Time Series Econometrics, Lecture notes of Introduction to Econometrics

Download Time Series Econometrics and more Lecture notes Introduction to Econometrics in PDF only on Docsity!

Session 3: Time Series

Econometrics

Prepared by Ziyodullo Parpiev, PhD

for Regional Summer School

September 21, 2016

Outline

1. Stochastic processes

2. Stationary processes

3. Nonstationary processes

4. Integrated variables

5. Random walk models

6. Unit root tests

7. Cointegration and error correction models

Some monthly U.S. macro and financial time

series

STOCHASTIC PROCESSES

• A random or stochastic process is a collection of random variables ordered

in time.

• If we let Y denote a random variable, and if it is continuous, we denote it as

Y ( t ), but if it is discrete, we denoted it as Yt. An example of the former is an

electrocardiogram, and an example of the latter is GDP, PDI, etc. Since most

economic data are collected at discrete points in time, for our purpose we

will use the notation Yt rather than Y ( t ).

• Keep in mind that each of these Y’s is a random variable. In what sense can

we regard GDP as a stochastic process? Consider for instance the GDP of

$2872.8 billion for 1970–I. In theory, the GDP figure for the first quarter of

1970 could have been any number, depending on the economic and

political climate then prevailing. The figure of 2872.8 is a particular

realization of all such possibilities. The distinction between the stochastic

process and its realization is akin to the distinction between population

and sample in cross-sectional data.

Stationary Stochastic Processes

(i) mean: E(Y

t

) = μ

(ii) variance: var(Y

t

) = E( Y

t

• μ)

= σ

(iii) Covariance: γ

k

= E[(Y

t

• μ)(Y

t-k

• μ)

where γk , the covariance (or autocovariance) at lag k , is the covariance

between the values of Yt and Yt + k , that is, between two Y values k

periods apart. If k = 0, we obtain γ 0, which is simply the variance of Y (=

σ 2); if k = 1, γ 1 is the covariance between two adjacent values of Y , the

type of covariance we encountered in Chapter 12 (recall the Markov

first-order autoregressive scheme).

Forms of Stationarity: weak, and strong

Stationarity vs. Nonstationarity

• A time series is stationary, if its mean, variance, and autocovariance (at

various lags) remain the same no matter at what point we measure them;

that is, they are time invariant.

• Such a time series will tend to return to its mean (called mean reversion )

and fluctuations around this mean (measured by its variance) will have a

broadly constant amplitude.

• If a time series is not stationary in the sense just defined, it is called a

nonstationary time series (keep in mind we are talking only about weak

stationarity). In other words, a nonstationary time series will have a time-

varying mean or a time-varying variance or both.

• Why are stationary time series so important? Because if a time series is

nonstationary, we can study its behavior only for the time period under

consideration. Each set of time series data will therefore be for a particular

episode. As a consequence, it is not possible to generalize it to other time

periods. Therefore, for the purpose of forecasting, such time series may be

of little practical value.

AU STR ALIA

C AN AD A

C H IN A

GER MAN Y

H ON GKON G

J APAN

KOR EA

MALAYSIA

SIN GAPOR E

TAIW AN

U K

U SA

Examples of Non-Stationary Time Series

“Unit Root” and order of integration

If a Non-Stationary Time Series Y

t

has to be

“differenced” d times to make it stationary,

then Y

t

is said to contain d “Unit Roots”. It is

customary to denote Y

t

~ I(d) which reads “ Y

t

is integrated of order d”

If Yt ~ I(0), then Y t is Stationary

If Yt ~ I(1), then Zt = Yt – Yt- 1 is Stationary

If Yt ~ I(2), then Zt = Yt – Yt- 1 – ( Yt – Yt- 2 )is Stationary

Unit Roots

• Consider an AR(1) process:

yt = a 1 yt- 1 + εt (Eq. 1)

t

~ N(0, σ

• Case #1: Random walk (a

y

t

= y

t- 1

t

Δy

t

t

Unit Roots

• H

: δ = 0 (there is a unit root)

• H

A

: δ ≠ 0 (there is not a unit root)

• If δ = 0, then we can rewrite Equation 2 as

Δy

t

= ε

t

Thus first differences of a random walk time series are

stationary, because by assumption, ε

t

is purely

random.

In general, a time series must be differenced d times to

become stationary; it is integrated of order d or I ( d).

A stationary series is I(0). A random walk series is I(1).

Tests for Unit Roots

• Dickey-Fuller test

• Estimates a regression using equation 2

• The usual t-statistic is not valid, thus D-F developed appropriate

critical values.

• You can include a constant, trend, or both in the test.

• If you accept the null hypothesis, you conclude that the time series

has a unit root.

• In that case, you should first difference the series before

proceeding with analysis.