Case 4.1 Fixed Effects estimator

Write a Matlab function‘FE.m’ that implements the FE estimator

Include an option to add time effects



In [1]:

    
using Distributions: TDist, ccdf

type FE_results
    beta
    SE
    tstat
    pval
end

function FE(Y, X, time_ind; time_effects=false)
    T = Int(maximum(time_ind)-minimum(time_ind)+1) #amount of timesteps
    nT = size(X) #total number of observations
    N = Int(nT[1]/T) #amount of individuals
    k = Int(nT[2]) #amount of parameters to be estimated
    D = kron(eye(N),ones(T,1))
    P_D = D*inv(D'*D)*D'
    M_D = eye(nT[1]) - P_D
    y_dev = M_D*Y #deviation from the mean for y
    x_dev = M_D*X #deviation from the mean for x
    
    if time_effects #time effects
        time_effect = kron(ones(N,1), eye(T)) #per individual there can be an effect per timestep
        x_dev = hcat(x_dev, time_effect)
        k = size(x_dev)[2]
        
    end
    dof = nT[1]-N-k #degrees of freedom left over after having individual effects and estimating the k explanatory variables
    β̂ = inv(x_dev'*x_dev)*x_dev'*y_dev
    residuals = y_dev - x_dev*β̂
    σ̂² = (residuals'*residuals)/dof
    SE = sqrt(diag(σ̂²[1]*inv(x_dev'*x_dev)))
    tstat = β̂ ./ SE
    pval  = 2 * ccdf(TDist(dof), abs(tstat))
    
    FE_results(β̂, SE, tstat, pval)
end









    Out[1]:





FE (generic function with 1 method)

Estimate the dynamic CigDem model in levels using the FE estimator



In [2]:

    
data, header = readcsv("Data_Baltagi.csv", header=true)
println.(header);









    



state
 year
 P/pack
 pop
 pop16+
 CPI
 ndi/capita
 C/capita
 Pn/pack
 ln C_it
 ln P_it
 ln Pn_it
 ln Y_it

Because we need to add a lagged value, we will need to delete the rows where year is the minimum value, since this can not be regressed on a lagged value, to do this we will first load the full data and then fill new matrices with the values needed for the lagged regression (hence without year 63 in this example).



In [3]:

    
Y_temp = data[:, 10]
X_temp = data[:, 11:13]
year_temp = data[:,2]
state_temp = data[:,1]
observations = size(Y_temp)[1]
T = maximum(year_temp) - minimum(year_temp)+1 #amount of timesteps
N = observations/T
#make empty arrays to fill
lagged_obs = Int(observations - N) #removing the N*1 observation to get the amount of lagged observations
Y = zeros(lagged_obs, 1)
Y_lag = zeros(lagged_obs,1)
X = zeros(lagged_obs, size(X_temp)[2])
year = zeros(lagged_obs)
state = zeros(lagged_obs);



In [4]:

    
#we fill the data-arrays with the first time period dropped
new_i = 1
lag_i = 1
for i = 1:observations
    if year_temp[i] != minimum(year_temp)
        Y[new_i] = Y_temp[i]
        X[new_i,:] = X_temp[i,:]
        year[new_i] = year_temp[i]
        state[new_i] = state_temp[i]
        new_i += 1
    end
    if year_temp[i] != maximum(year_temp)
        Y_lag[lag_i] = Y_temp[i]
        lag_i += 1
    end
end
X = hcat(Y_lag,X);

We estimate the model for the cigarette demand without time effect.



In [6]:

    
FE(Y,X,year; time_effects= false).beta









    Out[6]:





4×1 Array{Float64,2}:
  0.904419 
 -0.16664  
  0.0658706
 -0.0551062

We estimate the model for cigarette demand with a time effect in the form of time dummies. These are the within estimates as in the paper by Baltagi.



In [5]:

    
FE(Y,X,year; time_effects= true).beta









    Out[5]:





33×1 Array{Float64,2}:
  0.833383  
 -0.298598  
  0.0340454 
  0.100279  
  0.0263246 
  0.0575573 
  0.0406944 
  0.0448442 
  0.0292082 
  0.0257447 
  0.00460896
  0.0502825 
  0.0583618 
  ⋮         
 -0.0434015 
 -0.0524584 
 -0.0488399 
 -0.0443597 
 -0.0149016 
 -0.0179424 
 -0.0207784 
 -0.0264062 
 -0.0269659 
 -0.0284895 
 -0.00926797
  0.0190077

We see that the estimates are an exact match with those reported by Baltagi.

Statistical properties of the FE estimator in this model.

Here we have a lagged dependent variable as regressor, for large N and fixed T, this causes the estimates to be:

biased
inconsistent

Inconsistency however decreases with T, this is known as the Nickel-bias. This comes from the correlation of the lagged dependent variable with the lagged residual, which is transfered to the residual of the present period because of autocorrelation.

??? The most efficient since RE and pooled OLS are both inconsistent both for T and N to infinity ???

Why is it not necessary to estimate the first-differenced equation using FE?

This is not necessary because the first differenced transformation does exactly the same as the transformation used for the within estimator. Namely they both get rid of any individual-specific, time constant heterogeneity. The former does it by subtracting the previous period from a variable, the latter by subtracting the individual mean. So if the assumptions underlying the estimators are valid, they both give approximately the same estimates.



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