9.2 Fixed Effects Model

  • We can use the plm package to replicate the following investment equation as considered by Grunfeld (Grunfeld, 1958).

\[\begin{equation} I_{it}=\alpha+\beta_{1}F_{it}+\beta_{2}C_{it}+\varepsilon_{it} \tag{9.1} \end{equation}\]

The model in equation-1 is a one-way panel regression model which attempts to quantify the dependence of real gross investment (\(I_{it}\)) on the real value of the company (\(F_{it}\)) and real value of its capital stock (\(C_{it}\)).

  • Grunfeld (1958) studied 10 large manufacturing firms from the United States over 20 years (1935-954). A fixed effect estimation can be obtained with the following code
# using plm with 'within' estimator for fixed effects
fe1 = plm(inv ~ value + capital, data = pdata1, model = "within")
# the output can be summarised with summary
summary(fe1)
Oneway (individual) effect Within Model

Call:
plm(formula = inv ~ value + capital, data = pdata1, model = "within")

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

Residuals:
      Min.    1st Qu.     Median    3rd Qu.       Max. 
-184.00857  -17.64316    0.56337   19.19222  250.70974 

Coefficients:
        Estimate Std. Error t-value  Pr(>|t|)    
value   0.110124   0.011857  9.2879 < 2.2e-16 ***
capital 0.310065   0.017355 17.8666 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Total Sum of Squares:    2244400
Residual Sum of Squares: 523480
R-Squared:      0.76676
Adj. R-Squared: 0.75311
F-statistic: 309.014 on 2 and 188 DF, p-value: < 2.22e-16
  • The summary output for the fitted model object gives details about the fitted object.

  • The individual fixed effects can be obtained using the function , a summary method is also available as shown next

# individual fixed effects
fixef(fe1)
        1         2         3         4         5         6         7         8 
 -70.2967  101.9058 -235.5718  -27.8093 -114.6168  -23.1613  -66.5535  -57.5457 
        9        10 
 -87.2223   -6.5678 
# summary
summary(fixef(fe1))
    Estimate Std. Error t-value  Pr(>|t|)    
1   -70.2967    49.7080 -1.4142   0.15896    
2   101.9058    24.9383  4.0863 6.485e-05 ***
3  -235.5718    24.4316 -9.6421 < 2.2e-16 ***
4   -27.8093    14.0778 -1.9754   0.04969 *  
5  -114.6168    14.1654 -8.0913 7.141e-14 ***
6   -23.1613    12.6687 -1.8282   0.06910 .  
7   -66.5535    12.8430 -5.1821 5.629e-07 ***
8   -57.5457    13.9931 -4.1124 5.848e-05 ***
9   -87.2223    12.8919 -6.7657 1.635e-10 ***
10   -6.5678    11.8269 -0.5553   0.57933    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

References

Grunfeld, Yehuda. 1958. “The Determinants of Corporate Investment: A Study of a Number of Large Corporations in the United States.” PhD thesis, Department of Photoduplication, University of Chicago Library.