Problems with Incorrect Functional Form•You cannot compare R2 between two

different functional forms. ▫Why? TSS will be different.

•One should also remember that an incorrect functional form may work within sample but have large forecast errors outside of sample.

Linear Functional Form

Y = β0 + β1 X1 + β2 X2 + εSlope = β1 Impact of X1 on Y is

independent of the quantity of X2.

Elasticity = β1 * [X1/ Y]

Double-Log Functional Form What if you wished to estimate the

following model? Y = β0 X1 β1 X2

β2

To make this linear in the parameters InY = β0 + β1 InX1 + β2 InX2 + ε Slope = β1 = ΔlnY / ΔlnX1 = [ΔY / Y] /

[ΔX1 / X1] What is this? The elasticity, which is

constant across the sample.

What is the slope in a double-functional form?

Slope = β1 * (Y/X) =

[ΔY / Y] / [ΔX1 / X1] * (Y/X) =

ΔY / ΔX Impact of X1 on Y depends upon the

quantity of X2

In other words, the slope of X1 varies across the sample.

Why would this be a realistic property?

Other Functional FormSemi-log functional formPolynomial FormInverse FormKnow the equation and meaning

of β1 for each of these forms.More specifically, know the

calculation of slope and elasticity for each functional form.

Problems with Incorrect Functional Form You cannot compare R2 between two

different functional forms. Why? TSS will be different.

An incorrect functional form may work within sample but have large forecast errors outside of sample.

Violation of Classical Assumption I: The regression model is linear in the coefficients, is correctly specified, and has an additive error term.

Testing for Functional Form The Quasi-R2

Box-Cox Test The MacKinnon, White, Davidson Test

(MWD)

Quasi R2

1. Estimate a logged model and create a set of LnY^ (predicted logged dependent variable).

2. Transform LnY^ by taking the anti-log. In Excel (@exp) is the function needed.

3. Calculate a new RSS with the results of step 2.

4. Calculate the quasi-R2 with the results of step 3.

The Box Cox TestCalculate the geometric mean of

the dependent variable in the model. This can easily be calculated in Excel

Create a new dependent variable equal to Yi / Geometric Mean of Y

Re-estimate both forms of the model, with your new dependent variable. Compare the Residual Sum of Squares. Lowest value is the preferred functional form.

MWD Test1. Estimate the linear model an obtain the

predicted Y values (call this Yf^).2. Estimate the double-logged model an obtain

the predicted lnY values (call this lnf^).3. Create Z1 = ln(Yf^) – lnf^4. Regress Y on X’s and Z1. Reject Ho (Y is a

linear function of independent variables) if Z1 is statistically significant by the usual t-test.

5. Create Z2 = antilog of lnf^ - Yf^6. Regress log of Y on log of X’s and Z2. Reject

HA (double-logged model is best) if Z2 is statistically significant by the usual t-tests.

Intercept Dummies

• What if you thought season of the year impacted your sales?

• Your demand function would include three dummies (why three) to test the impact of seasons.

• This type of dummy variable is called an intercept dummy, since it changes the constant term but not the slopes of the other independent variables.

Criteria for choosing a specification

1. Occam’s razor or the principle of parsimony - model should be kept as simple as possible.

2. Goodness of fit3. Theoretical consistency4. Predictive power: Within sample

vs. Out of sample

If you leave out an important variable a bias

exists unless…• The true coefficient of the

omitted variables is zero.• Or, there is zero correlation

between the omitted variable(s) and the independent variables in the model.

• If these conditions don’t hold, ommitted variables will bias the coefficients in our model.

What to do?

• Add the missing variable. • What if you do not know which

variable is missing? In other words, what if you suspect something is left out – thus producing “strange” results – but you do not know what?

Irrelevant Variables

• Including an irrelevant variable will– Increase the standard errors of the

variables, thus reducing t-stats. (think back to how standard errors are calculated)

–Reduce adjusted R2

– It does not introduce bias in the estimated coefficients, but does impact our interpretation of what we found.

Four Important Specification Criteria

• Theory: Is the variable’s place in the equation unambiguous and theoretically sound?

• t-Test: Is the variable’s estimated coefficient significant in the expected direction?

• Adjusted R2: Does the overall fit of the equation improve when the variable is added to the equation?

• Bias: Do other variables’ coefficients change significantly when the variable is added to the equation?

Specification Searches: Other issues

• Good idea to rely on theory rather than statistical fit.

• Good idea to minimize the number of equations estimated.

• Bad idea to do sequential Searches or estimate an undisclosed number of regressions before settling on a final choice.

• Sensitivity Analysis: Are your results robust to alternative specifications? If not, maybe your not finding what you think you are finding.