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4. Using panel data. 4.1 The basic idea 4.2 Linear regression 4.3 Logit and probit models 4.4 Other models. 4.1 The basic idea. Panel data = data that are pooled for the same companies across time.

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## 4. Using panel data

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**4. Using panel data**• 4.1 The basic idea • 4.2 Linear regression • 4.3 Logit and probit models • 4.4 Other models**4.1 The basic idea**• Panel data = data that are pooled for the same companies across time. • In panel data, there are likely to be unobserved company-specific characteristics that are relatively constant over time. • I have already explained that it is necessary to control for this time-series dependence in order to obtain unbiased standard errors. • In STATA we can do this using the robust cluster () option**4.1 The basic idea**• The first advantage of panel data is that we are using a larger sample compared to the case where we have only one observation per company. • The larger sample permits greater estimation power, so the coefficients can be estimated more precisely. • Since the standard errors are lower (even when they are adjusted for time-series dependence), we are more likely to find statistically significant coefficients. • use "J:\phd\Fees.dta", clear • gen fye=date(yearend, "MDY") • format fye %d • gen year=year(fye) • sort year • gen lnaf=ln(auditfees) • gen lnta=ln(totalassets) • by year: reglnaflnta, robust cluster(companyid) • reglnaflnta, robust cluster(companyid)**4.1 The basic idea**• The second advantage of panel data is that we can estimate “dynamic” models. • For example, suppose we believe that audit fees depend not only on the company’s size but also its rate of growth • sort companyidfye • gen growth= lnta- lnta[_n-1] if companyid== companyid[_n-1] • reglnaflnta growth, robust cluster( companyid) • We find that audit firms offer lower fees to companies that are growing more quickly • If we had had only one year of data, we would not have been able to estimate this model.**4.1 The basic idea**• The third – and most important – advantage of panel data is that we are able to control for unobservable company-specific effects that are correlated with the observed explanatory variables • Let’s start with a simple regression model: • Let’s assume that the error term has an unobserved company-specific component that does not vary over time and an idiosyncratic component that is unique to each company-year observation:**4.1 The basic idea**• Putting the two together: • Recall that the standard error of will be biased if we do not adjust for time-series dependence • this adjustment is easy using the robust cluster () option • The OLS estimate of the coefficient will be unbiased as long as the unobservable company-specific component (ui) is uncorrelated with Xit**4.1 The basic idea**• Unfortunately, the assumption that ui is uncorrelated with Xit is unlikely to hold in practice. • If ui is correlated with Xit then it is also correlated with Xit • The OLS estimate of will be biased if it is correlated with Xit (recall our previous discussion and notes on omitted variable bias)**4.1 The basic idea**• An example can illustrate this bias. • Go to MySite • use "J:\phd\beatles.dta", clear • list • This dataset is a panel of four individuals observed over three years (1968-70) • In each year they were asked how satisfied they are with their lives • this is the lsat variable which takes larger values for increasing satisfaction • You want to test how age affects life satisfaction • reglsat age • It appears that they became slightly more satisfied as they got older.**4.1 The basic idea**• Suppose you now include dummy variables for each individual • tab persnr, gen(dum_) • Recall that you must omit either one dummy variable or the intercept in order to avoid perfect collinearity (see the previous notes about multicollinearity) • reglsat age dum_1 dum_2 dum_4 • reglsat age dum_1 dum_2 dum_3 dum_4, nocons • There now appears to be a highly significant negative impact of age on life satisfaction • What’s going on here?**4.1 The basic idea**• Recall that fitting a simple OLS model (lsat on age) is equivalent to plotting a line of best fit through the data • twoway (lfitlsat age) (scatter lsat age)**4.1 The basic idea**• I am now going to introduce a new command, separate , by() • separate lsat, by(persnr) • This creates four separate life satisfaction variables for each of the four individuals • Now graph the relationship between life satisfaction and age for each of the four people • twoway (lfit lsat1 age) (scatter lsat1 age) • twoway (lfit lsat2 age) (scatter lsat2 age) • twoway (lfit lsat3 age) (scatter lsat3 age) • twoway (lfit lsat4 age) (scatter lsat4 age)**It is clear that each of the four individuals became less**satisfied as they got older. • The simple OLS regression was biased because John and Ringo (who happened to be older) were generally more satisfied than Paul and George (who happened to be younger) • The multiple OLS regression controlled for these idiosyncratic differences by including dummy variables for each person • We can see this by plotting the simple OLS results and the multiple OLS results • reglsat age dum_1 dum_2 dum_3 dum_4, nocons • predict lsat_hat • separate lsat_hat, by(persnr) • twoway (line lsat_hat1-lsat_hat4 