Econometrics of Panel Data

Econometrics of Panel Data 1. Basics and Examples 2. The generalized least squares estimator 3. Fixed effects model 4. Random Effects model 1

1 Basics and examples We observes variables for N units, called the cross-sections, for T consecutive periods: (Y it, X it ) i = 1,...,N, with N the cross-sectional dimension. t = 1,...,T, with T the temporal dimension. panel of size N T. 2

Y it is the income of family i during year t, for 1 i 1000, and observed in years 2000, 2001, 2002, so T = 3. Y it is the unemployment rate for EU-country i, (1 i 15), observed monthly from 1998:01 up to 2001:12, so T = 48. Note that: T large, N small multiple time series T small, N large survey data on individuals/firms for a small number of waves. 3

Example 1: South American countries For 8 South-American countries we want to model the Real GDP per capita in 1985 prices (=Rgdl) in function of the following explicative variables. Population in 1000 s (Pop) Real Investment share of GDP, in % (I) Real Government share of GDP, in % (G) Exchange Rate with U.S. dollar (XR) Measure of Openness of the Economy (Open) You find the data in the file penn.wmf, already in Eviews format. We are in particular interested in the effect of Openness on economic growth. 4

1. Create a pool object in Eviews ( /Object/New object ). Give it a name and define the cross-section identifiers. These identifiers are those parts of the names of the series identifying the cross-section. 2. Open the XR-variables as a group and make a plot of them. Compute them in log-difference, using the PoolGenr menu of the pool object and logdifxr?=dlog(xr?). The? will be substituted by every cross-section identifier. Plot the transformed variables. 3. Compute the medians of the variable I? for the different countries (use View/descriptive statistics within the Pool object.) Compute now the average value of I? for every year. 4. Estimate the regression model for Brazil, using /Quick/estimate equation and specifying in Eviews the equation dlog(rgdp bra) c dlog(pop bra) i bra g bra dlog(xr bra) open bra 5. Now we want to pool the data of all countries, to increase the sample size. Use, within the pooled object, /Estimate, and specify: dependent variable=dlog(rgdp?); common coefficients=c dlog(pop?) i? g? dlog(xr?) open?. This is a pooled regression model. 5

6. Pooling the data ignores the fact that the data originate from different countries. Dummy variables for the different countries need to be added. This can be done by specifying the constant term as a cross section specific coefficient. We obtain a fixed effect panel data model. Discuss the regression output. 7. The fixed effect panel data model assumes that the effect of openness is the same of all countries. How could you relax this assumption? 8. Test whether all country effects are equal (to know how Eviews labels the coefficients, use View/Representation), using a Wald test. The country effects are called the fixed effects, and if there are significantly different then there is unobserved heterogeneity. 6

2 The Generalized Least Squares estimator Standard linear regression model: Y i = X iβ + ε i (i = 1,...,n) with Var(ε i ) = σ 2 is constant homoscedastic errors Cov(ε i, ε j ) = 0 for i j uncorrelated errors 7

At the standard model, the Ordinary Least Squares (OLS) estimator is Consistent, meaning that ˆβ β for n tending to infinity. Has the smallest variance among all estimators (for normal errors) and smallest variance among all linear estimators. One has that ˆβ OLS = ( n ) 1 ( n ) X i X i X i Y i. i=1 i=1 8

What if the the errors are not homoscedastic and uncorrelated? E.g. for panel data: Cross-sectional heteroscedasticity Correlation among cross sections Serial correlation within and across cross-sections... The Ordinary Least Squares (OLS) estimator is still consistent, but not optimal anymore. 9

General linear regression model: Y i = X iβ + ε i (i = 1,...,n) with Var(ε i ) = σ 2 i heteroscedastic errors Cov(ε i, ε j ) = σ ij for i j correlated errors. One can still use OLS (not even a bad idea), if one uses White standard errors (if heteroscedasticity) Newey-West standard errors (if correlated errors + heteroscedasticity) 10

The Generalized Least Squares (GLS) estimator will be consistent and optimal and is given by ˆβ GLS = n n w ij X i X j 1 n n w ij X i Y j, i=1 j=1 i=1 j=1 where the weights depends on the values of σ ij. More precisely: let Σ be the n n matrix with elements σ ij, then w ij = (Σ 1 ) ij. Unfortunately, the values in Σ are unknown. 11

The Feasible Generalized Least Squares (GLS) proceeds in 2 steps: 1. Compute ˆβ OLS and the residuals r OLS i = Y i X i ˆβ OLS. 2. Use the above residuals to estimate the σ ij. [This will require some additional assumptions on the structure of Σ] Compute then the GLS estimator with estimated weights w ij. The above scheme can be iterated fully iterated GLS estimator. 12

