1 Time series Introduction 2012 - Thus Spake VM · PDF file01-05-2011 · Enders, Walter (1995) Applied Econometric Time Series , John Wiley & Sons. Hamilton, James D (1994) Time Series

10/15/2012

1

15 October 2012Vijayamohan: CDS MPhil: Time Series 1 1

Time Series Time Series Time Series Time Series EconometricsEconometricsEconometricsEconometrics

1111

VijayamohananVijayamohananVijayamohananVijayamohananPillaiPillaiPillaiPillai NNNN


Essential Readings: Time series:

Enders, Walter (1995) Applied Econometric Time Series,John Wiley & Sons.

Hamilton, James D (1994) Time Series Analysis, PrincetonUniversity Press.

Hendry, David F. (1995) Dynamic Econometrics, OxfordUniversity Press.

Makridakis, S., Wheelwright, S. C. and McGee, V. E.(1983) Forecasting – Methods and Applications, Secondedition, John Wiley & Sons.

10/15/2012

2


Essential Readings: Time series:

My CDS Working Papers:

WP 312: Electricity Demand Analysis and Forecasting – TheTradition is Questioned!

WP 346: A Contribution to Peak Load Pricing: Theory andApplication


Essential Readings: Panel Data Analysis:

Baltagi, B. H. (2001) Econometric Analysis of PanelData, 2nd edition, John Wiley.

Cheng, Hsian (1986) Analysis of Panel Data,Cambridge University Press.

10/15/2012

3


A Time Series

Per Capita Net State Domestic

Product of Kerala at Constant (1993-94) Prices

(in Rs.)

1960 1965 1970 1975 1980 1985 1990 1995 2000

5000

6000

7000

8000

9000

10000

1960 1965 1970 1975 1980 1985 1990 1995 2000

5000

6000

7000

8000

9000

10000

Per Capita Per Capita Year NSDP (Rs.) Year NSDP (Rs.)1960-61 4313.79 1980-81 5392.591961-62 4214.82 1981-82 5404.591962-63 4206.08 1982-83 5472.791963-64 4269.36 1983-84 5186.761964-65 4286.69 1984-85 5438.951965-66 4441.45 1985-86 5566.171966-67 4476.00 1986-87 5367.131967-68 4607.29 1987-88 5523.341968-69 4976.05 1988-89 5993.691969-70 5006.78 1989-90 6305.481970-71 5175.36 1990-91 6683.071971-72 5352.05 1991-92 6830.411972-73 5391.20 1992-93 7324.321973-74 5272.47 1993-94 8063.361974-75 5235.12 1994-95 8780.751975-76 5365.44 1995-96 9149.741976-77 5211.88 1996-97 9494.411977-78 5194.39 1997-98 9517.951978-79 5262.51 1998-99 10024.561979-80 5410.00 1999-00 10502.18


The modern era in time series started in 1927 with George Udny Yule (Scottish statistician: 1871-1951)

"On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer'sSunspot Numbers", Philosophical Transactions of the Royal Society of London, Ser. A, Vol. 226 (1927), pp. 267–298.

10/15/2012

4


Yule showed such a series can be better described as a function of its past values.

Thus he introduced the concept of autoregression, even though he restricted himself to an order of four or fewer terms.


Yule's approach was extended by Sir Gilbert Thomas Walker ((((British physicist and statistician: British physicist and statistician: British physicist and statistician: British physicist and statistician: 1868186818681868----1958)1958)1958)1958) ::::

defined (1931) the general autoregressive scheme.

Gilbert Walker Gilbert Walker Gilbert Walker Gilbert Walker On Periodicity in Series of Related On Periodicity in Series of Related On Periodicity in Series of Related On Periodicity in Series of Related Terms,Terms,Terms,Terms, Proceedings of the Royal Society of Proceedings of the Royal Society of Proceedings of the Royal Society of Proceedings of the Royal Society of

London, Ser. A, Vol. 131, (1931) 518London, Ser. A, Vol. 131, (1931) 518London, Ser. A, Vol. 131, (1931) 518London, Ser. A, Vol. 131, (1931) 518--------532.532.532.532.

