1.) You are considering the relationship between annual returns on the S&P 500 index (January 31 to January 31) and annual changes in the unemployment rate. You define:

S = annual % change in the S&P500 (SPX)

U = annual % change in the unemployment rate

You consider the following univariate relationship:

S_i = b₀ + b₁U_i + _i

You have data on annual changes in the unemployment rate and the S&P500 (SPX) from 2002 through 2020 (19 observations) and you want to use the data to estimate the regression coefficients (intercept b₀ and slope b₁)

You are given the following statistics:

Statistic

Value

Cov(U,S)

-0.025043

Var(U)

0.034590

E(U)

-0.001530

E(S)

0.064267

TSS

0.627907

RSS

0.326368

a.Compute the estimated coefficients for the intercept (b₀) and slope (b₁) and the R²

b.So far in 2020, the unemployment rate has increased by 120% (from 3.6 percent to 7.9 percent).If the unemployment rate increase for the year ends up at 80% (with unemployment at 6.5 percent), what is the expected value of S (annual return on the S&P500)?

c.Once you have the forecast value for S, calculate a 95% confidence interval for your forecast.