Panel Data

Author

Song Lu

Quarto

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1. Load in data “Grunfeld”

In “Grunfeld”, the time component is “year” whereas the entity component is “firm”.

The data is balanced because “year” increases by 1 unit every year, while the firms staty the same.

library(plm)


data("Grunfeld", package = "plm")

df <- Grunfeld

2. OLS Regression and Coefficients Estimation

A OLS regression is run on “value” with respect to “inv” (gross investment) and “capital” (stock of plant and equipment).

The coefficients are statistically significant with the p-value for “investment” being lower than 0.001, “Investment” appears to be reasonable as it as a coefficient of 5.7598. while the p-value for “capital” is 0.00371 which indicates a modest level of statistical importance.

The p-value for “capital” is 0.00371 which indicates a modest level of statistical importance. In addition, “capital” seems unconventional with a modest -0.6153 coefficient, suggesting each unit increase in “Capital” decreases “Value” by 0.6153. There appears to be irregularities that is causing this issue potentially “omitted variable bias”.

OLS_model <- lm(df$value ~ df$inv + df$capital)

summary(OLS_model)


Call:
lm(formula = df$value ~ df$inv + df$capital)

Residuals:
     Min       1Q   Median       3Q      Max 
-2010.54  -339.50  -184.08    76.66  2707.84 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 410.8156    64.1419   6.405 1.08e-09 ***
df$inv        5.7598     0.2909  19.803  < 2e-16 ***
df$capital   -0.6153     0.2095  -2.937  0.00371 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 666.5 on 197 degrees of freedom
Multiple R-squared:  0.7455,    Adjusted R-squared:  0.7429 
F-statistic: 288.5 on 2 and 197 DF,  p-value: < 2.2e-16

3. Fixed Effect

\[ df$value = 410.8156 + 2.5694 * inv - 0.5885* capital + ε \]

Now our coefficients change again, and the statistical importance increases as well. The coefficient for “value” increases to 2.5694 while the coefficient for “capital” increases to -0.5885. This is because: by running a model that controls firm and time, we are able to reduce the effect of those two variables.

library(lfe)

Loading required package: Matrix


Attaching package: 'lfe'

The following object is masked from 'package:plm':

    sargan

fixed_effects_model_lfe <- felm(df$value ~ df$inv + df$capital | df$firm + df$year, data = df)

summary(fixed_effects_model_lfe)


Call:
   felm(formula = df$value ~ df$inv + df$capital | df$firm + df$year,      data = df) 

Residuals:
    Min      1Q  Median      3Q     Max 
-760.84 -105.29    4.94  129.14 1042.21 

Coefficients:
           Estimate Std. Error t value Pr(>|t|)    
df$inv       2.5694     0.3002   8.560 6.65e-15 ***
df$capital  -0.5885     0.1605  -3.666 0.000329 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 241.7 on 169 degrees of freedom
Multiple R-squared(full model): 0.9713   Adjusted R-squared: 0.9662 
Multiple R-squared(proj model): 0.3601   Adjusted R-squared: 0.2465 
F-statistic(full model):190.6 on 30 and 169 DF, p-value: < 2.2e-16 
F-statistic(proj model): 47.54 on 2 and 169 DF, p-value: < 2.2e-16

By running a plm() model that also controls “time” and “firm”, we saw the coefficients remain the same as the felm() model.

library(plm)

df <- pdata.frame(df)

fixed_effects_model_plm <- plm(df$value ~ df$inv+ df$capital, data = df, model = "within", effect = "twoways")
summary(fixed_effects_model_plm)

Twoways effects Within Model

Call:
plm(formula = df$value ~ df$inv + df$capital, data = df, effect = "twoways", 
    model = "within")

Balanced Panel: n = 10, T = 20, N = 200

Residuals:
     Min.   1st Qu.    Median   3rd Qu.      Max. 
-760.8351 -105.2869    4.9391  129.1363 1042.2079 

Coefficients:
           Estimate Std. Error t-value  Pr(>|t|)    
df$inv      2.56940    0.30015  8.5604 6.653e-15 ***
df$capital -0.58847    0.16051 -3.6664 0.0003292 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Total Sum of Squares:    15422000
Residual Sum of Squares: 9869100
R-Squared:      0.36006
Adj. R-Squared: 0.24646
F-statistic: 47.543 on 2 and 169 DF, p-value: < 2.22e-16