library(plm)
# Load the Grunfeld dataset
data("Grunfeld", package = "plm")# Overview of the Grunfeld dataset
summary(Grunfeld)## firm year inv value
## Min. : 1.0 Min. :1935 Min. : 0.93 Min. : 58.12
## 1st Qu.: 3.0 1st Qu.:1940 1st Qu.: 33.56 1st Qu.: 199.97
## Median : 5.5 Median :1944 Median : 57.48 Median : 517.95
## Mean : 5.5 Mean :1944 Mean : 145.96 Mean :1081.68
## 3rd Qu.: 8.0 3rd Qu.:1949 3rd Qu.: 138.04 3rd Qu.:1679.85
## Max. :10.0 Max. :1954 Max. :1486.70 Max. :6241.70
## capital
## Min. : 0.80
## 1st Qu.: 79.17
## Median : 205.60
## Mean : 276.02
## 3rd Qu.: 358.10
## Max. :2226.30# Check the balance of the panel
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:plm':
##
## between, lag, lead## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionGrunfeld %>%
group_by(firm) %>%
summarize(n = n()) %>%
distinct(n)## # A tibble: 1 × 1
## n
## <int>
## 1 20Since there is only one unique value (i.e. 20) listed, and it appears only once, this means every entity (like each firm in the Grunfeld dataset) has exactly 20 observations. Therefore, the dataset is balanced.
Estimating equation:
$$ \text{inv}_{it} = \beta_0 + \beta_1 \text{value}_{it} + \beta_2 \text{capital}_{it} + \epsilon_{it} $$
# OLS regression
ols_result <- lm(inv ~ value + capital, data = Grunfeld)
summary(ols_result)##
## Call:
## lm(formula = inv ~ value + capital, data = Grunfeld)
##
## Residuals:
## Min 1Q Median 3Q Max
## -291.68 -30.01 5.30 34.83 369.45
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -42.714369 9.511676 -4.491 1.21e-05 ***
## value 0.115562 0.005836 19.803 < 2e-16 ***
## capital 0.230678 0.025476 9.055 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 94.41 on 197 degrees of freedom
## Multiple R-squared: 0.8124, Adjusted R-squared: 0.8105
## F-statistic: 426.6 on 2 and 197 DF, p-value: < 2.2e-16\[ \text{inv}_{it} = -42.714369 + 0.115562 \times \text{value}_{it} + 0.230678 \times \text{capital}_{it} + \epsilon_{it} \]
Intercept (-42.714369): The negative intercept
suggests that when both value and
capital are zero, the predicted investment
is negative, which might not have practical meaning in this context but
indicates a baseline adjustment in the model.
Value Coefficient (0.115562): This coefficient is positive and statistically significant (p < 0.001), implying that an increase in firm value by one unit is associated with an increase in investment by approximately 0.116 units. This direction and magnitude make sense economically, as larger firms typically have more resources for investment.
Capital Coefficient (0.2306778): Similarly, the coefficient for capital is also positive and significant (p < 0.001), suggesting that an increase in capital stock is strongly associated with an increase in investment. This relationship is expected because firms with more capital assets are likely to invest more in their operations and growth.
Both the value and
capital coefficients are highly
statistically significant, indicating a strong linear relationship with
investment. The F-statistic is also extremely high, which supports the
model’s fit.
Omitted Variable Bias: Given the high R-squared (0.8124), the model explains a substantial portion of the variability in investment. However, the presence of omitted variable bias cannot be ruled out. Variables such as industry effects, firm age, market conditions, or other firm-specific factors like management quality could also impact investment but are not included in the model.
Reduction by Fixed Effects: Incorporating fixed effects into the model could help control for unobserved heterogeneity among firms that might be influencing investment decisions. For instance, firm-specific characteristics that do not change over time or are not directly measured can be accounted for by using firm-specific dummy variables or by using a within estimator in a fixed effects model.
# Fixed Effects Model
fe_model <- plm(inv ~ value + capital, data = Grunfeld, model = "within")
summary(fe_model)## Oneway (individual) effect Within Model
##
## Call:
## plm(formula = inv ~ value + capital, data = Grunfeld, 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-16OLS Estimates: From your OLS model,
value had a coefficient of approximately
0.115562 and capital had
0.230678.
Fixed Effects Estimates: In the fixed effects model,
value has a coefficient of 0.110124, and
capital has 0.310065.
These changes suggest that once you account for the unobserved individual (firm-specific) heterogeneity, the impact of capital on investment increases significantly, whereas the impact of value slightly decreases. This might indicate that some of the unobserved firm-specific characteristics could have been influencing the capital’s return on investment more strongly than was apparent in the OLS model.
The fixed effects in this model control for all time-invariant
characteristics of each entity (firm). This includes factors such as
management quality, company culture, or any firm-specific attributes
that do not change over time but might influence investment decisions.
The model essentially compares each firm to itself over time, thus
isolating the effect of changes in value
and capital.
Different specifications of fixed effects (e.g., including time effects, using different methods like LSDV or first differencing) might yield different coefficients. However, the within estimator (used here) and LSDV should theoretically provide similar results since both approaches aim to control for the same type of unobserved heterogeneity:
Least Squares Dummy Variable (LSDV) might directly
include dummy variables for each firm, which should lead to similar
estimates for value and
capital because it’s just a different way
of accounting for the firm-specific intercepts.
First differencing might be used if the panel is not balanced or if the focus is on removing effects that could be influenced by time trends. This approach would provide different estimates if the omitted variables are correlated differently over time.
Given the balanced nature of the Grunfeld data and the specific control for firm-level heterogeneity, the within and LSDV methods are expected to yield similar results. However, differences could still arise due to small sample adjustments, different handling of extreme values, or if the model assumptions hold differently under each method.