CHAPTER 8

Nonlinearity and Limited Dependent Variables

Invo tells the class that he is estimating a model of investment in plots of land scattered throughout the suburban areas of the city. He is wondering whether we could learn more about the interaction of a dummy variable, such as a certain neighborhood containing a lot, with a continuous variable, such as the land size. Prof. Metric says that this represents a nonlinear relation that will be discussed in this chapter, and once we finish with it, we will be able to:

Prof. Metric says that we first learn more about slope dummy variables and other nonlinear models in the next several sections. He reminds us that we are going to discuss only the cases of nonlinear in determinants instead of nonlinear in parameters such as a parameter c = a^b that requires econometric software other than Excel to estimate the knowledge of calculus to draw an interpretation.

Slope Dummy Variables

In the previous chapters, we were introduced to intercept dummy variables in a linear context and simple nonlinear models in logarithmic forms that can be made linear by transforming the models into new ones. We will learn slope dummy variables now.

Cross Sectional Dummies

They are called multiplicative dummy variables because the dummies are multiplied with the other variables in the model, instead of being adding to the intercept. This will change the slope of the regression line, and this change in slope by multiplying two variables together creates a nonlinear model. Invo provides an example of investment in land. Let D = 1 represent a piece of land in an agricultural neighborhood and D = 0 otherwise, then the land price is expressed as:

where ACRE is the land size in acres.

The interpretation is:

Alternatively, we can apply Equation (3.3b) from Chapter 3 to have:

Suppose a₂ = b = 1.5, and PRICE is in tens of thousands of dollars, then the total price will increase $35,000 per additional acre for a piece of land in an agricultural neighborhood (= 2*10,000 + 1.5*10,000) but only increase $20,000 per additional acre for a piece of land in a nonagricultural neighborhood (=2*20,000).

Prof. Metric emphasizes that we can incorporate both the intercept and the slope into the model. For example, we can add an intercept dummy to Equation (8.1) and have

Suppose d = 3, then having a piece of land in a fertile area increases the land value by $30,000 regardless of the land size.

Interaction of Qualitative Factors

We can also have an interaction of qualitative factors; for example, being a Hispanic (HIS) and female worker (FEM) results in different wages compared to a White worker even with the same experience (EXP).

Then,

Prof. Metric reminds us that we can only add three dummies to Equation (8.3) because the number of groups is G = 4, but we need to keep the constant term for comparison purposes. This method is equivalent to the simple dummy technique introduced in Chapter 4 but different from the least Squares Dummy Variable (LSDV) technique introduced in Chapter 6, where the dummies only serve to control for the different characteristics in a panel dataset. Hence, we only have three additional parameters (b₁, b₂, and b₃) in Equation (8.3).

We can use qualitative factors for different regions. An example is the various regions of a country, so the equation for wage could be:

Then

We can also add intercept dummies if we suspect that there are differences in both, the intercept and the slope.

Testing for the Equivalence of Two Regressions

Assuming no heteroscedasticity, the Chow test, introduced in Chapter 6, can be applied, in this chapter, for the equivalence of two regressions. For example, we can perform a regression of Equation (8.3) as the unrestricted model and the restricted model is the one without the slope dummies. The test for joint significance of all three dummies has the following hypotheses:

H₀ : b₁ = b₂ = b₃ = 0; H_a : They are jointly significant.

Suppose that N = 800, then N − K = 795, J = 3, and F_(0.99,3,795) = 3.8. Suppose also that the SSE_U = 26,437 and SSE_R = 30,176, then

In this case, we reject the null hypothesis, implying that race and gender are jointly significant, holding all other variables constant.

We can also add the five dummies in Equation (8.4) to Equation (8.3) and perform a regression of Equation (8.3) as the restricted model. We then add five variables with dummy EAST and perform an unrestricted model:

Suppose that F_stat = 4.98, N = 700, and we choose the confidence interval of 5 percent, then the critical value is F_c (0.95, 5,690) = 2.54. Hence, we reject the null hypothesis and conclude that there is evidence of the difference between the East and the West. This implies that we should estimate Equation (8.5) instead of Equation (8.3).

Controlling for Time

We can use seasonal dummies in regressions. Touro offers an example about the effect of different seasons on demand for airline tickets. Since the summer and winter months are usually considered to be the peak times for air travel, we can include two dummy variables:

Then we can see whether demand for airline tickets increases in summer and winter.