Theoretical Example Our sample of size n = 20 consists of two groups of equal size (e.g. men and women). There is no correlation among the observations, but we think that the variances of the error terms for men and women might be of different size. [The error terms contains the omitted and unobserved variables. We might indeed think that their size is different for women than for men, e.g. when regressing salary on individual characteristics] σ 2 i = σ ii = σ 2 M for i = 1,...,10 σ 2 i = σ ii = σ 2 F for i = 11,...,20 σ ij = 0 for i j. 13

Computation of the (Feasible) GLS estimator: 1. Compute the OLS estimator and the residuals r OLS i. 2. Estimate ˆσ 2 M = 1 10 10 i=1 (r OLS i ) 2 and ˆσ 2 F = 1 10 20 (r OLS i=11 i ) 2. Due to the simple structure of the matrix Σ, we have ŵ i = 1 ˆσ 2 M (i = 1,...,10) and ŵ i = 1 ˆσ 2 F (i = 11,...,20) ˆβ GLS = ( n ) 1 ( n ) w i X i X i w i X i Y i. i=1 i=1 14

Application to panel data regression Let ε it be the error term of a panel data regression model, with 1 i n, and 1 t T. Three different specifications are common: 1. V ar(ε it ) = σ 2 and all covariances between error terms are zero. OLS can be applied (no weighting). 2. V ar(ε it ) = σi 2 and all covariances between error terms are zero. We have cross-sectional heteroscedasticity. GLS can be applied (cross-section weights): 3. V ar(ε it ) = σi 2, Cov(ε it, ε jt ) = σ ij, all other covariances zero. We allow now for contemporaneous correlation between cross-sections. GLS can be applied (SUR weights). 15

Example South American (continued) 1. Have a look at the residuals (View/residuals/Graphs) within the pool object). Compute the covariance and the correlation matrix of the residuals (i) Is there cross-sectional heteroscedasticity? (ii) Is there contemporaneous correlation? 2. Estimate now the model with the appropriate GLS estimator. Are the results depending a lot on the weighting scheme? 3. Is there still serial correlation present in the residuals, i.e. (cross)-correlation at leads and lags? Hence, is the model capturing the dynamics in the data? 16

3 The Fixed Effects regression model Fixed effects Model: Y it = X itβ + α i + ε it with t = 1,...T time periods and i = 1...,N cross-sectional units. The α i contain the omitted variables, constant over time, for every unit i. The α i are called the fixed effects, and induce unobserved heterogeneity in the model. The X it are the observed part of the heterogeneity. The ε it contain the remaining omitted variables. 17

Testing for unobserved heterogeneity: H 0 : α 1 =... = α N := α (Test for redundant fixed effects) In case H 0 holds, there is no unobserved heterogeneity, and the model reduces to the pooled regression model: Y it = X itβ + α + ε it Ignoring unobserved heterogeneity may lead to severe bias of the estimated β, see figure: 18

15 Cross Section 1 Cross Section 2 10 Pooled Regression y 5 Cross Section 3 0 1 2 3 4 5 6 7 x 19

LSDV estimation LSDV=Least Squares Dummy Variable estimation Rewrite the model as Y it = α 1 D 1 i +... + α n D n i + X itβ + ε it, with D j i = 1 if i = j and zero if i j. Estimate model by OLS or GLS (weighting). If necessary, use White/Newey West type of Standard Errors (also if GLS is used, see later). 20

Within groups estimator Compute averages of X it and Y it within each group of cross-sectional unit X i. and Ȳi. Y it = X it β + α i + ε it Ȳ i. = X i. β + α i + ε i. (Y it Ȳi.) = (X it X i. ) β + (ε it ε i. ) Regress the centered Y it on the centered X it by OLS. By centering, the fixed effects are eliminated! One can show that the within group estimator is identical to LSDV. 21

Comments 1. If a variable X it is constant in time for all cross-sections, the FE model cannot be estimated. Why? 2. The fixed effects model can be rewritten with a common intercept included as Y it = X itβ + α + µ i + ε it, and µ 1 + µ 2 +... + µ N = 0. Obviously, we have α i = α + µ i, and α is the average of the fixed effects. 22

3. One can add time effects (or period effects) in the model: Y it = X itβ + α i + δ t + ε it, The δ t contain the omitted variables, constant over cross-sections, at every time point t. The time effects capture the business cycle. 23

4. If we think that the cross-sectional units are an i.i.d. sample (typical for micro-applications), but serial correlation or period heteroscedasticity is present (within each unit), then OLS can be made more precise/efficient: (a) V ar(ε it ) = σt 2 and all covariances between error terms are zero. We have period heteroscedasticity. GLS can be applied (Period weights): (b) V ar(ε it ) = σt 2, Cov(ε it, ε is ) = σ ts, all other covariances zero. We allow for serial correlation. GLS can be applied (Period weights). 24

