CHAPTER 7

Modeling Issues and Endogeneity

Touro shares an anecdote with us. His company sent him to China last week to carry out a study on demand for travel from Chinese residents. However, his colleagues in China warned him that China’s data have a great deal of measurement errors. He is wondering how we can control for this problem. Prof. Metric tells him not to worry, because we will discuss the issue this week. He says that upon finishing with this chapter, we will be able to:

We will discuss the modeling issues first because some of them are related to the endogeneity problem.

Modeling Issues

Model Specification

Unless we develop an econometric model based on a theoretical model, there will always be the possibility of having fewer or more variables than needed.

Omitted Variables

An omitted variable will significantly bias the regression coefficients. Given a model

If all coefficient estimates of this model are significant, but we accidentally omit x_i2, then there are two consequences:

For example, given the model:

where DUR is durable expenditures, WAGE is average weekly wage, ASSETP is average asset price, and DURP is durable price. If a₁, a₂, a₃, and a₄ are all significant, but we accidentally eliminate ASSETP, then there is an omitted-variable problem.

Irrelevant Variables

Including irrelevant variables will not significantly bias the regression coefficients, but the Ordinary Least Squares (OLS) procedure may provide incorrect variances of the coefficient estimates, so the tests are less reliable as discussed in Kmenta (2000). For example, if we accidentally add average stock price (STOCKP), having forgotten that we already included it in the ASSETP variable sometime in the past, then TOCKP is an irrelevant variable to Equation (7.2) and the model becomes:

The presence of the irrelevant variable, STOCKP, will not bias the coefficient estimates of the relevant variables, but their variances might be incorrect, so the test results will be less reliable.

We now see that choosing a correct model is crucial. Prof. Metric says that we might want to use a piecewise-downward approach starting from all theoretically possible variables with all available data. We then use F- and t-tests to eliminate the highly insignificant variables. He says that we can also use a piecewise-upward approach, which starts from a single explanatory variable. However, the downward approach is preferable, because this approach avoids the omitted variable problem that might arise if you use the piecewise-upward approach.

The Effect of Scaling the Data

Prof. Metric points out three cases of scaling the data.

We then work on an example of a spending model:

SPEND = 61.7 + 12.4 INCOMER² = 0.685

(se) (8.76)(3.52)

where income (INCOME) is in hundreds of dollars and spending (SPEND) is in dollars.

In this case, the two-tail t-statistics for INCOME is 3.52 (= 12.4/3.52).

Hence, spending changes by $12.40 when income changes by $100.

If income is in dollars instead of hundreds of dollars, then we have

SPEND = 61.7 + 0.124 INCOMER² = 0.685

(se)(8.76)(0.0352)

In this case, the two-tail t-statistics for INCOME is still 3.52 (= 0.124/0.0352), and spending changes by 12.4 cents when income change by $1 (the same as spending changes by $12.40 if income changes by $100).

Using Nonsample Information

Nonsample information is over and above the information contained in the sample observations. One popular application in econometrics is the use of Constant Returns to Scale (CRTS) to improve the precision of the estimated coefficients.

A production function (F) exhibits CRTS when it shows a constant ratio between inputs and outputs. In other words, CRTS occurs when output shows the same proportional change as all inputs proportional changes. Algebraically, for a constant c > 0, the equation for CRTS is

Y(cK, cL) = cY(K, L)

where Y denotes output, K is capital, and L is labor. An example of a Cobb-Douglas production function in CRTS is

Taking the logarithm of both sides yields:

LnY = Lnc + aLnL + bLnK = d + aLnL + (1 − a)LnK,where d = Lnc

LnY = d + aLnL + LnK − aLnK = d + a(LnL − LnK) + LnK

Hence, we can estimate the following equation:

Once we find a, we can calculate b = 1 − a.

Prof. Metric says that in addition to improving the precision of the estimated coefficients, the use of nonsample information might help correct other problems such as multicollinearity or volatility of the data because the logarithmic function is less volatile.

Cases of Endogeneity

The Issue

Prof. Metric says that in the previous chapters, we assume that the x’s are not random. If this assumption is violated, then we have an endogeneity problem, which is also called the random-regressor problem. Given the equation:

If x_i2 is random, then x_i2 might be correlated with e_i, that is, x_i2 is a random regressor and is said to be endogenous. In this case, the model has an endogeneity problem. There are two main consequences of the random-regressor problem:

Cases in Which x and e Are Correlated

There are many cases in which x and e are correlated.

Omitted Variables

A model might have an omitted variable or several omitted variables that will cause biased coefficient estimates. Omitted variable bias comes in two forms. The first one is called “theoretically driven bias” by Edwards (2014). Prof. Metric offers an example of this bias: We know that wage often depends on education (EDU)

However, IQ might affect WAGE_i, so IQ_i lies in the error term e_i. Also, since some people who have a higher IQ might also have a higher education, EDU_i is correlated with e_i.

The second case is called the “statistical omitted variable bias” in Edwards (2014). For example, giving an estimated model

In this case, w is a direct function of x and so is correlated with x and y. If we estimate Equation (7.7b) without w, then w lies in the error term e_i which will be correlated to x.

