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Part 22: Time Series 22-1/37
Econometrics I Professor William Greene
Stern School of Business
Department of Economics
Part 22: Time Series 22-2/37
Econometrics I
Part 22 – Time Series
Part 22: Time Series 22-3/37
Modeling an Economic Time Series
Observed y0, y1, …, yt,…
What is the “sample?” Realization of the
entire sequence?
Random sampling? Not really possible.
We are using a different type of statistics.
The “observation window”
Part 22: Time Series 22-4/37
Estimators
Functions of sums of correlated observations
Law of large numbers?
Non-independent observations
What does “increasing sample size” mean?
Asymptotic properties? (There are no finite
sample properties.)
Part 22: Time Series 22-5/37
Interpreting a Time Series
Time domain: A “process”
y(t) = ax(t) + by(t-1) + …
Regression like approach/interpretation
Frequency domain: A sum of terms
y(t) =
Contribution of different frequencies to the observed
series.
(“High frequency data and financial econometrics
– “frequency” is used slightly differently here.)
( ) ( )j jjCos t t
Part 22: Time Series 22-6/37
For example,…
Part 22: Time Series 22-7/37
In parts…
Part 22: Time Series 22-8/37
Studying the Frequency Domain
Cannot identify the number of terms
Cannot identify frequencies from the time series
Deconstructing the variance, autocovariances
and autocorrelations
Contributions at different frequencies
Apparent large weights at different frequencies
Using Fourier transforms of the data
Does this provide “new” information about the series?
Part 22: Time Series 22-9/37
Stationary Time Series
yt = b1yt-1 + b2yt-2 + … + bPyt-P + et
Autocovariance: γk = Cov[yt,yt-k]
Autocorrelation: k = γk / γ0
Stationary series: γk depends only on k, not on t
Weak stationarity: E[yt] is not a function of t, E[yt * yt-s] is not a function of t or s, only of |t-s|
Strong stationarity: The joint distribution of [yt,yt-1,…,yt-s] for any window of length s periods, is not a function of t or s.
A condition for weak stationarity: The smallest root of the characteristic polynomial: 1 - b1z
1 - b2z2 - … - bPzP = 0, is greater
than one.
The unit circle
Complex roots
Example: yt = yt-1 + ee, 1 - z = 0 has root z = 1/ , | z | > 1 => | | < 1.
Part 22: Time Series 22-10/37
Stationary vs. Nonstationary Series
Part 22: Time Series 22-11/37
The Lag Operator
Lxt = xt-1
L2 xt = xt-2
LPxt + LQxt = xt-P + xt-Q
Polynomials in L: yt = B(L)yt + et
A(L) yt = et
Invertibility: yt = [A(L)]-1 et
Part 22: Time Series 22-12/37
Inverting a Stationary Series
yt= yt-1 + et (1- L)yt = et
yt = [1- L]-1 et = et + et-1 + 2et-2 + …
Stationary series can be inverted
Autoregressive vs. moving average form of series
2 311 ( ) ( ) ( ) ...
1L L L
L
Part 22: Time Series 22-13/37
Autocorrelation
How does it arise?
What does it mean?
Modeling approaches Classical – direct: corrective
Estimation that accounts for autocorrelation
Inference in the presence of autocorrelation
Contemporary – structural Model the source
Incorporate the time series aspect in the model
Part 22: Time Series 22-14/37
Regression with Autocorrelation
yt = xt’b + et, et = et-1 + ut
(1- L)et = ut et = (1- L)-1ut
E[et] = E[ (1- L)-1ut] = (1- L)-1E[ut] = 0
Var[et] = (1- L)-2Var[ut] = 1+ 2u2 + … = u
2/(1- 2)
Cov[et,et-1] = Cov[et-1 + ut, et-1] =
=Cov[et-1,et-1]+Cov[ut,et-1] = u2/(1- 2)
Part 22: Time Series 22-15/37
Autocorrelation in Regression
Yt = b’xt + εt
Cov(εt, εt-1) ≠ 0
Ex. RealConst = a + bRealIncome + εt U.S. Data, quarterly, 1950-2000
Part 22: Time Series 22-16/37
Generalized Least Squares
Efficient estimation of and, by implication, the inefficiency of least squares b.
= (X*’X*)-1X*’y*
= (X’P’PX)-1 X’P’Py
= (X’Ω-1X)-1 X’Ω-1y
≠ b. is efficient, so by construction, b is not.
β̂
β̂ β̂
Part 22: Time Series 22-17/37
Autocorrelation
t = t-1 + ut
(‘First order autocorrelation.’ How does this come about?)
Assume -1 < < 1. Why?
ut = ‘nonautocorrelated white noise’
t = t-1 + ut (the autoregressive form)
= (t-2 + ut-1) + ut
= ... (continue to substitute)
= ut + ut-1 + 2ut-2 + 3ut-3 + ...
= (the moving average form)
Part 22: Time Series 22-18/37
Autocorrelation
2
t t t 1 t 1
i
t ii=0
22i 2 u
u 2i=0
t t 1 t t 1 t
2
t t 1 t t 1 t
Var[ ] Var[u u u ...]
= Var u
=1
An easier way: Since Var[ ] Var[ ] and u
Var[ ] Var[ ] Var[u ] 2 Cov[ ,u ]
=
2 2
t u
2
u
2
Var[ ]
1
Part 22: Time Series 22-19/37
Autocovariances
t t 1 t 1 t t 1
t 1 t 1 t t 1
t-1 t
2
u
2
t t 2 t 1 t t 2
Continuing...
