... (L)' t => {y} is difference

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Cointegration and Error Correction • Consider two or more difference-stationary series (y,x). • Sometimes a linear combination is stationary: Suppose

!

yt = " # xt +$t

and

!

"xt= # +u

t with stationary (ARMA) disturbances

!

"t and

!

ut.

• Linear combination has remarkable statistical properties: - Regressions of y on x yield super-consistent estimates for

!

" . Regressions would be “spurious” w/o cointegration.

- Intuition: vector (x,y) is driven by disturbances in

!

ut that determine the stochastic drift in both variables

- Practical issues: many ways of dealing with AR, heteroskedasticity, correlation between error terms. • Testing for cointegration: - Run “cointegrating regression.” Then run ADF regression on the error term, test for unit root in

!

"t.

Caution: Significance values are non-standard (consult Hamilton for details) - Special case: If

!

" is known, problem reduces to testing for a unit root in

!

"t.

• Implications for economic dynamics: Error Corrections representation. - Standard strategy for estimating the dynamics of a vector of time series: Estimate a Vector Autoregression (VAR)

!

yt

xt

"

# $

%

& ' = a0 + A1 (

yt)1

xt)1

"

# $

%

& ' + ..+ Ak (

yt)k

xt)k

"

# $

%

& ' +

uyt

uxt

"

# $

%

& ' , in levels or in differences.

- If series are cointegrated, deviations from cointegrating relationship are “corrected” => have predictive power. Estimate VAR in first differences with estimate deviations

!

"yt"xt

#

$ %

&

' ( = a0 + A1 )

"yt*1"xt*1

#

$ %

&

' ( + ..+ Ak )

"yt*k"xt*k

#

$ %

&

' ( +

+y+x

#

$ %

&

' ( )(yt * ,xt )+

uyt

uxt

#

$ %

&

' (

Error corrections idea suggests

!

"y # 0 and

!

"x# 0. Captures economic relevance.