Question

1. Tax efficiency is a measure- ranging from 0 to 100- of how much tax due to capital gains stock or mutual fund owners pay on their investments each year; the higher the tax efficiency, the lower is the tax.  Financial planners examined the relationship between investments in mutual fund portfolios and their associated tax efficiencies.  The following table shows percentage of investments in energy securities (x) and tax efficiency (y) for 10 mutual fund portfolios:

x	3.1	3.2	3.7	4.0	4.0	5.5	6.7	7.4	7.4	10.6
y	98.1	94.7	92.0	89.8	87.5	85.0	82.0	77.8	72.1	53.5

1. Give the simple linear regression model and assumptions.

2. Write down the least squares regression line (3 decimal places).  

3. Give the 95% confidence interval for the slope parameter.

4. Test for a significant linear relationship between x and y at the 5% significance level.  Give the hypotheses, test statistic, conclusion, and interpretation.

Answer 1

a)

Y=a+bx

Assumptions of Linear Regression

• The regression model is linear in parameters.
• The mean of residuals is zero.
• Homoscedasticity of residuals or equal variance.
• The X variables and residuals are uncorrelated.
• The variability in X values is positive.
• The regression model is correctly specified.
• No perfect multicollinearity.

.................

b)

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	55.6	832.5	53.424	1532.9	-278.69
mean	5.56	83.25	SSxx	SSyy	SSxy

sample size ,   n =   10          
here, x̅ = Σx / n=   5.56   ,     ȳ = Σy/n =   83.25  
                  
SSxx =    Σ(x-x̅)² =    53.4240          
SSxy=   Σ(x-x̅)(y-ȳ) =   -278.7          
                  
estimated slope , ß1 = SSxy/SSxx =   -278.7   /   53.424   =   -5.2166
                  
intercept,   ß0 = y̅-ß1* x̄ =   112.2541          
                  
so, regression line is   Ŷ =   112.25   +   -5.22   *x

.............

c)

SSE=   (SSxx * SSyy - SS²xy)/SSxx =    79.059
      
std error ,Se =    √(SSE/(n-2)) =    3.144
confidence interval for slope                  
α=   0.05              
t critical value=   t α/2 =    2.306   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    3.14363   /√   53.42   =   0.430
                  
margin of error ,E= t*std error =    2.306   *   0.430   =   0.992
estimated slope , ß^ =    -5.2166              
                  
                  
lower confidence limit = estimated slope - margin of error =   -5.2166   -   0.992   =   -6.2084
upper confidence limit=estimated slope + margin of error =   -5.2166   +   0.992   =   -4.2248

...............

d)

Ho:   ß1=   0          
H1:   ß1╪   0          
n=   10              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    3.144   /√   53.42   =   0.4301
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    -5.2166   /   0.4301   =   -12.1289
                  
  
Degree of freedom ,df = n-2=   8              
p-value =    0.0000              
decison :    p-value<α , reject Ho              
reject Ho and conclude  that linear relations exists between X and y

...............

THANKS

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