We can also define a yearly dummy. For example, the U.S. government offered a tax credit of $8,000 to all new home owners in 2008, so we can generate a dummy variable, YEAR08 = 1 if year is 2008, YEAR08 = 0 otherwise. We then can examine the different effects of this tax cut on the economy in the following years.

Other Nonlinear Models

Interaction of Continuous Variables

So far, we have only generated the interaction of a dummy variable with a continuous variable. In practice, we sometimes need to generate the interaction of two continuous variables. Prof. Metric offers an example. Vu and Noy (2016) estimated the impacts of disasters on investment in Vietnam. A simplified version of their equation is introduced here

where INV is firm investment, DMS is the damage caused by disasters, which are measured as the ratio of people killed (KILP), or injured (INJP), or affected (AFFP) over a population.

The authors found that disasters have positive impacts on investment overall and further wished to examine the different effects on rural and urban firms in detail. Since measures for urban versus rural disaster and investment are not available, but data for urban and rural population by subregion are available, they decided to write the coefficient of each disaster measure as a function of urban and rural population:

where URB_it and RUR_it are the values for urban and rural population, respectively. Substituting Equation (8.7) into Equation (8.6) yields:

In this case, the coefficient b₂ measures the change in investment per urban population due to a 1 percent change in the ratio of each disaster measure such as KILP, INJP, and so on.

The authors found that a 1 percent increase in the ratio of people killed to population, resulted in an increase of investment, per urban population, by 49.66 million of Vietnam Dong (VND), and a p-value of 0.007 indicates that this estimate is statistically significant. However, they found that a 1 percent increase in the ratio of people killed to population, there is a decrease of investment for a rural area by 15.09 million VND, and a p-value of 0.025 indicates that it is statistically significant (2.5 percent significance). Hence, the overall effect is only correct for urban population.

Continuous Variables in Log-Log Models

Taila offers another example from her company which plans to bring some projects on foreign direct investment (FDI) into a developing country. To make sure that they make profits in this country, her boss asked her to carry out research on the income growth and FDI inflows into various regions of this country over time. Her equation of estimation is:

where GDP is income, FDI is the realized value of real FDI invested in this country, and CAP is investment in capital, all in millions of dollars. The variable LAB is the country’s labor force in thousands of workers. She explains that FDI affects income growth mainly through the productivity of labor as discussed in the existing literature, so she allowed the variable LAB to interact with FDI during the transition period from a developing to a developed economy.

From Equation (8.9), the aggregate effect of FDI on the income growth-holding other variables unchanged-can be expressed as:

Solving for a₃ in this equation yields:

which approximates the percentage change of labor productivity due to a change in realized value of real FDI by a million dollars.

Prof. Metric praises her for an excellent example and says that the interaction of two continuous variables is very common in empirical study.

Dummies in Log-Linear Models

It is a surprise to us to realize that the interpretation of the log-linear model introduced in Chapter 4 is only an approximation. The exact calculation is slightly different and based on our new knowledge of a nonlinear model. Prof. Metric tells us to go back to Booka’s example on book sales in Chapter 4 but writing the equation in log-linear form:

where D = 1 for the paperback books, and D = 0 for the hardback books. In this case

In this case, we will have two solutions for the difference in sale values of the books. The first solution uses the approximation introduced in Chapter 4. Suppose the estimation results are:

then there is roughly a 25.26 percent differential between the paperback and the hardback books, and we can write this approximation as:

To obtain an exact calculation, we let PB stands for a paperback copy and HB denotes a hardback copy. Algebraically, we have:

The inverse function of the one in Equation (8.16) is written as:

Hence, the percentage difference between the paperback copy and the hardback copy is:

Combining Equations (8.17) and (8.18) yields:

For the previous example:

From Equation (8.20), a paperback copy reduces the sale value exactly by 22.32 percent compared to the sale value of a hardback copy. Since the approximation figure in Equation (8.15) is 25.26 percent, there is a 2.94 percent difference between the two methods of calculation.

Choice Models

In everyday lives, we face many situations when we have to make choices. For example, a college graduate has to decide to get a job or to go to graduate school, or a business owner has to decide to invest in a new storehouse, a dining hall for the employees, or a new computer system, and so on. In each of these cases, the dependent variable assumes some discrete values. To make it easy to understand, we start with a linear probability model first.

