# Expectation_ Unbiased And Variance for β1

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Sub: Statistics                                                                               Topic: Regression

Expectation, Unbiased and Variance for β1

Question:
Consider the standard simple regression model y= β0 + β1x +u under the gauss-Markov assumptions
SLR.1 through SLR.5. the usual OLs estimators                  and       are unbiased for their respective population

parameters. Let β1 be the estimator of β1 obtained by assuming the intercept is zero.

i) Find E (β1 )in terms of the xi, β0 and β1 . verify that β1 is unbiased for β1 when the population
intercept (β0 ) is zero. Are there other cases where β1 is unbiased?

ii) Find the variance of           .

iii) Show that var           ≤ var       .

Comment on the tradeoff between bias and variance when choosing between                                and        .

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Sub: Statistics                                                                                Topic: Regression

Solution:

(i)    If the original model is                                      , then the OLS estimate of parameters will be

obtained by solving the normal equations..

which comes out as                                            and

Now, since we have to fit the model without intercept, the model will be

The estimator will be obtained by solving the normal equation which is

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Sub: Statistics                                                                               Topic: Regression

Taking expectation over           keeping x as constant, we get

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Sub: Statistics                                                                               Topic: Regression

(iii) Since we know that                                          and from the hint                                                .

So,

=             .

(iv) For fixed n,         bias in                                                        which increases with increase

in                     i.e.    (keeping                    constant) since n is constant. But as                inceases, the

variance of         increaeses relative to variance of                    .The bias in            is also small when              is

small.Therefore, whether we prefer
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