Transcript iid N(m, s)
Random Series / White Noise Notation • WN (white noise) – uncorrelated • iid independent and identically distributed • Yt ~ iid N(m, s) Random Series • et ~ iid N(0, s) White Noise Data Generation • Independent observations at every t from the normal distribution (m, s) Yt Yt t 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 WN 35 40 45 50 Test of Randomness - 2 Correlogram Sample: 1 50 Included observations: 50 Autocorrelation .|. . |*. . |** **| . .|. .|. | | | | | | Partial Correlation .|. . |*. . |** **| . .|. .|. | | | | | | 1 2 3 4 5 6 AC PAC Q-Stat Prob -0.042 0.112 0.275 -0.215 0.036 0.047 -0.042 0.111 0.288 -0.218 -0.053 0.033 0.0952 0.7777 4.9515 7.5666 7.6427 7.7741 0.758 0.678 0.175 0.109 0.177 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 rk . By definition, r0 = 1. • Autocorrelation of a Random Series: If the series is random, rk = 0 for k = 1,... rk Process Correlogram 1 0 Lag, k -1 Sample Autocorrelation Coefficient Sample Autocorrelation at lag k. Y Y Y n ˆrk t 1 k t k t Y Y n 2 t t 1 Y Standard Error of the Sample Autocorrelation Coefficient • Standard Error of the sample autocorrelation if the Series is Random. s rk = n -k 1n n n +2 Z- Test of H0: rk = 0 z rk srˆ k Reject H0 if Z < -1.96 or Z > 1.96 Box-Ljung Q Statistic • Definition 1 ˆ2 QBL (m) n(n 2) r k k 1 n k m Sampling Distribution of QBL(m) | H0 • H0 : r1=r2=…rk = 0 • QBL(m) | H0 follows a c2 (DF=m) distribution Reject H0 if QBL > c2(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 – 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 Data Data Expected | Normal Expected | Normal Test of Normality - 2 Test Statistics n • Stand. Dev. • Skewness • Kurtosis sˆ Y t 1 t Y 2 n Yt Y 1 ˆ S n t 1 s n Yt Y 1 K n t 1 s n 4 3 The Jarque-Bera Test If the population is normal and the data are random, then: 2 1 2 n JB = S + K-3 6 4 follows approximately c2with the # 0f degrees of freedom 2. Reject H0 if JB > 6 Forecasting Random Series • Given the data Y1,...,Yn, the one step ahead forecast Y(n+1) is: Y t-coeff s 1+ 1 n or Approx. Y z-coeff s Forecasting a Random Series • If it is determined that Yt is RND N(m, s) a) The best point forecast of Yt = E(Yt) = m b) A 95% interval forecast of Yt = (m – 1.96 s, m+1.96 s) 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: Y n v t 2 t Y( t 1) 2 ( n 1) s 2 Y The non Neumann Ratio of the regression residual is the Durbin - Watson Statistic