The true regression is given by the following equation: However, you omit x2 and
ID: 1162654 • Letter: T
Question
The true regression is given by the following equation: However, you omit x2 and x3 and estimate the following regression instead: The relationship between the omitted variables and x1 is given by the following equations: (a) Find the estimated values of ?? and in terms ofthe true ?'s. What is the omitted variable bias term forA;? What is the omitted variable bias term for ßi? show your work. (b) Ify0, then how much bias in Bi is generated by omitting X?? Explain. (c) Suppose that the true effects of x2 and x3 on y are given by B2 0.6 and P3 -0.3. You observe that the relationships between these variables and x1 are given by o1 0.25 and 0.5. What is the predicted bias for B1? Explain your answer as clearly as possible.Explanation / Answer
A linear regression model with two predictor variables can be expressed with the following equation:
Y = B0 + B1*X1 + B2*X2 + e.
The variables in the model are:
The parameters in the model are:
One example would be a model of the height of a shrub (Y) based on the amount of bacteria in the soil (X1) and whether the plant is located in partial or full sun (X2).
Height is measured in cm, bacteria is measured in thousand per ml of soil, and type of sun = 0 if the plant is in partial sun and type of sun = 1 if the plant is in full sun.
Let’s say it turned out that the regression equation was estimated as follows:
Y = 42 + 2.3*X1 + 11*X2
Interpreting the Intercept
B0, the Y-intercept, can be interpreted as the value you would predict for Y if both X1 = 0 and X2 = 0.
We would expect an average height of 42 cm for shrubs in partial sun with no bacteria in the soil. However, this is only a meaningful interpretation if it is reasonable that both X1 and X2 can be 0, and if the data set actually included values for X1 and X2 that were near 0.
If neither of these conditions are true, then B0 really has no meaningful interpretation. It just anchors the regression line in the right place. In our case, it is easy to see that X2 sometimes is 0, but if X1, our bacteria level, never comes close to 0, then our intercept has no real interpretation.
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