Chapter 1 Introduction

1. Chapter 1Introduction • What are longitudinal and panel data? • Benefits and drawbacks of longitudinal data • Longitudinal data models • Historical notes

2. 1.1 What are longitudinal and panel data? • With regression data, we collect a cross-section of subjects. • The interest is comparing characteristics of the subject, that is, investigating relationships among the variables. • In contrast, with time series data, we identify one or more subjects and observe them over time. • This allows us to study relationships over time, the so-called dynamic aspect of a problem. • Longitudinal/panel data represent a marriage of regression and time series data. • As with regression, we collect a cross-section of subjects. • With panel data, we observe each subject over time. • The descriptor panel data comes from surveys of individuals; a panel is a group of individuals surveyed repeatedly over time.

3. Example 1.1 - Divorce rates • Figure 1.1 shows the 1965 divorce rates versus AFDC (Aid to Families with Dependent Children) for the fifty states. • The correlation is -0.37. • Counter-intuitive? - we might expect a positive relationship between welfare payments (AFDC) and divorce rates.

4. Example 1.1 - Divorce rates • A similar figure shows a negative relationship for 1975 (the correlation is -0.425) • Figure 1.2 shows both 1965 and 1975 data, with a line connecting each state • The line represents a change over time (dynamic), not a cross-sectional relationship. • Each line displays a positive relationship - as welfare payments increase so do divorce rates. • This is not to argue for a causal relationship between welfare payments and divorce rates. • The data are still observational. • The dynamic relationship between divorce and AFDC is different from the cross-sectional relationship.

5. Figure 1.2 1965 and 1975 Divorce rates versus AFDC

6. Some notation • Longitudinal/panel data - regression data with “double subscripts.” • Let yit be the response for the ith subject during the tthtime period. • We observe the ith subject over t=1, ..., Ti time periods, for each of i=1, ..., n subjects. • First subject - (y11, y12, ... , y1T1 ) • Second subject - (y21, y22, ... , y2T2 ) • . . . • . . . • The nth subject - (yn1, yn2, ... , ynTn)

7. Prevalence of panel data analysis • Importance in the literature • Panel data are also known as “cross-section time series” data in the social sciences • Referred to as “longitudinal data analysis” in the biological sciences • ABI/INFORM - 326 articles in 2002 and 2003. • The ISI Web of Science - 879 articles in 2002 and 2003. • Important panel data bases • Historically, we have: • Panel Survey of Income Dyanmics (PSID) • National Longitudinal Survey of Labor Market Experience (NLS) • Financial and Accounting • Compustat, CRSP, NAIC • Market scanner databases • See Appendix F

8. Appendix F. Selected Longitudinal and Panel Data Sets • Table F.1 – 20 International Household Panel Studies • Table F.2 – 5 Studies focused on youth and education • Table F.3 – 4 Studies focused on the elderly and retirement • Table F.4 – 7 miscellaneous studies, including • election data, • manufacturing data, • medical expenditure data and • insurance company data

9. 1.2 Benefits and drawbacks of longitudinal data • Several advantages of longitudinal data compared to • data that are either purely cross-sectional (regression) or • purely time series data. • Having longitudinal data allows us to: • Study dynamic relationships • Study heterogeneity • Reduce omitted variable bias • With longitudinal data, one can also argue • Estimators are more efficient • Addresses the causal nature of relationships • Main drawback - attrition

10. Dynamic relationships • Static versus dynamic relationships • Figure 1.1 showed a cross-sectional (static) relationship. • We estimate a decrease of 0.95 % in divorce rates for each $100 increase in AFDC payments. • Figure 1.2 showed a temporal (dynamic) relationship. • We estimate an increase of 2.9% in divorce rates for each $100 increase in AFDC payments. • From 1965 to 1975, AFDC payments increased an average of $59 and divorce rates increased 2.5%.

11. Historical approach • In early panel data studies, pooled cross-sectional data were analyzed by • estimating cross-sectional parameters using regression and • using time series methods to model the regression parameter estimates, treating the estimates as known with certainty. • Theil and Goldberger (1961) provide an early discussion on the advantages of estimating these two aspects simultaneously.

12. Dynamic relationships and time series analysis • When studying dynamic relationships, univariate time series methods are the most well-developed. • However, these methods do not account for relationships among different subjects. • Multivariate time series accounts for relationships among a limited number of different subjects. • Time series methods requires a fair number (generally, at least 30) observations to make reliable inferences.

13. Panel data as repeated time series • With panel data, we observe several (repeated) subjects for each time period. • By taking averages over subjects, • our statistics are more reliable • we require fewer time series observations to estimate dynamic patterns. • For repeated subjects, the model is yit=  + it, t=1, ..., Ti, i=1, ..., n. • Here,  is the overall mean and it represents subject-specific dynamic patterns. • “Unfortunately,” we don’t get identical repeated looks. • We hope to control for differences among subjects by introducing explanatory variables, or covariates. • A basic model is yit =  + xit´  + it, where xit is the explanatory variable. • Introducing explanatory variables leaves us with only subject-specific dynamic patterns, that is, yit - ( + xit´ = it

14. Heterogeneity • Subjects are unique. • In cross-sectional analysis, we use yit =  + xit´ + it • ascribe the uniqueness to " it ". • In panel data, we have an opportunity to model this uniqueness. • The model yit = i + xit´ + it is • unidentifiable in cross-sectional regression. • In panel data, we can estimate  and 1, .., n. • Subject-specific parameters, such as i, provide an important mechanism for controlling heterogeneity of individuals. • Vocabulary: • When {i} are fixed, unknown parameters to be estimated, we call this a fixed effects model. • When {i} are drawn from an unknown population, that is, random variables, we call this a model with random effects.

