Why is it important to test for non-stationarity in time series data before attempting to build an empirical model?Explain

Question 1 [24 Marks]
For ALL of the following statements, say whether each one is TRUE or FALSE. Then, justify your answer with a careful explanation. Please note that explanations may also involve mathematical and or/ graphical illustrations. Please also note that marks will depend on the accuracy of each answer provided. Each correct and fully explained answer is worth 8 marks.
a. If the variance of a stochastic process is finite, I can conclude that the process is weakly stationary.
b. If I fail to reject the null hypothesis of a Dickey Fuller test, I conclude that the process has no unit roots.

c. You can have more confidence in long term than short term forecasts.
Question 2 [23 Marks]
a. What might Ramsey’s RESET test be used for?
b. Consider the following graph:
[5 Marks]

Does this series look like it is stationary? Explain your answer.
[10 Marks]

c. Why is it important to test for non-stationarity in time series data before attempting to build an empirical model? [5 Marks]

d. What kind of variables are likely to be non-stationary? Give an example. [3 Marks]

Question 3 [30 Marks]
a. The Table 1 below reports the autocorrelation function (ACF) and partial autocorrelation function (PACF) for the nominal returns of S&P500 (the first difference of the log S&P500 index) using monthly data for the period 1970-2018. In the Table 1 below k represents the number of lags and SE stands for the standard error. Identify the model that you should utilize in your analysis by examining Table 1 below. Explain in detail your answer.

b. Table 2 below reports the estimated Akaike (AIC) and Schwarz Bayesian (SBC) criteria for various lags of order q (for the moving average part) and p (for the autoregressive part). The data utilized to estimate the criteria below are the nominal returns of S&P500 (the first difference of the log S&P500 index) using monthly data for the period 1970-2018. What is the suggested model(s) under both the AIC and SBC criteria? Explain in detail your answer.
[10 Marks]

c. Suppose that the best specification you found based on the criteria outlined in part (a) is provided by the AR(1) model: 𝑦𝑡 = 𝜃𝑦𝑡−1 + 𝜀𝑡. What conditions need to be imposed on the parameter 𝜃 for the model to be non-stationary?
[5 Marks]
d. Discuss how to test for the presence of unit roots after you estimate the AR(1) model in part (c).
[5 Marks]

Question 4 [23 Marks]
a. Test whether the following AR(2) model (eq. 2) for the time series {𝑦 }𝑇
𝑡 𝑡=1 is stationary:

𝑦𝑡 = 0.7𝑦𝑡−1 + 0.10𝑦𝑡−2 + 𝜀𝑡,
Show in detail your calculations.
[15 Marks]
b. Consider a simple model of the S&P500 stock price index (named “sp500price”). The data are daily over the period 2015 through 2020. We also generate the natural logarithm of the variable sp500price which is named ln_sp500price.

Suppose that you run the following command in Stata:
regress D.ln_sp500price
estat archlm, lags(1)
Table 3 below presents the results from the Engle’s LM test of the autoregressive conditional heteroskedasticity test: