You Have Collected Quarterly Canadian Data on the Unemployment and the Inflation

Essay

You have collected quarterly Canadian data on the unemployment and the inflation rate from 1962:I to 2001:IV. You want to re-estimate the ADL(3,1)formulation of the Phillips curve using a GARCH(1,1)specification. The results are as follows: $\widehat{\Delta I n f}_{t}=1.17-.56 \Delta I n f_{t-1}-.47 \Delta I n f_{t-2}-.31 \Delta I n f_{t-3}-.13 \text { Unempt-1 }$
(.48) (.08) $\quad$ (.10) $\quad$ (.09) $\quad$$(.06)$
$\hat{\sigma}_{t}^{2}=.86+.27 u_{t-1}^{2}+.53 \sigma_{t-1}^{2} .$
$(.40)(.11)$$\quad$ (.15)
(a)Test the two coefficients for $u _ { t - 1 } ^ { 2 }$ and $\sigma _ { t - 1 } ^ { 2 }$ in the GARCH model individually for statistical significance.
(b)Estimating the same equation by OLS results in $\widehat{\Delta I n f}_{t}=1.19-.51 \Delta I n f_{t-1}-.47 \Delta I n f_{t-2}-.28 \Delta I n f_{t-3}-.16 U_{n e m p t-1}$
(.54) (.10) $\quad$ (.11) $\quad$ (.08) $\quad$$(.07)$
Briefly compare the estimates. Which of the two methods do you prefer?
(c)Given your results from the test in (a), what can you say about the variance of the error terms in the Phillips Curve for Canada?
(d)The following figure plots the residuals along with bands of plus or minus one predicted standard deviation (that is, ± $\widehat{\sigma _ { t }}$ )based on the GARCH(1,1)model. Describe what you see.