This question is based on daily log returns for Microsoft Corporation stock from $10/19/2020$ to $\frac{10}{15}\frac{?}{2021}.$ You are interested in modeling the log returns using an AR$(1)$ process. Use the SAS output provided and answer the following questions using the $5\%$ significance level when performing an hypothesis test. Round to four decimals when reporting and$/$or using values from the SAS output.

a$)$ What are the estimates if you regress the log return on its lag and on a constant?

b$)$ Test whether the null hypothesis that the log returns autoregressive parameter is zero against the alternative that it is different from zero. Test whether the null hypothesis that the intercept is zero against the alternative that it is different from zero.

c$)$ Test whether the null hypothesis that the log returns autoregressive parameter is one against the alternative that it is different from one.

d$)$Test whether the log returns are normally distributed.

e$)$ Test whether all autocorrelations up to lag $12$ of the residuals from the AR$1$ fit are all zeros.

Parameter Estimates

$\backslash$table$[[$Variable DF$,$Estimate,Standard Error,t Value,Approx Pr $>|t|,],[$Intercept$,1,0\mathrm{.}001403,0\mathrm{.}000863,1\mathrm{.}62,0\mathrm{.}1055]]$

Estimates of Autocorrelations

$\backslash$table$[[$Lag$,$Covariance,Correlation,$-1,0,-0\mathrm{.}5,0\mathrm{.}0,0\mathrm{.}5],[0,0\mathrm{.}0002,1\mathrm{.}0000,,,,8],[1,-0\mathrm{.}0000042,-0\mathrm{.}0225,,,1,]]$

Estimates of Autoregressive Parameters

$\backslash$table$[[$Lag$,$Coefficient,Standard Error,t Value$],[1,0\mathrm{.}022520,0\mathrm{.}063484,0\mathrm{.}35]]$

'Microsoft $-$ Daily prices, Oct $19,2020$ to Oct $15,2021\text{'}$

$\text{'}$AR$(1)\text{'}$

The ARIMA Procedure

Name of Variable $=$ logreturn

Mean of Working Series $0\mathrm{.}001403$

Standard Deviation $,0\mathrm{.}013625$

Number of Observations $250$

Autocorrelation Check of Residuals

$\backslash$table$[[$To Lag,Chi$-$Square,DF$,$Pr $>$ ChiSq,Autocorrelations,$],[6,7\mathrm{.}15,5,0\mathrm{.}2095,-0\mathrm{.}000,-0\mathrm{.}018,-0\mathrm{.}140,0\mathrm{.}041,-0\mathrm{.}068,0\mathrm{.}043],[12,11\mathrm{.}22,11,0\mathrm{.}4247,0\mathrm{.}042,0\mathrm{.}057,-0\mathrm{.}029,-0\mathrm{.}072,0\mathrm{.}017,-0\mathrm{.}065]]$