The distribution of the JarqueBera test statistic

8 points) The price of an asset changes from 195.35 $ to 209.05 $ during a period of 762 days.
(a) Give the simple net return and the log return for the period.
(b) Give the annualized simple net return.

2. (10 points) The skewness of a sample of 240 monthly returns is -0.03 and the kurtosis is 4.41.
   Under the null hypothesis H0 that the returns are iid Gaussian, the distribution of the JarqueBera test statistic follows a chi-squared distribution with two degrees of freedom,
   JB = Sˆ2
   N/6
   +
   (Kˆ − 3)2
   24/N ∼ χ
   2
   (2).
   (a) Is the test with the Jarque-Bera statistic one-sided or two-sided? Justify.
   (b) Test H0 with a size of 5 % using the Jarque-Bera test statistic.
   F(x) 0.010 0.025 0.050 0.100 0.500 0.900 0.950 0.975 0.990
   x 0.020 0.051 0.103 0.211 1.386 4.605 5.992 7.378 9.210
   Table 1: Some values of the distribution function of a law χ
   2
   (2).
3. (26 points) Consider the following ARMA(2,1) process, where t
   is Gaussian white noise:
   rt = 1.3 rt−1 − 0.4 rt−2 + t + 0.6 t−1, t ∼ N (0, 0.2).
   (a) Show that one of the two roots of the characteristic polynomial is 2 and find the other. Is
   the process stationary? Justify.
   (b) Find the values ψ1 and ψ2 of the MA(∞) expansion of the process:
   rt = t + ψ1t−1 + ψ2t−2 + ψ3t−3 + . . .
   (c) Give the system of equations that allows to determine the autocovariance coefficients γ0,
   γ1 and γ2 of the process and explain how to obtain the coefficients γl
   for l = 3, 4, . . . as a
   function of the coefficients γk, k = 1, . . . , l − 1.
4. (20 points) You consider an ARMA(1,2) model,
   rt = φ0 + φ1rt−1 + t − θ1t−1 − θ2t−2, t ∼ D(0, σ2
   ),
   where t
   is a white noise process, with the objective of predicting the return of a financial asset.
   You get the following point estimates for the fitted model:
   φˆ
   0 = 0.65 ˆθ1 = −0.40 ˆσ = 1.25
   φˆ
   1 = 0.55 ˆθ2 = 0.25
   (a) Compute the point estimate ˆµ of the unconditional mean µ ≡ E(rt).
   (b) Suppose that the value of the last two observations is rn = 2.15 and rn−1 = 3.95 and
   that the estimated value of the corresponding innovations is ˆn = 2.51 and ˆn−1 = −0.68.
   Calculate the point predictions ˆrn(1) and ˆrn(2) as well as their standard deviation.
   (c) The table below contains four point predictions ˆrk−1(1) obtained with an ARMA(1,2) model
   and an AR(1) model in a prediction exercise with moving window. The observed values of
   rt are listed on the last line. Propose a selection criterion and indicate which of the two
   models is favored according to those prediction results.
   k t + 1 t + 2 t + 3 t + 4
   ARMA(1,2) 1.26 -0.56 1.04 -0.69
   AR(1) 1.33 -0.61 1.18 -0.72
   rk 1.23 -0.52 1.16 -0.76
   Table 2: Results of the prediction exercise