15. Heterogeneity bias • Suppose that a data analyst mistakenly uses the model yit =  + xit´ + it when yit = i + xit´ + it is the true model. • This is an example of heterogeneity bias, or a problem with aggregation with data. • Similarly, one could have different (heterogeneous) slopes yit =  + xit´i + it • or different intercepts and slopes yit = i + xit´i + it

17. Omitted variables • Panel data serves to reduce the omitted variable bias. • When omitted variables are time constant, we can still get reliable estimates. • Consider the “true” model yit =  + xit´ + zi´ + it. • Unfortunately, we cannot (or not thought to) measure zi. • It is “lurking” or “latent.” By considering the changes yit*= yit - yi,t-1 = ( + xit´ + zi´ + it) - ( + xit-1´ + zi´ + it-1) = (xit- xit -1 )´ + it- it-1) = xit* ´ + it* • we do not need to worry about the bias that ordinarily arises from the latent variable, zi. • Introducing the subject-specific variable i, accounts for the presence of many types of latent variables.

18. Efficiency of Estimators • Subject-specific variables i also account for a large portion of the variability in many data sets • This reduces the mean square error • Increases the efficiency (or reduces the standard errors) of our parameter estimators. • With panel data, we generally have more observations than with time series or regression. • A longitudinal data design may yield more efficient estimators than estimators based on a comparable amount of data from alternative designs. • Suppose that the interest is in assessing the average change in a response over time, such as the divorce rate. • A repeated cross-section yields • Longitudinal data design yields

19. Causality and correlation • Three ingredients necessary for establishing causality, taken from the sociology literature: • A statistically significant relationship is required. • The association between two variables must not be due to another, omitted, variable. • The “causal” variable must precede the other variable in time. • Longitudinal data are based on measurements taken over time and thus address the third requirement of a temporal ordering of events. • Moreover, longitudinal data models provide additional strategies for accommodating omitted variables that are not available in purely cross-sectional data.

20. Drawbacks: Sampling Design (attrition) • Selection bias • may occur when a rule other than simple random sampling is used to select observational units • Example – “endogeneous” decisions by agents to join a labor pool or participate in a social program. • Missing data • Because we follow the same subjects over time, nonresponse typically increases through time. • Example: US Panel Study of Income Dynamics (PSID): • In the first year (1968), the nonresponse rate was 24%. • By 1985, the nonresponse rate was about 50%.

21. 1.3 Longitudinal data models • Types of inference • Primary. We are interested in the effect that an (exogenous) explanatory variable has on a response, controlling for other variables (including omitted variables). • Forecasting. We would like to predict future values of the response from a specific subject. • Conditional means. • We would like to predict the expected value of a future response from a specific subject. • Here, the conditioning is on latent (unobserved) characteristics associated with the subject. • Types of applications - many

22. Social science statistical modeling • A model based on data characteristics is known as a sampling based model. The model arises from a data generating process. • In contrast, a structural model is a statistical model that represents causal relationships, as opposed to relationships that simply capture statistical associations. • Why bother with an extra layer of theory when considering statistical models? Manski (1992) offers : • Interpretation - the primary purpose of many statistical analyses is to assess relationships generated by theory from a scientific field. • Structural models utilize additional information from an underlying functional field. If this information is utilized correctly, then in some sense the structural model should provide a better representation than a model without this information. (explanation) • Particularly for public policy analysis, the goal of a statistical analysis is to infer the likely behavior of data outside of those realized (extrapolation).

23. Modeling issues • With subject-specific parameters, there can be many parameters that describe the model • “Fixed” versus “random” effects models • Incorporating dynamic structure is important • Econometric “dynamic” models (lagged endogenous) versus serial correlation approach • Linear versus nonlinear (generalized linear) models • Marginal versus hierarchical estimation approaches • Parametric versus semiparametric models • We wish to separate the effects of: • the mean • the cross-sectional variance and • serial correlation structure

24. 1.4 Historical notes • The term ‘panel study’ was coined in a marketing context when Lazarsfeld and Fiske (1938) • Considered the effect of radio advertising on product sales. • People buy a product would be more likely to hear the advertisement, or vice versa. • They proposed repeatedly interviewing a set of people (the ‘panel’) to clarify the issue. • Econometrics • Early economics applications include Kuh (1959), Johnson (1960), Mundlak (1961) and Hoch (1962). • Biostatistics • Wishart (1938), Rao (1959, 1965), Potthoff and Roy (1964) – used multivariate analysis to consider the problem of polynomial growth curves of serial measurements from a single group of subjects. • Grizzle and Allen (1969) – introduced covariates