econometrics i 25

7/25/2019 Econometrics I 25

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Part 25: Time Series5-1/25

Econometrics I

Professor William Greene

Stern School of Business

Department of Economics


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Econometrics I

Part 25 Time Series


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Modeling an Economic Time Series

!ser"ed #$% #&% '% #t%'

What is the (sample) *andom sampling+

The (o!ser"ation ,indo,)


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Estimators

-unctions of sums of o!ser"ations

.a, of large num!ers+ /onindependent o!ser"ations

What does (increasing sample si0e) mean+

1s#mptotic properties+ There are no finite

sample properties34


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Interpreting a Time Series

Time domain: 1 (process) #t4 a6t4 7 !#t8&4 7 '

*egression li9e approachinterpretation

-re;uenc# domain: 1 sum of terms #t4


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-or e6ample%'


7/25Part 25: Time Series5-7/25

In parts'



Stud#ing the -re;uenc# Domain



1utocorrelation in *egression Yt = bxt + t

Cov(t, t-1) 0

Ex. RealConst= a + bRealIncome + t U.. !ata, "#a$te$l%,

1&'0-000



1utocorrelation

=o, does it arise+ What does it mean+ Modeling approaches

direct: corrective Estimation that accounts for autocorrelation Inference in the presence of autocorrelation

structural

Model the source Incorporate the time series aspect in the model



Stationar# Time Series

0t !t8&7 !2#t827 ' 7 !P#t8P7 et 1utoco"ariance: @9



Stationar# "s3 /onstationar# Series

T

-5.73

-1.15

3.43

8.01

12.59

-10.31

20 40 60 80 1000

YT ET

Va

ria

b

le



The .ag perator

.6t 6t8&

.26t 6t82

.P6t7 .H6t 6t8P7 6t8H

Pol#nomials in .: #t B.4#t7 et

1.4 #t et

In"erti!ilit#: #t A1.48&et



In"erting a Stationar# Series

#t #t8&7 et&8 .4#t et

#t A&8 .8&et et7 et8&7

2et827 '

Stationar# series can !e in"erted

1utoregressi"e "s3 mo"ing a"erage form of series

2 31 1 ( ) ( ) ( ) ...1

L L LL

= + + + +



*egression ,ith 1utocorrelation

#t 6t! 7 et% et et8&7 ut

&8 .4et utet &8 .48&ut

EAet EA &8 .48&ut &8 .4

8&EAut $

JarAet &8 .482JarAut &7

2u27 ' u

2&8 24



.S "s3 G.S

.S ?n!iased+



+----------------------------------------------------+

| Ordinary least squares regression |

| LHS=REALCONS Mean = 2999!"# || Auto$orrel %ur&in-'atson Stat = (92(!)( |

| R*o = $ore,e-./0 = 91"9#( |

+----------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|3aria&le | Coe44i$ient | Standard Error |t-ratio |5|6|7t0 | Mean o4 8|

+---------+--------------+----------------+--------+---------+----------+

Constant -)("1!!)) .!"(1)1.1 -1#. ((((

REAL%5 92.#)1# ((").1 2")(1! (((( ""!.!19)

| Ro&ust 3C Ne:ey-'est, 5eriods = .( |

Constant -)("1!!)) !.2"92.! -.92# (111

REAL%5 92.#)1# (.1("1.# #."(2 (((( ""!.!19)+---------------------------------------------+

| AR./ Model; et/ = r*o < et-./ + ut/ |

| inal >alue o4 R*o = 99))2 |

| ter= #, SS= ..)"#((, Log-L=-9!.".9.! |

| %ur&in-'atson; et/ = ((2!"# |

| Std %e>iation; et/ = !9(1#9.( |

| Std %e>iation; ut/ = 2!2(#92# |

| %ur&in-'atson; ut/ = .99!91 |

| Auto$orrelation; ut/ = ((212. |

| N(,.0 used 4or signi4i$an$e le>els |+---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|3aria&le | Coe44i$ient | Standard Error |&?StEr|5|@|70 | Mean o4 8|

+---------+--------------+----------------+--------+---------+----------+

Constant .(.9"2#)( !....1# 2!9 (."2

REAL%5 #"!2". ("9219" .#912 (((( ""!.!19)

RHO 99)).). (("!#""2 2))")9 ((((



Detecting 1utocorrelation

?se residuals Dur!in8Watson d

1ssumes normall# distri!uted distur!ances strictl#

e6ogenous regressors

Jaria!le addition Godfre#4 #t 6t 7 Kt8&7 ut

?se regression residuals etand test $1ssumes consistenc# of !3

2

2 1

2

1

( )2(1 )

T

t t t

T

t t

e er

e

=

=



1 ?nit *oot+

=o, to test for &+

B# construction: Kt> Kt8& 8 &4Kt8&7 ut

Test for @ 8 &4 $ using regression+ Jariance goes to $ faster than &T3 /eed a ne, ta!leL

cant use standard t ta!les3

Dic9e# > -uller tests

?nit roots in economic data3 1re there+4 /onstationar# series

Implications for con"entional anal#sis



*einterpreting 1utocorrelation

1

1 1 1 1

t

t

Regression form

' ,

Error Correction Form'( ) ( ' ) , ( 1)

' the equilibrium

The model describes d!ustment of " to equilibrium #hen

$ chnges.

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

= + = +

= + + =

=



Integrated Processes Integration of order P4 ,hen the Pth differenced

series is stationar#

Stationar# series are I$4 Trending series are often I&43 Then #t> #t8& #t

is I$43 AMost macroeconomic data series3

1ccelerating series might !e I243 Then#t> #t8&48 #t> #t8&4 2#tis I$4 AMone# stoc9 in

h#perinflationar# economies3 Difficult to find

man# applications in economics


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Di"ergent Series+

O bs erv.#

4.30

8.48

12.67

16.85

21.04

.12

20 40 60 80 1000

YT %T

Variable


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