Evgeny Evgenievich (or Eugen) Slutsky(Russian Statistician, Economist:(Russian Statistician, Economist:(Russian Statistician, Economist:(Russian Statistician, Economist: 1880188018801880----1948)1948)1948)1948)

presented (1937) the moving-average scheme.

10/15/2012

5


Herman Ole Andreas Wold(1908- 1992)

(Swedish Statistician)

provided (1938; 1954)the theoretical foundation of combined ARMA processes.


George Edward Pelham Box and Gwilym Meirion Jenkins (1970; 1976) codified the applied univariate time series ARIMA modelling

Box, George and Jenkins, Gwilym(1970) Time series analysis: Forecasting and control, San Francisco: Holden-Day.

10/15/2012

6


Stationarity

A time series is stationary

if it fluctuates if it fluctuates if it fluctuates if it fluctuates around a constant around a constant around a constant around a constant mean. mean. mean. mean.

A nonnonnonnon----stationarystationarystationarystationary series

includes a longer-term secular trend.


Stationarity

But, a large number of actual time series

are not stationary;

however, not an insolvable problem,

several methods to transform a non-stationary series into a stationary one.

10/15/2012

7


YEAR MONTHS WPI All Commodities

1971-72 april 101.7may 102.3june 104.7july 106august 107.2september 107.5october 106.6november 105.1december 103.9january 107february 107march 108.1

1972-73 april 108.7may 109.7….. ……….. ……

1987-88 april 381.2may 390.3june 394july 400.6august 409.6september 408.9october 409.5november 411.1december 410.4january 416february 415.8march 417.6

Wholesale Price Index

(All Commodities)


Notation for the time series

� A time series variable: yt.

Yt = the random variable Y at period t.

�A time series: (yt, yt−1, yt−2, . . . , y1, y0).

� T = standard number of

observations in a time series.

10/15/2012

8


Time series process

a stochastic process

A sequence of random variables

ordered in time.

Random (stochastic) variable:

takes values in a certain range

with probabilities,

specified by a pdf.


Time series: Stochastic Difference Equation.

Difference equations:

Mathematics of Economic Dynamics.

deal with time as a discrete variable –changing only from period to period –

DE expresses the value of a variable as a function of its own lagged values, time and other variables:

yt = a +b yt – 1:

a linear first order difference equation.

10/15/2012

9


If a random variable, ut , added,

yt = a + b yt–1 + ut :

Stochastic DE : Time Series.

First order Autoregressive, AR(1), process.

ut : equilibrium error, disturbance,

: information shock →→→→ ‘innovation’ →→→→

the only new information to yt.

: assumed to be a ‘white noise process’.

Noise, white noise: from signal theory, physics


Noise

Filter

Noise

Signal (Input)

Output

Noise? Disturbance : Error

Filtering =

Decomposing output into input (signal) and noise.

10/15/2012

10


Signal (Input)

Output

Noise

Filter

Noise

FilteringFilteringFilteringFiltering = = = = DecomposingDecomposingDecomposingDecomposing output output output output into into into into input (input (input (input (signalsignalsignalsignal) and ) and ) and ) and noisenoisenoisenoise....

Time series = Signal (input) + Noise;

= Regression function + Noise.

Filters: Regression; Auto Regression; Moving Average; ….


White noise

analogous to white light which contains all frequencies.

a random signal (or process) with a flat power spectral density.

The signal's power spectral density has equal power in any band.

10/15/2012

11


Inte

nsit

y (d

b)

Frequency (Hz)

White NoiseCalculated spectrum of a generated approximation of white noise

An example realization of a

white noise process.


