Random Series / White Noise

Notation

• WN (white noise) – uncorrelated

• iid independent and identically distributed

• Yt ~ iid N(, ) Random Series

• t ~ iid N(0, ) White Noise

Data Generation

• Independent observations at every t from

the normal distribution (, )

t

YtYt

Identification of WN Process

How to determine if data are from WN process?

Tests of Randomness - 1

• Timeplot of the Data

Check trend

Check heteroscedasticity

Check seasonality

Generating a Random Series Using Eviews

• Command: nrnd generates a RND N(0, 1)

-3

-2

-1

0

1

2

3

5 10 15 20 25 30 35 40 45 50

WN

Test of Randomness - 2 Correlogram

Sample: 1 50 Included observations: 50

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

. | . | . | . | 1 -0.042 -0.042 0.0952 0.758 . |*. | . |*. | 2 0.112 0.111 0.7777 0.678 . |** | . |** | 3 0.275 0.288 4.9515 0.175 **| . | **| . | 4 -0.215 -0.218 7.5666 0.109 . | . | . | . | 5 0.036 -0.053 7.6427 0.177 . | . | . | . | 6 0.047 0.033 7.7741 0.255

Scatterplot and Correlation Coefficient - Review

Y

X

Autocorrelation Coefficient

• Definition:

The correlation coefficient between Yt and Y(t-k) is called the autocorrelation coefficient at lag k and is denoted as k . By definition, 0 = 1.

• Autocorrelation of a Random Series:

If the series is random, k = 0 for k = 1,...

Process Correlogram

Lag, k

k

1

-1

Sample Autocorrelation Coefficient

Sample Autocorrelation at lag k.

ˆ k Yt Y Yt k Y

t1 k

n

Yt Y 2

t 1

n

Standard Error of the Sample Autocorrelation Coefficient

• Standard Error of the sample autocorrelation

if the Series is Random.

s k = n - k

n n + 2 1

n

Z- Test of H0: k = 0

ˆk

kzs

Reject H0 if Z < -1.96 or Z > 1.96

Box-Ljung Q Statistic

• Definition

2

1

1ˆ( ) ( 2)

m

BL kk

Q m n nn k

Sampling Distribution of QBL(m) | H0

• H0 : 1=2=…k = 0

• QBL(m) | H0 follows a 2 (DF=m) distribution

Reject H0 if QBL > 2(95%tile)

Test of Normality - 1Graphical Test

• Normal Probability Plot of the Data

Check the shape: straight, convex, S-shaped

Construction of a Normal Probability Plot

• Alternative estimates of the cumulative relative frequency of an observation

– pi = (i - 0.5)/ n– pi = i / (n+1)– pi = (i - 0.375) / (n+0.25)

• Estimate of the percentile | Normal– Standardized Q(pi) = NORMSINV(pi)– Q(pi) = NORMINV(pi, mean, stand. dev.)

Non-Normal Populations

Flat Skewed

Expected | Normal

Data

Expected | Normal

Data

Test of Normality - 2Test Statistics

• Stand. Dev.

• Skewness

• Kurtosis

2

1ˆ

n

tt

Y Y

n

3

1

1ˆn

t

t

Y YS

n

4

1

1 nt

t

Y YK

n

The Jarque-Bera Test

If the population is normal and the data are random, then:

follows approximately with the # 0f degrees of freedom 2.

Reject H0 if JB > 6

JB = n6

S 2+ 14

K-3 2

Forecasting Random Series

• Given the data Y1,...,Yn, the one step ahead forecast Y(n+1) is:

or Approx.

Y t-coeff s 1 + 1n

Y z-coeff s

Forecasting a Random Series

• If it is determined that Yt is RND N(, )

a) The best point forecast of Yt = E(Yt) =

b) A 95% interval forecast of Yt =

( – 1.96 , +1.96 )

for all t (one important long run implication of a stationary series.)

The Sampling Distribution of the von-Neumann Ratio

The vN Ratio | H0 follows an approximate normal with:

Expected Value of v: E(v) = 2

Standard Error of v: SE (v) = 4 (n - 2)n2 - 1

Appendix:

The von Neumann Ratio

• Definition:

The non Neumann Ratio of the regression residual is the Durbin - Watson Statistic

2

( 1)2

2( 1)

n

t tt

Y

Y Yv

n s