Betas greater than one

[Huyen Nguyen VUW]

Hi Statalist member,

I have run a regression of cash flow volatility (s_oina_v1_w01) on several independent variables. The coefficients of the independent variables are much greater than 1. After a while searching for the reasons, this issue might be caused by multicollinearity. I use vif command to check and found that if I exclude year fixed effect, the vif will decrease to normal. However, the betas of independent variables are still greater than 1. The question is: can betas be greater than 1 or they should be bounded between 0 and 1?

My first model including year fixed effects

reg s_oina_v1_w01 vc pe $CONTROLSF i.fyear if nomiss==1, cluster (gvkey_n)

Linear regression                               Number of obs     =     42,240
F(64, 5187)       =      55.64
Prob > F          =     0.0000
R-squared         =     0.3537
Root MSE          =     23.421

(Std. Err. adjusted for 5,188 clusters in gvkey_n)

Robust
s_oina_v1~01       Coef.   Std. Err.      t    P>t     [95% Conf. Interval]

vc     2.60776   .5108771     5.10   0.000     1.606225    3.609294
pe    -.431211   .4109789    -1.05   0.294    -1.236903    .3744809
MB2_w01    2.588564   .1564188    16.55   0.000     2.281917     2.89521
fsize_w01   -2.044109     .13061   -15.65   0.000    -2.300159   -1.788058
tang_w01   -4.856372   .6371705    -7.62   0.000    -6.105495    -3.60725
prof2_w01    -23.1196   1.482164   -15.60   0.000    -26.02527   -20.21394
LnRnD_w01      22.105   1.338913    16.51   0.000     19.48017    24.72984
capex_w01    15.53585   2.653995     5.85   0.000      10.3329     20.7388
abne_w01    1.614974   .6722279     2.40   0.016     .2971238    2.932824
indlev_w01     -4.6235   .6954019    -6.65   0.000    -5.986781   -3.260219
            
fyear
1963     -2.70082   1.855219    -1.46   0.146    -6.337831    .9361915
1964     -4.36412    2.44979    -1.78   0.075    -9.166741    .4385007
1965    -3.483502   2.541331    -1.37   0.171    -8.465583    1.498578
1966    -2.861024   2.190433    -1.31   0.192    -7.155195    1.433147
1967    -3.777086   2.748357    -1.37   0.169    -9.165025    1.610852
1968    -1.183331   3.818591    -0.31   0.757    -8.669378    6.302716
1969    -4.279388   2.535129    -1.69   0.091     -9.24931    .6905337
1970    -2.396924   1.959319    -1.22   0.221    -6.238015    1.444168
1971    -5.224563   2.179666    -2.40   0.017    -9.497626      -.9515
1972    -5.423982   2.626562    -2.07   0.039    -10.57315   -.2748138
1973    -3.160456   2.739316    -1.15   0.249    -8.530669    2.209758
1974    -1.109184   2.648728    -0.42   0.675    -6.301808    4.083439
1975    -1.772796    2.62446    -0.68   0.499    -6.917842    3.372251
1976    -1.968919   2.604779    -0.76   0.450    -7.075383    3.137546
1977    -3.536893   2.602726    -1.36   0.174    -8.639332    1.565547
1978    -3.240465   2.626672    -1.23   0.217    -8.389849     1.90892
1979     -3.14485   2.587022    -1.22   0.224    -8.216504    1.926803
1980    -3.663743   2.586793    -1.42   0.157    -8.734947    1.407461
1981    -4.014471    2.67717    -1.50   0.134    -9.262852     1.23391
1982    -4.120776   2.651059    -1.55   0.120    -9.317968    1.076416
1983    -5.077192   2.616589    -1.94   0.052    -10.20681    .0524262
1984    -1.935351   2.664314    -0.73   0.468    -7.158529    3.287827
1985     -2.99127   2.628142    -1.14   0.255    -8.143537    2.160997
1986    -1.640733   2.643156    -0.62   0.535    -6.822432    3.540965
1987    -1.961486   2.624165    -0.75   0.455    -7.105956    3.182984
1988      -2.4146   2.619251    -0.92   0.357    -7.549435    2.720235
1989     -2.59247   2.617404    -0.99   0.322    -7.723685    2.538745
1990    -1.634035   2.621029    -0.62   0.533    -6.772357    3.504287
1991    -2.309193   2.621048    -0.88   0.378    -7.447551    2.829166
1992    -2.145213   2.631998    -0.82   0.415    -7.305038    3.014611
1993    -4.298855   2.602566    -1.65   0.099    -9.400981    .8032715
1994    -3.740801   2.596153    -1.44   0.150    -8.830355    1.348752
1995     -2.85765   2.593843    -1.10   0.271    -7.942675    2.227376
1996    -2.550483   2.595195    -0.98   0.326    -7.638159    2.537193
1997    -2.886435   2.598277    -1.11   0.267    -7.980153    2.207284
1998    -.6481403   2.612095    -0.25   0.804    -5.768947    4.472667
1999    -1.143714   2.627518    -0.44   0.663    -6.294757    4.007328
2000     2.853986   2.648816     1.08   0.281     -2.33881    8.046782
2001     1.088894   2.631785     0.41   0.679    -4.070513    6.248301
2002     .4298029   2.626765     0.16   0.870    -4.719764     5.57937
2003    -3.096295   2.617106    -1.18   0.237    -8.226926    2.034336
2004    -1.838908   2.626983    -0.70   0.484    -6.988901    3.311085
2005    -2.909433   2.629826    -1.11   0.269       -8.065    2.246134
2006    -1.747426   2.633339    -0.66   0.507     -6.90988    3.415028
2007    -2.096727   2.643802    -0.79   0.428    -7.279694    3.086239
2008     1.854633    2.66005     0.70   0.486    -3.360186    7.069451
2009     2.525076   2.653766     0.95   0.341    -2.677423    7.727576
2010     2.350179   2.672366     0.88   0.379    -2.888786    7.589143
2011     1.023631   2.682469     0.38   0.703    -4.235139    6.282401
2012      .217227   2.656954     0.08   0.935    -4.991522    5.425976
2013    -3.136067   2.644907    -1.19   0.236    -8.321199    2.049064
2014     -.691762   2.678946    -0.26   0.796    -5.943625    4.560101
2015     .3019233   2.697211     0.11   0.911    -4.985747    5.589594
2016     .1598025   2.719114     0.06   0.953    -5.170807    5.490412
            
