Multiple Regression Analysis. y = b 0 + b 1 x 1 + b 2 x 2 + . . . b k x k + u 4. Further Issues. Redefining Variables.
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y = b0 + b1x1 + b2x2 + . . . bkxk + u
4. Further Issues
log(wage)= 0 +1(educ)+2(exper)+3 (exper)2
In this particular specification we have an example of a log specification with a quadratic term--both are examples of nonlinearities that can be introduced into the standard linear regression model
1. If the model is ln(y) = b0 + b1ln(x) + u, then b1 is an elasticity. e.g. if we obtained an estimate of 1.2, this would suggest that a 1 percent increase in x causes y to increase by 1.2 percent.
2. If the model is ln(y) = b0 + b1x + u, then b1*100 is the percent change in y resulting from a unit change in x. e.g. if we obtained an estimate of 0.05, this would suggest that a 1 unit increase in x causes a 5% increase in y.
3. If the model is y = b0 + b1ln(x) + u, then b1/100 is the unit change in y resulting from a 1 percent change in x. e.g. if we obtained an estimate of 20, this would suggest that a 1 percent increase in x causes a 0.2 unit increase in y.
What types of variables are often used in log form?
*Variables in positive dollar amounts
*Variables measuring numbers of people
-school enrollments, population, # employees
*Variables subject to extreme outliers
What types of variables are often used in level form?
*Anything that takes on a negative or zero value
*Variables measured in years
Now the effect of an extra unit of x on y depends in part on the value of x. Suppose b1 is positive. Then if b2 is positive, an extra unit of x has a larger impact on y when x is big than when x is small. If b2 is negative, an extra unit of x has a smaller impact on y when x is big than when x is
Example: suppose the model can be written: y = b0 + b1x1 + b2x2 + b3x1x2 + u
2. y=0+1x+2x2+ u
We say that (1) is nested in (2); alternatively, (1) is a special case of (2). With a t-test on 2 we can choose between these two models (if reject null of 2=0, we pick model 2). For multiple exclusion restrictions can use F-test.
2. y=0+1x+ 2x2+
and we want to obtain a prediction of y for specific value of the x’s. In general we can obtain predictions of y
by plugging values of the actual x’s into our fitted model.
1. Variance due to the sampling error in prediction (because yhat based on estimated coefficients)
2. Variance in the error of the population