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Eco 205: Econometrics

Any questions?

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population data point

Y

X

Observed Value

of Y for X3

X3

Population Linear Regression Model

Yi = b 0 + b 1X i + u

u

b 0X11

Y11

u11

pop slope = b 1

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chosen in sample

not chosen in sample

estimated error for X3(residual)

Y

X

Y3

Yi

=b0

+b1

Xi

X3

estimated slope =estimated

intercept =

Sample Regression Equation

u

u

b 0

b1

pop slope = b 1

b 0

Y3

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The OLS estimator solves

b 0 , b 1min[Yi - ( b 0

i=1

n

+ b 1Xi)]2

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CaliforniaTest Score/Class Size data

Interpretations

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Predicted values & residuals:

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OLS regression: STATA output

regress testscr str, robust

Regression with robust standard errors Number of obs = 420

F( 1, 418) = 19.26

Prob > F = 0.0000

R-squared = 0.0512

Root MSE = 18.581

-------------------------------------------------------------------------| Robust

testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval]

--------+----------------------------------------------------------------

str | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671

_cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057

-------------------------------------------------------------------------

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Measures of Fit

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TheStandard Error of theRegression (SER)

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Root Mean Squared Error (RMSE)

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R2 and SER Example

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The Least Squares Assumptions

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LSA #1: E(u|X = x) = 0

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LSA #2: (Xi,Yi), i= 1,,n are i.i.d.

LSA #3: E(X4) < and E(Y4) <

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OLS can be sensitive to an outlier

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Sampling Distribution of1b

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Some Preliminary Algebra

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b 1 - b 1 =

(Xi - X)

i=1

n

(ui - u )

(Xi - X)2

i=1

n

=

(Xi - X)

i=1

n

ui

(Xi - X)2

i=1

n

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Now we can calculate E( ) and var( )1b1b

E[ b 1 ] = E[b 1] + E (Xi-

X)i=1

n

ui

(Xi - X)2

i=1

n

E[ b 1 ] = b 1 + E(X

i- X)E[

i=1

n

u iXi ]

(Xi - X)2

i=1

n

= ?

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Next calculate var( )1

b

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The larger the variance of X, thesmaller the variance of

1

b

There are the same number of black and blue dots using whichwould you get a more accurate regression line?

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What is the sampling distribution of ?1

b

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We are now ready to turn to hypothesis tests & confidence

intervals