1

Time Series Analysis

Definition of a Time Series processAR, MA, ARMA, ARIMAVector Autoregression

Impulse ResponseForecasting

2

tttttty 2,0~ NIDt

tttt 11 2,0~ NIDt

ttt 1 2,0~ NIDt

tsttt 11 .. 2,0~ wt NID

*,

,*

1,

1,*,

,

cossin

sincos

tj

tj

tj

tj

jj

jj

tj

tj

Tt

sj

,....,1

2/,...,1

**

1

1,* cossin

sincos

t

t

t

tj

cc

cc

t

t

Tt ,....,1

10

ttvt 1 2,0~ NIDt

Four Components of a Time SeriesTime Series

Trend

Season

Cycle

Random

(refer STAMP manual p.140)

3

tvtyty 1

121 tvtyty

332 tvtyty

443 tvtyty

. … …

ntvntynty 1 (5)

From substitution

tvtyty 1 =

tvtvtv

tyty tt vy

122

12

tvtv

tyty

23

tvtvty tttttt vvyvvy 122

33

1232

Iterative Substitution in AR(1) Model

4

tvtvtvtyntn

ntyn

ty 12

23

3...11

(6)

In the limit the term ntn y becomes close to zero as

n .

Rearranging (6) we can write ty in terms of current

and past values of error terms

ntn

tvtvtvtvtvty 1...4

43

32

21

(7)

AR(1) Time Series as a Function of Past Innovations (Impulses or Shocks)

5

What is the mean of ty in (7)?

ntEn

tvE

tvE

tvE

tvEtvEtyE 1...

44

33

22

1

Because of assumption

2,0~ vNtv ; 0

tyE

What is the variance of ty ?

ntn

tv

tv

tv

tvtvVartyVar 1...

44

33

22

1if 1

and

if there is no autocorrelation among the random terms 01

tvtvE

2.2.....222vtvvvvtyVar

Thus the variance of Y term increases with time. This makes this series non stationary.

Rule of thumb : A series is non-stationary if 1 .

A series is stationary if 1 .

Time Dependent Variance

6

tvtyty 1

tvtyty 11 ;

Random Walk:

tvtyty 1

Random Walk with a drift (intercept):

tvtyty 10

Trend stationary process

tvtytty 110

Augmented Dicky Fuller Test

tvm

iityiatytty

1

110

Dicky-Fuller and Augmented Dicky-Fuller Tests

Null hypotheses: There is unit root and time series in non-stationary=0 (1-)=0Alternative hypothesis:There is no unit root and time series is stationary <0 (1-)<0 <1

7

11 tetetY tyE

112)11var(var etetety

12)1)(()1cov( etytyEtYtY

A u t o c o r r e l a t i o n f u n c t i o n : i t t a p e r s o f f a f t e r k l a g s

2112

21

var1cov

e

e

tytyty

k

S o m e e x a m p l e s o f M A ( 1 ) p r o c e s s :

18.0 tetetY

18.0 tetetY

Moving Average-MA Process

8

211 2 tetetetY tyE

2

221

12)211var(var 2 etetetety

211

2)1)(()1cov( etytyEtYtY

2)2)(()2cov( 2 etytyEtYtY 0)3cov( tYtY

22

211

211

var1cov

1

tytyty

22

211

1var

2cov2

tytyty ; 0k

M A ( 2 ) p r o c e s s h a s t o w p e r i o d l o n g m e m o r y .

MA(2) Process

9

tetytY 11

ktyEtyEtyE .....1

tetyEYE y 11

1 ; 1

1

tetyty 11varvar = >

211

22

ey

21)1)(()1cov( ytyEtytyEtyEtYtY

S om e exam ples:

tetytY 18.0

tetytY 18.0

C onvergence occurs if 11 . T he series is ca lled

sta tionary .

Autoregressive Process

10

1111 tetetytY

2

11112var tetetyEtyEty

=

11112221

20

21 tetyEee

ARMA(1,1) Process

11

If two economic variables have long-run equilibrium relationship linear combination of these variables may be stationary even if the individual series may be non stationary. These two variables are said to be co-integrated to each other.

Suppose ty is consumption and tX is disposable income.

tXtYte 21

Even if ty and tX are I(1) te is I(0).

tvt

ete 10

If is zero then series te is stationary and ty and tX are

I(1).

Co-integration

12

tvtXtYty

2121

The term in the parenthesis is the error term and thecoefficient 2 governs the speed of adjustment towardslong-run equilibrium.

Error Correction Model

13

t

P

jjtmmj

r

jjtj

P

jjtnnj

P

jjtjt exbxbyayaay 1

1,1

1,111

1,1

1,11110,1 ....

.

.

.

nt

P

jjtmmj

r

jjtj

P

jjtnnnj

P

jjtjnntn exbxbyayaay

1,1

1,111

1,

1,110, ....

