A written report with answers to the questions of the homework, plus code, should be submitted via Canvas. Use Python Jupyter Notebook or Google Colab to code, but other coding methods will not be penalized. Use built-in functions or packages whenever possible.
In ppi.xls you will find monthly data for the Producer Price Index. We need a quarterly model for the PPI, so that we can come up with forecasts (up to 4 quarters ahead). There is no single right answer for forecasting. Well-trained econometricians can end up choosing different specifications even though they are confronted with the same sample. However, there definitely are some wrong answers. This is a classical economic forecast problem. The methods used here can be easily adapted to other time series analysis such as retail sales forecast.
Part A
We look for a stationary version of this series. Using the entire sample and extract end-of-quarter data for all following analysis. Make following plots:
i. Plot PPI in levels.
ii. Plot ∆PPI (first-order difference of PPI).
iii. Plot ∆ln(PPI).
Part B
Which version of the series looks stationary to you and why? Let’s call the stationary version 𝑦𝑡 = 𝑓(PPI𝑡).
Part C
Plot ACF of yt for 12 quarters. What do you conclude? If the ACF converges very slowly, re-think whether yt really is stationary.
Part D
Plot the PACF of yt for 12 quarters. What do you conclude?
Part E
On the basis of ACF and PACF, select four different ARMA(p, q) model specifications (p and q) to model yt. For each model:
i. Using entire sample, estimate coefficients and standard errors. Check stationarity (hint: does coefficients meet characteristic root constraint).
ii. Plot ACF and report Q-stat (Ljung-Box) of residuals for 12 quarters. Also plot residual distribution. Do residuals represent white noise?
iv. Calculate AIC. Select a preferred model on the basis of these diagnostics. Explain your choice.
Part F (Optional, no penalty if skip; extra bonus if complete with right answer)
Re-estimate your four models using only data up to 2000.IV and compute the MSPE (mean squared prediction error) on the remainder of the sample for one-quarter ahead forecasts:

where H is the length of the hold-back (test) sample, and ei is the one-step ahead prediction error. Also report the MSPE assuming there is no predictability in yt (hint: no predictability means that your best forecast of future is the latest value available). What do you conclude?
Guideline and References
The following links are external online examples. You may find some codes in those examples useful. Meanwhile, I encourage you to use Google search as a way of self-learning beyond the materials provided by professor.
• https://medium.com/@stallonejacob/time-series-forecast-a-basic-introduction-using-python- 414fcb963000
• https://kanoki.org/2020/04/30/time-series-analysis-and-forecasting-with-arima-python/
• https://machinelearningmastery.com/time-series-forecasting-methods-in-python-cheat- sheet/
• https://blog.usejournal.com/time-series-analysis-in-python-3b815b5f28f2
• https://www.statsmodels.org/stable/tsa.html

Sample Solution