Question $4.[15$pt$]$

We have quarterly data for GDP growth $(dlgd{p}_t)$ and CPI inflation in $\{$:$f{l}_t)$ for country

A$.$ Assume that both series are stationary. In Figure $3$ we estimate a VAR model.

$(1)$ What is the number of lags in the model? Write down the exact form of the model

we have estimated. $[5$pt$]$Vector Autoregression Estimates

Date: $05/20/18$ Time: $21$:$37$

Sample $($adjusted$)$: $2009$Q$12018$Q$1$

Included observations: $37$ after adjustments

Standard errors in $()$ & t$-$statistics in $[]$

Figure $3$: Results from a VAR model$(2)$ We would like to test whether output growth Granger causes inflation. Using your

notations in $(1),$ write down the null hypothesis, and describe the testing procedure.

$(3)$ Suppose we are interested in the effect on the economy over time from an output

shock. What tool would you propose? Describe it briefly. $[5$pt$]$Question $3.[40$pt$]$

Suppose we are doing a study on the daily closing price $p_t$ of a stock, which is observed

over $t=1,$dots,$100.$ We take the log difference of the closing price to get the log return,

and assume that the $log$ return $r_t$ follows the $AR(2)$ model

$r_t=0\mathrm{.}01+0\mathrm{.}02r_{t-1}+0\mathrm{.}008r_{t-2}+{}_t$

where ${}_t$ is a Gaussian white noise series with mean zero and variance $0\mathrm{.}04.$

$(1)$ Explain the reason to take log differences rather than to take first differences of the

stock price. $[5$pt$]$

$(2)$ Argue that the above AR$(2)$ model is stationary. $[5$pt$]$

$(3)$ Find the expected value of $r_t.[5$pt$]$

$(4)$ Transform the model to an equivalent AR model with mean zero. $[5$pt$]$

$(5)$ If you use OLS to estimate the model, what is the effective sample size? $[5$pt$]$

$(6)$ Now assume that we have some recent observations $r_{100}=-0\mathrm{.}1,r_{99}=0\mathrm{.}05,$ and

$r_{98}=0\mathrm{.}03.$ Compute the forecasts of the return series at $l=101$ and $102.[5$pt$]$

$(7)$ What are the associated standard deviations of the forecast error in $(6)?[5$pt$]$

$(8)$ In studying stock return, researchers often employ an ARCH model. Describe the

model, and explain what it tries to capture. $[5$pt$]$