Question

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 + ε_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

1. Compute the estimated coefficients for the intercept (b₀) and slope (b₁) and the R²
2. So far in 2020, the unemployment rate has increased by 280% (from 3.6 percent to 13.3 percent). If the unemployment rate increase for the year ends up at 100%, what is the expected value of S (annual return on the S&P500)?
3. Once you have the forecast value for S, calculate a 95% confidence interval for your forecast.

Answer 1

Solution a) The slope of the regression line (b1) can be calculated by dividing the covariance of X and Y by the variance of X.

where Y = Dependent Variable

X = Independent Variable

Following, the regression line is given:

Si = b0 + b1Ui + εi

here, Si = Dependent Variable

and Ui = Independent Variable

b0 = Y-intercept

b1 = Slope coefficient

Hence, slope of the regression line = Cov(Si,Ui)/Var(Ui)

It is given that

Cov(Ui,Si)	-0.025043
Var(Ui)	0.034590

Putting these value in equation, slope is calculated as:

Slope = -0.025043/0.034590 = -0.724

The Y-intercept (b0) of a regression line can be calculated by subtracting the product of the slope and X mean from the Y mean: b = Y mean - m * X mean.

E(Ui) = X mean	-0.001530
E(Si) = Y mean	0.064267

Putting these value, we get b0 = 0.064267 - (-0.724)*(-0.001530)

= 0.064267 - 0.0011077

= 0.063159 = 0.0632

Coefficient of determination (R2) is calculated as:

We have given following values:

TSS	0.627907
RSS	0.326368

Thus, R2 = 1 - 0.326368/0.627907 = 1 - 0.519771 = 0.480229

Solution b) Based on the values calculated in part (a), the regression line becomes:

Si = 0.0632 - 0.724Ui

Annual percentage change in Unemployment rate = 100%

Correspondingly, annual percentage change in S&P500 (SPX) Si can be calculated from the regression line as:

Si = 0.0632 - 0.724*100% = 0.0632 - 0.724*1 = 0.0632 - 0.724 = -0.6608 or -66.08%

Thus, there will be fall in the S&P500 value by 66.08%

Solution c) The general formula to calculate the confidence intervals of the sample estimate is:

Sample estimate ± (t-multiplier × standard error)

Number of observations (n) = 19

Degree of freedom = Number of observations - 1

= 19 - 1 = 18

For 95% confidence level, the t-stats is calculated as:

t-stat = 2.101

Standard Error = Standard Deviation/Number of observations

Standard Deviation = sqrt(Variance)

= sqrt(0.034590) = 0.185984

Lower limit = Sample estimate - (t-multiplier × standard error)

=  -0.6608 - (2.101*0.185984)

= -0.6608 - 0.390752 = -1.05155

Upper limit = Sample estimate + (t-multiplier × standard error)

=  -0.6608 + (2.101*0.185984)

= -0.6608 + 0.390752 = -0.27005