time series chap21 (2)

8/4/2019 Time Series Chap21 (2)

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Time Series Data Analysis

Main Reading: Gujarati, Chapter 21.

Hamid Ullah


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Key Concepts

Stationarity and non-stationarity.

Autoregressive and moving average

processes.

Unit roots.

Dickey fuller test.

Cointegration and spurious regressions.

Testing for cointegration.


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Univariate Time Series Models

Aim to describe the behaviour of a

variable in terms

of its past values

Why use these models?

SimpleLack of theoryUseful for forecasting


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Autoregressive processes:

An AR(1) process is written as

yt = yt-1 + t

where t ~ IID(0,2)

ie. the current value of yt is equal to times its previous

value plus an unpredictable component t

This can be extended to an AR(p) process

yt = 1yt-1 + 2yt-2+..+pyt-p + t

where t ~ IID(0,2)

ie. the current value of yt depends on p past values plus

an unpredictable component t


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Moving Average processes:

A MA(1) process is written as

yt = t + t-1

where t ~ IID(0,2)

ie. the current value is given by an unpredictable

component t and times the previous periods error

This can also be extended to an MA(q) process

yt = t + 1t-1 +..+ qt-q

where

t ~ IID(0,

2

)


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STOCHASTIC PROCESSES

A random or stochastic process is acollection of random variables ordered in

time and is often denoted by Yt, where t =1,,T (where the subscript t representstime).

Time series data has a temporal ordering,unlike cross-section data.


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What is Stationarity?

A stochastic process is said to be stationary if its

mean and variance are constant over time and the

value of the covariance between the two time

periods depends only on the distance or gap

between the two time periods and not the actual

time at which the covariance is computed


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StationarityStationarity may be strong or weak (covariance)

E(yt) is independent of time;

Var(yt) is a finite, positive constant and independent

of time;

Cov(yt, yk) is a finite function of t-k, but not of t or k

The whole distribution of the

variable does not depend on time


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Thus, this weaker form of Stationarity

requires only that the mean and variance are

constant across time, and the covariancebetween the two time periods depends only

on the distance or gap or lag between the

two time periods and not the actual time atwhich the covariance is computed.

if a time series is stationary, its mean,

variance, and auto-covariance (at variouslags) remain the same no matter at what

point of time we measure them; that is, they

are time invariant.


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Non Stationarity

a non stationary time series will have a time-varying mean or a time-varying variance or both.

our interest is in stationary time series, one oftenencounters non-stationary time series, the classic

example being the random walk model (RWM).

stock prices or exchange rates, follow a random

walk; that is, they are non stationary.


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Non-stationary TS

Non-stationary series can be due todeterministic trend

tt etY

ttteYY

1

or stochastic trend (has a unit root)

or both!


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Random Walks

A random walk is an AR(1) model wherer1 = 1,

meaning the series is not weakly dependent.

With a random walk, the expected value ofyt is aconstant (it doesnt depend on t).

Var(yt) = e2t, so it increases with t.

We say a random walk is highly persistent sinceE(yt+h|yt) =ytfor all h 1


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Types of random walks

There are two types of RWM

(1) Random walk without drift (i.e., no constant

or intercept term)

(2) random walk with drift (i.e., a constant term is

present.)


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Random Walk without Drift.

The value of Y at time t is equal to its value attime (t 1) plus a random shock; thus it is

an AR(1) model.

Yt = Yt1 + ut Efficient capital market hypothesis argue that stock prices

are essentially random and therefore there is no scope for

profitable speculation in the stock market: If one could

predict tomorrows price on the basis oftodays price, wewould all be millionaires.


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Y1 = Y0 + u1

Y2 = Y1 + u2 = Y0 + u1 + u2

Y3 = Y2 + u3 = Y0 + u1 + u2 + u3

In general, if the process started at some time 0 with a

value of Y0, we have

Yt = Y0 + ut

the mean of Y is equal to its initial, or starting, value,which is constant, but as t increases, its variance increases

Indefinitely, thus violating a condition of stationarity.