age) (lfitlsat age) (scatter lsat1-lsat4 age)**4.1 The basic idea**• What does all this have to do with panel data being advantageous? • Without panel data we would not have been able to control for the idiosyncracies of the four individuals. • If we had data for only one year, we would not have known that the age coefficient was biased in the simple regression. • We can demonstrate this by running a regression of lsat on age for each year in the sample • sort time • by time: reglsat age • Without panel data, we would have incorrectly concluded that people get happier as they get older**4.1 The basic idea**• In the multiple regression, we include dummy variables (dum_1 dum_2 dum_3 dum_4) which control for the individual-specific effects (ui) • Without including the person dummies, our estimate of would be biased because the dummies are correlated with age. • The person dummies “explain” all the cross-sectional variation in life satisfaction across the four individuals. • The only variation that is left is the change in satisfaction within each person as he gets older. • Therefore, the model with dummies is sometimes called the “within” estimator or the “fixed-effects” model.**4.1 The basic idea**• In small datasets like this, it is easy to create dummy variables for each person (or each company). • In large datasets, we may have thousands of individuals or companies. • The number of variables in STATA is restricted due to memory limits. • Also it is not very inconvenient to have results for thousands of dummy variables (just imagine how long your log file would be!).**4.1 The basic idea**• Instead of including dummy variables, we can control for idiosyncratic effects by transforming the Y and X variables. • Taking averages of eq. (1) over time gives: • Subtracting eq. (2) from eq. (1) gives: • The key thing to note here is that the individual-specific effects (ui) have been “differenced out” so they will not bias our estimate of .**4.1 The basic idea**• Another transformation that will do the same trick is to take differences rather than subtract means • Lagging by one period • Subtracting eq. (2) from eq. (1) gives: • Again the individual-specific effects (ui) have been “differenced out” so they will not bias our estimate of .**Class exercise 4a**• Estimate the following models, where Y = life satisfaction and X = age. • Compare the age coefficients in these models to the age coefficient in the untransformed model with person dummies (ignore the standard errors of the age coefficients because they are biased)**4.2 Linear regression using panel data (xtreg, fe i())**• Fortunately, STATA has a command that: • allows us to avoid creating dummy variables for each person • corrects the standard errors • xt is a prefix that tells STATA we want to estimate a panel data model • The fe option tells STATA we want to estimate a fixed effects model • in OLS this is equivalent to including dummy variables to control for person-specific effects • The i() term tells STATA the variable that identifies each unique person • xtreglsat age , fei(persnr)**Note that the age coefficient and t-statistic are exactly**the same as in the OLS model that includes person dummies • reg lsat age dum_1 dum_2 dum_3 dum_4, nocons • There are 12 person-years, 4 persons, and the minimum, average and maximum number of observations per person is 3.**Since we are estimating a within-effects model, it is the**within R2 that is directly relevant (93.2%). • If we used the same independent variables to estimate a “between-effects” model, we would have an R2 of 88.4% (I will explain later what we mean by the “between-effects” model). • If we used the same independent variables to estimate a simple OLS model, we would get an R2 of 16.5%. (reglsat age) • The F-statistic is a test that the coefficient(s) on the X variable(s) (i.e., age) are all zero.**sigma_u is the standard deviation of the estimates of the**fixed effects, ui (u) • sigma_e is the standard deviation of the estimates of the residuals, eit (e) • rho = u2 / (u2 + e2) = 4.932 / (4.932 + 0.472) = 0.99**The correlation between uit and Xit is -0.83.**• This correlation appears to be high confirming our prior finding that the fixed effects are correlated with age. • The F-test allows us to reject the hypothesis that there are no fixed effects. • If we had not rejected this hypothesis, we could estimate a simple OLS instead of the fixed-effects model.**4.2 Linear regression (predict)**• After running the fixed-effects model, we can obtain various predicted statistics using the predict command • predict , xb • predict , u • predict , e • predict , ue**4.2 Linear regression (predict)**• For example: • xtreglsat age , fei(persnr) • drop lsat_hat • predict lsat_hat, xb • predict lsat_u, u • predict lsat_e, e • predict lsat_ue, ue • Checking that lsat_ue = lsat_u + lsat_e • list lsat_ulsat_elsat_ue • Checking that the correlation between uit and Xitis -0.83 • corrlsat_hatlsat_u**4.2 Linear regression**• I have explained that there are three main advantages of panel data: • The larger sample increases power, so the coefficients are estimated more precisely • We can estimate models that incorporate dynamic variables (e.g., the effect of growth on audit fees) • We can control for unobservable fixed effects (e.g., company-specific or person-specific characteristics) by estimating fixed-effects models.