First differencing vs Fixed effects interpretation

[Dana Baade]

Hi all,

After reading the forum regarding my question, I am still wondering about the interpretation. I have a dataset similar to the grunfeld dataset and therefore I am using the grufield set in order to get familiar with my question. I was namely wondering what the interpretation would be if I estimate my model by first differencing. I know the interpretation stays the same when I use first differencing and fixed effects with two time periods. I was wondering however what the interpretation would be when I a: I only first difference the dependent variable and my other variables stay in levels and b: what the interpretation would be if I additionally lag my explanatory variables. I would be very greatful if someone could help me with the interpretation. Moreover under a and b should I include company fixed effects?

This quote says interpretation should be the same is you first difference both. But after reading I am not sure about mine.

The Stata syntax is correct. The interpretation of variables remains the same. That is, a doubling of the tariff (-> d.ln_tariffs = 1) leads to a b1 percent change in ln_employment (d.ln_emp = +b1). I personally find the wording in growth misleading, as that implies a permanent effect (to me, but YMMV). That's only the case if emp is in differences but tariff is in levels. Whereas if they are both in differences, then the effect of the change in tariffs dies out immediately in the next period. If you do not find this plausible, you could include lags and leads of the tariff variable, or switch to the local projection method (Jorda, 2005).

 webuse grunfeld

. keep if year==1951 | year==1952
(180 observations deleted)

. xtset company year
       panel variable:  company (strongly balanced)
        time variable:  year, 1951 to 1952
                delta:  1 year

. xtreg invest mvalue kstock i.time, fe

Fixed-effects (within) regression               Number of obs     =         20
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.8413                                         min =          2
     between = 0.6347                                         avg =        2.0
     overall = 0.6358                                         max =          2

                                                F(3,7)            =      12.37
corr(u_i, Xb)  = -0.4483                        Prob > F          =     0.0035

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0675063    .066438     1.02   0.343    -.0895947    .2246073
      kstock |   .5776233    .115057     5.02   0.002     .3055567      .84969
     18.time |  -12.81064   10.53266    -1.22   0.263    -37.71642    12.09515
       _cons |  -128.3884   97.66454    -1.31   0.230    -359.3284    102.5515
-------------+----------------------------------------------------------------
     sigma_u |   186.5057
     sigma_e |  16.083172
         rho |  .99261855   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 7) = 85.90                       Prob > F = 0.0000

. reg D.(invest mvalue kstock) i.time, nocons

      Source |       SS           df       MS      Number of obs   =        10
-------------+----------------------------------   F(3, 7)         =     12.37
       Model |  19201.1351         3  6400.37836   Prob > F        =    0.0035
    Residual |  3621.35789         7  517.336841   R-squared       =    0.8413
-------------+----------------------------------   Adj R-squared   =    0.7733
       Total |   22822.493        10   2282.2493   Root MSE        =    22.745

------------------------------------------------------------------------------
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |
         D1. |   .0675063    .066438     1.02   0.343    -.0895947    .2246073
             |
      kstock |
         D1. |   .5776233    .115057     5.02   0.002     .3055567      .84969
             |
     18.time |  -12.81064   10.53266    -1.22   0.263    -37.71642    12.09515
------------------------------------------------------------------------------



As you can see here it is the same

Now I only use FD on the dependent variable

 reg D.invest mvalue kstock i.time, nocons

      Source |       SS           df       MS      Number of obs   =        10
-------------+----------------------------------   F(3, 7)         =     27.58
       Model |  21042.1245         3  7014.04151   Prob > F        =    0.0003
    Residual |  1780.36844         7  254.338349   R-squared       =    0.9220
-------------+----------------------------------   Adj R-squared   =    0.8886
       Total |   22822.493        10   2282.2493   Root MSE        =    15.948