A flat power spectral density =

Zero mean

Constant variance

Zero autocovariance = No autocorrelation.

A white noise process

10/15/2012

12


So, in yt = a + b yt–1 + ut :

ut : White noise

(a) E(ut) = E(ut - 1) = …. = 0

(b) Var(ut) = Var(ut - 1) = …. = σ2

(c) Cov(ut; ut–k ) = Cov(ut–s; ut–s–k) = …. = 0,

for all k and s; k ≠≠≠≠ 0:

‘No memory’.

(d) ut is normally distributed. (This assumption

not essential.)


White noise process = stationary process

Not vice versa!

For a stationary process,

Mean, variance, and covariance : all constant;

independent of t:

Termed covariance stationarity.

(or second order,

weak,

wide sense stationarity).

10/15/2012

13


Stationarity in two senses:

1. First-order (Strict) Stationarity

2. Second-order (Weak) Stationarity



The time series process Xt is completely (strictly)

stationary, if its joint pdf is not affected by a time

translation.

i.e., at whatever point in time it starts, a sample of

T successive observations will have the same pdf

for all t :

Time invariant probabilistic structure.

The process is in ‘stochastic equilibrium’.

10/15/2012

14



Formally:

The process {X(t)} is strictly stationary

if, for any admissible t1, t2, …, tn, and

any interval k,

the joint pdf of {X(t1), X(t2), …, X(tn)} is

identical with the joint pdf of

{X(t1+k), X(t2+k), …, X(tn+k)}.



A process Yt is weakly stationary if :

1. E(Yt) = (constant mean )

2. Cov(Yt, Yt-k) = (depends only on k not on t)

For k = 0, the second condition implies

a constant .

i.e., the first and second order moment structure of Yt is

constant over time .

∞∞∞∞<<<<µµµµ

∞∞∞∞<<<<kγγγγ

2)( σσσσ====tYVar

10/15/2012

15



i.e., the first and second order moment structure of Yt

is constant over time .

‘Wide-sense stationarity’ or ‘ covariance stationarity’ :

No constant mean condition is too restrictive for

most economic time series ( having clear trends ).

We restrict our attention to weak stationarity and

use `stationarity' to mean weak stationarity .


In

yt = a + b yt–1 + ut ;

ut : White noise

But yt may NOT be stationary.

Depends on the magnitude of

b, the root of DE.

10/15/2012

16


Consider a DE: yt = b yt–1 ;

Its solution, time path:

yt = y0 bt ;

Nature of the time path yt,

whether it converges or not as t → ∞,

depends on the sign and magnitude of b.

Consider the following cases, with y0 = 1:

yt = bt .


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

b > 1| b |>1

2t 1 2 4 8 16 32 64

10/15/2012

17


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

b = 1| b | = 1

1t 1 1 1 1 1 1 1


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

b < 1| b |<1

(1/2)t 1 1/2 1/4 1/8 1/16 1/32 1/64

10/15/2012

18


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

-1< b < 0| b | < 1

(-1/2)t 1 -1/2 1/4 -1/8 1/16 -1/32 1/64


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

b = -1| b | = 1

-1t 1 -1 1 -1 1 -1 1

10/15/2012

19


Value of b bt

Time path of yt

t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

b < -1| b |>1

-2t 1 -2 4 -8 16 -32 64


Thus given a first order DE, yt – byt–1 = 0,

Its time path yt may be

ocillatory,

or non-oscillatory,

converging(damped)

or diverging (explosive)

What are the conditions?

10/15/2012

20


Time path yt will be

ocillatory, if b < 0, (negative root) and

non-oscillatory, if b > 0 (positive root).



converging, if 0 < b < 1, or −−−−1 < b < 0,

that is, −−−−1 < b < 1, or | b | < 1,

and

Non-converging, if b ≥≥≥≥ 1, or b ≤≤≤≤ −−−−1,

that is, −−−−1 ≥≥≥≥ b ≥≥≥≥ 1, or | b | ≥≥≥≥ 1.

