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NOTATION & ASSUMPTIONS 2 Y i = 1 + 2 X 2i + 3 X 3i + U i Zero mean value of U i No serial correlation Homoscedasticity Zero covariance between U i and X i No specification bias No exact collinearity between the X variables

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ESTIMATION 3

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BETA COEFFICIENTS 4 Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x

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VARIANCE AND STANDARD ERROR 5

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THE T TEST 6

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ONE-SIDED ALTERNATIVES 7 y i = 0 + 1 x i1 + … + k x ik + u i H 0 : j = 0 H 1 : j > 0 c 0 Fail to reject reject

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TWO-SIDED ALTERNATIVES 8 y i = 0 + 1 X i1 + … + k X ik + u i H 0 : j = 0 H 1 : j > 0 c 0 -c reject fail to reject

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THE COEFFICIENT OF DETERMINATION A MEASURE OF “GOODNESS OF FIT” 9 Verbally, R-square measure the proportion or percentage of the total variation in Y explained by the regression model. Verbally, R-square measure the proportion or percentage of the total variation in Y explained by the regression model.

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ADJUSTED R-SQUARED 10 Recall that the R2 will always increase as more variables are added to the model. The adjusted R2 takes into account the number of variables in a model, and may decrease.

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CONT.. 11 It’s easy to see that the adjusted R2 is just (1 – R2)(n – 1) / (n – k – 1), but most packages will give you both R2 and adj-R2. You can compare the fit of 2 models (with the same y) by comparing the adj-R2. You cannot use the adj-R2 to compare models with different y’s (e.g. y vs. ln(y)).

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THE F STATISTIC 12 0 c f( F ) F reject fail to reject Reject H 0 at significance level if F > c

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