An independently pooled cross section 4This type of data is obtained by sampling randomly from a population at different points in time (usually in different years) 4You can pool the data from different year and run regressions. 4However, you usually include year dummies. 2
Panel data 4This is the cross section data collected at different points in time. 4However, this data follow the same individuals over time. 4You can do a bit more than the pooled cross section with Panel data. 4You usually include year dummies as well. 3
Pooling independent cross sections across time. 4As long as data are collected independently, it causes little problem pooling these data over time. 4However, the distribution of independent variables may change over time. For example, the distribution of education changes over time. 4To account for such changes, you usually need to include dummy variables for each year (year dummies), except one year as the base year 4Often the coefficients for year dummies are of interest. 4
Example 1 4Consider that you would like to see the changes in fertility rate over time after controlling for various characteristics. 4Next slide shows the OLS estimates of the determinants of fertility over time. (Data: FERTIL1.dta) 4The data is collected every other year. 4The base year for the year dummies are year 1972. 5
6 Dependent variable =# kids per woman
4The number of children one woman has in 1982 is 0.49 less than the base year. Similar result is found for year 1984. 4The year dummies show significant drops in fertility rate over time. 7
Example 2 4CPS78_85.dta has wage data collected in 1978 and 1985. 4we estimate the earning equation which includes education, experience, experience squared, union dummy, female dummy and the year dummy for 1985. 4Suppose that you want to see if gender gap has changed over time, you include interaction between female and 1985; that is you estimate the following. 8
Log(wage)= β 0 +β 1 (educ) +β 2 (exper)+β 3 (expersq)+β 4 (Union) +β 5 (female) +β 6 (year85) +β 7 (year85)(female) 4You can check if gender wage gap in 1985 is different from the base year (1978) by checking if β 7 is equal to zero or not. 4The gender gap in each period is given by: -gender gap in the base year (1978) = β 5 -gender gap in 1985= β 5 + β 7 9
10 Coefficient for the interaction term (y85)(Female) is positive and significant at 10% significance level. So gender gap appear to have reduced over time. gender gap in 1978 =-0.319 gender gap in 1985=-0.319+0.088 =-0.231
Policy analysis with pooled cross sections: The difference in difference estimator 4I explain a typical policy analysis with pooled cross section data, called the difference-in-difference estimation, using an example. 11
Example: Effects of garbage incinerator on housing prices 4This example is based on the studies of housing price in North Andover in Massachusetts 4The rumor that a garbage incinerator will be build in North Andover began after 1978. The construction of incinerator began in 1981. 4You want to examine if the incinerator affected the housing price. 12
4Our hypothesis is the following. Hypothesis: House located near the incinerator would fall relative to the price of more distant houses. 4For illustration define a house to be near the incinerator if it is within 3 miles. 4So create the following dummy variables nearinc =1 if the house is `near’ the incinerator =0 if otherwise 13
4Most naïve analysis would be to run the following regression using only 1981 data. price = β 0 +β 1 (nearinc)+u where the price is the real price (i.e., deflated using CPI to express it in 1978 constant dollar). Using the KIELMC.dta, the result is the following 14 But can we say from this estimation that the incinerator has negatively affected the housing price?
4To see this, estimate the same equation using 1979 data. Note this is before the rumor of incinerator building began. 15 Note that the price of the house near the place where the incinerator is to be build is lower than houses farther from the location. So negative coefficient simply means that the garbage incinerator was build in the location where the housing price is low.
4Now, compare the two regressions. 16 Year 1978 regression Year 1981 regression Compared to 1978, the price penalty for houses near the incinerator is greater in 1981. Perhaps, the increase in the price penalty in 1981 is caused by the incinerator This is the basic idea of the difference-in-difference estimator
4The difference-in-difference estimator in this example may be computed as follows. I will show you more a general case later on. The difference-in-difference estimator : = (coefficient for nearinc in 1981) ‒ (coefficient for nearinc in 1979) = ‒ 30688.27 ‒ ( ‒ 18824.37)= ‒ 11846 17 So, incinerator has decreased the house prices on average by $11846.
4Note that, in this example, the coefficient for (nearinc) in 1979 is equal to 18 Average price of houses near the incinerator Average price of houses not near the incinerator ‒ This is because the regression includes only one dummy variable: (Just recall Ex.1 of the homework 2). Therefore the difference in difference estimator in this example is written as. This is the reason why the estimator is called the difference in difference estimator.