------------------------------------------------------------------------------
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0259089   .0063405     4.09   0.005      .010916    .0409019
      kstock |   .0077066   .0231235     0.33   0.749    -.0469717    .0623849
     18.time |  -11.79278   8.227355    -1.43   0.195    -31.24739    7.661817

now I lag my independent variables

 reg D.invest L.mvalue L.kstock i.time, nocons

      Source |       SS           df       MS      Number of obs   =        10
-------------+----------------------------------   F(3, 7)         =     30.02
       Model |  21176.6158         3  7058.87192   Prob > F        =    0.0002
    Residual |  1645.87721         7  235.125316   R-squared       =    0.9279
-------------+----------------------------------   Adj R-squared   =    0.8970
       Total |   22822.493        10   2282.2493   Root MSE        =    15.334

------------------------------------------------------------------------------
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |
         L1. |   .0258598   .0053399     4.84   0.002     .0132329    .0384867
             |
      kstock |
         L1. |   .0120669   .0221206     0.55   0.602    -.0402401    .0643739
             |
     18.time |  -11.93865   7.886201    -1.51   0.174    -30.58655    6.709256

[Carlo Lazzaro]

Dana:
welcome to this forum.
I would simply say that, while your two first codes actually run a panel data regression with -fe- specification, the remaining two don't (and I cannot get any useful information from them).

[Dana Baade]

Thank you Carlo for your response,

So you would say you can never mix the two and therfore they are not useful? I was just interested at the different interpretations and not so much is the output. I was just wondering what the interpreation would be with FD-Level, FD-lagged level vs FD-FD

With the first code I would say all else equal, an increase of 1 unit in mvalue increases invest with 0.068. The second code would probaly be the same as it gives the same output, but I was not sure about third and fourth one.

Initially I ran

  
 xtreg invest L.mvalue L.kstock i.time, fe

My supervisor advised me to tranform invest and L.mvalue into first differencing to account for the persistency of the first two variables in stead of using a dynamic panel data model. I ran (A)

 
 reg D.invest D.L.mvalue L.kstock i.time, nocons

and (B)

  xtreg D.invest D.L.mvalue L.kstock i.time, fe

I am however confused about interpretations and if it is even correct, which one I should use (A or B).

[Carlo Lazzaro]

Dana:
as far as I know, a dynamic panel data model needs a different [XT] command, such as -xtabond-.
Probably, your supervisor's advice related to some time-series characteristics of your variable (stationarity?).
Otherwise, I fail to get the reason why of your 3rd and 4th codes.

[Andrew Musau]

The within-transformation, first-differencing and dummy variable regression are just different estimation methods aimed at controlling for the possibly correlated, time-invariant heterogeneity in panel data. With \(T=2\), all 3 methods will result in identical coefficient estimates. If your regression is

xtreg invest L.mvalue L.kstock i.time, fe

you need to difference all the variables if estimating with the FD estimator

xi: regress D.(invest L.mvalue L.kstock i.time), nocons robust

and the LSDV command is

regress invest L.mvalue L.kstock i.time i.company

For the interpretetion in all 3: Assuming the variables mvalue and invest are measured in dollars, then a dollar increase in lagged market value increases investment by "_b[mvalue]" dollars, holding lagged capital stock fixed and controlling for time-invariant firm effects and firm-invariant time effects.

[Dana Baade]

Thank you both Carlo and Andrew,

Andrew, thank you. With two time periods I understand the equivalents.


@Carlo, my advisor said there is some non stationarity in the dependent variable and called it persistency and said that I should either include a lagged dependent variable which induces bias with fixed effects. My advisor told me to stay away from xtabond2 etc. I was always told that using first differencing on everything works with non stationarity, but did not know about including a lagged dependent variable would also fix this.
Then my advisor said I should perhaps first difference only those variables that show non stationary and keep the others in levels and stay away from including a lagged dependent variable. That is how I came up with the equations in #3. My N is about 4000 and T is 8 btw.

