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## 10.1 Panel Data

### Notation for Panel Data

In contrast to cross-section data where we have observations on $n$ subjects (entities), panel data has observations on $n$ entities at $T\geq2$ time periods. This is denoted

$(X_{it},Y_{it}), \ i=1,\dots,n \ \ \ \text{and} \ \ \ t=1,\dots,T$ where the index $i$ refers to the entity while $t$ refers to the time period.

Sometimes panel data is also called longitudinal data as it adds a temporal dimension to cross-sectional data. Let us have a look at the dataset Fatalities by checking its structure and listing the first few observations.

# load the package and the dataset
library(AER)
data(Fatalities)
# obtain the dimension and inspect the structure
is.data.frame(Fatalities)
#> [1] TRUE
dim(Fatalities)
#> [1] 336  34
str(Fatalities)
#> 'data.frame':	336 obs. of  34 variables:
#>  $state : Factor w/ 48 levels "al","az","ar",..: 1 1 1 1 1 1 1 2 2 2 ... #>$ year        : Factor w/ 7 levels "1982","1983",..: 1 2 3 4 5 6 7 1 2 3 ...
#>  $spirits : num 1.37 1.36 1.32 1.28 1.23 ... #>$ unemp       : num  14.4 13.7 11.1 8.9 9.8 ...
#>  $income : num 10544 10733 11109 11333 11662 ... #>$ emppop      : num  50.7 52.1 54.2 55.3 56.5 ...
#>  $beertax : num 1.54 1.79 1.71 1.65 1.61 ... #>$ baptist     : num  30.4 30.3 30.3 30.3 30.3 ...
#>  $mormon : num 0.328 0.343 0.359 0.376 0.393 ... #>$ drinkage    : num  19 19 19 19.7 21 ...
#>  $dry : num 25 23 24 23.6 23.5 ... #>$ youngdrivers: num  0.212 0.211 0.211 0.211 0.213 ...
#>  $miles : num 7234 7836 8263 8727 8953 ... #>$ breath      : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 1 1 1 ...
#>  $jail : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 2 2 2 ... #>$ service     : Factor w/ 2 levels "no","yes": 1 1 1 1 1 1 1 2 2 2 ...
#>  $fatal : int 839 930 932 882 1081 1110 1023 724 675 869 ... #>$ nfatal      : int  146 154 165 146 172 181 139 131 112 149 ...
#>  $sfatal : int 99 98 94 98 119 114 89 76 60 81 ... #>$ fatal1517   : int  53 71 49 66 82 94 66 40 40 51 ...
#>  $nfatal1517 : int 9 8 7 9 10 11 8 7 7 8 ... #>$ fatal1820   : int  99 108 103 100 120 127 105 81 83 118 ...
#>  $nfatal1820 : int 34 26 25 23 23 31 24 16 19 34 ... #>$ fatal2124   : int  120 124 118 114 119 138 123 96 80 123 ...
#>  $nfatal2124 : int 32 35 34 45 29 30 25 36 17 33 ... #>$ afatal      : num  309 342 305 277 361 ...
#>  $pop : num 3942002 3960008 3988992 4021008 4049994 ... #>$ pop1517     : num  209000 202000 197000 195000 204000 ...
#>  $pop1820 : num 221553 219125 216724 214349 212000 ... #>$ pop2124     : num  290000 290000 288000 284000 263000 ...
#>  $milestot : num 28516 31032 32961 35091 36259 ... #>$ unempus     : num  9.7 9.6 7.5 7.2 7 ...
#>  $emppopus : num 57.8 57.9 59.5 60.1 60.7 ... #>$ gsp         : num  -0.0221 0.0466 0.0628 0.0275 0.0321 ...
# list the first few observations
#>   state year spirits unemp   income   emppop  beertax baptist  mormon drinkage
#> 1    al 1982    1.37  14.4 10544.15 50.69204 1.539379 30.3557 0.32829    19.00
#> 2    al 1983    1.36  13.7 10732.80 52.14703 1.788991 30.3336 0.34341    19.00
#> 3    al 1984    1.32  11.1 11108.79 54.16809 1.714286 30.3115 0.35924    19.00
#> 4    al 1985    1.28   8.9 11332.63 55.27114 1.652542 30.2895 0.37579    19.67
#> 5    al 1986    1.23   9.8 11661.51 56.51450 1.609907 30.2674 0.39311    21.00
#> 6    al 1987    1.18   7.8 11944.00 57.50988 1.560000 30.2453 0.41123    21.00
#>       dry youngdrivers    miles breath jail service fatal nfatal sfatal
#> 1 25.0063     0.211572 7233.887     no   no      no   839    146     99
#> 2 22.9942     0.210768 7836.348     no   no      no   930    154     98
#> 3 24.0426     0.211484 8262.990     no   no      no   932    165     94
#> 4 23.6339     0.211140 8726.917     no   no      no   882    146     98
#> 5 23.4647     0.213400 8952.854     no   no      no  1081    172    119
#> 6 23.7924     0.215527 9166.302     no   no      no  1110    181    114
#>   fatal1517 nfatal1517 fatal1820 nfatal1820 fatal2124 nfatal2124  afatal
#> 1        53          9        99         34       120         32 309.438
#> 2        71          8       108         26       124         35 341.834
#> 3        49          7       103         25       118         34 304.872
#> 4        66          9       100         23       114         45 276.742
#> 5        82         10       120         23       119         29 360.716
#> 6        94         11       127         31       138         30 368.421
#>       pop  pop1517  pop1820  pop2124 milestot unempus emppopus         gsp
#> 1 3942002 208999.6 221553.4 290000.1    28516     9.7     57.8 -0.02212476
#> 2 3960008 202000.1 219125.5 290000.2    31032     9.6     57.9  0.04655825
#> 3 3988992 197000.0 216724.1 288000.2    32961     7.5     59.5  0.06279784
#> 4 4021008 194999.7 214349.0 284000.3    35091     7.2     60.1  0.02748997
#> 5 4049994 203999.9 212000.0 263000.3    36259     7.0     60.7  0.03214295
#> 6 4082999 204999.8 208998.5 258999.8    37426     6.2     61.5  0.04897637
# summarize the variables 'state' and 'year'
summary(Fatalities[, c(1, 2)])
#>      state       year
#>  al     :  7   1982:48
#>  az     :  7   1983:48
#>  ar     :  7   1984:48
#>  ca     :  7   1985:48
#>  co     :  7   1986:48
#>  ct     :  7   1987:48
#>  (Other):294   1988:48

