I am using panel data consisting of macroeconomic variables on country level and for the time period between 2009 and 2019. I have a balanced panel data set.
I performed a Chow test, a Hausman test and a Lagrange Multiplier test to select the best panel data approach and it turns out to be fixed effects. Now, I am wondering how to formally determine if I should use only individual-fixed effects or only time-fixed effects or both. Is there a formal test to solve this issue?
Formally test which fixed effects to use with panel data
fixed-effects-modelpanel datatime series
Related Solutions
You can and should use a well-specified random effects model. Always.
The Hausman test is said to suggest fixed effects models, but can and should be viewed "as a standard Wald test for the omission of the variables $\widetilde{\mathbf{X}}$" (Baltagi 2008, §4.3), where $\widetilde{\mathbf{X}}$ is a matrix of deviations from group means. If you do not omit $\widetilde{\mathbf{X}}$, a random effects model gives you the same population (fixed) effects as a fixed effects model, and the individual effects.
Mundlak (1978) argues that there is a unique estimator for the model $$\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{Z}\boldsymbol{\alpha}+\mathbf{u}\qquad\qquad \mathbf{Z}=\mathbf{I}_{N}\otimes\mathbf{e}_T$$ where $\mathbf{I}_{N}$ is an identity matrix, $\otimes$ denotes Kronecker product, $\mathbf{e}_T$ is a vector of ones, so $\mathbf{Z}$ is the matrix of individual dummies, and $\boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_N)$.
If $\alpha_i=\overline{\mathbf{X}}_{i*}\boldsymbol{\pi}+w_{i}$, $\boldsymbol{\pi}\ne\mathbf{0}$, averaging over $t$ for a given $i$, the model can be written as $$\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{P}(\mathbf{X}\boldsymbol{\pi}+\mathbf{w})+\mathbf{u}\qquad\qquad \mathbf{P}=\mathbf{I}_N\otimes\bar{\mathbf{J}}_T$$ where $\mathbf{P}$ is a matrix which averages the observations across time for each individual (Baltagi 2008, §2.1). Under the fixed effects model, the within estimator is $$\hat{\boldsymbol{\beta}}_{w}=(\mathbf{X'QX})^{-1}\mathbf{X'Qy}\tag{1}$$ where $\mathbf{Q}=\mathbf{I}-\mathbf{P}$ is a matrix which obtains the deviations from individual means. Mundlak argues that under the random effects model, to get the same estimates the estimator should be $$\begin{bmatrix} \hat{\boldsymbol{\beta}} \\ \hat{\boldsymbol{\pi}}\end{bmatrix}= \left(\begin{bmatrix}\mathbf{X}' \\ \mathbf{X'P}\end{bmatrix}\boldsymbol{\Sigma}^{-1}\begin{bmatrix}\mathbf{X}&\mathbf{XP} \end{bmatrix}\right)^{-1}\begin{bmatrix}\mathbf{X}' \\ \mathbf{X'P} \end{bmatrix}\boldsymbol{\Sigma}^{-1}\mathbf{y}\tag{2}$$ where $\boldsymbol{\Sigma}^{-1}$ is the variance of the error term, while the "usual" estimator (the so-called "Balestra-Nerlove estimator") is $$\hat{\boldsymbol{\beta}}=(\mathbf{X}'\boldsymbol{\Sigma}^{-1}\mathbf{X})^{-1}\mathbf{X}'\boldsymbol{\Sigma}^{-1}\mathbf{y}$$ which is biased. According to Mundlak, since $(1)$ and $(2)$ obtain the same estimates for $\boldsymbol{\beta}$, $(2)$ is the within estimator, i.e. $(1)$ is the unique estimator and does not depend on the knowledge of the variance components.
However, the models $$\begin{align} \mathbf{y}&=\mathbf{X}\boldsymbol{\beta}+\mathbf{P}(\mathbf{X}\boldsymbol{\pi}+\mathbf{w})+\mathbf{u}\tag{FE} \\ \mathbf{y}&=\mathbf{X}\boldsymbol{\beta}+\mathbf{P}\mathbf{X}\boldsymbol{\pi}+(\mathbf{Pw}+\mathbf{u})\tag{RE} \end{align}$$ are formally equivalent (Hsiao 2003, §4.3), so a random effects model obtains the same estimates ... as long as you do not omit $\widetilde{\mathbf{X}}$! Let's try.
