Gauss exercises II:

Gauss
exercises II:

1. Generate a set of three orthogonal regressors, X (including the constant) and define a true
3x1 parameter vector, beta, of your choice.

a) Generate normal errors, u, and compute the regressand, y=Xβ+u. Then
regress y only on the first two columns of X and retain your estimate b. Repeat
these steps (generation of errors and y, regression) 500 times using a loop
command, e.g. “for”. Looking at the average of your 500 estimates, does the
regression seem to produce a bias?

b) Redo exercise a) but now start from a set of
three regressors that are not orthogonal to each other any more. Specifically,
make sure that x₂ and x₃ are correlated (though not
perfectly). Do your results indicate that there is a bias problem now?

c) In the previous setup, estimate the
parameter on the third regressor, β₃, using “residual regression”,
i.e. regress the residuals from regressing y upon x₁ and x₂ (ºy*) on the residuals from regressing x₃ upon
x₁ and x₂ (ºx*₃). Do you obtain the same
estimate as in the case, where you simply regress y on all three columns of X?
What happens when you regress y on x*₃? Finally, what happens when
you regress y* on x₃?

d) Now add two further columns to the regressor matrix
used in b), but keep generating y only as a linear function of the first three
columns of X. What happens to your OLS estimates when, instead of regressing y
upon the first three columns of X, you now regress on all 5 columns of X?

2. When a certain sequence of commands or data
manipulations is required several times in a program it can be useful to
“outsource” this sequence of commands to a so-called procedure. Whenever the
respective sequence of commands is needed, instead of entering every single
command in your program you then simply call the procedure, which has to be
defined only once, e.g. at the end of your code. The structure of a procedure
in Gauss is as follows:

proc (# outputs) = myprocedure(x,y,z,….); /*x,y,z,…. are data inputs */

local a,b,c,…; /*
defines local variables that are used in the procedure */

a=x*y;b=z-a;…….. /*
commands that manipulate the data in the desired way */

output1=a;output2=b^2;….; /*
define which output you want the proc to return */

retp(output1,output2,output3….); /* return output */

endp;

To call the procedure in your program, simply type

{output1,output2,output3…}=myprocedure(x,y,z,….);

Try to write a procedure for OLS estimation that returns b, the
corresponding standard errors, R² and adjusted R² as
output, using X and y as inputs.

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