Autocorrelation in Lagged-Dependent Model

In Chapter 5, we learned about the model with lagged-dependent variables:

y_t = ay_t−1 + e_t, so y_t−1 = ay_t−2 + e_t−1

If the error has an autocorrelation problem, then:

Since y_t−1 is correlated with e_t−1, which is also correlated to e_t as shown in equation (7.8), y_t−1 is correlated with e_t as well.

Measurement Error (Attenuation Bias)

Prof. Metris says that we now come to the case asked by Touro at the beginning of the chapter. Suppose the estimated equation includes x* which is measured with some error:

We do not know x*, so we write the correct variable, x, as:

where u is some error.

Combining Equations (7.9) and (7.10) yields

where e = (v − a₂u)

Because x* includes x and some error u, which belongs to a composite error e, x* is correlated with e, that is, the covariance of x and e is different from zero. Hence, the measurement error creates an endogeneity problem called attenuation bias.

Simultaneous Equation Bias

In the supply-demand model, Q_d = Q_s = Q in equilibrium, so the demand function can be written as having quantity (Q) dependent on price (P) and income (Y):

Alternatively, the inverted demand function has P dependent on Q and Y

However, the supply function has Q dependent on P and input price, for example, wage (W):

In practice, we should estimate these two equations in a system of simultaneous equations as follows:

It is clear from Equations (7.12b) and (7.12c) that P and Q are not exogenous (they depend on each other and must be determined simultaneously). Hence, the system has an endogeneity problem called the simultaneous equation bias.

Dealing with Endogeneity

Detecting Endogeneity

We learn that in order to detect endogeneity, we need to perform a Hausman test. Prof. Metric says that to perform the original Hausman test, we need to master knowledge of matrix algebra and know how to operate a sophisticated econometric package. Thus, he would rather teach us the modified Hausman test discussed in Kennedy (2008). The theoretical concept of the modified Hausman test is as follows:

The remaining part of x_i2 is explained by the residual v_i from the estimation:

Once an estimation of v_i is obtained from estimating Equation (7.14), it can be added to Equation (7.6) in the subsequent regression:

A t-test on the coefficient estimate of v_i is then performed. If this coefficient estimate is not significantly different from zero, then x_i2 is exogenous, and the model does not have an endogeneity problem.

Figure 7.1 Diagram for the modified Hausman test

Hence, the steps to perform the test are as follows:

Prof. Metric then gives us an example, “Suppose regressing Equation (7.15) in Step (ii) yields estimated coefficient of as ĉ = 3.2 with its standard error se (ĉ) = 4.5, and N − K = 34. Let’s find out whether the model has an endogeneity problem.”

We proceed with the problem and calculate t_STAT = 3.2/4.5 = 0.71. We choose α = 0.05 and look at the t-table for t_C = t_{(0.975, 34)} = 2.032. Since |t_STAT| < t_C, we do not reject the null and so c = 0, meaning that x_i2 is not correlated with the error and implying that the model does not have an endogeneity problem.

Correcting Endogeneity

We learn that that when an endogeneity problem exists, we can use an alternative method of estimation, the Method of Moments (MM). When all classic assumptions in linear regressions are satisfied, the MM procedure leads us to the OLS estimators. When one or more of the explanatory variables are random, the MM procedure leads us to the Instrumental Variable (IV) estimators that are asymptotically consistent in large samples.

The purpose of the MM estimation is to find a variable as a substitute for the endogenous variable x. Theoretically, the MM procedure will lead to estimators that satisfy the condition that cov(w,e) = 0. Empirically, this procedure will lead to estimators that almost satisfies this condition, that is, the method will minimize cov(w,e) as much as possible. The variable w is called the IV.

Suppose that x_i2 continues to be the endogenous variable as in the previous section, then:

The IV = w must satisfy two conditions:

Prof. Metric also says that in cross-sectional data, it is very difficult to choose an IV that satisfies both conditions. In time-series or panel data, we can choose a lagged value that is correlated to the endogenous variable as the IV so that the first condition is satisfied. Assuming no serial correlation, the second condition is automatically satisfied by the classic assumptions, that is, the IV is in a lagged period, so it is not correlated to the error of the current period. The IV estimation is performed in two stages and so is also called two-stage least squares (2SLS) estimation. We learn that we can have more than one IV in performing a 2SLS procedure.

Prof. Metric also reminds us that the lags must be close enough in time to serve as acceptable IVs. For instance, if data are collected every five years, it is unlikely that the first lag will be a good instrument for the current variable because the lagged dataset and the current dataset were collected five years apart. However if the data are collected annually, then it’s possible that the lagged variable will be a good instrument for the current variable.

Using 2SLS in Single Equation Estimations

For a single equation model, the procedure is as follows:

First stage: Perform a regression of the endogenous variable x_i2 on all exogenous variables, including the IV, which is w_i:

Prof. Metric says that Equation (7.16) is called the reduced-form equation.

Second stage: Estimate Equation (7.6), which is called the structural equation, with x_i2 replaced by the predicted value of x_i2, which is obtained from the reduced-form estimation of Equation (7.16).