Data generation (R code):
set.seed(1234)
N <- 25 # individuals
T <- 5 # time
In <- diag(N) # identity matrix of order N
Int <- diag(N*T) # identity matrix of order N*T
Jt <- matrix(1, T, T) # matrix of ones of order T
Jtm <- Jt / T
P <- kronecker(In, Jtm) # averages the obs across time for each individual
s2a <- 0.3 # sigma^2_\alpha
s2u <- 0.6 # sigma^2_u
w <- rep(rnorm(N, 0, sqrt(s2a)), each = T)
u <- rnorm(N*T, 0, sqrt(s2u))
b <- c(1.5, -2)
p <- c(-0.7, 0.8)
X <- cbind(runif(N*T, 2, 5), runif(N*T, 4, 8))
XPX <- cbind(X, P %*% X) # [ X PX ]
y <- XPX %*% c(b,p) + (P %*% w + u) # y = Xb + PXp + Pw + u
ds <- data.frame(id=rep(1:N, each=T), wave=rep(1:T, N), y, split(X, col(X)))
Under a fixed effects model we get:
> fe.1 <- plm(y ~ X1 + X2, data=ds, model="within")
> summary(fe.1)$coefficients
Estimate Std. Error t-value Pr(>|t|)
X1 1.435987 0.07825464 18.35019 1.806239e-33
X2 -1.916447 0.06339342 -30.23100 1.757634e-51
while under a random effects model...
> re.1 <- plm(y ~ X1 + X2, data=ds, model="random")
> summary(re.1)$coefficients
Estimate Std. Error t-value Pr(>|t|)
(Intercept) 1.830633 0.51687109 3.541759 5.638216e-04
X1 1.405060 0.07927271 17.724390 1.505521e-35
X2 -1.874784 0.06372731 -29.418846 3.076414e-57
bias!
But what if we do not omit $\widetilde{\mathbf{X}}=\mathbf{QX}$?
> Q <- diag(N*T) - P
> X1.mean <- P %*% ds$X1
> X1.dev <- Q %*% ds$X1
> X2.mean <- P %*% ds$X2
> X2.dev <- Q %*% ds$X2
> re.2 <- plm(y ~ X1.mean + X1.dev + X2.mean + X2.dev, data=ds, model="random")
> summary(re.2)$coefficients
Estimate Std. Error t-value Pr(>|t|)
(Intercept) -0.04123108 2.30907450 -0.01785611 9.857833e-01
X1.mean 0.81279279 0.38146339 2.13072292 3.515287e-02
X1.dev 1.43598746 0.07824535 18.35236883 1.239171e-36
X2.mean -1.23071499 0.26379329 -4.66545216 8.072196e-06
X2.dev -1.91644653 0.06338590 -30.23458903 5.809240e-58
The estimates for X1.dev and X2.dev are equal to the within estimates for X1 and X2 (no room for Hausman tests!), and you get much more. You get what you need.
However this is just the tip of the iceberg. I recommend that you read at least Bafumi and Gelman (2006), Snijders and Berkhof (2008), Bell and Jones (2014).
References
Baltagi, Badi H. (2008), Econometric Analysis of Panel Data, John Wiley & Sons
Bafumi, Joseph and Andrew Gelman (2006), Fitting Multilevel Models When Predictors and Group Effects Correlate, http://www.stat.columbia.edu/~gelman/research/unpublished/Bafumi_Gelman_Midwest06.pdf
Bell, Andrew and Kelvyn Jones (2014), "Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data", Political Science Research and Methods, http://dx.doi.org/10.7910/DVN/23415
Hsiao, Cheng (2003), Analysis of Panel Data, Cambridge University Press
Mundlak, Yair (1978), "On the Pooling of Time Series and Cross Section Data", Econometrica, 43(1), 44-56
Sniiders, Tom A. B. and Johannes Berkhof (2008), "Diagnostic Checks for Multilevel Models", in: Jan de Leeuw and Erik Meijer (eds), Handbook of Multilevel Analysis, Springer, Chap. 3
The plm package in R provides a a function for the poolability test in just three steps:
# 1. Run a normal OLS model with fixed effects (model="within")
plm_model<- plm(y ~ x, data= dataset, model= "within")
# 2. Run a variable coefficients model with fixed effects (model="within")
pvcm_model<- pvcm(y ~ x, data= dataset, model= "within")
# Run the poolability test
pooltest(plm_model, pvcm_model)
The null hypothesis is that the dataset is poolable (i.e. individuals have the same slope coefficients), so if p<0.05 you reject the null and you need a variable coefficients model.
Best Answer
In this case, you should test for heteroskedasticity and autocorrelation primarily. One key reason to use any fixed effect model is to solve for heterogeneity, which is not exactly as heteroskedasticity, but close. Don't follow R2 or F-statistic to much, as two-fixed effect could win either way.