[Andrew Musau]

You have a wide panel \(N>>T\), so you need not focus on stationarity. It is usually a problem in long panels \(T>N\). You can estimate both the FE regression and the FD regression and compare the coefficient estimates. There is some discussion in the chapter on advanced panel data models in Introductory Econometrics: A Modern Approach by Jeff Wooldridge. Here is a relevant quote from this chapter:

When T is large, and especially when N is not very large (for example, N = 20 and T = 30), we must exercise caution in using the fixed effects estimator. Although exact distributional results hold for any N and T under the classical fixed effects assumptions, inference can be very sensitive to violations of the assumptions when N is small and T is large. In particular, if we are using unit root processes—see Chapter 11—the spurious regression problem can arise. First differencing has the advantage of turning an integrated time series process into a weakly dependent process. Therefore, if we apply first differencing, we can appeal to the central limit theorem even in cases where T is larger than N. Normality in the idiosyncratic errors is not needed, and heteroskedasticity and serial correlation can be dealt with as we touched on in Chapter 13. Inference with the fixed effects estimator is potentially more sensitive to nonnormality, heteroskedasticity, and serial correlation in the idiosyncratic errors.

[Dana Baade]

Thank you Andrew. I actually have read the discussions about N>T and brought them up. My advisor told me however to still do something about the non stationarity to see if it changes results at least. That is why I am really trying to learn.

When plotting Y and lag Y, there is a high correlation and I was always told that this could be a good indicator for non stationarity. Does it therefore mean that when Y depends on the lag of Y there is always non stationarity? So in a dynamic panel model there is always non stationarity?
my advisor told me: y (and some controls) likely show non stationarity and are therefore highly persistent. In order to deal with this you can:
1. include a lagged dependent variable and stay away from xtabond. This will induce Nickell bias
2. use first differencing of your dependent variable and only those controls which show non stationarity
3. Use first differencing of your dependent variable and include a lagged dependent variable which would be similar to a Dickey Fueller test.

[Andrew Musau]

You have panel data, so I do not follow the reference to a Dickey Fuller test. For testing for unit roots in panel data, see

help xtunitroot

Also, you should ask your advisor for literature references of what he/she suggests. As #7 suggests, a comparison of FD estimates and FE estimates should be sufficient even in the presence of I(1) variables. If you happen to get such references, post them here.

[Dana Baade]

Thank you Andrew,

I know about the xtunitroot.

I will ask my advisor about his advise given and post his answer. If I want to compare FE and FD I should use first differencing on all my variables you say?

xtreg Y L.i.xi L.c.x2 i.year, fe cluster(panelid)

and

regress D.(Y L.i.x1 L.c.x2 i.year), nocons cluster(panelid)

I also have an interaction term in the form of L.i.xi*L.c.x2. As no factor notation and time series are allowed I now have created the interaction term by hand.

gen interaction=i.x1*c.x2

and I ran

regress D.(Y L.interaction L.i.x1 L.c.x2 i.year), nocons cluster(panelid)

The coefficients of my main variables (interaction term) now turn out to be insignificant. What could this mean? Could it mean the effects were previously picking up some of the non stationarity? How can there be a difference between FD and FE

[Andrew Musau]

L.i.xi

This is confusing. Is xi binary 0/1? Furthermore, if xi has more than 2 categories

L.i.xi*L.c.x2

defines more than one variable. If it is binary, treat it as continuous. It's better to show your actual commands and the Stata output rather than showing pseudo-code. Note that in the FD regression, you are not using factor variable notation as this cannot be combined with time-series operators. Instead, you are using the -xi- prefix, and here, "c." is not recognized. See

help xi

[Dana Baade]

Sorry Andrew. I indeed made a big mistake in notation.
It is indeed a binary variable (0 or 1) and the intercation term is thus c.L.x1*i.L.x2
I am sady not allowed to post my data. But i have tried to replicate. In the case below, Dummy becomes significant. In my case it is the other way around. In my case the interaction term becomes insignificant when moving from FE to FD (so with the data I cannot post). I have read that FD can be more efficient when there is a random walk of the error terms. Why hoewever do when then cluster with FE?

