Endogeneity Bias

Module 4 Day 1 Session 1

Meaning• In traditional usage, a variable is said to be endogenous if it is

determined within the context of the model

• An Explanatory Variable is said to be endogenous if it is correlated with u

• If, however, E(u/x) = 0 holds, then the explanatory variables are exogenous

• Intuitive definition of Endogeneity• A situation where the “ceteris paribus condition” is not fulfilled

whenever the independent variable of interest is changed.

Source• In applied Econometrics, endogeneity can arise in three possible

ways (Wooldridge, 2002):

– Omitted Variable

– Measurement Error

– Simultaneity

• Note: the distinction between the three possible sources of bias is not sharp, generally an equation can exhibit more than one source of endogeneity

Omitted Variable Bias

• Omitted Variables appear when we would like to control for one or more additional variables but because of data unavailability we cannot include them in the regression model

• Omitted variable could be

– Observable

– Unobservable

Example• Impact of Monetary Policy shocks on Stock Returns

• When estimating the effect of monetary policy on stock prices, estimates may be severely biased due to omitted variable issues

• Caused by factors that influence both policy rates and stock prices which are commonly excluded from regression analysis

• Changes in policy rates and stock prices may be caused by other important news released earlier in the period which are omitted from regression, giving rise to the omitted variable bias.

• E.g. employment, inflation or economic activity reports etc.

Omitted Observable Variables

• Solution : Include all factors that are I. Important in explaining the dependent variable II. Associated with one of the explanatory variables, into the structural

equation. • In practical terms, this will usually require the researcher to conduct a thorough

review of the extant theoretical and empirical literatures to identify these joint determinants. (Note :Factors that are associated with both dependent and one or more explanatory variables are said to be ‘joint determinants’.)

• Once included in the structural model, the disturbance term is purged from the source of its correlation with the explanatory variables and the estimation of the parameters of interest should no longer be affected by endogeneity.

Omitted Unobservable Variables

• Since the researcher will not be able to gather data on omitted variables that are unobservable, the solution of including any joint determinants will no longer work.

• Two Ways to address to the problem

(1) Self-Selection(2) Firm-Specific Heterogeneity

Self-Selection• Self or sample-selection arises if the probability that a firm is included into the sample

and the dependent variable are both affected by an (omitted unobservable) variable.

• Sample is no longer random

• Example: whether a commitment to increased disclosure, as required under international standards, has tangible benefits in the form of lower cost-of-capital?

• Increased disclosure is equivalent to switching reporting regime

• Firms will decide on disclosure based on the expected consequences with regard to their cost-of-capital

• the factors that determine the disclosure choice are likely to also affect the dependent variable, current cost-of-capital

Example

• In most corporate governance studies, researchers study the largest listed companies

• this induces a sample selection bias in the model• The average characteristics of the sample will differ from that

of the population• Leads to incorrect measure of the relationship between

corporate governance and performance as the impact of corporate governance is confounded with the impact of the parameters that determine selection

Firm-Specific Heterogeneity.

• Unobserved omitted variables often represent features of the firm that are given and do not change over the period in question.

• Specifically, firm characteristics like managerial ability, structural arrangements, and employee skills can be thought of as roughly constant over time.

• If these firm characteristics impact on both the dependent variable and one or more explanatory variables, the structural disturbance will be correlated with those explanatory variables.

Example of Unobserved Firm Heterogeneity

• Consider two firms that are identical except that the owner of one of the firms has access to a superior monitoring technology.

• Under the optimal contracting regime, the owners with access to the superior monitoring technology will choose a lower level of managerial ownership to align incentives

• This firm will have a higher valuation because fewer resources will be diverted to managerial perquisites.

• If measures of the quality of the monitoring technology are omitted from the specification, a regression of firm value on managerial ownership will spuriously (and falsely) indicate a negative relation, because ownership is a negative proxy for the quality of monitoring technology.

Example of Unobserved Firm Heterogeneity

• Intangible assets provide an example of unobserved firm heterogeneity.

• Suppose two firms are identical except that one of the firms operates with a higher fraction of its assets in the form of intangibles.

• Under the optimal contracting regime, the owners of this firm will require a higher level of managerial ownership to align incentives because the intangible assets are hard to monitor

• This firm will also have a higher Q value because the market will value intangibles in the numerator (market value), but the book value of assets in the denominator will understate the value of intangibles (because Tobin's Q is measured as the ratio of the market value of the firm's outstanding debt and equity divided by the book value of assets).

• In this example, the unobserved level of intangibles induces a positive correlation between managerial ownership and Tobin's Q, but this relation is spurious, not causal.

Solution

• Proxy variable: for the firm characteristic and plug this into the structural equation

• Instrumental variable (IV) estimation: instruments might be available for those explanatory variables that are correlated with the unobservable firm characteristic

• Panel Data Techniques: fixed effects to account for firm-specific heterogeneity

Proxy Variable

• In corporate finance literature, moral hazard is a factor to be accounted for in a model measuring performance

• Why?• How do we measure moral hazard?• Common proxies for moral hazard include • Firm size• Capital intensity- hard capital• R&D intensity• Advertising intensity• Cash flow• Investment rate

Soft Capital

Instrumental Variable• The instrument used should have a high correlation with the explanatory

variable• The instrument and the error term should be uncorrelated• Slightest correlation between them would lead to a bias > bias in case of

OLS with endogeneity• In practise, it is extremely difficult to find such an instrument• Most empirical studies use “imperfect instruments”• Imperfect Instruments:

– either exogenous but having low correlation with the endogenous variable, termed as weak instruments

– Or having a high correlation with the endogenous variable, but not exogenous, termed as semi-exogenous or quasi-instrumental

Types of Instruments• Both instruments have their disadvantages

• Weak instruments often lead to inefficient as well as inconsistent estimators

• Example of weak instruments??

