Types of Data I

Types of Data I

• Panel data: combines these dimensions: individual i at time t

Panel Data I

Panel Data II

Panel Data II

Panel Data: Our Motivating Example

The Data I

glimpse(phones)

## Observations: 306
## Variables: 8
## $ year          <fct> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2…
## $ state         <fct> Alabama, Alaska, Arizona, Arkansas, California, Colorad…
## $ urban_percent <dbl> 30, 55, 45, 21, 54, 34, 84, 31, 100, 53, 39, 45, 11, 56…
## $ cell_plans    <dbl> 8135.525, 6730.282, 7572.465, 8071.125, 8821.933, 8162.…
## $ cell_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
## $ text_ban      <fct> 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
## $ deaths        <dbl> 18.075232, 16.301184, 16.930578, 19.595430, 12.104340, …
## $ year_num      <dbl> 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2007, 2…

The Data II

phones %>%
  count(state)

phones %>%
  count(year)

The Data III

phones %>%
  distinct(state)

phones %>%
  distinct(year)

The Data IV

phones %>%
  summarize(States = n_distinct(state),
            Years = n_distinct(year))

The Data: With plm

# install.packages("plm")
library(plm)
pdim(phones, index=c("state","year"))

## Balanced Panel: n = 51, T = 6, N = 306

Pooled Regression I

• What if we just ran a standard regression:

^Yit=β0+β1Xit+uit

• N number of i groups (e.g. U.S. States)
• T number of t periods (e.g. years)

• This is a pooled regression model: treats all observations as independent

Pooled Regression II

pooled<-lm(deaths~cell_plans, data=phones)
summary(pooled)

## 
## Call:
## lm(formula = deaths ~ cell_plans, data = phones)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.0951 -2.6411 -0.2893  2.2755 11.2665 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 17.3371034  0.9753845  17.775  < 2e-16 ***
## cell_plans  -0.0005666  0.0001070  -5.297 2.26e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.279 on 304 degrees of freedom
## Multiple R-squared:  0.0845,    Adjusted R-squared:  0.08148 
## F-statistic: 28.06 on 1 and 304 DF,  p-value: 2.264e-07

Pooled Regression III

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=18)

Pooled Regression III

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point()+
  geom_smooth(method = "lm", color = "red")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=18)

Recap: Assumptions about Errors

Biases of Pooled Regression

^Yit=β0+β1Xit+ϵit

• Assumption 3: cor(ui,uj)=0∀i≠j

• Pooled regression model is biased because it ignores:

  • Multiple observations from same group i
  • Multiple observations from same time t

• Thus, errors are serially or auto-correlated; cor(ui,uj)≠0 within same i and within same t

Biases of Pooled Regression: Our Example

^Deathsit=β0+β1Cell Phonesit+uit

• Multiple observations from same state i

  • Probably similarities among u for obs in same state

• Multiple observations from same year t

  • Probably similarities among u for obs in same year

Look at 5 States

phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths)+
  geom_point(aes(color = state))+
  geom_smooth(method = "lm", color = "red")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=18)+
  theme(legend.position = "top")

Look at 5 States

phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+
  geom_point()+
  geom_smooth(method = "lm")+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=18)+
  theme(legend.position = "top")

Look at 5 States

phones %>%
  filter(state %in% c("District of Columbia",
                      "Maryland", "Texas",
                      "California", "Kansas")) %>%
ggplot(data = .)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=18)+
  theme(legend.position = "none")+
  facet_wrap(~state, ncol=3)

Look at All States

ggplot(data = phones)+
  aes(x = cell_plans,
      y = deaths,
      color = state)+ 
  geom_point()+ 
  geom_smooth(method = "lm")+ 
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed")+
  theme(legend.position = "none")+
  facet_wrap(~state, ncol=7)

The Bias in our Pooled Regression

^Deathsit=β0+β1Cell Phonesit+uit

• Cell Phonesit is endogenous:

cor(uit,cell phonesit)≠0E[uit|cell phonesit]≠0

• Things in uit correlated with Cell phonesit:

  • infrastructure spending, population, urban vs. rural, more/less cautious citizens, cultural attitudes towards driving, texting, etc

• A lot of these things vary systematically by State!

