7.) Specification: Choosing a Functional Form Flashcards
What usually dictates the form of a regression model?
Theoretical considerations
The basic technique in deciding on a functional form is…
To choose the shape that best exemplifies the expected underlying economic or business principles and then to use the mathematical form that produces that shape.
An estimate of beta naught has at least three components…
- ) The true beta naught
- ) the constant impact of any specification errors (an omitted variable, for example), and
- ) THe of the error term for the correctly specified equation (if not equal to zero)
Suppressing the constant term leads to a violation of the Classical Assumptions. This is because …
Classical Assumption II (that the error term has an expected value of zero) can be met only if the constant term absorbs any nonzero mean that the observations of the error term might have in a given sample.
If you omit the constant term…
then the impact of the constant is forced into the stimates of the other coefficients, causing potential bias.
There are at least the two following reason that the intercept term should not be relied on for purposes of analysis or inference.
- ) The error term is generated, in part, by the omission of a number of marginal independent variables, the mean effect of which is placed in the constant term.
- ) The constant term is the value of the dependent variable when all the independent variables and the error term are zero, but the variables used for economic analysis are usually positive.
An equation is linear in the coefficients only if…
the coefficients (the betas) appear in their simplest form - they are not raised to any powers (other than one), are not multiplied or divided by other coefficients, and do not themselves include some sort of function (like logs or exponents).
Linear regression analysis can be applied to an equation that is nonlinear in the variables as long as…
the equation is linear in the coefficients.
The linear regression model is based on the assumption ..
that the slope of the relationship between the independent variable and the dependent variable is constant.
Since the slope is constant, the elasticity of Y with respect to X…
(the percentage change in the dependent variable cause by a 1-percent increase in the independent variable, holding the other variables I the equation constant be calculated fairly easily.
The double-log form is the most common functional form that is…
nonlinear in the variables while still being linear in the coefficients.
In the a double-log form…
the natural log of Y is dependent variable and the natural log of X is the independent variable.
The double-log form is often used because…
a researcher has specified that the elasticities of the model are constant and the slopes are not.
In a linear model the slopes are constant but..
the elasticities are not.
Only run double log models when…
logged variables are positive.
The semilog functional form Is…
a variant of the double-log equation in which some but not all of the variables (dependent and independent) are expressed in terms of their natural logs.
The semilog function should be used when…
the relationship btw X1 and Y is hypothesized to have this “increasing at a decreasing rate form.”
An example of the application of the semilog function would be…
consumption functions bc they tend to increase at a decreasing rate past some level of income. These Engel Curves tend to flatten out because as incomes get higher, a smaller percentage of income goes to consumption and a greater percentage goes to saving. S
The lnY semilog function is perfect for any model
in which the dependent variable adjusts in percentage terms to a unit change in an independent variable.
The alternative semilog function is to have…
the log on the left hand side of the equation. This would mean that the natural log of Y would be a function of unlogged values of the Xs.
Polynomial functional forms express Y as…
a function of independent variables, some of which are raised to powers other than 1.
With polynomial regressions, the interpretation of the individual regression coefficients becomes …
difficult, and the equation may produce unwanted results for particular ranges of X.
The inverse functional form expresses Y as…
a function of the reciprocal (or inverse) of one or more of the independent variables.
The best way to choose a functional form for a regression model is..
to choose a specification that matches the underlying theory of the equation.
The inverse (or reciprocal) functional form …
should be used when the impact of a particular independent variable is expected to approach zero as that independent variable approaches infinity.
What is a lag?
The length of time between a change in the independent variable and the resulting change in the dependent variable.
The estimated coefficient of a lagged X measures…
the change in this year’s Y attributed to a one-unit increase in last year’s X (holding constant the other Xs in the equation).
What is a distributed lag?
If the lag structure is hypothesized to take place over more than one time period, or if a lagged dependent variable is included on the right-hand side of an equation, the question becomes significantly more complex.
Use a dummy variable or intercept dummy to…
change the constant or intercept term, depending on the whether the qualitative condition is met.
The omitted condition..
Is the event not explicitly represented by a dummy variable and forms the basis against which the included conditions are compared.
An interaction term is an…
independent variable in a regression equation that is the multiple of two or more independent variables.
Interaction terms are used when…
the change in Y with respect to one independent variable depends on the level of another independent variable.
Slope Dummy Variables
allow the slope of the relationship between the dependent variable and an independent variable to be different depending on whether the condition specified by a dummy variable is met.
