The seminal paper about ARCH by Robert Engle was concerned with the variance of UK inflation. The data file \(u k\) contains seasonally adjusted data on the UK consumer price index (UKCPI) for the sample period 1957M6 to 2006M6.

a. Compute the monthly rate of inflation (y) for the sample period 1957M7 to 2006M6 using the formula

\[y_{t}=100\left[\frac{U K C P I_{t}-U K C P I_{t-1}}{U K C P I_{t-1}}\right]\]

b. Estimate a T-GARCH-in-mean model and check that you obtain the following results:

\[\begin{gather*}\hat{y}_{t}=-0.407+1.983 \sqrt{h_{t}} \\(t) \quad(-2.862)(5.243) \\\hat{h}_{t}=0.022+\left(0.211-0.221 d_{t-1}\right) e_{t-1}^{2}+0.782 \hat{h}_{t-1} \\(t) \quad(4.697) \quad(8.952)(-8.728) \tag{27.677}\end{gather*}\]

c. The negative asymmetric effect (-0.221) suggests that negative shocks (such as falls in prices) reduce volatility in inflation. Is this a sensible result for inflation?

d. What does the positive in-mean effect (1.983) tell you about inflation in the UK and volatility in prices?