  • cor(uit1,uit2)≠0
  • Error in State i during t1 correlates with error in State i during t2 (things in State that don't change over time)

Fixed Effects: Decomposing uit

• Much of the endogeneity in Xit can be explained by systematic differences across i (groups)

• Exploit the systematic variation across groups with a fixed effects model

• Decompose the error term into two parts:

uit=αi+ϵit

Fixed Effects: αi

uit=αi+ϵit

• αi are group-specific fixed effects

  • group i tends to have higher or lower ˆY than other groups given regressor(s) Xit
  • estimate a separate αi for each group i
  • essentially, a separate constant (or intercept), (^β0) for each group

• This includes all factors that do not change within group (i) over time

Fixed Effects: ϵit

uit=αi+ϵit

• ϵit is the remaining random error

  • As usual in OLS, assume the four assumptions about this error:
  • E[ϵit]=0, var[ϵit]=σ2ϵ, cor(ϵit,ϵjt)=0, cor(ϵit,Xit)=0

• Includes all other factors affecting Yit not contained in group effect αi

  • i.e. differences within each group that change over time

• Xit can still be endogenous from non-group-specific effects!

Fixed Effects: New Regression Equation

ˆYit=β0+β1Xit+αi+ϵit

• We've pulled αi out of the original error term into the regression

• Essentially we'll estimate an intercept for each group (minus one, which is β0)

• Must have multiple observations (over time) for each group (i.e. panel data)

Fixed Effects: Our Example

^Deathsit=β0+β1Cell phonesit+αi+ϵit

• αi is the State fixed effect

  • Captures everything unique about each state i that does not change over time
  • culture, institutions, history, geography, climate, etc!

• There could still be factors in ϵit that are correlated with Cell phonesit!

  • things that do change over time within States
  • Perhaps a State has a cell phone ban only for some years in our data

Estimating Fixed Effects Models

ˆYit=β0+β1Xit+αi

• Two methods to estimate fixed effects models:

1. Least Squares Dummy Variable (LSDV) approach

2. De-meaned data approach

Least Squares Dummy Variable Approach

^Yit=β0+β1Xit+β2D1i+β3D2i+⋯+βND(N−1)i

• A dummy variable Di={1 if observation is from group i0 if observation is not from group i for each possible group =1 if observation it is from group i, otherwise =0

• If there are N groups:

  • Include N−1 dummies (to avoid dummy variable trap) and β0 is the reference category¹

• So we are estimating different intercepts for each group

• Sounds like a lot of work, automatic in R

• This soaks up anything in the error term fixed within groups over time!

¹ If we do not estimate β0, we could include all N dummies. In either case, β0 takes the place of one category-dummy.

Least Squares Dummy Variable Approach: Our Example

Our Example in R I

^Deathsit=β1Cell Phonesit+Alabamai+Alaskai+⋯+Wyomingi

• If state is a factor variable, just include it in the regression

• R automatically creates N−1 dummy variables and includes them in the regression

  • Keeps intercept and leaves out first group dummy

Our Example in R II

fe_reg_1<-lm(deaths~cell_plans+state, data = phones)
summary(fe_reg_1)