**4.2 Linear regression**• Are there any disadvantages? • Yes, unfortunately we cannot investigate the effect of explanatory variables that are held constant over time. • From a technical point of view, this is because the time-invariant variable would be perfectly collinear with the person dummies. • From an economic point of view, this is because fixed-effect models are designed to study what causes the dependent variable to change within a given person. A time-invariant characteristic cannot cause such a change.**4.2 Linear regression**• For example, suppose that the height of the four persons is constant over the three years. • Let’s create a height variable and test the effect of height on life satisfaction • gen height=185 if dum_1==1 • replace height=180 if dum_2==1 • replace height=175 if dum_3==1 • replace height=170 if dum_4==1 • list persnr height • Note that the height variable is a constant for each person. • We can estimate the effect of height as long as we do not control for unobservable person-specific effects • reglsat age height**4.2 Linear regression**• If we try to control for person-specific effects by including dummy variables: • reg lsat age height dum_1 dum_2 dum_3 dum_4, nocons • Note that STATA has to throw away either a dummy variable or the height variable. • Why is that? • The only way we can include dummies for each person is if we do not include the height variable. • reg lsat age dum_1 dum_2 dum_3 dum_4, nocons • If we try to estimate the effect of height using the xtreg, fei() command, STATA will inform us that there is a problem of perfect collinearity • xtreglsat age height, fei( persnr)**4.2 Linear regression**• Note that the height coefficient can be estimated if there is some variation over time for one or more persons. • The fixed-effects estimator can exploit this time variation to estimate the effect of height on life satisfaction. • For example, suppose that each person became 1cm taller in 1970. • replace height= height+1 if time==1970 • xtreglsat age height, fei( persnr)**The xtreg, fei() command estimates the following**fixed-effects model: • Recall that we derived this model by taking averages: • The averages model is sometimes called the “between” estimator because the comparison is cross-sectional between persons rather than over time. • Like OLS, the between estimator provides unbiased estimates of only if the unobservable company-specific component (ui) is uncorrelated with Xit • If we wanted to estimate the “between effects” model, the command in STATA is xtreg , be i() • xtreglsat age, be i( persnr)**Note that the age coefficient is positive**• the reason is that we are not controlling for person-specific effects, which are correlated with age. • therefore, the between-effects estimate of the age coefficient is biased. • Since we are estimating a between-effects model, it is the between R2 that is relevant (88.4%). • Note that this is also the between-effects R2 that was previously reported using the fixed-effects model. • Note that the R2 for the between-effects model is high despite that the age coefficient is severely biased. Again, this reinforces the fact that a high R2 does not imply that the model is well specified.**The between estimator is also less efficient than simple OLS**because it throws away all the variation over time in the dependent and independent variables. • In fact the between estimator is equivalent to estimating an OLS model on the averages for just one year • Recall that we have already created averages for the lsat and age variables (avlsatavage) • regavlsatavage if time==1968 • regavlsatavage if time==1969 • regavlsatavage if time==1970 • xtreglsat age height, be i( persnr) • Since we actually have three years of data, it seems silly (and it is silly) to throw data away**4.2 Linear regression (xtreg)**• Normally, then, we would never be interested in estimating a between-effects model: • The estimates are biased if the person-specific effects are correlated with the X variables • The estimates are inefficient because we are ignoring any time-series variation in the data • The fixed effects estimator is attractive because it controls for any correlation between ui and Xit • An unattractive feature is that it is forced to estimate a fixed parameter for each person or company in the data • you can think of these parameters as being the coefficients on the person dummy variables**4.2 Linear regression (xtreg)**• An alternative is the “random effects” model in which the ui are assumed to be randomly distributed with a mean of zero and a constant variance (ui ~ IID(0, 2u) rather than fixed. • Intuitively, the random effects model is like having an OLS model where the intercept varies randomly across individuals i. • Like simple OLS, the random effects model assumes that there is zero correlation between ui and Xit • If ui and Xit are correlated, the random-effects estimates are biased.**4.2 Linear regression (xtreg)**• If we want to estimate a random effects model, the command is xtreg , re i() • For example: • xtreglsat age, re i( persnr) • Note that because we have controlled for (random) unobserved person effects, the age coefficient is estimated with the correct negative sign.