Interacting Fixed Effects

[Frank Giaquinto]

Dear Professors,

I would greatly appreciate your advice on the estimation strategy I intend to use for my analysis.

I'm working with a dataset that contains observations for different industries in the same country across multiple points in time. Specifically, I have data for 7 industrial sectors, with aggregate observations for each sector spanning 24 years. My initial specification is as follows:

y_st = x_st β + θ_s + ξ_t + u_st

with s=1,...,7 and t=1,...,24. Here, x_st is a row vector of controls, and θ_s and ξ_t denote sector and time fixed effects, respectively.

I am considering using the two-way fixed effects (TWFE) estimator, but I have two concerns that I would like your input on:

i) My current specification assumes that shocks affect all industries uniformly over time. Would it be advisable to relax this assumption and instead estimate a model with interacted fixed effects, such as: y_st = x_st β + θ_s * ξ_t + u_st ;
ii) The time dimension in my dataset is large relative to the cross-sectional dimension (industries). Is that a problem?

Thank you in advance for your help!

[Carlo Lazzaro]

Frank:
i) if you use the -fe- estimator, I would nit relax that assumption;
ii) the issue here i9s ti switch from -xtreg,fe- (I assume that yiu dependent variable is continuous, to -xtregar,fe-m as you are dealing with a T>N panel dataset.

[Frank Giaquinto]

Dear Professor Lazzaro,
thank you for your reply and your advice. The dependent variable is continuous. If I may ask out of curiosity, is it an issue to use xtreg when T > N ? Additionally, could you explain the difference between xtreg and xtregard for the TWFE estimator, which I plan to use in my case?
Thank you very much for your time!

[Carlo Lazzaro]

Frank (please call me Carlo):
1) when T>N, the are issues to deal with (say, AR) that do not creep up when N>T.
2) if you want to go two-way -fe- with -xtregar,fe-, you may want to take a look at the following toy-example:

. webuse grunfeld

. xtset company time

Panel variable: company (strongly balanced)
 Time variable: time, 1 to 20
         Delta: 1 unit

. xtregar invest mvalue kstock i.year, fe

FE (within) regression with AR(1) disturbances  Number of obs     =        190
Group variable: company                         Number of groups  =         10

R-squared:                                      Obs per group:
     Within  = 0.6513                                         min =         19
     Between = 0.7838                                         avg =       19.0
     Overall = 0.7798                                         max =         19

                                                F(20,160)         =      14.94
corr(u_i, Xb) = -0.1416                         Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      mvalue |   .0938865   .0104378     8.99   0.000     .0732729    .1145001
      kstock |   .4076907   .0373936    10.90   0.000      .333842    .4815393
             |
        year |
       1936  |   22.60707   15.23516     1.48   0.140    -7.480873    52.69502
       1937  |   29.02835   20.63253     1.41   0.161    -11.71886    69.77557
       1938  |   32.19837   23.69301     1.36   0.176    -14.59298    78.98973
       1939  |   17.30833   25.51947     0.68   0.499    -33.09011    67.70676
       1940  |   50.88403   27.01934     1.88   0.061    -2.476513    104.2446
       1941  |   79.24101   27.76837     2.85   0.005     24.40121    134.0808
       1942  |   73.97942   28.25514     2.62   0.010      18.1783    129.7805
       1943  |   56.66199   28.31439     2.00   0.047     .7438482    112.5801
       1944  |   58.95475   28.56968     2.06   0.041     2.532449     115.377
       1945  |   49.74642   28.40551     1.75   0.082    -6.351667    105.8445
       1946  |   75.78518   28.05301     2.70   0.008     20.38324    131.1871
       1947  |   57.76681   26.94116     2.14   0.034     4.560676    110.9729
       1948  |   51.01013   26.04522     1.96   0.052    -.4266119    102.4469
       1949  |   20.19058   24.97134     0.81   0.420    -29.12536    69.50651
       1950  |   18.35979   23.75434     0.77   0.441     -28.5527    65.27228
       1951  |   36.28607   21.77871     1.67   0.098    -6.724731    79.29688
       1952  |   32.36863   18.82811     1.72   0.088    -4.815034    69.55229
       1953  |   30.71474   14.03026     2.19   0.030     3.006356    58.42313
       1954  |          0  (omitted)
             |
       _cons |  -124.2761   10.59512   -11.73   0.000    -145.2004   -103.3518
-------------+----------------------------------------------------------------
      rho_ar |  .68127018
     sigma_u |  96.029876
     sigma_e |  39.732352
     rho_fov |  .85383317   (fraction of variance because of u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9,160) = 12.27                      Prob > F = 0.0000

.

[Frank Giaquinto]

Thanks, Carlo