. webuse grunfeld

. xtset company year
       panel variable:  company (strongly balanced)
        time variable:  year, 1935 to 1954
                delta:  1 year

. generate Dummy=0

. replace Dummy=1 if kstock<200
(95 real changes made)

. xtreg invest c.L.mvalue##i.L.Dummy i.time, fe cluster(company)

Fixed-effects (within) regression               Number of obs     =        190
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.5579                                         min =         19
     between = 0.8597                                         avg =       19.0
     overall = 0.7900                                         max =         19

                                                F(9,9)            =          .
corr(u_i, Xb)  = -0.2038                        Prob > F          =          .

                                    (Std. Err. adjusted for 10 clusters in company)
-----------------------------------------------------------------------------------
                  |               Robust
           invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
           mvalue |
              L1. |   .1779941   .0316367     5.63   0.000     .1064269    .2495612
                  |
          L.Dummy |
               1  |   80.11598   51.66307     1.55   0.155    -36.75399     196.986
                  |
L.Dummy#cL.mvalue |
               1  |  -.0889403   .0181073    -4.91   0.001    -.1299018   -.0479788
                  |
             time |
               3  |  -5.744876   24.07154    -0.24   0.817    -60.19847    48.70872
               4  |  -75.17961   40.68834    -1.85   0.098     -167.223    16.86381
               5  |  -51.73819   23.99772    -2.16   0.059    -106.0248    2.548425
               6  |  -50.13372   35.62508    -1.41   0.193    -130.7233    30.45581
               7  |  -26.22903   38.33263    -0.68   0.511    -112.9435    60.48541
               8  |  -47.86291    39.5978    -1.21   0.258    -137.4394    41.71354
               9  |  -17.90656   11.19933    -1.60   0.144    -43.24122    7.428088
              10  |  -32.91038   21.22332    -1.55   0.155    -80.92087     15.1001
              11  |  -34.03976   23.81303    -1.43   0.187    -87.90857    19.82906
              12  |  -14.55189   25.56881    -0.57   0.583    -72.39257    43.28878
              13  |  -28.41291   42.08109    -0.68   0.517    -123.6069    66.78113
              14  |   12.88209   20.63941     0.62   0.548    -33.80749    59.57168
              15  |   5.364969   14.98397     0.36   0.729    -28.53112    39.26106
              16  |   12.57217   15.16982     0.83   0.429    -21.74434    46.88869
              17  |   53.56299   30.44524     1.76   0.112    -15.30894    122.4349
              18  |    39.6748   29.95183     1.32   0.218    -28.08095    107.4305
              19  |   86.14643   55.14445     1.56   0.153    -38.59899    210.8918
              20  |   47.41338   44.07351     1.08   0.310    -52.28783    147.1146
                  |
            _cons |  -41.51193    35.7804    -1.16   0.276    -122.4528    39.42896
------------------+----------------------------------------------------------------
          sigma_u |  78.260659
          sigma_e |  76.455962
              rho |  .51166295   (fraction of variance due to u_i)
-----------------------------------------------------------------------------------

. generate interaction=Dummy*mvalue

. xi: reg D.(invest L.mvalue L.Dummy L.interaction i.time), nocons cluster(company)
i.time            _Itime_1-20         (naturally coded; _Itime_1 omitted)
note: D._Itime_20 omitted because of collinearity

Linear regression                               Number of obs     =        180
                                                F(9, 9)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.3099
                                                Root MSE          =     50.064