The problem is solved because we use w_i as IV for x_i2, and w_i is not correlated with e_i, so is not correlated with e_i either, that is, , is exogenous.

Another issue is the standard errors, which are calculated from the residuals of the regression on Equation (7.17)

This error term is incorrect because it is different from the original in Equation (7.6)

Most econometric software will automatically provide the correct standard errors based on e_i, as long as we select the command “2SLS” or “IV” estimation. Unfortunately, Excel does not have this feature, so Prof. Empirie will teach us how to obtain these correct standard errors in the Data Analysis section.

Using 2SLS in System of Equations Estimations

There are two modifications needed for using 2SLS procedure on a system of equations. The first is called the “identification condition,” which requires that there is at least (M − 1) variables excluded from the other equation in a system of M equations. Detailed explanations of the theoretical intuition behind this condition are in Hill, Griffiths, and Lim (2011). Going back to System (7.13), we see that the first equation has the variable Y excluded from the second equation, which has the variable W excluded from the first equation, so System (7.13) satisfies the identification condition.

The second modification is that the reduced form in the first stage includes all exogenous variables from both equations, in addition to, the IVs for the system. Hence, in System (7.13), the reduced form is:

where IV_j, j = 1,2,…, j, is one of the IVs for the system.

Obtaining the predicted value of P from estimating Equation (7.18), we should substitute it into each of the two structural equations in System (7.13) to perform the second stage of the IV estimation.

Prof. Metric also tells us that the 2SLS procedure does not yield a reliable R² value, so Greene (2012) recommends that we use the Root Mean Square Error (RMSE) as the alternative measure of goodness-of-fit. This is true for all 2SLS procedures, regardless of a single equation or a system of equations estimations. He reminds us that discussions of the RMSE concept are in Chapter 6 of this textbook.

Data Analyses

Nonsample Information

Prof. Empirie says that we will learn to use nonsample information to correct the problem of multicollinearity between two explanatory variables. She tells us to go the file Ch07.xls, Fig.7.2, where data for a production process include logs of capital (LnK), labor (LnL) and output (LnY) are in millions of dollars. We first examine the correlation between LnK and LnL by performing the following steps:

Figure 7.2 Correlation analyses: Original results and correction

The results are displayed in Cells J1 through L3 of Figure 7.2 and reveal that there is a multicollinearity problem between the two variables.

We then use the nonsample information of CRTS to change one variable to Ln(K/L) and perform another correlation analysis between Ln(K/L) and LnL:

The results are displayed in Cells N1 through P3 of Figure 7.2 and reveal that there is no multicollinearity problem between Ln(K/L) and LnL. Hence, we can proceed to perform the regression of the following model:

Once the coefficient b is obtained from the regression, the coefficient a can be calculated by writing a = 1 − b.

Endogeneity

Testing Endogeneity

Data on sale values (DEM_t), values of promotion (PROM_t), income (INC_t), and advertisement expenditures (ADS_t), are from the file Ch07.xls, Fig.7.3-7.5. The model is:

DEM_t = a₁ + a₂INC_t + a₃PROM_t + e_t

We suspect that PROM_t is endogenous and wish to use ADS_t-1 as an instrument variable (IV) for PROM_t. Since most companies use promotion sales to support their advertisements, PROM_t is likely correlated with ADS_t-1 and will be examined later. Since ADS_t-1 is in period (t − 1), it is not correlated with e_t by the classic assumptions. This might make ADS_t-1 an acceptable IV for PROM_t. To perform the modified Hausman test for endogeneity, we first perform a regression of PROM on ADS and INC:

Next, we perform the regression of the original equation with the Residuals added:

Figure 7.3 Qualification of ADS_t−1 as instrument variable

Figure 7.4 Main results for modified Hausman test

The main results are displayed in Figure 7.4.

The results reveal that the coefficient estimate of , called “Residuals,” is weakly significant with a p-value of 0.06, so endogeneity problem exists although it is not too serious.

Correcting the Endogeneity Problem

The results in Figure 7.3 also support our argument that ADS_t−1 is correlated with PROM_t because the coefficient estimate of ADS_t−1 has the p-value = 0.0003 < 0.05. Additionally, by the classic assumptions, ADS_t−1 is not correlated to e_t. Therefore, ADS_t−1 satisfies both conditions to be an acceptable IV.

From the results, the estimated equation is

Figure 7.5 Regression results for endogeneity correction

As discussed in the theoretical section, we still must obtain the correct standard errors. For your convenience, we have copied and pasted the original data into the file Ch07.xls, Fig.7.6-7.7. In this file, we also have the aforementioned results in Cells I1 through O17. You should perform the following steps to obtain the correct standard errors for the slope estimates:

The results are reported in Figure 7.6.

Figure 7.6 Obtaining correct standard error for the model

Figure 7.7 Obtain correct standard errors and p-values for slopes

The results are reported in Figure 7.7 with correct values in Cells K20 through M21.

We now can use these correct values for the t-tests, F-tests, interval estimations, and interpretations of the coefficient estimates.

Exercises