What is spurious regression? Explain. $(10)$

$($a$)$ Assuming $\backslash (\backslash$rho$=1\backslash ),$ randomly generate $1000$ observations of variables $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash )$ using the following equations:

$\backslash [$

$\backslash$begin$\{$array$\}\{$l$\}$

X$\_\{$t$\}=\backslash$rho X$\_\{$t$-1\}+$e$\_\{$t$\}\backslash \backslash$

Y$\_\{$t$\}=\backslash$rho Y$\_\{$t$-1\}+$e$\_\{$t$\}(20)$

$\backslash$end$\{$array$\}$

$\backslash ]$

Report the graphs of $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash )$ that you have generated. $($Note: Your graph of $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash )$ will be very different from the ones I generated in the video lectures, because they are randomly generated. In fact, each time you randomly generate $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash ),$ they will look different.$)$

$($b$)$ In part $($b$)$ of this question you generated data on $\backslash ($ X $\backslash )$ and $\backslash ($ Y $\backslash ).$ Assume generated $\backslash ($ Y $\backslash )$ as dependent variable and $\backslash ($ X $\backslash )$ as independent variable. Then run the regression: $\backslash ($ Y$\_\{$t$\}=\backslash$alpha$+\backslash$beta X$\_\{$t$\}+$e$\_\{$t$\}\backslash ).$ Report the estimated results. $(10)$

$($c$)$ What did you expect about the magnitudes $\backslash (\backslash$beta $\backslash )$ and R $-$square? How are the estimated values different from your expectations? $(10)$

$($d$)$ In HW $2$ problem $1,$ you estimated a regression of personal consumption expenditures $($PCE$)$ on personal disposable income $($PDI$).$ Both PCE and PDI were time series data. Explain why that regression was not spurious. $(10)$

Problem $2$: Using data set marco.xls $($posted under Week $6)$ conduct unit$-$root test of the nominal interest rate and the exchange rate using the following methods:

$($a$)$ Graphical approach $(10)$

$($b$)$ ADF test $(15)$

$($c$)$ ACF test $(15)$

Note: Graph the nominal interest rate and the exchange rate separately. Then you will see that the exchange looks like unit$-$root $($non$-$stationary$).$ Also, to confirm if these variables are unit$-$root or not you need to conduct these tests on the first difference of these variables.

Problem $3$:

$($a$)$ Explain the four steps of Box$-$Jenkin's methods of forecasting $(25)$

$($b$)$ Using Box$-$Jenkin's four$-$step method, forecast the US quarterly GDP for the second quarter of $2015($Using GDP data posted under week $8)(25)($Note: Show all your work when you conduct forecast for GDP$.$ Show the details of each step.$)$