Time fixed effects are present but insignificant in regression model

[Sonal Barve]

Hi,

I am trying to estimate the impact of climate variables on suicide rates. I am working with panel data of 10 years for each of 25 states.
As given below, srate denotes the annual suicide rate in a given state in a given year and yearlytemp refers to the annual temperature in a given state in a given year.

I am using this guide for reference: Panel Data Analysis Fixed and Random Effects using Stata (v. 4.2) https://www.princeton.edu/~otorres/Panel101.pdf

From what I have read so far on this forum as well as other sources, OLS results (although significant in terms of p-values) are invalid here because observations are almost never independent for panel data and that is a necessary requirement if one wants to use an OLS regression model.

I first tried a fixed effects model, followed by a random effects model.

I then do a Hausman test which gives me a Prob > chi2 value of 0.5043, i.e. greater than 0.05. This suggests that I follow a random effects model instead of a fixed effects model, and even I think that a random effects model is more suitable here since state fixed effects (like population, area etc) are not likely to affect the annual temperature of that state.

Now I also test for time-fixed effects as follows:

xtreg srate yearlytemp i.year, fe

Fixed-effects (within) regression               Number of obs      =       247
Group variable: state1                          Number of groups   =        25

R-sq:  within  = 0.1097                         Obs per group: min =         8
       between = 0.2039                                        avg =       9.9
       overall = 0.1075                                        max =        10

                                                F(10,212)          =      2.61
corr(u_i, Xb)  = -0.5011                        Prob > F           =    0.0052

------------------------------------------------------------------------------
       srate |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  yearlytemp |  -.5129098   .5935633    -0.86   0.388    -1.682952    .6571322
             |
        year |
       1991  |   .4169206    .633913     0.66   0.511    -.8326595    1.666501
       1992  |  -.0940757   .6223848    -0.15   0.880    -1.320931     1.13278
       1993  |   .2369076   .6330399     0.37   0.709    -1.010951    1.484767
       1994  |   .6744753   .6505162     1.04   0.301    -.6078333    1.956784
       1995  |   .8625228   .6680059     1.29   0.198    -.4542617    2.179307
       1996  |   1.483321   .6610198     2.24   0.026      .180308    2.786335
       1997  |   .7698841   .6239403     1.23   0.219    -.4600377    1.999806
       1998  |   1.803748   .7162068     2.52   0.013     .3919493    3.215547
       1999  |   2.583697    .726554     3.56   0.000     1.151501    4.015892
             |
       _cons |   21.12578   14.17553     1.49   0.138    -6.817256    49.06883
-------------+----------------------------------------------------------------
     sigma_u |  7.8994315
     sigma_e |  2.1997248
         rho |  .92803684   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(24, 212) =    84.20             Prob > F = 0.0000

. testparm i.year

 ( 1)  1991.year = 0
 ( 2)  1992.year = 0
 ( 3)  1993.year = 0
 ( 4)  1994.year = 0
 ( 5)  1995.year = 0
 ( 6)  1996.year = 0
 ( 7)  1997.year = 0
 ( 8)  1998.year = 0
 ( 9)  1999.year = 0

       F(  9,   212) =    2.45
            Prob > F =    0.0114

As you can see, the Prob > F value of the test shows that time-fixed effects are present, however including them in the regression model gives insignificant coefficients for almost all years.

So my questions are:
1. If time-fixed effects are present, why are they not significant in my (random effects) regression model?

2. The R-squared values without and with year dummies are as given below:

xtreg srate yearlytemp, re
R-sq:  within  = 0.0173                         Obs per group: min =         8
       between = 0.2141                                        avg =       9.9
       overall = 0.1959                                        max =        10

xtreg srate yearlytemp i.year, re
R-sq:  within  = 0.0977                         Obs per group: min =         8
       between = 0.2219                                        avg =       9.9
       overall = 0.1887                                        max =        10

So this means that more variation in suicide rates is explained when year dummies are included - and they should be included as suggested by the test for time-fixed effects - but at the same time, they yield insignificant coefficients in the regression model.
How do I make sense of this?

3. What do we mean when we say that observations are not independent in panel data? I did a serial correlation test and obtained no serial correlation results. Does that not mean that one observation is not dependent (correlated) on the previous one?

[Carlo Lazzaro]

Sonal:
welcome to this forum.
1) you should look at -year- joint significance;
2) under -xtreg,re- specification, you should look at R-sq between; including -i.year- seems to make things a bit better;
3) as it (almost) always the case, observations belonging to the same panel are more similar that the ones belonging to a different panel. All in all, a panel can be figured out as a group of observations belonging to the same unit (which is measured repeatedly on the same outcome at scheduled and theoretically equally spaced time-points). For instance, Carlo's related observations will share Carlo's hidden features (that will be collected in ui component of the error) which are, in all likelihood, different from somebody else's hidden features. Put differently, let's assume that, on a tennis court two players are repeatedly mesured on their backhand stroke winning points (a categorical dependent variable). Let's also assume that the predictors are: years of tennis practice, wind speed, temperature on the court. Let's eventually assume that one of those player (let's call it A) ill-handles her racket: this feature (the ui error component), althoung not included in the set of predictors, will affect all shots: in this toy-example, this feature makes player A's observations non-independent.

[Sonal Barve]

Thanks a lot, Carlo Lazzaro!

Well, I used the 'testparm' command in Stata, which is a joint test to see if dummies for all the years are zero. The contradiction I mentioned in my question 1 above is not observed for the random effects model (which is the model suggested by Hausman test as well).

So given that time-fixed effects really are insignificant, I suppose R-sq values without year dummies are decent enough. Am I correct?

Thank you for the neat explanation!

[Carlo Lazzaro]

Sonal:
- if -hausman- points you to -re- specification, it's wise to go that way;
- insignificant -i.time- does not necessarily mean that years do not deserve to remain in your regression model: they might be part of the data generating process but your sample does not show evidence of a significant effect;
- I would say that between R-sq values are similar.