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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