Difference in difference estimator: More general case. 4The difference-in-difference estimator can be estimated by running the following single equation using pooled sample. price = β 0 +β 1 (nearinc) +β 2 (year81)+δ 1 (year81)(nearinc) 19 Difference in difference estimator
20 Difference in difference estimator This form is more general since in addition to policy dummy (nearinc), you can include more variables that affect the housing price such as the number of bedrooms etc. When you include more variables, cannot be expressed in a simple difference-in- difference format. However, the interpretation does not change, and therefore, it is still called the difference-in-difference estimator
Natural experiment (or quasi-experiment) 4The difference in difference estimator is frequently used to evaluate the effect of governmental policy. 4Often governmental policy affects one group of people, while it does not affect other group of people. This type of policy change is called the natural experiment. 4For example, the change in spousal tax deduction system in Japan which took place in 1995 has affected married couples but did not affect single people. 21
4The group of people who are affected by the policy is called the treatment group. 4Those who are not affected by the policy is called the control group. 4Suppose that you want to know how the change in spousal tax deduction has affected the hours worked by women. Suppose, you have the pooled data of workers in 1994 and 1995. 4The next slide shows the typical procedure you follow to conduct the difference-in-difference analysis. 22
4Step 1: Create the treatment dummy such that D treat =1 if the person is affected by the policy change =0 otherwise. 4Step 2: Run the following regression. (Hours worked)= β 0 +β 1 D treat + β 0 (year95) +δ 1 (Year95)(D treat )+u 23 Difference in difference estimator. This shows the effect of the policy change on the women’s hours worked.
Two period panel data analysis 4Motivation: Remember the effects of employee training grant on the scrap rate. You estimated the following model for the 1987 data. 24 You did not find the evidence that receiving the grant will reduce scrap rate.
4The reason why we did not find the significant effect is probably due to the endogeneity problem. 4The company with low ability workers tend to apply for the grant, which creates positive bias in the estimation. If you observe the average ability of the workers, you can eliminate the bias by including the ability variable. But since you cannot observe ability, you have the following situation. 25 where ability is in the error term v. v=(β 3 ability+u) is called the composite error term.
4Because ability and grant are correlated (negatively), this causes a bias in the coefficient for (grant). 4We predicted the direction of bias in the following way. 26 True effect of grant Bias term Effect of ability on scrap rate Sign is determined by the correlation between ability and grant The true negative effect of grant is cancelled out by the bias term. Thus, the bias make it difficult to find the effect.
4Now you know that there is a bias. Is there anything we can do to correct for the bias? 4When you have a panel data, we can eliminate the bias. 4I will explain the method using this example. I will generalize it later. 27
Eliminating bias using two period panel data 4Now, go back to the equation. 28 4The grant is administered in 1988. Suppose that you have a panel data of firms for two period, 1987 and 1988. 4Further assume that the average ability of workers does not change over time. So (ability) is interpreted as the innate ability of workers, such as IQ.
4When you have the two period panel data, the equation can be written as: 29 4 i is the index for i th firm. t is the index for the period. 4Since ability is constant overtime, ability has only i index. Now, I will use a short hand notation for β 4 (ability) i. Since (ability) is assumed constant over time, write β 4 (ability) i =a i. Then above equation can be written as:
4a i is called, the fixed effect, or the unobserved effect. If you want to emphasize that it is the unobserved firm characteristic, you can call it the firm fixed effect as well 4u it is called the idiosyncratic error. 4Now the bias in OLS occurs because the fixed effect is correlated with (grant). 4So if we can get rid of the fixed effect, we can eliminate the bias. This is the basic idea. 4In the next slide, I will show the procedure of what is called the first-differenced estimation. 30
4First, for each firm, take the first difference. That is, compute the following. 31 4It follows that, The first differenced equation.