**The rest of the output is similar to the fixed-effects model**except: • We use a Wald statistic instead of an F statistic to test the significance of the independent variables. Here we can reject the hypothesis that age is insignificant. • The Wald statistic is used because only the asymptotic properties of the random-effects estimator are known. • The output explicitly tells us that we have imposed the assumption that ui and Xit are uncorrelated. • This is the key difference between the random-effects and fixed-effects models.**We can test whether ui and Xit are correlated.**• If they are correlated, we should use the fixed-effects model rather than OLS or the random-effects model (otherwise the coefficients are biased). • If they are not correlated, it is better to use the random-effects model (because it is more efficient). • The test was devised by Hausman • if ui and Xit are correlated, the random-effects estimates are biased (inconsistent) while the fixed-effects coefficients are unbiased (consistent) • In this case, there will be a large difference between the random-effects and fixed-effects coefficient estimates • if ui and Xit are uncorrelated, the random-effects and fixed-effects coefficients are both unbiased (consistent); the fixed-effects coefficients are inefficient while the random-effects coefficients are efficient. • In this case, there will not be a large difference between the random-effects and fixed-effects coefficient estimates • The Hausman test indicates whether the two sets of coefficient estimates are significantly different**Null hypothesis (H0): ui and Xit are uncorrelated**• The Hausman statistic is distributed as chi2 and is computed as • If the chi2 statistic is positive and statistically significant, we can reject the null hypothesis. This would mean that the fixed-effects model is preferable because the coefficients are consistent. • If the chi2 statistic is not positive and statistically significant, we cannot reject the null hypothesis. This would mean that the random-effects model is preferable because the coefficients are consistent and efficient. • NB: The (Vc-Ve)-1 matrix is guaranteed to be positive only asymptotically. In small samples, this asymptotic result may not hold in which case the computed chi2 statistic will be negative.**4.2 Linear regression (estimates store, hausman)**• The procedure for executing a Hausman test is as follows: • Save the coefficients that are consistent even if the null is not true: • xtreglsat age, fei( persnr) • estimates store fixed_effects • Save the coefficients that are inconsistent if the null is not true: • xtreglsat age, re i( persnr) • estimates store random_effects • The command for the Hausman test is: • hausmanname_consistentname_efficient • hausmanfixed_effectsrandom_effects**b is the fixed-effects coefficient while B is the**random-effects coefficient. • The (Vc-Ve)-1 matrix has a negative value on the leading diagonal and, as a result, the square root of the leading diagonal is undefined. This is why the Chi2 statistic is negative. • Since the Chi2 statistic is not significantly positive, we might decide that we cannot reject the null hypothesis (see p. 57 of the STATA reference manual for the Hausman test). • On the other hand, this result is not very reliable because the asymptotic assumption fails to hold in this small sample.**If we reject the null hypothesis that ui and Xit are**uncorrelated, the fixed-effects model is preferable to the OLS and random-effects models. • If we cannot reject the null hypothesis that ui and Xit are uncorrelated, we need to determine whether the ui are distributed randomly across individuals. • Recall that the random-effects model is like having an OLS model where the constant term varies randomly across individuals i. • Therefore, we need to test whether there is significant variation in ui across individuals.**rho = u2 / (u2 + e2)**= 1.032 / (1.032 + 0.472) = 0.83 • u2 captures the variation in ui across individuals. • If u2 is significantly positive, the random-effects model is preferable to the OLS model. • The Breusch and Pagan (1980) Lagrange multiplier test is used to investigate whether u2 is significantly positive.**We perform the Breusch-Pagan test by typing xttest0 after**xtreg, re • Our estimate of u2 is 1.067 (note that u is estimated to be 1.032 which is the same as sigma_u on the previous slide). • We are unable to reject the hypothesis that u2 = 0. Therefore, we cannot conclude that the random-effects model is preferable to the OLS model. • NB: Our Hausman and LM tests lack power because the sample consists of only 12 observations. In larger samples, we are more likely to reject the hypothesis that u2 = 0 and we are more likely to reject the hypothesis that ui and Xitare uncorrelated.**Class exercise 4b**• Estimate models in which the dependent variable is the log of audit fees. • Estimate the models using: • OLS without controlling for ui • Fixed-effects models • Random-effects models • How do the coefficient estimates vary across the different models? • Which of these models is preferable?**4.2 Linear regression**• Compared to economics and finance, there are not many accounting studies that exploit panel data in order to control for unobserved company-specific effects (ui). • Most studies simply report OLS estimates on the pooled data. • Some studies even fail to adjust the OLS standard errors for time-series dependence • this can be a very serious mistake especially when the panels are long (e.g., the sample period covers many years).

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