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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
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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" $) %t is stationary. • Conclude: 1. Deterministic time trends do not add uncertainty. 2. Distinction between trend-stationary and difference-stationary processes is crucial for projections.
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General Stationary Processes: 1. Trend-stationary stochastic process with Moving Average representation:
!
yt = µ +"t + # jj=0$
% &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#
$ <# .
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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.
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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
!
" = # $1
- Phillips-Perron approach (PP): Estimate without lags, then compute corrected standard errors. • Theoretical problems: If
!
yt"1 is non-stationary, usual asymptotic theory does not apply.
- Critical values for
!
" depend on inclusion/exclusion of trend and constant => use Dickey Fuller Tables. • Caution about the interpretation: - Failure to reject a unit root does not prove a unit root; question of power - KPSS test for null hypothesis of stationarity available but less commonly used.
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Cointegration and Error Correction • Consider two or more difference-stationary series (y,x). • Sometimes a linear combination is stationary: Suppose
!
yt = " # xt +$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.