Linear Probability Model

In this binary model, the dependent variable y takes a value of 1 if the event occurs and 0 other wise. Booka offers an example: her boss wants to make a decision on whether to buy a pickup truck or a van. Prof. Metric takes this example and tells us to write the following conditions:

If the probability of buying a van is p, then the probability of buying a pickup truck is (1−p). There are many factors that affect these probabilities. We just make it simple by assuming only one factor, the difference in prices of the two vehicles, so the explanatory variable is:

x = (Price of the van–Price of the pickup truck)

We now can build a linear probability model based on our knowledge that y = p + e, so:

Prof. Metric says that the error term of a linear probability model is heteroskedastic, so we should use Generalized Least Squares (GLS) estimation by transforming the model as so that we can estimate this transformed model by Ordinary Least Squares (OLS). He reminds us that the heteroskedastic issue was discussed in Chapter 4.

Probit and Logit Models

When we calculate predicted values of Equation (8.22), sometimes we obtain probability values that are less than 0 or greater than 1. This occurs because the slope in this model is a constant, so when x increases, the probability of buying a van continues to increase at a constant rate. This does not make sense, so an alternative approach is to use a probit or logit model.

The probit model confines the relationship within a standard normal cumulative distribution function, that is, an S-shape curve that is bounded between 0 and 1. Unfortunately, probit model estimation is complicated because it is based on normal distribution, and the knowledge of its cumulative distribution function requires calculus to understand. An alternative approach is to use the logit model.

This model only has a slightly different S-shape curve from the probit model but has a much simpler distribution function. If F is a logistic random variable, then its cumulative distribution function is:

Using the information from Equation (8.22), the probability that y takes the value of 1 is:

Equation (8.24) can be extended into the case for more than two choices and is called a multinomial logit model. Prof. Metric says that the probit and logit models are estimated using maximum likelihood estimations, which need special econometric software. For this reason, he only provides us a brief introduction, hoping that we will be curious enough to look for a book on the subject such as the one by Maddala (1986).

Polynomial Models

A general form of the polynomial model is

Since we can always generate w₂ = x², w₃ = x³, and so on, we will not have any problem using Excel to perform a linear regression of the model

Prof. Metric says that we only discuss the case of quadratic relationships that are used extensively in econometrics and that are easy enough to interpret without any knowledge of calculus. An example of a quadratic function is the marginal cost curve in a production process that implies that the production cost is increasing at an increasing rate due to the scarcity of the recourses. For the heuristic purpose, we focus on the quadratic term only. In this case, the estimated equation is

where MC is marginal cost and Q is quantity of production. Since we do not have knowledge of calculus, Prof. Metric tells us just remember to place number 2 in front of a₂, which is then multiplied by Q itself, for quadratic function to come up with the interpretation

As long as a₂ > 0, larger quantity of production will have larger slope and a larger estimated costs per additional quantity of production. For example, if a₂ = 0.015, the estimated cost per additional quantity for a $20,000 production level is $600 (= 2*0.015*20,000), and for a $40,000 production level is $1,200 (= 2*0.015*40,000).

Prof. Metric also reminds us to refer back to Chapter 7 on the omitted variable bias case for quadratic function and try to include x in addition to x². If coefficient estimate of x is also statistically significant, we need to involve both variables to avoid omitted variable bias.

Piecewise and Linear Spline Models

Prof. Metric says that nonlinear relationships sometimes are difficult to fit with a single parameter model or a polynomial model because the slope keeps changing in many directions. For example, we try to estimate how education expenditures depend on per capita income and find that the data seem to have changing slopes at three income levels: $40,000, $60,000, and $90,000. In this case, a piecewise model can be written as

where EDU is education expenditures and INC is per capita income, and each income level is entered into Equation (8.28) in the same manner. The dummy variable (INC > 40,000) takes on the value of 1 when INC > 40,000 and zero otherwise. Hence, for observations where INC exceeds 40,000, the model becomes

The point of separation in the piecewise regression system is called a knot, and the technique is called the piecewise regression. We can see that Equation (8.28) has three knots at (INC > 40,000), (INV > 60,000) and (INC > 90,000). The piecewise regression accounts for changes in slopes, each of which starts a new line. Nevertheless, the lines need not join at the knots. To force the lines to join, we need to eliminate several intercept-difference parameters. Giving a general model

y = a₁ + a₂ x, we can define a system with k knots as b1 . . . bk, then

Estimating this equation will yield a joined transition of the lines. Each of the terms of the form w = (x−b_(.)) will have the value w if w is positive, and 0 otherwise, and the technique is called linear spline regression.

