Consider the time series modelWhere (i) Is this a stati

Question

Consider the time series model

$y_{1}=10+0.5y_{t-1}-0.8y_{t-2}+\varepsilon _{1}$

Where $\varepsilon _{1}-N\left [ 0,1 \right ]$
(i) Is this a stationary time series?
(ii) What are the mean and variance of the time series?
(iii) Calculate the autocorrelation function.
(iv) Plot the correlogram.

08 Sep 2023

Answer :

Word Count : 581

Let's analyze the given time series model step by step:

The given time series model is:

$\small \[y_{t} = 10 + 0.5y_{t-1} - 0.8y_{t-2} + \varepsilon_{1}\]$

Where $\small $\varepsilon_{1}$$ is a normally distributed random variable with mean $\small $\mu = 0$$ and variance $\small $\sigma^{2} = 1$$ .

(i) To determine if this time series is stationary, we need to check if its properties (mean, variance, and autocorrelation) remain constant over time. Stationarity is a desirable property for time series analysis.

First, let's express the equation in terms of $\small $y_{t}$$ :

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