Lesson 8: Testing for IID Hypothesis with the correlogram

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1 Lesson 8: Testing for IID Hypothesis with the correlogram Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it

2 Testing for i.i.d. Hypothesis Given a time series {x 1, x 2,..., x T }, we want establish if it can be considered a realization of an i.i.d. process x t i.i.d.(0, σ 2 ) An i.i.d. process is a sequence of independent and identically distributed (i.i.d.) random variables with zero mean and variance σ 2

3 Testing for i.i.d. Hypothesis We want to test the null hypothesis H 0 : ρ k = 0

4 Testing for i.i.d. Hypothesis The decision rule could be: Reject H 0 if ˆρ k > c where c is a constant. How do we can choose the constant c?

5 We can choose c such that Now, we have This implies that P( ˆρ k > c H 0 ) = 0.05 P( ˆρ k > c H 0 ) = 1 P( ˆρ k c H 0 ) = 0.05 P( ˆρ k c H 0 ) = P ( c T T ˆρ k c ) T = 0.95

6 Testing for i.i.d. Hypothesis with the correlogram If x t i.i.d.(0, σ 2 ) then T ˆρk N(0, 1) This means that the standard normal distribution provides a good approximation to the true distribution of T ˆρ k for large T.

7 Testing for i.i.d. Hypothesis with the correlogram It follows that P ( c T T ˆρ k c ) T = 0.95 if and only if and hence c T = 1.96 c = 1.96 T

8 Testing for i.i.d. Hypothesis with the correlogram Reject H 0 if ˆρ k > 1.96 T that is if ˆρ k / [ 1.96, 1.96 ] T T

9 Testing for i.i.d. Hypothesis with the correlogram If the data {x 1,..., x T } were really generated by an i.i.d. process, then about 95% of the sample autocorrelations ˆρ 1, ˆρ 2,...ˆρ n should fall between the bounds ± 1.96 T. In other terms, if the considered process is i.i.d., we would expect 5% of sample autocorrelations to lie outside the blue dashed lines.

10 Testing for i.i.d. Hypothesis with the correlogram For example if we calculate the first 40 values of ˆρ k, then one expects only two values which fall outside these limits.

11 Testing for i.i.d. Hypothesis with the correlogram Consider the following time series

12 Testing for i.i.d. Hypothesis with the correlogram The correlogram for the data of this example is We see that 2 of the first 40 values of ˆρ k lie just outside the bounds ±1.96/ T. As these occur not at relevant time lags, we conclude that there is no evidence to reject the hypothesis that the observations are independently distributed.

13 Testing for i.i.d. Hypothesis with the correlogram If we compute the sample autocorrelations up to lag 40 and find that more than two or three values fall outside the bounds, or that one value falls far outside the bounds, we reject the i.i.d. hypothesis.

14 Testing for i.i.d. Hypothesis with the correlogram Consider the following time series

15 Testing for i.i.d. Hypothesis with the correlogram We reject the i.i.d. hypothesis.

16 Testing for i.i.d. Hypothesis with the correlogram Consider the following time series

17 Testing for i.i.d. Hypothesis with the correlogram In this case, only one value of ˆρ k lies outside the bounds ±1.96/ T. However, this occurs at lag 1, a relevant time lag. Thus, we reject the i.i.d. hypothesis. As we will see, in this case, an MA(1) model could be appropriate.

18 Test of the random walk hypothesis for financial data Are the prices of financial assets random walk?

19 Test of the random walk hypothesis for financial data The process {x t ; t Z} is a random walk if x t = x t 1 + u t where u t i.i.d.(0, σ 2 ). The increments, or first differences of x, are independently and identically distributed (i.i.d.). Thus the increments are unpredictable.

20 Test of the random walk hypothesis for financial data Usually, to investigate whether the data are RW, the first difference data t = x t x t 1 are used. The difference data should be i.i.d. (0, σ 2 ) if the system is a RW.

21 Test of the random walk hypothesis for financial data We examine the logarithm of the daily close prices of IBM stock from 3 January 2000 to 1 October 2002.

22 Test of the random walk hypothesis for financial data The graph of the differenced (the returns) series is

23 Test of the random walk hypothesis for financial data The corellogram is given by We accept the random walk hypothesis

24 Test of the random walk hypothesis for financial data Here, we consider monthly returns on Bank of New York stock from through

25 Test of the random walk hypothesis for financial data The corellogram is given by We accept the random walk hypothesis

26 Squared Returns Consider the series of the squared returns

27 Test of the random walk hypothesis for financial data The corellogram is given by We conclude that the the squared returns are not i.i.d.

28 Test of the random walk hypothesis for financial data Whereas the sample autocorrelations of the returns are close to zero, the correlogram of the squared returns shows quite a different picture: the squared return seems significantly correlated.

29 Portmanteau testing for i.i.d. processes In addition to assess the individual significance of sample autocorrelogram, at a specific lag, the researchers are often interested to the joint significance of a set of sample autocorrelations. H 0 : ρ k = 0 for k = 1,..., K

30 Portmanteau testing for i.i.d. processes If x t is an i.i.d. sequence with mean zero and finite variance, then for T large and K < T, the random variable Q K = T K k=1 is approximately chi-square with K degrees of freedom. ˆρ 2 k

31 Portmanteau testing for i.i.d. processes Thus, the joint statistical significance of ˆρ 1,..., ˆρ K may be tested using the Box-Pierce Portmanteau statistic. Q K = T We reject the i.i.d. hypothesis K k=1 ˆρ 2 k H 0 : ρ 1 = ρ 2 =... = ρ K = 0 at level α if Q K > χ 2 1 α,k, where χ2 1 α,k is the 1 α quantile of the chi-squared distribution with K degrees of freedom. The value K is chosen, somewhat arbitrarily, equal to 20.

32 Portmanteau testing for i.i.d. processes A refinement of this test, formulated by Ljung and Box (1978), is obtained replacing Q K with Q LB K = T (T + 2) K k=1 ˆρ 2 k /(T k) whose distribution is better approximated by the chi-squared distribution with K degrees of freedom. Large values of QK LB to a rejection of the null hipothesis. lead