Quadratic spline regression fits quadratic functions as in Equation (8.28) that are joined at a series of k knots. These functions will look smoother than the one in Equation (8.31) and follows the following form

Note again that our models only need linear in parameter, that is, we do not deal with the case of a model with a parameter c = a^b, so the case in Equation (8.31) is considers “linear” and belongs to the linear spline models.

Prof. Metric says that applications of the spline models will be discussed in Chapter 11 of this volume.

Data Analysis

Prof. Empirie tells us to go the file Ch08.xls, Fig.8.1, where data on revenues (REV) and (PRICE) of agricultural products were provided by three companies in different regions of the city: North, South, and West. We generate the interactions of the South Region dummy (D_s) and the North Region dummy (D_N) with PRICE to obtain D_S*PRICE and D_N*PRICE. We then perform the regression of REV on D_S*PRICE and D_N*PRICE:

The main results are displayed in Figure 8.1.

Figure 8.1 Interaction of dummies and continuous variables: Main results

Since revenue equals the product of quantity and price, we see that those agricultural products appear to be inelastic because increases in prices raise revenue, implying that the negative changes in quantities are less than the positive changes in prices. All coefficients are highly significant. The coefficient estimate of PRICE is for the base group, in this case the West Region. The coefficient estimate for the South Region is 4.67 (= 2.26 + 2.41), implying that the agricultural products of the South are even more inelastic than those of the West. The coefficient estimate for the North Region is 0.70 (= 2.26 − 1.56), implying agricultural products of the North are more elastic than those of the base group. However, it is still inelastic because revenue still increases.

We then examine a model with the interaction of continuous variables in a log-log model. Taila offers a dataset from her research on capital and profits for her company, which has been investing in several developing countries. The dataset is in the file Ch08.xls, Fig.8.2 and contains data on log of profit (Ln PRO), log of capital (Ln CAP), and an interaction term of interest rate (INT) and Ln CAP. The interaction term reflects a fact that, in these developing countries, the central government has retained some control on the capital market.

LnPRO_it = a₁ + a₂LnCAP_it + a₃INT * LnCAP_it + e_it

We perform the regression as follows:

Figure 8.2 Interaction of two continuous variables: Main results

The main results are displayed in Figure 8.2.

Again, all coefficients are highly significant. The coefficient estimate of LnCAP shows that capital accounts for roughly 11 percent of the change in profits. The effect is positive and statistically significant.

The indirect effect of CAP on the company’s profits, holding other variables unchanged, can be expressed as:

lnPRO = a₃INT * LnCAP | holding others constant

Solving for a₃ in this equation yields:

which approximates the percentage change of capital productivity due to a 1 percent change in real interest rate. Since the coefficient estimate is negative and statistically significant, it implies that a 1 percent increase in the real interest rate decreases capital productivity by 2.89 percent.

Finally, we perform a linear probability model. Prof. Empirie tells us to go to the file Ch08.xls, Fig.8.3 provided by Booka on her company’s decision on buying (BUY) a pickup truck or a van as written in Equation (8.21). We again make it simple by assuming only one factor, the difference (DIF) in prices of the two vehicles, affect this decision:

DIF = (Price of the van–Price of the pickup truck)

Hence, the linear probability model based on our knowledge is:

BUY = a₁ + a₂DIF + e

To save time, Booka has transformed the model by dividing all variables by to correct for the heteroskedastic problem so that we can estimate this transformed model by OLS.

We proceed to perform the regression as follows:

The main results are displayed in Figure 8.3.

From this figure, the probability is positive and statistically significant, implying that an increase in the difference in price between the van and the pickup truck increases the probability of buying the pickup truck. Touro exclaims, “I see a negative probability in Cell G21 and a greater than one probability in Cell G23.” Prof. Empirie praises him for a keen observation and says that these are invalid values for a probability function. However, if Excel is the only econometric tool we have at hand, at least the results give us an approximation of the probability. For example, if the unit of the prices is in thousands of dollars, then a $1,000 increase in the difference in price between the van and the pickup truck increases the probability of buying the pickup truck roughly by 1.3 percent for Booka’s company.

Figure 8.3 Linear probability model: Main results

Exercises

Table 8.1 Estimation results for Exercise 3

	Coefficients	Standard error	t Stat
Intercept	4.0536	1.1165	5.477
Squaft	0.0671	0.0203	3.3051
Volcan	−0.1056	−0.0329	3.201