Cov[ , ] = Cov[ u , ]
= Cov[ , ] Cov[u , ]
= Var[ ] Var[ ]
=(1 )
Cov[ , ] = Cov[ u , ]
t 1 t 2 t t 2
t t 1
2 2
u
2
= Cov[ , ] Cov[u , ]
= Cov[ , ]
= and so on.(1 )
Part 22: Time Series 22-20/37
Autocorrelation Matrix
2 1
2
22 2 3
2
1 2 3
1
1
11
1
(Note, trace = n as required.)
Ω
Ω
T
T
u T
T T T
Part 22: Time Series 22-21/37
Generalized Least Squares
2
1 /2
2
1
2 11 /2
3 2
T T 1
1 0 0 ... 0
1 0 ... 0
0 1 ... 0
... ... ... ... ...
0 0 0 0
1 y
y y
y y
...
y
Ω
Ω y =
Part 22: Time Series 22-22/37
The Autoregressive Transformation
t t t t 1 t
t 1 t 1
t t 1 t t 1
t t 1 t
y u
y
y y ( ) + ( )
y y ( ) + u
(Note the first observation is lost.)
t
t-1
t t-1
t t-1
x 'β
x 'β
x x 'β
x x 'β
Part 22: Time Series 22-23/37
Unknown
The problem (of course), is unknown. For now, we will consider two methods of estimation:
Two step, or feasible estimation. Estimate first, then do GLS. Emphasize - same logic as White and Newey-West. We don’t need to estimate . We need to find a matrix that behaves the same as (1/n)X-1X.
Properties of the feasible GLS estimator
Maximum likelihood estimation of , 2, and all at the same time.
Joint estimation of all parameters. Fairly rare. Some generalities…
We will examine two applications: Harvey’s model of heteroscedasticity and Beach-MacKinnon on the first order autocorrelation model
Part 22: Time Series 22-24/37
Sorry to bother you again, but an important issue has come up. I am using
LIMDEP to produce results for my testimony in a utility rate case. I have a
time series sample of 40 years, and am doing simple OLS analysis using a
primary independent variable and a dummy. There is serial correlation
present. The issue is what is the BEST available AR1 procedure in LIMDEP
for a sample of this type?? I have tried Cochrane-Orcott, Prais-Winsten, and
the MLE procedure recommended by Beach-MacKinnon, with slight but
meaningful differences.
By modern constructions, your best choice if you are comfortable
with AR1 is Prais-Winsten. No one has ever shown that iterating it is better
or worse than not. Cochrance-Orcutt is inferior because it discards
information (the first observation). Beach and MacKinnon would be best, but
it assumes normality, and in contemporary treatments, fewer assumptions is
better. If you are not comfortable with AR1, use OLS
with Newey-West and 3 or 4 lags.
Part 22: Time Series 22-25/37
OLS vs. GLS
OLS
Unbiased?
Consistent: (Except in the presence of a lagged
dependent variable)
Inefficient
GLS
Consistent and efficient
Part 22: Time Series 22-26/37
The Newey-West Estimator
Robust to Autocorrelation
n 2
0 i i ii 1
L n
1 l t t l t t l t l tl 1 t l 1
l
Heteroscedasticity Component - Diagonal Elements
1e
n
Autocorrelation Component - Off Diagonal Elements
1w e e ( )
n
lw 1 = "Bartlett weight"
L 1
1Est.Var[ ]=
n
S x x '
S x x x x
Xb
1 1
0 1[ ]n n
'X X'XS S
Part 22: Time Series 22-27/37
Part 22: Time Series 22-28/37
Part 22: Time Series 22-29/37
Detecting Autocorrelation
Use residuals
Durbin-Watson d=
Assumes normally distributed disturbances strictly
exogenous regressors
Variable addition (Godfrey)
yt = ’xt + εt-1 + ut
Use regression residuals et and test = 0
Assumes consistency of b.
2
2 1
2
1
( )2(1 )
T
t t t
T
t t
e er
e
Part 22: Time Series 22-30/37
A Unit Root?
How to test for = 1?
By construction: εt – εt-1 = ( - 1)εt-1 + ut
Test for γ = ( - 1) = 0 using regression?
Variance goes to 0 faster than 1/T. Need a new table;
can’t use standard t tables.
Dickey – Fuller tests
Unit roots in economic data.
Nonstationary series
Implications for conventional analysis
Part 22: Time Series 22-31/37
Integrated Processes
Integration of order (P) when the P’th differenced
series is stationary
Stationary series are I(0)
Trending series are often I(1). Then yt – yt-1 = yt
is I(0). [Most macroeconomic data series.]
Accelerating series might be I(2). Then
(yt – yt-1)- (yt – yt-1) = 2yt is I(0) [Money stock in
hyperinflationary economies. Difficult to find
many applications in economics]
Part 22: Time Series 22-32/37
Cointegration: Real DPI and Real Consumption
Part 22: Time Series 22-33/37
Cointegration – Divergent Series?
Part 22: Time Series 22-34/37
Cointegration
X(t) and y(t) are obviously I(1)
Looks like any linear combination of x(t) and y(t) will also
be I(1)
Does a model y(t) = bx(t) + u(u) where u(t) is I(0) make
any sense? How can u(t) be I(0)?
In fact, there is a linear combination, [1,-] that is I(0).
y(t) = .1*t + noise, x(t) = .2*t + noise
y(t) and x(t) have a common trend
y(t) and x(t) are cointegrated.
Part 22: Time Series 22-35/37
Cointegration and I(0) Residuals
Part 22: Time Series 22-36/37
Reinterpreting Autocorrelation
1
1 1 1 1
t
t
Regression form
' ,
Error Correction Form
'( ) ( ' ) , ( 1)
' the equilibrium
The model describes adjustment of y to equilibrium when
x changes.
t t t t t t
t t t t t t t
t
y x u
y y x x y x u
x
Part 22: Time Series 22-37/37