_cons     24.0461   2.597978     9.26   0.000     18.95297    29.13923

and check the variable inflation factor

.    vif

    Variable    VIF    1/VIF    
            
    vc    1.34    0.746548
    pe    1.17    0.856759
    MB2_w01    1.27    0.787382
    fsize_w01    1.79    0.557760
    tang_w01    1.46    0.683080
    prof2_w01    2.01    0.497613
    LnRnD_w01    1.82    0.548491
    capex_w01    1.45    0.691691
    abne_w01    1.07    0.930430
    indlev_w01    1.65    0.606256
    fyear    
    1963    2.11    0.473790
    1964    2.33    0.428682
    1965    2.89    0.346256
    1966    3.33    0.300107
    1967    4.11    0.243356
    1968    4.22    0.236961
    1969    4.55    0.219642
    1970    5.00    0.200127
    1971    5.44    0.183805
    1972    6.55    0.152660
    1973    22.83    0.043792
    1974    33.10    0.030214
    1975    34.71    0.028808
    1976    34.05    0.029371
    1977    35.25    0.028365
    1978    34.82    0.028723
    1979    36.90    0.027101
    1980    38.31    0.026101
    1981    41.38    0.024169
    1982    52.71    0.018970
    1983    57.15    0.017497
    1984    77.92    0.012834
    1985    85.74    0.011663
    1986    90.22    0.011083
    1987    103.02    0.009707
    1988    110.87    0.009020
    1989    105.88    0.009445
    1990    103.77    0.009637
    1991    101.83    0.009821
    1992    108.93    0.009180
    1993    127.18    0.007863
    1994    143.60    0.006964
    1995    156.90    0.006373
    1996    172.69    0.005791
    1997    191.50    0.005222
    1998    187.52    0.005333
    1999    172.17    0.005808
    2000    168.06    0.005950
    2001    166.50    0.006006
    2002    154.39    0.006477
    2003    140.76    0.007104
    2004    132.61    0.007541
    2005    130.84    0.007643
    2006    126.53    0.007903
    2007    122.74    0.008147
    2008    118.64    0.008429
    2009    111.12    0.008999
    2010    106.88    0.009356
    2011    105.61    0.009469
    2012    104.55    0.009565
    2013    103.71    0.009643
    2014    108.16    0.009246
    2015    110.81    0.009025
    2016    109.64    0.009120
            
    Mean VIF    72.47

The results after I exclude year fixed effects

reg s_oina_v1_w01 vc pe $CONTROLSF if nomiss==1, cluster(gvkey_n)

Linear regression                               Number of obs     =     42,240
F(10, 5187)       =     326.72
Prob > F          =     0.0000
R-squared         =     0.3497
Root MSE          =     23.479

(Std. Err. adjusted for 5,188 clusters in gvkey_n)

Robust
s_oina_v1~01       Coef.   Std. Err.      t    P>t     [95% Conf. Interval]

vc    2.786415   .5083468     5.48   0.000     1.789841    3.782989
pe   -.4794882    .399109    -1.20   0.230     -1.26191    .3029337
MB2_w01    2.578851   .1551001    16.63   0.000     2.274789    2.882912
fsize_w01   -1.711538   .1023789   -16.72   0.000    -1.912244   -1.510832
tang_w01   -4.678255   .6267758    -7.46   0.000       -5.907    -3.44951
prof2_w01   -24.49453   1.459964   -16.78   0.000    -27.35667   -21.63238
LnRnD_w01    22.10426   1.336092    16.54   0.000     19.48495    24.72356
capex_w01    11.85221   2.584446     4.59   0.000     6.785603    16.91881
abne_w01    1.551582   .6725154     2.31   0.021     .2331683    2.869995
indlev_w01   -4.388744     .61205    -7.17   0.000     -5.58862   -3.188868
_cons    21.09565    .791491    26.65   0.000       19.544    22.64731

