Functional Forms
(many relationships we want to measure are not linear)

Reciprocal Transformation: Y = b 1 + b 2(1/X) + u
Good for demand curves. Y approaches b 1 as X becomes large and you can see if it crosses the X axis.

Polynomial: Y = b 1 + b 2 X +b 3 X2 + ….. +b k+1 Xk
k=2: quadratic; k=3: cubic
Drawbacks: lose degree of freedom for each extra term & high correlation between Xs.

Interaction Terms: Marginal Effect of one explanatory variable depends on another variable.
b 1 depends on another variable. b 1 = a + a 2Z
So, Y = b 0 + b 1 X = b 0 + (a + a 2Z) X = b 0 + a X+ a 2Z X.
Null Hypothesis that there is no interaction: H0: a 2=0. Do a t-test on a 2.

Spurious Non-Linearities: If you have a model with several independent variables (Xs), it's dangerous to graph Y against a single X to determine the relationship.

Lags: Y is a function of past Xs or past Ys
Lags may lead to biased results because including them violates assumptions 3.4 and 3.6.
If you lag the first value of a series, you may need to reset the sample range.
Leads to high degree of multicollinearity.

Log-Linear: Yt = (1+g) Yt-1 è Yt = Y0 (1+g)t or lnYt = lnY0 + t*ln(1+g) = a + b t
a =lnY0 and b =ln(1+g). Knowing this, you can solve for Y0 and g.
Use for growth rates (when you want to estimate g)
The coefficient on t is the %change in Y that a unit change in t causes.

R2: Do NOT compare R2 values when the dependent variables are different. Need to make the correction (see book or notes).

Log-Log: lnY = a + b lnX
b is the elasticity of Y with respect to X (the %change in Y due to a %change in X)
Use for: cobb-douglass production functions, demand functions.
To test if the elasticity = 1 …. H0: b =1. tc=(b ^ - 1)/SEB^

Logit: Use when the dependent variables need to be between 0 and 1.
 

Other things to study:
Properties of Logs
Be able to calculate slopes and elasticities.
 

Model Selection Methods

General to Simple (Hendry/LSE approach)
Start with a big general model that includes lags, logs, squares, etc and then do databased simplification.
Use T-tests and F-tests.
Start with alternative (the big general model) and ask whether the null is preferred.
 

Simple to General:
Good because if start with general, potentially start with lots of mulitcollinearity.
Uses LM test. (This test will be applied to other situations in the future)
Starts with the null applied (small model) and asks whether the alternative is preferred.

Simple Model: Y = b 0 + b 1 X1 + b 2 X2+ b 3 X3 + u
General Model: Y =b 0 + b 1 X1 + b 2 X2+ b 3 X3 + b 4 X4 + b 5 X5+ b 6 X6 + v
H0: b 4 = b 5 = b 6 = 0

Estimate the simple model. Obtain residuals (uhat).
Estimate uhat = b 0 + b 1 X1 + b 2 X2+ b 3 X3 + b 4 X4 + b 5 X5+ b 6 X6 (auxiliary regression)
Do LM test: LM = nR2 ~ X2 with the degrees of freedom equal to the number of restrictions.
If LM > LM* (which you can look up in the book), then Reject the null and conclude that at least one of the other variables (X4, X5, X6) should be included in the model.
Add variables to your simple model if their p-val < .5 in the auxiliary regression.
(At this point, be sure to keep all variables from your original simple model)
Do data based model simplification. (Eliminate variable with highest p-val until all are significant)

The book prefers Hendry/LSE approach because it is surer and does not depend on the arbitrary 0.5 rule of selecting variables from the auxiliary regression.

Parsimonious Specification is good because:
1.  Estimates are more precise because of reduced multicollinearity.
2.  Estimates are more reliable because there are more degrees of freedom.
3.  Power of the tests is greater.
4. A simpler model is more easily understood than a complex one.
 

 

DISCLAIMER – DISCLAIMER -- DISCLAIMER
These are the notes that I took as I was reading the chapter. They are not intended to be your sole study source. Rather, they can be used to highlight and outline the main topics presented in Ch. 6. If something doesn't make sense, be sure to read about it in the chapter.

Good luck studying and good luck on Thursday!