I really need some help with reverse regressions. I'm trying to solve the following exercise:
a) Consider the following Conditional Expectation Function model:
$Y = alpha + beta X + epsilon , E[Y|X] = alpha + beta X$
Suppose that we have transformed the above equation in the following way:
$X = -(alpha /beta) + (1/beta)Y – (1/beta)epsilon $
Show that this equation does not satisfy the definition of the CEF model.
b) Consider now the best linear predictor equation:
$Y = alpha_L + beta_L X + epsilon_L$
Consider as well the reverse best linear predictor equation:
$X = alpha^*_L + beta^*_L Y + epsilon^*_L$.
Show that in the Best Linear Predictor Setting $beta_L ne 1/beta^*_L$
c) Assume there is an Instrument Z. Show that $beta^*_L = 1/beta^*_L$ in the IV setting.
I would be really thankful if someone could explain me the idea behind the problem about reverse Regression and how it is related to the IV approach.
Best Answer
I'm sorry: I don't understand what you mean by point c. I'll reply to points a and b.
a) $X = -(alpha /beta) + (1/beta)Y – (1/beta)epsilon rightarrow E[X_i|Y_i]=-(alpha /beta) + (1/beta)Y_i$, because $X perp epsilon$. To satisfy the CEF model, we need: $E[- (1/beta)epsilon_i | Y_i]= 0 rightarrow – (1/beta)E[epsilon_i | Y_i] = 0 rightarrow (beta!=0, E[epsilon_i | Y_i] = 0)$. Let's focus on the second condition: $E[epsilon_i | Y_i]=0$, and define $W_i=epsilon_i*Y_i$. By the Law of Total Expectation, we have: $E[epsilon_i | Y_i]=0 rightarrow E[W_i]=E_{Y_i}[E[W_i|Y_i]]=E[Y_i*E[epsilon_i|Y_i]]=E[Y_i*0]=0rightarrow Cov(epsilon,Y)=E[epsilon*Y]-E[epsilon]E[Y]=0-0=0.$ But, given $Y = alpha + beta X + epsilon$, then: $Cov(epsilon,Y)=beta*Cov(X,epsilon)+Cov(epsilon,epsilon)=beta*0+Var(epsilon)=sigma^2>0$. Thus, the condition $E[epsilon_i | Y_i]=0$ does not hold, so the CEF model is not satisfied. To be precise, the condition would only be satisfied in the degenerate case, i.e.: $sigma^2=0rightarrow epsilon_i=0 forall i$, i.e. in the deterministic setting.
b) Let's suppose $beta_L=1/beta^*_L$. Then, we get: $Cov(X,Y)/Var(Y)=Var(X)/Cov(X,Y)rightarrow Cov(X,Y)^2=Var(X)*Var(Y)rightarrow [Corr(X,Y)*SD( X)*SD(Y)]^2=Corr(X,Y)^2*Var(X)*Var(Y)=Var(X)*Var(Y)rightarrow Corr(X,Y)=+1 bigvee Corr(X,Y)=-1.$ Again, the condition only holds in case the relationship between $X$ and $Y$ is deterministic.
In fact, Galton (1886) showed that, if you normalize $X$ and $Y$ to have the same variance, then the coefficient in regressing $Y$ on $X$ or viceversa are the same, and equal to the coefficient of correlation (thus, between $-1$ and $1$). In case of perfect association and same variance of $X$ and $Y$, the coefficient would be $+1 bigvee -1$ indeed. This link may help: http://davegiles.blogspot.it/2014/11/reverse-regression-follow-up.html
Also, you may be interested in these Qs&As:
What is the difference between linear regression on y with x and x with y? Effect of switching response and explanatory variable in simple linear regression
Galton, Francis (1886): “Regression Towards Mediocrity in Hereditary Stature,” The Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246-263.
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