Interesting question on Linear regression

[monsieuruzairo3]

Dear David,

I went through the question and found it quite challenging as no possible solution approach was coming to my mind. Could you kindly help me get the solution? Thanks

Consider the following linear regression model: Y = a + b*X + e. Suppose a = 0.05, b = 1.2, Std(Y) = 0.26, Std(e) = 0.1, what is the correlation between X and Y?
Choose one answer.
a. 0.852
b. 0.701
c. 0.923
d. 0.462

 

[ShaktiRathore]

hi proceed as
Std(Y) ^2=(b*Std(X))^2+Std(e)^2 [see how it comes below******]
=>(b*Std(X))^2=Std(Y) ^2-Std(e)^2=.26^2-.1^2=.0676-.01=.0576
=>(b*Std(X))^2=.0576
=>b*Std(X)=.24=>Std(X)=.24/b=.24/1.2=.20
we know b=Cov(X,Y)/std(X)^2 b is the beta [@@@ below derivation]
Cov(X,Y)=b*std(X)^2=1.2*.2^2=1.2*.04=.048
correlation formula is corr(X,Y)=Cov(X,Y)/Std(X)*Std(Y)=.048/.2*.26=.923(tell me if the answer is right)

[*********from regression equation Y = a + b*X + e take Variances on both sides,
Var(Y)=VaR(a + b*X + e )=VaR(a)+VaR(b*X)+VaR(e)+2Cov(a,b*X)+2Cov(a,e)+2Cov(b*X,e)
Var(a)=0 as vaariance of a constant is 0.
correlation of constant with variable is 0 =>2Cov(a,b*X)=2Cov(a,e)=0
we also assume that correlation of e errors terms with the independent variable is 0 which is one of assumtion of regression(not any specific correlation given in question)=>2Cov(b*X,e)=0
after considering all this,
Var(Y)=VaR(a + b*X + e )=0+VaR(b*X)+VaR(e)+0+0+0
Var(Y)=VaR(b*X)+VaR(e)
Var(Y)=b^2*Std(X)^2+Std(e)^2]
[@@@ from regression Y = a + b*X + e take CoVariances on both sides w.r.t X,
Cov(X,Y)=Cov[X,(a + b*X + e )=>Cov(X,Y)=Cov[X,a]+Cov[X, b*X ]+Cov[X,e]
Covariance of constant a with variable X is 0=>Cov[X,a]=0 also from assumtion of regression we assume that otherwise not specified oin the Question correlation of independent variable wr.t the error terms is 0=>Cov[X,e]=0
Cov(X,Y)=0+Cov[X, b*X ]+0=>Cov(X,Y)=b*Cov[X, X ]=b*VaR(X,X)=b*Std(X)^2=>b=Cov(X,Y)/Std(X)^2]

thanks

 

[monsieuruzairo3]

Thanks a lot Shakti. Could you help me in some tricky Quant questions via mail?