                               (Std. Err. adjusted for 10 clusters in company)
------------------------------------------------------------------------------
             |               Robust
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |
         LD. |    .060379   .0065486     9.22   0.000     .0455649     .075193
             |
       Dummy |
         LD. |   67.88266   31.67366     2.14   0.061    -3.768128    139.5335
             |
 interaction |
         LD. |  -.0564086   .0117466    -4.80   0.001    -.0829813   -.0298358
             |
    _Itime_2 |
         D1. |  -147.1528   84.86367    -1.73   0.117    -339.1277    44.82218
             |
    _Itime_3 |
         D1. |   -121.915   83.79901    -1.45   0.180    -311.4815    67.65151
             |
    _Itime_4 |
         D1. |  -168.0648   96.23358    -1.75   0.115    -385.7603    49.63064
             |
    _Itime_5 |
         D1. |  -175.2221   99.22726    -1.77   0.111    -399.6898    49.24554
             |
    _Itime_6 |
         D1. |  -148.2817   94.73846    -1.57   0.152     -362.595    66.03156
             |
    _Itime_7 |
         D1. |  -118.9144   92.55379    -1.28   0.231    -328.2856    90.45681
             |
    _Itime_8 |
         D1. |  -139.5851   96.97332    -1.44   0.184     -358.954    79.78376
             |
    _Itime_9 |
         D1. |   -133.244   82.73653    -1.61   0.142     -320.407    53.91904
             |
   _Itime_10 |
         D1. |  -135.6103    84.0055    -1.61   0.141     -325.644    54.42332
             |
   _Itime_11 |
         D1. |  -133.4315   84.62405    -1.58   0.149    -324.8643    58.00143
             |
   _Itime_12 |
         D1. |  -101.5664   75.24948    -1.35   0.210    -271.7926    68.65975
             |
   _Itime_13 |
         D1. |   -113.914   88.49397    -1.29   0.230    -314.1013    86.27326
             |
   _Itime_14 |
         D1. |  -98.05225   84.16103    -1.17   0.274    -288.4377    92.33323
             |
   _Itime_15 |
         D1. |  -109.7901   77.40066    -1.42   0.190    -284.8825    65.30238
             |
   _Itime_16 |
         D1. |  -99.81639   71.31979    -1.40   0.195     -261.153    61.52018
             |
   _Itime_17 |
         D1. |  -52.94649   65.41134    -0.81   0.439    -200.9172    95.02424
             |
   _Itime_18 |
         D1. |  -38.97327   59.27652    -0.66   0.527    -173.0661    95.11953
             |
   _Itime_19 |
         D1. |   13.49406   22.70726     0.59   0.567    -37.87332    64.86144
             |
   _Itime_20 |
         D1. |          0  (omitted)
------------------------------------------------------------------------------

[Andrew Musau]

Do not cluster your standard errors if you have less than 30 clusters. In some cases, you need even more than this number of clusters. A better dataset that depicts your sample (4000 units, 8 time periods) is Stata's nlswork dataset.

webuse nlswork, clear
keep if id<=5000
bys id (year): replace year=_n
keep if year<=8
gen dummy= rnormal(1,2)<2
g wage= exp(ln_w)
xtset id year
xtreg wage c.L1.tenure##c.L.dummy i.year, fe cluster(id)
gen LdumXLten= c.L.dummy#c.L.tenure
xi: regress D.(wage L.tenure L.dummy LdumXLten i.year), nocons robust

Below, lagged tenure does not appear to have much of an effect in both regressions and the dummy and its interaction are comparable across both. Now, if it happens that the P-value is 0.09 in one and 0.12 in another, do not make much of a deal of it. They are both comparable. Also, after differencing, just specify the -robust- option in FD instead of -cluster()-.

. xtset id year
       panel variable:  idcode (unbalanced)
        time variable:  year, 1 to 8
                delta:  1 unit

.
. xtreg wage c.L1.tenure##c.L.dummy i.year, fe cluster(id)

Fixed-effects (within) regression               Number of obs     =     18,793
Group variable: idcode                          Number of groups  =      4,032

R-sq:                                           Obs per group:
     within  = 0.0394                                         min =          1
     between = 0.0127                                         avg =        4.7
     overall = 0.0251                                         max =          7

                                                F(9,4031)         =      49.57
corr(u_i, Xb)  = 0.0175                         Prob > F          =     0.0000