## 
## Call:
## lm(formula = deaths ~ cell_plans + state, data = phones)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5617 -0.6577 -0.1353  0.5997  3.7087 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               25.5076799  1.0176400  25.066  < 2e-16 ***
## cell_plans                -0.0012037  0.0001013 -11.881  < 2e-16 ***
## stateAlaska               -2.4841648  0.6745076  -3.683 0.000282 ***
## stateArizona              -1.5105774  0.6704570  -2.253 0.025109 *  
## stateArkansas              3.1926629  0.6664384   4.791 2.83e-06 ***
## stateCalifornia           -4.9786687  0.6655468  -7.481 1.21e-12 ***
## stateColorado             -4.3445535  0.6654735  -6.529 3.59e-10 ***
## stateConnecticut          -6.5951855  0.6654429  -9.911  < 2e-16 ***
## stateDelaware             -2.0983936  0.6666483  -3.148 0.001842 ** 
## stateDistrict of Columbia  6.3557900  1.2897173   4.928 1.50e-06 ***
## stateFlorida              -1.0347656  0.6656584  -1.554 0.121310    
## stateGeorgia              -2.1693995  0.6661316  -3.257 0.001281 ** 
## stateHawaii               -2.9915131  0.6662205  -4.490 1.08e-05 ***
## stateIdaho                -2.4608133  0.6721521  -3.661 0.000306 ***
## stateIllinois             -5.0620677  0.6657586  -7.603 5.59e-13 ***
## stateIndiana              -5.2284659  0.6696566  -7.808 1.53e-13 ***
## stateIowa                 -3.2495889  0.6705450  -4.846 2.19e-06 ***
## stateKansas               -1.4581629  0.6654571  -2.191 0.029345 *  
## stateKentucky              1.0915751  0.6671138   1.636 0.103023    
## stateLouisiana             3.9178819  0.6715685   5.834 1.64e-08 ***
## stateMaine                -4.6283292  0.6691173  -6.917 3.73e-11 ***
## stateMaryland             -4.2430981  0.6697410  -6.335 1.07e-09 ***
## stateMassachusetts        -7.7934371  0.6672061 -11.681  < 2e-16 ***
## stateMichigan             -5.3689561  0.6657520  -8.064 2.91e-14 ***
## stateMinnesota            -7.5721267  0.6657464 -11.374  < 2e-16 ***
## stateMississippi           1.5615987  0.6689602   2.334 0.020357 *  
## stateMissouri             -2.2813452  0.6656121  -3.427 0.000711 ***
## stateMontana               4.0032162  0.6692246   5.982 7.46e-09 ***
## stateNebraska             -4.3224747  0.6667641  -6.483 4.66e-10 ***
## stateNevada               -1.8496986  0.6655435  -2.779 0.005856 ** 
## stateNew Hampshire        -6.1273821  0.6660296  -9.200  < 2e-16 ***
## stateNew Jersey           -5.7073658  0.6688922  -8.533 1.31e-15 ***
## stateNew Mexico           -1.7191586  0.6716722  -2.560 0.011062 *  
## stateNew York             -4.7548279  0.6694121  -7.103 1.23e-11 ***
## stateNorth Carolina       -1.5292152  0.6654724  -2.298 0.022378 *  
## stateNorth Dakota          0.5958807  0.6659609   0.895 0.371758    
## stateOhio                 -4.6832026  0.6654475  -7.038 1.82e-11 ***
## stateOklahoma             -0.0654323  0.6660894  -0.098 0.921824    
## stateOregon               -4.2716778  0.6667266  -6.407 7.16e-10 ***
## statePennsylvania         -1.9224145  0.6658671  -2.887 0.004223 ** 
## stateRhode Island         -6.5332075  0.6655747  -9.816  < 2e-16 ***
## stateSouth Carolina        2.5662807  0.6685038   3.839 0.000156 ***
## stateSouth Dakota         -0.8890418  0.6673510  -1.332 0.183990    
## stateTennessee             0.5437220  0.6675007   0.815 0.416085    
## stateTexas                -1.2153059  0.6654310  -1.826 0.068972 .  
## stateUtah                 -6.3933504  0.6723736  -9.509  < 2e-16 ***
## stateVermont              -7.0703850  0.6803788 -10.392  < 2e-16 ***
## stateVirginia             -4.4686467  0.6660141  -6.710 1.26e-10 ***
## stateWashington           -6.2367572  0.6654881  -9.372  < 2e-16 ***
## stateWest Virginia         2.1062917  0.6744981   3.123 0.001999 ** 
## stateWisconsin            -5.3569895  0.6712323  -7.981 5.02e-14 ***
## stateWyoming               0.7736131  0.6660728   1.161 0.246547    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.153 on 254 degrees of freedom
## Multiple R-squared:  0.9055,    Adjusted R-squared:  0.8865 
## F-statistic: 47.72 on 51 and 254 DF,  p-value: < 2.2e-16

Our Example in R II

De-meaned Approach I

• Alternatively, we can hold group fixed effects constant without directly estimating them

• We simply de-mean the data for each group

• For each group i, find the means (over time, t): ˉYi=β0+β1ˉXi+ˉαi+ˉϵi

• ˉYi: average value of Yit for group i
• ˉXi: average value of Xit for group i
• ˉαi: average value of αi for group i ($=\alpha_i$)
• ˉϵi=0, by assumption

De-meaned Approach II

^Yit=β0+β1Xit+uitˉYi=β0+β1ˉXi+ˉαi+ˉϵi

• Subtract the means equation from the pooled equation to get:

Yi−ˉYi=β1(Xit−ˉXi)+˜ϵit˜Yit=β1˜Xit+˜ϵit

• Within each group i, the de-meaned variables ˜Yit and ˜Xit's all have a mean of 0¹

• Variables that don't change over time will drop out of analysis altogether

• Removes any source of variation across groups to only work with variation within each group

¹ Recall Rule 4 from the class notes on the Summation Operator: (\sum(X_i-\bar{X})=0)

