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Lecture 5

# Lecture 5 - LECTURE 5 THE CLASSICAL LINEAR REGRESSION MODEL...

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LECTURE 5: THE CLASSICAL LINEAR REGRESSION MODEL Suppose that we are interested in estimating β 0 and β 1 in the following model: Y i = β 0 + β 1 X i + ε i We may estimate the unknown β 0 and β 1 by OLS, that is, by forming X β ˆ Y β ˆ 1 0 = = i 2 i i i i 1 ) X X ( ) Y Y )( X X ( β ˆ If we want to say something about the stochastic properties of the OLS estimators of β 0 and β 1 (for example, to calculate their bias, variance, sampling distribution so that we can do statistical inference), we need to make specific assumptions about how the error terms are generated in the PRF. The following assumptions constitute the Classical Linear Regression Model (CLRM), which is also known as the Simple Linear Regression Model (SLRM) . ASSUMPTIONS OF THE CLRM (aka SLRM) 1. Linearity in the parameters. Y is generated as: Y i = β 0 + β 1 X i + ε i 2. X i variable(s) is nonstochastic (i.e., its value is a fixed number in repeated samples): Or If X is stochastic, then, in the sample not all X i ’s are equal to some constant 3. The expected, or mean, value of the disturbance term ε i is zero (if X is nonstochastic): E( ε i ) = 0

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OR The error term ε has zero conditional mean given X: E( ε i |X i ) = 0 4. The variance of each ε i is constant for all i, that is, ε i , is homoskedastic : If X is nonstochastic: var( ε i ) = σ 2 OR If X is stochastic: var( ε i |X i ) = σ 2 5. There is no correlation between two error terms. This is the assumption of no autocorrelation or no serial correlation : cov( ε i , ε j ) = 0 j i This will also be true if we assume that the pairs (X i , Y i ) are independent and identically distributed across i = 1, …, n. This assumption is equivalent to assuming that (X i , ε i ) are i.i.d. FROM NOW ON, WE WILL ASSUME THAT THE X’s ARE NON- STOCHASTIC.
EXPECTATION, VARIANCE AND STANDARD ERRORS OF OLS ESTIMATORS OLS Estimators are random variables (values change from sample to sample). We

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