                                   (Std. Err. adjusted for 4,032 clusters in idcode)
------------------------------------------------------------------------------------
                   |               Robust
              wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------------+----------------------------------------------------------------
            tenure |
               L1. |     .02751   .0220916     1.25   0.213    -.0158017    .0708217
                   |
             dummy |
               L1. |  -.1484001   .0679662    -2.18   0.029    -.2816514   -.0151488
                   |
cL.tenure#cL.dummy |   .0584529   .0182039     3.21   0.001     .0227632    .0941425
                   |
              year |
                3  |   .3409486   .0549649     6.20   0.000      .233187    .4487103
                4  |   .5651989   .0637944     8.86   0.000     .4401266    .6902712
                5  |   .7792176   .0767702    10.15   0.000     .6287056    .9297295
                6  |   1.047363   .0946832    11.06   0.000     .8617315    1.232994
                7  |   1.220583   .1094077    11.16   0.000     1.006083    1.435082
                8  |     1.4339   .1262445    11.36   0.000     1.186391     1.68141
                   |
             _cons |    5.37104   .0597579    89.88   0.000     5.253881    5.488198
-------------------+----------------------------------------------------------------
           sigma_u |  3.3078309
           sigma_e |   2.706313
               rho |  .59902659   (fraction of variance due to u_i)
------------------------------------------------------------------------------------

.
. gen LdumXLten= c.L.dummy#c.L.tenure
(4,906 missing values generated)

.
. xi: regress D.(wage L.tenure L.dummy LdumXLten i.year), nocons robust
i.year            _Iyear_1-8          (naturally coded; _Iyear_1 omitted)
note: D._Iyear_8 omitted because of collinearity

Linear regression                               Number of obs     =     14,608
                                                F(9, 14599)       =      18.39
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0108
                                                Root MSE          =        3.4

------------------------------------------------------------------------------
             |               Robust
      D.wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      tenure |
         LD. |  -.0175918   .0576551    -0.31   0.760     -.130603    .0954194
             |
       dummy |
         LD. |  -.1776017   .0691343    -2.57   0.010    -.3131137   -.0420898
             |
   LdumXLten |
         D1. |   .0557949   .0217507     2.57   0.010     .0131607     .098429
             |
    _Iyear_2 |
         D1. |   -1.96119   .3201332    -6.13   0.000    -2.588692   -1.333689
             |
    _Iyear_3 |
         D1. |  -1.594312   .2815682    -5.66   0.000    -2.146221   -1.042403
             |
    _Iyear_4 |
         D1. |  -1.314081   .2166745    -6.06   0.000    -1.738791   -.8893717
             |
    _Iyear_5 |
         D1. |  -1.047875   .1810437    -5.79   0.000    -1.402743   -.6930061
             |
    _Iyear_6 |
         D1. |  -.7062866   .1465698    -4.82   0.000     -.993582   -.4189912
             |
    _Iyear_7 |
         D1. |  -.3810403   .0958527    -3.98   0.000    -.5689236   -.1931569
             |
    _Iyear_8 |
         D1. |          0  (omitted)
------------------------------------------------------------------------------

.

[Dana Baade]

Thank you Andrew. That is indeed a better dataset. In my case the p value of the interaction term was 0.025 and becomes 0.2 however. That is quite a large increase…
Additionally, can I conclude from this that entering some variables in levels would make no sense and would not be interpretable?

[Andrew Musau]

Additionally, can I conclude from this that entering some variables in levels would make no sense and would not be interpretable?

You have some model from theory and you estimate it. As I stated in #5, FD and the within-estimator are just different estimation methods of the same model. If you need to difference, for comparison with FE, you need to difference all variables involved. If you arbitrary lag some variables in one of the regressions, you are not estimating the same model.

That is indeed a better dataset. In my case the p value of the interaction term was 0.025 and becomes 0.2 however

As I cannot see your data, models and estimates, you will need to discuss this with your advisor and determine whether the difference is of importance. In any case, you need to look at the overall effect of the main effects and interaction for comparison, e.g., using margins.