De-meaned Approach III

˜Yit=β1˜Xit+˜ϵit

• Yields identical results to dummy variable approach

• More useful when we have many groups (would be many dummies)

• Demonstrates intuition behind fixed effects:

  • Converts all data to deviations from the mean of each group
  • All groups are "centered" at 0
  • Fixed effects are often called the "within" estimators, they exploit variation within groups, not across groups

De-meaned Approach IV

• We are basically comparing groups to themselves over time

  • apples to apples comparison
  • e.g. Maryland in 2000 vs. Maryland in 2005

• Ignore all differences between groups, only look at differences within groups over time

De-Meaning the Data in R I

# get means of Y and X by state
means_state<-phones %>%
  group_by(state) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))
# look at it
means_state

De-Meaning the Data in R II

ggplot(data = means_state)+
  aes(x = fct_reorder(state,avg_deaths),
      y = avg_deaths,
      color = state)+
  geom_point()+
  geom_segment(aes(y=0, yend=avg_deaths, x=state, xend=state))+
  coord_flip()+
  labs(x = "Cell Phones Per 10,000 People",
       y = "Deaths Per Billion Miles Driven",
       color = NULL)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=10)+
  theme(legend.position = "none")

Visualizing "Within Estimates" for the 5 States

Visualizing "Within Estimates" for All 51 States

De-meaned Approach in R I

• The plm package is designed for panel data

• plm() function is just like lm(), with some additional arguments:

  • index="group_variable_name" set equal to the name of your factor variable for the groups
  • model= set equal to "within" to use fixed-effects (within-estimator)

#install.packages("plm")
library(plm)
fe_reg_1_alt<-plm(deaths ~ cell_plans,
                  data = phones,
                  index = "state",
                  model = "within")

De-meaned Approach in R II

summary(fe_reg_1_alt)

## Oneway (individual) effect Within Model
## 
## Call:
## plm(formula = deaths ~ cell_plans, data = phones, model = "within", 
##     index = "state")
## 
## Balanced Panel: n = 51, T = 6, N = 306
## 
## Residuals:
##     Min.  1st Qu.   Median  3rd Qu.     Max. 
## -3.56170 -0.65772 -0.13533  0.59971  3.70868 
## 
## Coefficients:
##               Estimate  Std. Error t-value  Pr(>|t|)    
## cell_plans -0.00120374  0.00010131 -11.882 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    524.94
## Residual Sum of Squares: 337.41
## R-Squared:      0.35724
## Adj. R-Squared: 0.22818
## F-statistic: 141.169 on 1 and 254 DF, p-value: < 2.22e-16

Two-Way Fixed Effects

• So far, we've looked at a one-way fixed effects model that estimates a fixed effect for groups

• Two-way fixed effects model estimates a fixed effect for both the groups and the time periods ^Yit=β0+β1Xit+αi+θt+νit

• αi: group fixed effects

  • accounts for time-invariant differences across groups

• θt: time fixed effects

  • acounts for group-invariant differences over time

Two-Way Fixed Effects: Our Example

^Deathsit=β0+β1Cell phonesit+αi+θt+ϵit

• αi: State fixed effects

  • differences across states that are stable over time
  • e.g. geography, culture, (unchanging) state laws

• θt: Year fixed effects

  • differences over time that are stable across states
  • e.g. economy-wide macroeconomic changes, federal laws passed

Visualizing Year Effects I

# find averages for years
means_year<-phones %>%
  group_by(year) %>%
  summarize(avg_deaths = mean(deaths),
            avg_phones = mean(cell_plans))
means_year

Visualizing Year Effects II

ggplot(data = means_year)+
  aes(x = year,
      y = avg_deaths)+
  geom_point(data = phones,
             aes(x = year,
                 y = deaths,
                 color = year))+
  geom_point(size=3)+
  geom_path()

Estimating Two-Way Fixed Effects

ˆYit=β0+β1Xit+αi+θt

• As before, several equivalent ways to estimate two-way fixed effects models:

1. Least Squares Dummy Variable (LSDV) Approach: add dummies for both groups and time periods (separate intercepts for groups and times)

2. Fully De-meaned data: ˜Yit=β1˜Xit+˜νit

where each ~variableit=variableit−¯variablet−¯variablei

1. Hybrid: de-mean for one effect (groups or times) and add dummies for the other effect (times or groups)

LSDV Method

fe2_reg_1<-lm(deaths~cell_plans+state+year, data = phones)
summary(fe2_reg_1)

