TS Module 14 Model diagnostics

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Time series practice problems model diagnostics

*Question 14.1: Selecting the model

The sample autocorrelations for lags of 1, 2, 3, 4, and 5 from a time series of 900 observations are

25%, 24%, 24.5%, 23.5%, 22.5%

Which of the following is the most likely model (of the following five) for the time series?

A.     non-stationary

B.     white noise

C.     stationary moving average of order 1

D.     stationary autoregressive of order 1

E.      stationary ARMA(1,1)

Answer 14.1: A

Statement B: If this were a white noise process, the sample autocorrelations would be close to zero, with a standard deviation of 1/30 = 3.3%.  The observed autocorrelations are too high.

Statement C: A stationary moving average process of order 1 has white noise autocorrelations for lags 2 and higher.  They would be close to zero with a standard deviation of 3.3%.

Statement D: A stationary autoregressive process of order 1 has geometrically declining autocorrelations for lags 2 and higher; there is no decline in the observed autocorrelations.

Statement E: An ARMA(1,1) model is similar to an AR(1) model for the autocorrelations.

Statement A: For a non-stationary process, the autocorrelations may stay high.

*Question 14.2: Diagnostics

An actuary examines a the stock price of ABC Corporation for the 252 trading days in 20X7. How might the actuary test the hypothesis that the stock prices form a process y_t = 1.01 × y_t-1 × ó, where ó is a lognormally distributed random variable?

A.     Assume the time series is a white noise process and test if the residuals are normally distributed.

B.     Assume the logarithm of the time series is a white noise process and test if the residuals are normally distributed.

C.     Assume the first differences of the logarithm of the time series is a white noise process and test if the residuals are normally distributed.

D.     Assume the second differences of the logarithm of the time series is a white noise process and test if the residuals are normally distributed.

E.      Diagnostic tests are used if the error term is normally distributed; thay are not used if the error term in the original time series is lognormally distributed.

Answer 14.2: C

The first differences of the logarithms of the original time series would be a white noise process with a normally distributed error term.

*Question 14.3: Autocorrelation Significance

We are testing whether the observed autocorrelations of a time series are significant at the 5% level.  We have 1,067 observations, and the critical t value at a 5% level of significance is 1.96.  The null hypothesis is that the autocorrelations are zero.

A sample autocorrelation is significant at the 5% level of significance if its absolute value is larger than which of the following?

A.     0.011

B.     0.030

C.     0.022

D.     0.060

E.      0.090

Answer 14.3: D

If the expected autocorrelations are zero, the observed values (sample autocorrelations) are randomly distributed about zero.  Their standard deviation is 1/√T = 1/1,067^½ = 0.031.

1.96 standard deviations is 1.96 × 0.031 = 0.060

Jacob: If an observed sample autocorrelation is greater (in absolute value) than 0.060, do we assume that the true autocorrelation is non-zero?

Rachel: If we examine 100 sample autocorrelations, we expect 5 of them to be greater (in absolute value) than 0.060.  If 5 or fewer are greater than 0.060, we assume they are all zero, and these sample autocorrelations stem from fluctuations.  Even if 7 or 8 of them are greater than 0.060, we might attribute this to chance.  But if the first 3 autocorrelations are greater than 0.060 (especially if they are much greater than 0.060) and none of the subsequent sample autocorrelations are greater than 0.060, we might assume a MA(3) process.

*Question 14.4: Bartlett’s test

A time series of 144 observations, t=1, 2, …, 144, has ∑ _t =0 and =

           324 for k = 0

           108 for k = 1

           96 for k = 2

           84 for k = 3

           72 for k = 4

           60 for k = 5

           48 for k = 6

           36 for k = 7

           24 for k = 8

           12 for k = 9

We are testing the null hypotheses ρ_k = 0 against the alternative hypotheses ρ_k ≠ 0 using a separate test for each value of k.  What is the lowest value of k for which we would not reject the null hypothesis at a 5% level of significance, for which the critical z-value is about 2?