Macroeconomists have also noticed that interest rates change following oil price jumps. Let \(R_{t}\) denote the interest rate on three-month Treasury bills (in percentage points at an annual rate). The distributed lag regression relating the change in \(R_{t}\left(\Delta R_{t}\right)\) to \(O_{t}\) estimated over 1960:Q1-2017:Q4 is

\[ \begin{aligned} \widehat{\Delta R_{t}}= & 0.03+0.013 O_{t}+0.013 O_{t-1}-0.004 O_{t-2}-0.024 O_{t-3}-0.000 O_{t-4} \\ & (0.05)(0.010) \quad(0.010) \quad(0.008) \quad(0.015) \\ & +0.006 O_{t-5}-0.005 O_{t-6}-0.018 O_{t-7}-0.004 O_{t-8} \\ & (0.015) \quad(0.015) \quad(0.010) \end{aligned} \]

a. Suppose that oil prices jump \(25 \%\) above their previous peak value and stay at this new higher level (so that \(O_{t}=25\) and \(O_{t+1}=O_{t+2}=\cdots=0\) ). What is the predicted change in interest rates for each quarter over the next two years?

b. Construct \(95 \%\) confidence intervals for your answers to (a).

c. What is the effect of this change in oil prices on the level of interest rates in period \(t+8\) ? How is your answer related to the cumulative multiplier?

d. The HAC \(F\)-statistic testing whether the coefficients on \(O_{t}\) and its lags are 0 is 1.92 . Are the coefficients significantly different from 0 ?