## 
## Call:
## lm(formula = deaths ~ cell_plans + state + year, data = phones)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7361 -0.5110 -0.0242  0.5182  3.0994 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)               18.9304707  1.4511324  13.045  < 2e-16 ***
## cell_plans                -0.0002995  0.0001723  -1.738 0.083400 .  
## stateAlaska               -1.4998292  0.6241083  -2.403 0.016986 *  
## stateArizona              -0.7791715  0.6113519  -1.275 0.203672    
## stateArkansas              2.8655345  0.5985063   4.788 2.90e-06 ***
## stateCalifornia           -5.0900897  0.5956293  -8.546 1.30e-15 ***
## stateColorado             -4.4127242  0.5953925  -7.411 1.95e-12 ***
## stateConnecticut          -6.6325835  0.5952934 -11.142  < 2e-16 ***
## stateDelaware             -2.4579830  0.5991822  -4.102 5.55e-05 ***
## stateDistrict of Columbia -3.5044964  1.9710939  -1.778 0.076633 .  
## stateFlorida              -1.1904908  0.5959900  -1.998 0.046859 *  
## stateGeorgia              -2.4422504  0.5975175  -4.087 5.89e-05 ***
## stateHawaii               -3.2811475  0.5978043  -5.489 9.96e-08 ***
## stateIdaho                -1.6145030  0.6167130  -2.618 0.009389 ** 
## stateIllinois             -5.2488047  0.5963135  -8.802 2.30e-16 ***
## stateIndiana              -4.5580058  0.6088090  -7.487 1.22e-12 ***
## stateIowa                 -2.5117858  0.6116310  -4.107 5.45e-05 ***
## stateKansas               -1.4042964  0.5953392  -2.359 0.019106 *  
## stateKentucky              1.5143772  0.6006792   2.521 0.012324 *  
## stateLouisiana             3.1093174  0.6148710   5.057 8.27e-07 ***
## stateMaine                -4.0022270  0.6070912  -6.592 2.57e-10 ***
## stateMaryland             -4.9202431  0.6090776  -8.078 2.84e-14 ***
## stateMassachusetts        -8.2276855  0.6009756 -13.691  < 2e-16 ***
## stateMichigan             -5.1840812  0.5962925  -8.694 4.80e-16 ***
## stateMinnesota            -7.3888658  0.5962744 -12.392  < 2e-16 ***
## stateMississippi           2.1741873  0.6065903   3.584 0.000407 ***
## stateMissouri             -2.1422872  0.5958403  -3.595 0.000390 ***
## stateMontana               4.6383935  0.6074334   7.636 4.80e-13 ***
## stateNebraska             -3.9461735  0.5995548  -6.582 2.73e-10 ***
## stateNevada               -1.9595530  0.5956188  -3.290 0.001147 ** 
## stateNew Hampshire        -5.8751509  0.5971883  -9.838  < 2e-16 ***
## stateNew Jersey           -6.3140122  0.6063734 -10.413  < 2e-16 ***
## stateNew Mexico           -0.9037640  0.6151985  -1.469 0.143079    
## stateNew York             -5.4055532  0.6080308  -8.890  < 2e-16 ***
## stateNorth Carolina       -1.4618984  0.5953890  -2.455 0.014758 *  
## stateNorth Dakota          0.3585402  0.5969666   0.601 0.548650    
## stateOhio                 -4.7265870  0.5953082  -7.940 6.95e-14 ***
## stateOklahoma              0.1990835  0.5973813   0.333 0.739218    
## stateOregon               -3.9007021  0.5994343  -6.507 4.17e-10 ***
## statePennsylvania         -1.7070557  0.5966639  -2.861 0.004582 ** 
## stateRhode Island         -6.4092098  0.5957196 -10.759  < 2e-16 ***
## stateSouth Carolina        3.1378079  0.6051330   5.185 4.47e-07 ***
## stateSouth Dakota         -0.4374085  0.6014408  -0.727 0.467745    
## stateTennessee             0.0748011  0.6019211   0.124 0.901201    
## stateTexas                -1.2268813  0.5952548  -2.061 0.040332 *  
## stateUtah                 -5.5331389  0.6174110  -8.962  < 2e-16 ***
## stateVermont              -5.8044650  0.6422793  -9.037  < 2e-16 ***
## stateVirginia             -4.7176009  0.5971384  -7.900 8.95e-14 ***
## stateWashington           -6.1580557  0.5954397 -10.342  < 2e-16 ***
## stateWest Virginia         3.0901095  0.6240787   4.951 1.36e-06 ***
## stateWisconsin            -4.5709779  0.6138081  -7.447 1.56e-12 ***
## stateWyoming               0.5124453  0.5973278   0.858 0.391775    
## year2008                  -1.0106325  0.2165233  -4.668 4.98e-06 ***
## year2009                  -1.6830082  0.2458856  -6.845 5.93e-11 ***
## year2010                  -2.0721270  0.2751070  -7.532 9.20e-13 ***
## year2011                  -2.2176559  0.3187735  -6.957 3.06e-11 ***
## year2012                  -1.8728162  0.3425095  -5.468 1.11e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.031 on 249 degrees of freedom
## Multiple R-squared:  0.9259,    Adjusted R-squared:  0.9092 
## F-statistic: 55.53 on 56 and 249 DF,  p-value: < 2.2e-16