10/15/2012

21


To recap,

Negative root ⇒⇒⇒⇒ Oscillatory

Positive root ⇒⇒⇒⇒ Non-oscillatory

Absolute value of the root < 1 ⇒⇒⇒⇒

Convergence.

Absolute value of the root ≥≥≥≥ 1 ⇒⇒⇒⇒

Non-convergence.

So, with unit root, |b| = 1,

No convergence: yt unstable; non-stationary.


With unit root, b = 1, yt = yt–1 + ut ;

Its time path yt = ut + ut−−−−1 + …. = ΣΣΣΣui

for i = 1, 2, …, t;

cumulation (from past to the present) of all

random shocks: stochastic trend

That is, the shock persists;

the process is non-stationary:

unit root problem = non-stationarity problem

Non-stationarity :

10/15/2012

22


Thus with unit root , b = 1 :

E(yt) = E(ΣΣΣΣ ui) = 0, but

var(yt) = var(ΣΣΣΣui) = tσσσσu2, for i = 1, 2, .., t,

cov(yt yt + k) = E{ΣΣΣΣui ΣΣΣΣui + k ) = tσσσσu2,

both functions of time.

Non-stationarity

yt = byt–1 + ut = ∑∑∑∑−−−−

====−−−−

1

0

t

iit

iub



converging if −−−−1 < b < 1, or | b | < 1.

yt stationary

Stationarity

10/15/2012

23


With | b | < 1 in yt = byt–1 + ut ;

1. E(Yt) = 0 ( = a, with an intercept)

2. Var(Yt) = σu2 /(1– b2)

3. Cov(Yt; Yt-k) = bk σu2 /(1– b2):

all constant;

independent of t:

yt : stationary process.

Stationarity :


A Stationary Time Series

White noise ut

Stationary Processes: Some Examples:

10/15/2012

24


Stationary Processes: Some Examples:

1. White Noise (Purely random process):

Simplest form of a time series.

The white noise process is a zero mean, constant variance collection of random variables which are uncorrelated over time.


1. White Noise (Purely random process):

Yt is a white noise process if Yt = εt, where:

(a) E(εt) = 0 (zero mean assumption)(zero mean assumption)(zero mean assumption)(zero mean assumption)

(b) Var(εt) = σ2 (constant variance assumption)(constant variance assumption)(constant variance assumption)(constant variance assumption)

(c) Cov(εt; εt±±±±k) = 0 for all k; t; k ≠≠≠≠ 0: (independence of (independence of (independence of (independence of errors assumption): errors assumption): errors assumption): errors assumption): ‘No memory’ .

(d) εt, are normally distributed. are normally distributed. are normally distributed. are normally distributed. (This assumption not essential.)(This assumption not essential.)(This assumption not essential.)(This assumption not essential.)

0 50 100 150 200 250 300

-3

-2

-1

0

1

2

0 50 100 150 200 250 300

-3

-2

-1

0

1

2

10/15/2012

25


A Stationary Time SeriesAR(1)

yt = 0.6yt–1 +ut

A Stationary Time Series

MA(1)yt = ut + 0.9ut–1


Time series: approximated to

Autoregressive (AR) process

Moving Average (MA) process

Combination of AR and MA process – ARMA.

AR(1) process: yt = a + b yt–1 + ut .

MA(1) process: yt = ut + d ut – 1 .

ARMA(1, 1) process:

yt = a + b yt–1 + ut + d ut – 1 .

Stationary processes

10/15/2012

26

15 October 2012 Vijayamohan: CDS MPhil: Time Series 1

51

Documents

1 Time series Introduction 2012 - Thus Spake VM · PDF file01-05-2011 · Enders, Walter (1995) Applied Econometric Time Series , John Wiley & Sons. Hamilton, James D (1994) Time Series