Structure of a VAR Model

tttttt exbxbyayaay 12121112,212111110

tttttt exbxbyayaax 22221,1212,222112120

Simple Example

14

t

t

t

t

t

t

t

t

e

e

x

x

bb

bb

y

y

aa

aa

x

y

2

1

2

11

2221

1211

22

11

2221

1211

t

t

t

t

t

t

t

t

e

e

x

x

y

y

bb

bb

aa

aa

x

y

2

1

2

11

22

11

2221

1211

2221

1211

UAIBXAIY 11

UBIAYBIX 11

1

00Y

1

011 AIY

1

0111

12 AIAIYAIY

Impulse Response Analysis in a VAR Model

15

Estimation sample is 1971. 2 - 2000. 1. (T = 116, n = 111).Log-Likelihood is 250.781 (-2 LogL = -501.563).Prediction error variance is 0.0106071Summary statistics ER Std.Error 0.10299 Normality 9.9490 H( 37) 0.58124 r( 1) 0.0039775 r( 9) -0.10584 DW 1.9721 Q( 9, 6) 7.2307 Rs^2 -0.41360ER = Trend + Trigo seasonal + Expl vars + Irregular

Eq 3 : Estimated coefficients of final state vector.Variable Coefficient R.m.s.e. t-valueLvl 1.2519 0.26875 4.6583 [ 0.0000]Slp -0.0056233 0.0082428 -0.68221 [ 0.4965]Sea_ 1 0.0026817 0.0081813 0.32779 [ 0.7437]Sea_ 2 0.0017017 0.0081876 0.20784 [ 0.8357]Sea_ 3 -0.00036460 0.0040829 -0.089298 [ 0.9290]Eq 3 : Estimated coefficients of explanatory variables.Variable Coefficient R.m.s.e. t-valueER_1 0.22213 0.096664 2.298 [ 0.0234]ER_2 -0.070019 0.099049 -0.70692 [ 0.4811]ER_3 0.027285 0.099054 0.27545 [ 0.7835]ER_4 0.035090 0.096723 0.36279 [ 0.7174]Eq 3 : Seasonal analysis (at end of period).Seasonal Chi^2( 3) test is 0.173462 [0.9818]. Seas 1 Seas 2 Seas 3 Seas 4 Value 0.0023171 0.0020663 -0.0030463 -0.0013371

Stamp Program for Time Series Analysis

16

1985 1990 1995 2000 2005

1.25

1.50

1.75

2.00F-ER Forecast

1985 1990 1995 2000 2005

1.25

1.50

1.75

2.00F-ER F-TrendX_ER

1985 1990 1995 2000 2005

-0.002

0.000

0.002 F-Seas_ER

Forecasting of the Exchange Rate

17

References• Burns, A and W. Michell, (1946), “Measuring Business Cycles” NBER, New York.• Campbell J. Y. and R.J. Shiller (1987) Cointegration and Tests of Present Value Models, Journal of Political Economy, 95, 5, pp. 1062-1087.• Cooley and Thomas F. and S.F. LeRoy (1985) Atheoretical Macroeconometrics, Journal of Monetary Economics, North Holland 16: 283-308.• Dickey D.A. and W.A. Fuller (1979) Distribution of the Estimator for Autoregressive Time Series with a Unit Root, Journal of the American Statistical

Association, June. • Dickey D.A. and W. A. Fuller (1981) Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrics, 49:4 July, 1057-1071.• Engle R E and C.W.J. Granger (1987) Co-integration and Error Correction: Representation, Estimation and Testing. Econometrica, vol. 55, No. 2, pp.

251-276.• Enders W. (1995) Applied Econometric Time Series, John Wiley and Sons • Fair R.C.(1984) Specification, Estimation, and Analysis of Macroeconomic Models, Harvard.• Garratt A., K. Lee, M.H. Pesaran and Y. Shin (2003) A Structural Cointegration VAR Approach to Macroeconometric Modelling, Economic Journal.• Cooly Thomas F (1995) Frontiers of Business Cycle Research, Princeton.• Doornik J.A and D.F. Hendry (2003) Econometric Modelling Using PCGive Volumes I, II and II, Timberlake Consultant Ltd, London.• Hendry D.F. (1997) Dynamic Econometrics, Oxford University Press.• Harris R. and R. Sollis (2003) Applied Time Series Modelling and Forecasting, John Willey. • Holly S and M Weale Eds.(2000) Econometric Modelling: Techniques and Applications, pp.69-93, the Cambridge University Press.• Johansen Soren (1988) Estimation and Hypothesis Testing of Cointegration Verctors in Gaussian Vector Autoregressive Models, Econometrica, 59:6,

1551-1580.• Johansen Soren (1988) Statistical Analysis of Cointegration Vectors, Journal of Economic Dynamics and Control 12 231-254, North Holland.• Nelson C. R. and C. I. Plosser (1982) Trends and Random Walks in Macroeconomic • Time Series: Some Evidence and Implications, Journal of Monetary Economics.• Pagan A. and M. Wickens (1989) A Survey of Some Recent Econometric Methods, Economic Journal, 99 pp. 962-1025.• Phillips P.C.B. (1987) Time Series Regression with an Unit Root, Econometrica, vol. 55, No. 2, 277-301.• Prescott, E.C. (1986), “Theory Ahead of Business Cycle Measurement,” Federal Reserve Bank of Minneapolis, Quarterly Review; Fall. • Pindyck R.S and Robinfeld D.L. (1998) Econometric Models and Economic Forecasts, 4th edition, McGraw Hill.• Quah, D.T., (1995), “Business Cycle Empirics: Calibration and Estimation,” The Economic Journal 105 (November) 1594-1596• Sims Christopher A (1980) Macroeconomics and Reality, 48:1 January, pp. 1-45. • Wallis KF. (1989) Macroeconomic Forecasting: A Survey, Economic Journal, 99, March, pp 28-61.• Wallis Kenneth (1980) Econometric Implications of the Rational Expectations • Hypothesis, Econometrics 48:1, pp, 48-71.