With plm

fe2_reg_2<-plm(deaths~cell_plans, index=c("state", "year"), model="within", data = phones)
summary(fe2_reg_2)

## Oneway (individual) effect Within Model
## 
## Call:
## plm(formula = deaths ~ cell_plans, data = phones, model = "within", 
##     index = c("state", "year"))
## 
## Balanced Panel: n = 51, T = 6, N = 306
## 
## Residuals:
##     Min.  1st Qu.   Median  3rd Qu.     Max. 
## -3.56170 -0.65772 -0.13533  0.59971  3.70868 
## 
## Coefficients:
##               Estimate  Std. Error t-value  Pr(>|t|)    
## cell_plans -0.00120374  0.00010131 -11.882 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    524.94
## Residual Sum of Squares: 337.41
## R-Squared:      0.35724
## Adj. R-Squared: 0.22818
## F-statistic: 141.169 on 1 and 254 DF, p-value: < 2.22e-16

## Oneway (individual) effect Within Model
## 
## Call:
## plm(formula = deaths ~ cell_plans, data = phones, model = "within", 
##     index = c("state", "year"))
## 
## Balanced Panel: n = 51, T = 6, N = 306
## 
## Residuals:
##     Min.  1st Qu.   Median  3rd Qu.     Max. 
## -3.56170 -0.65772 -0.13533  0.59971  3.70868 
## 
## Coefficients:
##               Estimate  Std. Error t-value  Pr(>|t|)    
## cell_plans -0.00120374  0.00010131 -11.882 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    524.94
## Residual Sum of Squares: 337.41
## R-Squared:      0.35724
## Adj. R-Squared: 0.22818
## F-statistic: 141.169 on 1 and 254 DF, p-value: < 2.22e-16

Adding Covariates I

Adding Covariates II

fe2_controls_reg<-plm(deaths~cell_plans+text_ban+urban_percent+cell_ban, data=phones,index=c("state","year"), model="within", effect="twoways") 
summary(fe2_controls_reg)

## Twoways effects Within Model
## 
## Call:
## plm(formula = deaths ~ cell_plans + text_ban + urban_percent + 
##     cell_ban, data = phones, effect = "twoways", model = "within", 
##     index = c("state", "year"))
## 
## Balanced Panel: n = 51, T = 6, N = 306
## 
## Residuals:
##     Min.  1st Qu.   Median  3rd Qu.     Max. 
## -2.73515 -0.51252 -0.01354  0.47735  3.14133 
## 
## Coefficients:
##                  Estimate  Std. Error t-value Pr(>|t|)  
## cell_plans    -0.00034037  0.00017294 -1.9682  0.05017 .
## text_ban1      0.25592616  0.22219230  1.1518  0.25051  
## urban_percent  0.01313477  0.01119861  1.1729  0.24197  
## cell_ban1     -0.67979565  0.40294912 -1.6871  0.09286 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    267.89
## Residual Sum of Squares: 259.07
## R-Squared:      0.032939
## Adj. R-Squared: -0.199
## F-statistic: 2.09472 on 4 and 246 DF, p-value: 0.082085

Comparing Models

library(huxtable)
huxreg("Pooled" = pooled,
       "State Effects" = fe_reg_1,
       "State and Year Effects" = fe2_reg_1,
       "With Controls" = fe2_controls_reg,
       coefs = c("Intercept" = "(Intercept)",
                 "Cell phones" = "cell_plans",
                 "Cell Ban" = "cell_ban1",
                 "Texting Ban" = "text_ban1",
                 "Urbanization Rate" = "urban_percent"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 4)

Comparing Models