Consider the time series model \[ \mathrm{y}_{\mathrm{t}}=10+0.5 \mathrm{y}_{\mathrm{t}-1}-0.8 \mathrm{y}_{\mathrm{t}-2}+\varepsilon_{\mathrm{t}} \] where \(\varepsilon_{\mathrm{t}} \sim \mathrm{N}[0,1]\) (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 ...

Let's analyze given time series model step by step. The model is: \[ \mathrm{y}_{\mathrm{t}}=10+0.5 \mathrm{y}_{\mathrm{t}-1}-0.8 \mathrm{y}_{\mathrm{t}-2}+\varepsilon_{\mathrm{t}} \] where \(\varepsilon_{\mathrm{t}} \sim \mathrm{N}(0,1)\). ### (i) Checking for Stationarity For an AR(2) process \(\mRead more

Let’s analyze given time series model step by step. The model is:

\[

\mathrm{y}_{\mathrm{t}}=10+0.5 \mathrm{y}_{\mathrm{t}-1}-0.8 \mathrm{y}_{\mathrm{t}-2}+\varepsilon_{\mathrm{t}}

\]

where \(\varepsilon_{\mathrm{t}} \sim \mathrm{N}(0,1)\).

### (i) Checking for Stationarity

For an AR(2) process \(\mathrm{y}_{\mathrm{t}} = \delta + \phi_1 \mathrm{y}_{\mathrm{t}-1} + \phi_2 \mathrm{y}_{\mathrm{t}-2} + \varepsilon_{\mathrm{t}}\), we identify:

– \(\delta = 10\)

– \(\phi_1 = 0.5\)

– \(\phi_2 = -0.8\)

The conditions for stationarity in an AR(2) process are:

1. \(-1 < \phi_2 < 1\) 2. \(\phi_1 + \phi_2 < 1\) 3. \(\phi_1 - \phi_2 < 1\) Checking these conditions: - \( -1 < -0.8 < 1 \) - \( 0.5 - 0.8 = -0.3 < 1 \) - \( 0.5 + 0.8 = 1.3 > 1 \)

The last condition is not satisfied, hence the process is **not stationary**.

### (ii) Mean and Variance of the Time Series

Since the process is non-stationary, the concepts of mean and variance do not have the usual interpretations as they would in a stationary context. Non-stationary data can have a mean and variance that change over time.

### (iii) Autocorrelation Function

For non-stationary processes, the traditional autocorrelation function is not typically calculated, as the mean and variance are not constant. However, if the series were stationary, the autocorrelation function for an AR(2) could be calculated using Yule-Walker equations.

### (iv) Plotting the Correlogram

Plotting the correlogram for a non-stationary time series wouldn’t be meaningful as the autocorrelation function would not correctly reflect the time-dependent structure in the data.

### Conclusion

Given time series model is non-stationary, and hence the methods typically used to analyze stationary time series (like calculation of mean, variance, autocorrelation, and plotting correlograms) are not appropriate in this case.

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