Example: Grunfeld data We consider investment data for 10 American firms from 1935-1954, and consider the model INV it = β i1 V AL it + β i2 CAP it + α i + ε it for 1 i N = 10, and 1 t T = 20. The variables are Gross investment for the firm (INV) Value of the firm (VAL) Real Value of the Capital stock (plant and equipment) (CAP) The data are in the excel file grunfeld2.xls. 25

1. Have a look at the data in the Excel File. Write up the number of observations, the number of variables, and the upper left cell of the data matrix. Close the Excel file, create an unstructured Workfile and read in the data (Proc/Import/Read Text Lotus Excel). 2. To apply a panel structure, double click on the Range: line at the top of the workfile window, or select Proc/Structure/Resize Current Page. Select Dated Panel, and enter the appropriate variables as Date Series and as Cross Section ID series. 3. Open the investment series. Explore the Descriptive Statistics and tests menu. 26

4. Use View/Graph to (i) Make a line plot of the time series for every cross section (ii) Make boxplots of the distribution of investment over the different cross sections and over time. 5. Use Quick/Estimate Equation to estimate the fixed effects model. Specify the equation inv c cap value and use Panel Options to indicate that you use fixed effects. 6. Interpret your outcome. Would it be useful to add period effects? Test whether they this is necessary with View/Fixed Random Effects testing. 7. Select an appropriate weighting scheme within Panel Options. Interpret your outcome. 27

4 Random Effects model Model Y it = c + X itβ + ε it where the error term is decomposed as ε it = α i + v it. α i is a random effect N(0, σα). 2 It is the permanent component of the error term. v it a noise term N(0, σv). 2 It is the idiosyncratic component of the error term. 28

(The v it are uncorrelated among cross-sections, are serially uncorrelated at all leads and lags, within and across cross sections. The random effects are uncorrelated among cross-sections.) At the price of one extra parameter σα 2, the random effects model allows for correlation within cross-section units: For every i and t s: Cov(ε it, ε is ) = Cov(α i + v it, α i + v is ) = σ 2 α The following Variance decomposition holds: Var(ε it ) = Var(α i + v it ) = σ 2 α + σ 2 v. 29

Within groups/cross sections correlation: ρ = Corr(ε it, ε is ) = σ2 α σ 2 α + σ 2 v. The larger the value of ρ, the more unobserved heterogeneity. One estimates β by Generalized Least Squares, and obtains the RE-estimator. Different methods are existing to make GLS feasible. 30

Testing for correlated random effects: The random effect α i needs to be uncorrelated with the X-variables. This is a strong assumption. If not, there is an endogeneity problem, and the RE-estimator is inconsistent. H 0 : Corr(α i, X it ) = 0 The Hausman test compares two estimators: the FE (always consistent) and the RE estimator (consistent under H 0 ). One rejects H 0 if the difference between the two estimators is large. 31

Using Fixed or random effects? In econometrics, the fixed effects model seems to be the most appropriate (H O not needed). If N is large, and T is small, and the cross-sectional units are a random sample from a population, then random effects model becomes attractive: It is a parsimonious model, that captures within group-correlation. (For N large, FE requires estimation of many parameters) Random effects is popular for modeling grouped data: (i) Sample of 1000 children coming from 30 different schools (ii) Sample of 1000 persons from 20 different villages... 32

Robust Standard Errors: For RE no weighted versions are available. Using robust standard errors (or coefficient covariance) might be appropriate. This only affects the SE, not the estimators. 1. White cross section: robust to V ar(ε it ) = σi 2 and Cov(ε it, ε jt ) = σ ij. [robust to cross-section heteroscedasticity and contemporenous correlation among cross sections; appropriate if N << T.] 2. White period: robust to V ar(ε it ) = σ 2 t and Cov(ε it, ε is ) = σ ts. [robust to serial correlation within cross-section and changing variances over time; appropriate if cross-sections are random sample and T << N.] 3. White diagonal: robust to V ar(ε it ) = σ 2 it [robust to all forms of heteroscedasticity, but not robust for any type of correlation over time of across cross-section.] Can also be used for FE. 33

Exercise Consider the grunfeld data in grundfeld2.wf1. The model was: INV it = β i1 V AL it + β i2 CAP it + α i + ε it 1. Estimate the model as a random effects model. 2. What is the within-group correlation? 3. Perform the Hausman test. (View/Fixed random effects testing/correlated random effects) 4. Compute different types of robust SE. How is this affecting the results? 34