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7. Specification

The formulation of a satisfactory econometric model is very important if we are to draw any conclusion from it. Sometimes the underlying theory of the model gives us guidelines on how it should be specified, but in other cases we have to rely on statistical tests. In this chapter we will discuss the most important issues related to the formulation and specification of an econometric model.

7.1. Choosing the functional form

To choose a correct functional form is very important since it has implication on the interpretation of the parameters that are estimated. When formulating the model we need to know how the coefficients are to be interpreted, and how the marginal effect and elasticity looks like. Below we will go through the most basic functional forms and describe when they can be used.

7.1.1. The linear specification

When talking about a linear specification we have to remember that all models that we are talking about in this text are linear in their parameters. Any deviation from linearity may therefore only be related to the relation between the variables. The linear specification is appropriate when Y and X has a linear relation. The econometric model would then be expressed in this way:

For simplicity reasons we express the model as the simple regression model. With this specification the interpretation of the slope coefficient coincide with the marginal effect, which is

This means that when X increases by 1 unit, Y will change by B1 units. Since it is expressed in units it is important to remember in what unit form the data is organized. If Y represents the yearly disposable income expressed in thousands of Euro and X represents age given in years, we have to understand that a unit change in X represents a year and the corresponding effect in the yearly disposable income is in thousands of Euros.

Since the unit change usually is dependent on the level of the dependent variable the marginal effect has some limitation. That is, the effect of a unit change in X may be different whether the level of X is 10 or if it is 10,000. Therefore economists usually prefer to analyze the elastic ties instead of the marginal effect, and in the linear model the elasticity is given by:

which usually is expressed using mean values of X and Y. The elasticity denoted by e is not expressed in terms of units, but instead expressed in relative terms. A 1 percent increase in X will result in e percent change in Y.

Example 7.1

Calculate the marginal effect and the elasticity using the regression results in Table 7.1 received from a sample of data using model (7.1).

Regression results from a model with a linear specification

Table 7.1 Regression results from a model with a linear specification

The marginal effect from the variable x on y using the linear specification is received directly from the parameter estimate. In this example we receive

Hence, when x increase by 1 unit, y increases by 8.6 units. The more important measure of elasticity is here given by:

That is, when x increases by 1 percent, the dependent variable y increases by 0.98 percent.

7.1.2. The log-linear specification

In the log linear specification the relationship between x and y is no longer linear and is written as

Several factors could motivate this specification. This specification is widely used in the human capital literature where economic theory suggests that earnings should be in logarithmic form when estimating the return to education on earnings. This could be motivated in the following way: assume that the rate of return to an extra year of education is denoted by r. Given an initial period earnings, w0 , the first year of schooling would generate an earnings equal to wi = (1 + r )w0. After 5 years of schooling we would have an earnings equal to: ws = (1 + r)s w0 . Taking the logarithms of this we receive

which is a log-linear relationship between years of schooling and earnings. With a similar motivation we could include several other variables, such as age and years of work experience which the theory states are important factors for the earnings generation. By including an error term we form a statistical model in the form of (7.4). How do we interpret the slope parameter? It is important to remember that we are primarily interested in the effect on earnings not the logarithm of the earnings. Hence, it is not possible or meaningful to say that a unit increase in 5 will result in a unit change in the logarithm of earnings. Taking the derivative of earnings with respect to schooling gives us the following expression:

Expression (7.6) shows us that the slope coefficient should be interpreted as the relative change in earnings as a ratio of the absolute change in schoolings. In other words, if schooling increase by one year, earnings will change by B1 x100 percent.

Using (7.6) we see that the marginal effect is given by:

Hence, the marginal effect is an increasing function of earnings itself. That is, if schooling increases by one year, earnings will change by Bl x ws units. Hence the response on the dependent variable will change in terms of unit, but is constant in relative terms.

Using (7.7) we can derive the earnings elasticity with respect to years of schooling:

The earnings elasticity is an increasing function of the number of years of schooling. Hence the longer you have studied the larger is the elasticity.

7.1.3. The linear-log specification

In the linear-log model it is the explanatory variable that is expressed and transformed using the logarithmic transformation which appears as follows

Taking the derivative of y with respect to x we receive:

Hence, the parameter estimate of the slope coefficient is the absolute change in y over the relative change in X, which is to say that if x increase by 1 percent, the dependent variable y will change by B1/100 units.

Using the expression for the coefficient we may write the elasticity as follows:

Hence the elasticity is a function of the dependent variable, and the larger the dependent variable is the smaller become the elasticity, everything else equal.

7.1.4. The log-log specification

The log-log specification is another important and commonly used specification that can be motivated by the economic model. The so called Cobb-Douglass functions are often used as production functions in economic theories. They are usually expressed as follows:

(7.10) is a commonly used production function that is a function of two variables; labor (L) and capital (K). This model is multiplicative and non linear in nature which makes it difficult to use. However, there is an easy way to make this model linear and that is by taking the logarithm of both sides. Doing that and adding an error term we receive:

Hence, the so called log-log specification requires that both left hand and right hand side of the equation are in logarithmic form, and that is what we have in (7.11). Furthermore, it is also linear in the parameters which make it easy to estimate statistically. How do we interpret these parameters? Let us focus on B1 when answering this question. Take the derivative of q with respect to l and receive:

The coefficients of the log-log model are conveniently expressed as elasticities. So the elasticity and the coefficients coincide. Remember that the elasticity is expressed in percentage and not in decimal form and hence should not be multiplied by 100.

The marginal effect of the log-log model can be received from (7.12) and equals:

which is a function of both q and l. Hence the marginal effect is increasing in q and decreasing in L, everything else equal.

 
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