Consider the linear regression model Y_i = beta_0 + beta_1 X_i + u_i (a) Let Y*

Question

Consider the linear regression model Y_i = beta_0 + beta_1 X_i + u_i (a) Let Y* = aY and X* = bX where a and b are scaling factors. Suppose you run the regression of Y* on X* and obtain estimators beta*_0 and beta*_1 What is the relationship between beta*_0 and beta*_0, and between beta_1 and beta*_1? (b) Now assume that beta_0 = 0, derive the OLS estimator of beta_1 (c) Suppose you realize that the statistical package that you are using does not allow you to drop the intercept. In other words, the software estimates regression models only with a constant. To get around this issue, you come up with the following idea. For each data point (Y_i, X_i), you also add (-Y_i, -X_i) to the dataset and then you estimate the model Y_i = alpha_0 + alpha_1 X_i + epsilon using this new dataset. i. Show that this idea works, that is, show that the OLS estimator of alpha_1 is the same as the OLS estimator of beta that you derived in part (b). ii. Show that sigma_j=1^2n epsilon_j^2 = 2 simga_i=1^n u_i^2

Explanation / Answer

Solution

Part (a)

Back-up Theory

1.If cap and cap are respectively the least square estimates of and of the linear regression model,

Y = + X + , then

cap = Mean Y - cap.Mean X and cap = Cov(X,Y)/Var(X)……............ (1)

2. If U = aX and V = bY, then

Var(U) = a^2.Var(X); Var(V) = b^2.Var(Y); Cov(U,V) = ab Cov(X,Y). ….. (2) and

Mean U = aMeanX; MeanV = bMeanY ……....................................... (3)

Now, to answer the question,

Given Y = 0 + 1X + u, least square estimates of 0 and 1 are respectively,

0cap = Mean Y – 1cap.Mean X and 1cap = Cov(X,Y)/Var(X) [vide (1)]

Similarly, least square estimates of 0* and 1* are respectively,

0cap* = Mean Y* - 1cap*.Mean X* and 1cap* = Cov(X*,Y*)/Var(X*).

Now, 1cap* = Cov(X*,Y*)/Var(X*) = ab.Cov(X,Y)/b^2.Var(X) [vide (2)]

                  = (a/b)Cov(X,Y)/Var(X)

                   = (a/b).1cap ANSWER 1

Also,

0cap* = Mean Y* - 1cap*.Mean X* = a.Mean Y - (a/b).1cap.b.Mean X [vide (3)]

              = a{Mean Y - 1cap.Mean X}

         = a.0cap ANSWER 2

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