$8\mathrm{.}6$ Continuing with modeling the DJIA data in Exercises $8\mathrm{.}4$ and $8\mathrm{.}5.$ A simplified

version of the so$-$called random walk model of stock prices states that the best

prediction of the stock index at Day $t$ is the value of the index at Day $t-1.$ In

regression model terms it would mean that for the models fitted in Exercises

$8\mathrm{.}4$ and $8\mathrm{.}5$ the constant term is $0,$ and the regression coefficient is $1.$

$($a$)$ Carry out the appropriate statistical tests of significance. $($Test the values

of the coefficients individually and then simultaneously.$)$ Which test is

the appropriate one: the individual or the simultaneous?

$($b$)$ The random walk theory implies that the first differences of the index $($the

difference between successive values$)$ should be independently normally

distributed with zero mean and constant variance. Examine the first

differences of DJIA and $log($DJIA$)$ to see if this hypothesis holds.

$($c$)$ DJIA is widely available. Collect the latest values available to see if the

findings for $1996$ hold for the latest period.