Week 3 Tutorial Exercises
Review Questions (these may or may not be discussed in tutorial classes)
The minimum requirement for OLS to be carried out for the data set {(
x
i
,
y
i
), i=1,…,
n
} with the
sample size n > 2 is that the sample variance of
x
is positive. In what circumstances is the sample
variance of
x
zero?
When all observations on
x
have the same value, there is no variation in
x
.
The OLS estimation of the simple regression model has the following properties:
a)
the sum of the residuals is zero;
b)
the sample covariance of the residuals and
x
is zero.
Why? How would you relate them to the “least squares” principle?
These are derived from the first
‐
order conditions for minimising the sum of squared residuals
(SSR).
Convince yourself that the point
)
,
(
y
x
, the sample means of
x
and
y
, is on the sample
regression function (SRF), which is a straight line.
This can be derived from the first of the first
‐
order conditions for minimising SSR.
How do you know that SST = SSE + SSR is true?
The dependent variable values can be expressed as the fitted value plus the residual. After
taking the average away from both side of the equation, we find
ሺݕ
െݕ
തሻ ൌ ሺݕ
ො
െݕ
തሻݑ
ො
.
The required “SST = SSE + SSR” is obtained by squaring and summing both sides of the above
equation, noting
ሺݕ
ො
െݕ
തሻൌߚ
መ
ଵ
ሺݔ
െݔҧሻ
, and using the second of the first
‐
order conditions of
minimising SSR.
Which of the following models is (are) nonlinear model(s)?
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 One '11
 yang
 var, Yi, birth weight, SRF

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