In short, the RWM without drift is a non stationary

stochastic process. (Yt Yt1) = Yt = ut

its first difference is stationary. In other words, the first

differences of a random walk time series are stationary. So

it is Difference Stationarity Process (DPS)


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Random Walk with Drift.Yt = + Yt1 + ut

where is known as the drift parameter. The name drift comes

from the fact that if we write the preceding equation as

it shows that Yt drifts upward or downward, depending on

being positive or negative. Note that above model is also anAR(1) model. var (Yt ) = t2

RWM with drift the mean as well as the variance increases over

time, again violating the conditions of (weak) Stationarity.

The above model is a DSP process because the non Stationarity in

Yt can be eliminated by taking first differences of the time series.


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Trending Series The general tendency of the time series data to increase or

decrease during a long period of time is called the seculartrend or long term trend or simply trend.

The concept of trend does not include short range

oscillations.

Trend may have either upward or downward movement,such as production, prices, income are upward trend while a

downward trend is noticed in the time series relating to

deaths, epidemics etc.

A trending series cannot be stationary, since the mean ischanging over time.

If a series is weakly dependent and is stationary about its

trend, we will call it a trend-stationary process.


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Types of TrendsLinear or Straight Line Trend:

If we get a straight line, when the values are plotted on a graph,then it is called a linear trend.

yt = 0 + 1t + et, t = 1,2,

Non-Linear Trend:

If we get a curve after plotting the time series values then it is

called non-linear or curvilinear trend.

a. Another possibility is an exponential trend, which can be

modeled as

log(yt) = 0 + 1t + et, t = 1,2,

b. Another possibility is a quadratic trend, which can be modeled

as

yt = 0 + 1t + + 2t2 + et, t = 1,2,


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Unit Root

We can write the AR(1) model as follows:

More precisely, if is 1 [a random walk], we facewhat is known as a unit root problem. In fact, this isexactly the random walk (without drift) described

above. Note, the variance is not stationary.


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Non-stationary to stationary

Add a trend (i.e. make it trend stationary)

Difference the series (i.e. make it stochastic

difference stationary) Stationary series are I(0) i.e. integrated of order 0.

For I(1) series, difference once to make stationary; For I(2)

series, difference twice to make stationary, etc.

Or, add a trend as well as difference of the series


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Graphical test for Stationarity

One simple test of stationarity is based on the so-

called autocorrelation function (ACF). The ACF at

lag k, denoted by k, is defined as

This can be estimated by the sample ACF


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In other words, the autocorrelation function k

is the correlation between a variable (say, Y)and Y lagged kperiods.


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Testing for Unit Roots

Consider an AR(1): yt= +ryt-1 + et

Let H0:r= 1, (assume there is a unit root).

Define =r1 and subtractyt-1 from both sidesto obtain Dyt= + yt-1 + et.

Unfortunately, a simple t-test is inappropriate,

since this is an I(1) process.

A Dickey-Fuller Test uses the t-statistic, but

different critical values.


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Testing for Unit Roots (cont.d.)

We can addp lags ofDytto allow for more dynamics

in the process.

Still want to calculate the t-statistic for .

Now its called an augmented Dickey-Fuller test, but

still the same critical values.

The lags are intended to clear up any serial

correlation, if too few, test wont be right.


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Testing for Unit Roots w/ Trends

If a series is clearly trending, then we need to adjust

for that or might mistake a trend stationary series for

one with a unit root. Can just add a trend to the model.

Still looking at the t-statistic for q, but the critical

values for the Dickey-Fuller test change.


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The Augmented DickeyFuller

(ADF) Test The ADF test consist of estimating the following

equation

If1 = 0 then no deterministic trend

If = 0 then the series has stochastic trend

For example, = ( -1) for AR(1) model (where ifyou recall, unit root means =1).


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AR(p) model

And its generalization with p-lags can be writtenas:

tptptt eYYtY 31321 ...

tmtmttteYYYtY DDD

...

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which can be written as:

The deterministic trend with stationary AR(1)

component can be written as:

ttt eYtY 1321

time series chap21 (2)

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