• Semi-exogenous instruments include lagged value of the endogenous variable, e.g. in the corporate governance model, for instance, lagged corporate rating could be used as an instrument

• Problem: performance in period (t-1) is related to corporate governance in period (t-1) and performance in period (t-1) is related to performance in period t. Degree of exogeneity depends on how much past performance affect current performance

Characteristics of a Good Instrument• Uncontroversial instruments are those which are determined from outside the

firm but affect the firms (Borsh-Supan and Koke 2002)

• Again using the corporate governance example, a good candidate would be country-level factors, like, country score on corporate governance practices, number of recommendations published for corporate governance practices in a year

• This measure should be highly correlated with the endogenous variable, assuming that implementation in the country is strong

• Exogeneity ensured by the fact that companies do not generally influence such recommendations

• Size or wealth effect may have an effect though

Example

• Is nominal exchange rate affected by the sterilized interventions in foreign exchange market?

• The answer to this question cannot be conclusively obtained due to endogeneity problems

• Source 1 : simultaneity between the intervention decision and contemporaneous exchange rate

• Intervention affects current spot rate, current spot rate triggers intervention

• Source 2: omitted variable• What should be the valid controls when the dependent variable is daily

nominal exchange rate?– Macroeconomic variables are unlikely to have relevance on a daily horizon

Instrument

• To avoid simultaneity, one may model an interventionary reaction function

• Explanatory factors include: change in nominal exchange rate, other factors that explain intervention

• Same problem: valid instruments may not be available

• Plausible valid instrument: customer trades data from the market microstructure finance data

Another Solution• Recall the omitted variable bias problem in the problem of monetary

shock effecting stock returns• Solution: • the omitted variable issue can be resolved by introducing a new high

frequency data set consisting of intraday changes around policy announcement time in, for instance, S&P500 futures and spot-month federal funds futures.

• S&P500 futures can be used to capture the stock market response to policy actions,

• Federal funds futures can be used to measure policy shocks. • The new data set decreases the likelihood that other common factors

impact both policy rates and stock prices during the tight intervals around announcements, thus reducing the omitted variable bias.

Caution

• The mere fact that some variable represents a decision (or choice) to the firm or, more generally, an economic agent, is not in itself sufficient for ‘econometric endogeneity’ to arise.

• Only if the factors that impact on the decision by the

economic agent, whether observable or not, are also inter-related with the dependent variable will endogeneity exist.

Measurement Error

• We would like to measure the effect of a variable, but we can observe only an imperfect measure of it

• Replacing the actual measure by the imperfect measure creates a measurement error

• Magnitude of the measurement error depends on the correlation between the actual and imperfect measure

• Measurement error can arise due to misreporting

Measurement Error & Proxy Variable Problem

Proxy Variable• One is looking for a variable

which is somehow related to the unobserved variable

• The partial effect of the omitted variable is rarely of central interest

Measurement Error• The variable we do not

observe has well-defined, quantitative meaning

• Mismeasured independent variable is of central interest

Measurement Error in Dependent Variable

Saving-Income Example• Suppose savings is a function of income, size, education and age

• Actual Saving may deviate from reported saving measurement error

• Will the measurement error be correlated with the explanatory variables ?

• In general not

• However, one might expect families with higher incomes or more education to report their saving more accurately

Measurement Error in Independent Variable

Example: Beta Estimation

Measurement Error in Beta

Bias in Beta Estimation

Solution to Measurement Error

Simultaneity• Simultaneity arises when at least one of the explanatory variables is

determined simultaneously along with the dependent variable

• The concerned explanatory variable and the disturbance term would generally be correlated

• Example: In Modelling financial intermediary development and sources of growth, endogenity may arise due to simultaneous determination of financial intermediary development and

• (i) private savings rates, • (ii) capital accumulation, • (iii) total factor productivity growth, and • (iv) overall real per capita GDP growth.

EXAMPLE• Revisiting the corporate governance and performance relation• The direction of causation is important• Board Composition could affect firm performance and firm performance could also

cause a change in board composition• This reverse causality leads to biased OLS estimators• Direction of bias depends on whether performance positively or negatively impact

governance• Expected Relationship: poor performing companies increase the number of

independent board members to improve performance• Reverse causality has a negative impact• Coefficient of corporate governance is underestimated• OLS estimate will lead to downward biased estimates

Structural Equation

• To capture instantaneous feedback relations, researchers write a system of equations

• It consists of separate structural equations for each endogenous variable.

• When variables y1 impacts on y2 and vice versa, – y2 would be included as an explanatory variable in the

structural– equation for y1; y1, in turn, is an explanatory variable in

the structural equation of y2.

Rank and Order Condition

General Order Condition

Estimation

• Single Equation Methods– Indirect least squares– Two-stage least squares– LIML

• System Methods– Three-stage least squares– FIML

Example• Estimation in the Corporate governance and performance problem

• Instrument variable method in 3 stages:

• First stage: Selection Equation• Probit (in the sample) = f(variable in selection process)

• Second stage: Reduced form Regression• Endogenous Variable = f(instruments, inverse mills ratio, control variables)

• Third stage: Structural Equation• Dependent Variable = f (stage 2 predicted value, inverse mills ratio, control

variables)

Shifting Supply Function to trace Demand Function

S1

S2

S3Price

Quantity

D