4So, by taking the first difference, you can eliminate the fixed effect. 32 If ∆u it is not correlated with ∆( grant) it, estimating the first differenced model using OLS will produce unbiased estimates. If we have controlled for enough time-varying variables, it is reasonable to assume that they are uncorrelated. 4 Note that this model does not have the constant. 4Now, estimate this model using JTRAIN.dta
33 Now, the grant is negative and significant at 10% level. When you use ‘nocons’ option, the stata omits constant term.
4Note that, when you use this method in your research, it is a good idea to tell your audience what the potential fixed effect would be and whether it is correlated with the explanatory variables. In this example, unobserved ability is potentially an important source of the fixed effect. 4Off course, one can never tell exactly what the fixed effect is since it is the aggregate effects of all the unobserved effects. However, if you tell what is contained in the fixed effect, your audience can understand the potential direction of the bias, and why you need to use the first- differenced method. 34
General case 4First differenced model in a more general situation can be written as follows. Y it = β 0 +β 1 x it1 +β 2 x it2 +…+β k x itk +a i +u it If a i is correlated with any of the explanatory variables, the estimated coefficients will be biased. So take the first difference to eliminate a i, then estimate the following model by OLS. ∆Y it =∆ β 1 x it1 + ∆ β 2 x it2 +…+ ∆ x itk + ∆ u it 35 Fixed effect
4Note, when you take the first difference, the constant term will also be eliminated. So you should use `nocons’ option in STATA when you estimate the model. 4When some variables are time invariant, these variables are also eliminated. If the treatment variable does not change overtime, you cannot use this method. 36
First differencing for more than two periods. 4You can use first differencing for more than two periods. 4You just have to difference two adjacent periods successively. 4For example, suppose that you have 3 periods. Then for the dependent variable, you compute ∆y i2 =y i2 -y i1, and ∆y i3 =y i3 -y i2. Do the same for x-variables. Then run the regression. 37
Exercise 4The data ezunem.dta contains the city level unemployment claim statistics in the state of Indiana. This data also contains information about whether the city has an enterprise zone or not. 4The enterprise zone is the area which encourages businesses and investments through reduced taxes and restrictions. Enterprise zones are usually created in an economically depressed area with the purpose of increasing the economic activities and reducing unemployment. 38
4Using the data, ezunem.dta, you are asked to estimate the effect of enterprise zones on the city-level unemployment claim. Use the log of unemployment claim as the dependent variable Ex1. First estimate the following model using OLS. log(unemployment claims) it =β0+β1(Enterprise zone) it +β(year dummies) it +v it Discuss whether the coefficient for enterprise zone is biased or not. If you think it is biased, what is the direction of bias? Ex2. Estimate the model using the first difference method. Did it change the result? Was your prediction of bias correct? 39
40 OLS results
41 First differencing
The do file used to generate the results. tsset city year reg luclms ez d81 d82 d83 d84 d85 d86 d87 d88 gen lagluclms =luclms -L.luclms gen lagez =ez -L.ez gen lagd81 =d81 -L.d81 gen lagd82 =d82 -L.d82 gen lagd83 =d83 -L.d83 gen lagd84 =d84 -L.d84 gen lagd85 =d85 -L.d85 gen lagd86 =d86 -L.d86 gen lagd87 =d87 -L.d87 gen lagd88 =d88 -L.d88 reg lagluclms lagez lagd81 lagd82 lagd83 lagd84 lagd85 lagd86 lagd87 lagd88, nocons 42
The assumptions for the first difference method. Assumption FD1 : Linearity For each i, the model is written as y it = β 0 +β 1 x it1 +…+β k x itk +a i +u it 43
Assumption FD2 : We have a random sample from the cross section Assumption FD3 : There is no perfect collinearity. In addition, each explanatory variable changes over time at least for some i in the sample. 44
Assumption FD4. Strict exogeneity E(u it |X i,a i )=0 for each i. Where X i is the short hand notation for ‘all the explanatory variables for i th individual for all the time period’. This means that u it is uncorrelated with the current year’s explanatory variables as well as with other years’ explanatory variables. 45
The unbiasedness of first difference method 4Under FD1 through FD4, the estimated parameters for the first difference method are unbiased. 46
Assumption FD5 : Homoskedasticity Var(∆u it |X i )=σ 2 Assumption FD6 : No serial correlation within ith individual. Cov(∆u it,∆u is )=0 for t≠s Note that FD2 assumes random sampling across difference individual, but does not assume randomness within each individual. So you need an additional assumption to rule out the serial correlation. 47