Random Series / White Noise

Notation WN (white noise) – uncorrelated iid independent and identically distributed Y t ~ iid N( ,  ) Random Series  t ~ iid N(0,  ) White Noise 

Data Generation Independent observations at every t from the normal distribution ( ,  ) t YtYt 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)

Test of Randomness - 2 Correlogram

Scatterplot and Correlation Coefficient - Review Y X

Autocorrelation Coefficient Definition: The correlation coefficient between Y t 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k 1 

Sample Autocorrelation Coefficient Sample Autocorrelation at lag k.

Standard Error of the Sample Autocorrelation Coefficient Standard Error of the sample autocorrelation if the Series is Random.

Z- Test of H 0 :  k = 0 Reject H 0 if Z 1.96

Box-Ljung Q Statistic Definition

Sampling Distribution of Q BL (m) | H 0 H 0 :  1 =  2 =…  k = 0 Q BL (m) | H 0 follows a  2 (DF=m) distribution Reject H 0 if Q BL >  2 (95%tile)

Test of Normality - 1 Graphical 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 –p i = (i - 0.5)/ n –p i = i / (n+1) –p i = (i ) / (n+0.25) Estimate of the percentile | Normal –Standardized Q(p i ) = NORMSINV(p i ) –Q(p i ) = NORMINV(p i, mean, stand. dev.)

Non-Normal Populations Flat Skewed Expected | Normal Data Expected | Normal Data

Test of Normality - 2 Test Statistics Stand. Dev. Skewness Kurtosis

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 H 0 if JB > 6

Forecasting Random Series Given the data Y1,...,Yn, the one step ahead forecast Y(n+1) is: or Approx.

Forecasting a Random Series If it is determined that Y t is RND N( ,  ) a) The best point forecast of Y t = E(Y t ) =  b) A 95% interval forecast of Y t = (  – 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:

Appendix: The von Neumann Ratio Definition: The non Neumann Ratio of the regression residual is the Durbin - Watson Statistic