(a) Write the conditions for the weak stationarity and strict stationarity, respectively. (5points)
(b) Derive the expectation, variance and first order autocorrelation for AR(1) and MA (2)process in the stationary condition, respectively
(a) Weak stationarity means that a time series has a constant mean, constant variance, and autocovariance that only depends on the time lag. This basically means that the statistical properties of the time series don't change over time. Strict stationarity is a more restrictive condition that requires the joint distribution of any finite set of time series observations to be invariant to shifts in time. This means that the statistical properties of the time series are completely time-invariant. (b) For AR(1) and MA(2) processes in the stationary condition, the expectation, variance, and first order autocorrelation can be calculated as follows: AR(1): - Expectation: E(Yt) = 0 / (1-φ)
- Variance: Var(Yt) = σ^2 / (1-φ^2)
- First order autocorrelation: ρ1 = φ
MA(2): - Expectation: E(Yt) = 0
- Variance: Var(Yt) = σ^2(1 + θ1^2 + θ2^2 + 2θ1ρ1 + 2θ2ρ2)
- First order autocorrelation: ρ1 = θ1 / (1 + θ1^2 + θ2^2)
The first order autocorrelation measures the linear dependence between adjacent observations in the time series. It's an important statistic to look at when analyzing the time series, because it gives an idea of how much influence the past values have on the future values. The expectation and variance are also important because they tell us about the center and spread of the time series.
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