We find that the dataset consists of 336 observations on 34 variables. Notice that the variable state is a factor variable with 48 levels (one for each of the 48 contiguous federal states of the U.S.). The variable year is also a factor variable that has 7 levels identifying the time period when the observation was made. This gives us $7\times48 = 336$ observations in total. Since all variables are observed for all entities and over all time periods, the panel is balanced. If there were missing data for at least one entity in at least one time period we would call the panel unbalanced.

#### Example: Traffic Deaths and Alcohol Taxes

We start by reproducing Figure 10.1 of the book. To this end we estimate simple regressions using data for years 1982 and 1988 that model the relationship between beer tax (adjusted for 1988 dollars) and the traffic fatality rate, measured as the number of fatalities per 10000 inhabitants. Afterwards, we plot the data and add the corresponding estimated regression functions.

# define the fatality rate
Fatalities$fatal_rate <- Fatalities$fatal / Fatalitiespop * 10000 # subset the data Fatalities1982 <- subset(Fatalities, year == "1982") Fatalities1988 <- subset(Fatalities, year == "1988") # estimate simple regression models using 1982 and 1988 data fatal1982_mod <- lm(fatal_rate ~ beertax, data = Fatalities1982) fatal1988_mod <- lm(fatal_rate ~ beertax, data = Fatalities1988) coeftest(fatal1982_mod, vcov. = vcovHC, type = "HC1") #> #> t test of coefficients: #> #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 2.01038 0.14957 13.4408 <2e-16 *** #> beertax 0.14846 0.13261 1.1196 0.2687 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 coeftest(fatal1988_mod, vcov. = vcovHC, type = "HC1") #> #> t test of coefficients: #> #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 1.85907 0.11461 16.2205 < 2.2e-16 *** #> beertax 0.43875 0.12786 3.4314 0.001279 ** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 The estimated regression functions are \begin{align*} \widehat{FatalityRate} =& \, \underset{(0.15)}{2.01} + \underset{(0.13)}{0.15} \times BeerTax \quad (1982 \text{ data}), \\ \widehat{FatalityRate} =& \, \underset{(0.11)}{1.86} + \underset{(0.13)}{0.44} \times BeerTax \quad (1988 \text{ data}). \end{align*} # plot the observations and add the estimated regression line for 1982 data plot(x = Fatalities1982beertax,
y = Fatalities1982$fatal_rate, xlab = "Beer tax (in 1988 dollars)", ylab = "Fatality rate (fatalities per 10000)", main = "Traffic Fatality Rates and Beer Taxes in 1982", ylim = c(0, 4.5), pch = 20, col = "steelblue") abline(fatal1982_mod, lwd = 1.5)  # plot observations and add estimated regression line for 1988 data plot(x = Fatalities1988$beertax,
y = Fatalities1988\$fatal_rate,
xlab = "Beer tax (in 1988 dollars)",
ylab = "Fatality rate (fatalities per 10000)",
main = "Traffic Fatality Rates and Beer Taxes in 1988",
ylim = c(0, 4.5),
pch = 20,
col = "steelblue")

abline(fatal1988_mod, lwd = 1.5)

In both plots, each point represents observations of beer tax and fatality rate for a given state in the respective year. The regression results indicate a positive relationship between the beer tax and the fatality rate for both years. The estimated coefficient on beer tax for the 1988 data is almost three times as large as for the 1988 dataset. This is contrary to our expectations: alcohol taxes are supposed to lower the rate of traffic fatalities. As we known from Chapter 6, this is possibly due to omitted variable bias, since both models do not include any covariates, e.g., economic conditions. This could be corrected for using a multiple regression approach. However, this cannot account for omitted unobservable factors that differ from state to state but can be assumed to be constant over the observation span, e.g., the populations’ attitude towards drunk driving. As shown in the next section, panel data allow us to hold such factors constant.