1 Question 1 (10 marks) Time series descriptive statistics can be different from the ones we use for non-time series data. Given the inflation series in Figure 1, answer questions Q1.1 - Q1.5, using what have learnt in the subject. US Inflation 15 10 Annualised quarterly inflation 5 0 1954 1964 1974 1984 1994 2004 2014 2024 Date Figure 1: Inflation series for Question 1 1. (1 mark) A central banker wants to understand how inflation has changed over time. Why is it not a good idea to report a sample average and a sample variance? 2. (1 + 1 marks) You would like to use a rolling-window method to compute the local mean. How would such an estimate look like? What happens if you change the window size? 3. (2 marks) You suspect there is also time variation in the variance of inflation. Describe how you can use a rolling window procedure to visualise the time-varying pattern of inflation variance. 4. (2 marks) Considering that inflation is observed monthly or quarterly, there might exist both long-run movement and short-run movement at the business cycle frequency. HP filter can help us extract these movements. Suppose you have obtained the cyclical component of inflation. How can you make use of it when studying the relationship between inflation and some other macroeconomic variables? 5. (2 + 1 marks) Inflation It is computed as Tt = log pt - log Pt-1, where pe is the index of price level (aggregate price), and log(.) denotes the natural logarithm. Most national aggregate prices in logarithm are integrated of order 2, or I(2). Explain what this means to someone without much knowledge in mathematics or statistics. Discuss how we can use the augmented Dicky-Fuller test to check if log pt is indeed I(2)