Variable    VIF    1/VIF     
        
prof2_w01    1.88    0.531979
LnRnD_w01    1.80    0.554424
tang_w01    1.42    0.704543
indlev_w01    1.35    0.738034
vc    1.33    0.751402
capex_w01    1.33    0.752730
MB2_w01    1.25    0.799959
fsize_w01    1.23    0.815327
pe    1.15    0.868929
abne_w01    1.07    0.937740
        
Mean VIF    1.38

P.s: I don't know why the tables look like that. Is there any way to re-align them?

Regards,
Huyen

[Carlo Lazzaro]

Huyen:
1) betas can be >1 (or<0).
2) It's clear that -year- is problematic, has you can see from the width of related 95% CIs (that are far more informative that the -estat vif- outcome).
3) I'm not sure whether, as you have a time serie, -regress- (and not some other procedure available under -ts- suite) is actually the way to go.

[Huyen Nguyen VUW]

Hi Carlo,
1. Thank you
2. Do you mean that 95%CIs of -year- is quite large? I think they are a bit larger compared with 95% CIs of other control variables. But how to consider as narrow or large 95% CIs?
3. My data is a panel, not a time series. I re-try to use -xtreg-, the results are quite identically. I didn't get your idea about "under -ts- suite", could you please clarify this? Also, do you have any suggestions about other suitable models?
4. You said betas can be greater than 1. Just for future similar issues.
I found somewhere saying that to correct the multicollinearity problem, I can :
- leave the variable causing it out of the model. I excluded -year- out of my model
or:
- use -orthog- command. I did use this command to create a new range of control variables and re-run regression. The results aren't much different
or:
- use the partial least square regression. I read from STATA and it seems that this one is similar to -orthog- command. Is that right? Do you know how to use that in STATA?

Regards
Huyen

[Carlo Lazzaro]

Huyen:
1) There's non hard and fast rule to split 95% CIs in wide and narrow. In your case 95% CIs for years were almost all actually wide and a suspect of multicollinearity was legal.
2) If you dealt with a panel data set, why reporting -regress- results? For -ts- suite of commands, type -help time series-.
3) Multicollinearity issue, when really relevant, can be easily managed just omitting the culprit variable(s).
4) I've never used -orthog- hence I cannot advise on it.

[Huyen Nguyen VUW]

Thank you Carlo.

[Huyen Nguyen VUW]

Hi Carlo,

In the model above, I would like to know how the relationship of VC backing (variable "vc(0/1)") and cash flow volatility (s_oina_v1_w01) during the first 10 years since listing. Then I add age dummy variable ( age1(0/1) -age10(0/1) ) and their interaction with vc(0/1). However, when I run the model, age1(0/1) and its interaction term with vc(0/1) and pe(0/1) are omitted "because of collinearity". Is it normal if I exclude these variables out of my model?

. xtreg s_oina_v1_w01 age1-age10 int_av1-int_av10 int_ap1-int_ap10 $CONTROLSF    if    nomiss==1,    vce(cluster    gvkey_n)
note: age1 omitted because of collinearity
note: int_av1 omitted because of collinearity
note: int_ap1 omitted because of collinearity

Random-effects GLS regression                   Number of obs     =     42,240
Group variable: gvkey_n                         Number of groups  =      5,188

R-sq:                                           Obs per group:
within  = 0.0593                                         min =          1
between = 0.3499                                         avg =        8.1
overall = 0.3394                                         max =         52

Wald chi2(35)     =    1886.86
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

(Std. Err. adjusted for 5,188 clusters in gvkey_n)

Robust
s_oina_v1~01       Coef.   Std. Err.      z    P>z     [95% Conf. Interval]

age1           0  (omitted)
age2    2.442264   .6953139     3.51   0.000     1.079474    3.805055
age3     .011724   .6044512     0.02   0.985    -1.172979    1.196427
age4    .1201383   .6199582     0.19   0.846    -1.094957    1.335234
age5   -.4386206   .5811833    -0.75   0.450    -1.577719    .7004777
age6   -.8541236    .560806    -1.52   0.128    -1.953283     .245036
age7   -.6683499   .5774217    -1.16   0.247    -1.800076    .4633759
age8   -1.406466   .5703668    -2.47   0.014    -2.524365    -.288568
age9   -1.751251   .5559751    -3.15   0.002    -2.840942   -.6615601
age10   -1.307863   .5573396    -2.35   0.019    -2.400229    -.215498
int_av1           0  (omitted)
int_av2    6.420925   1.077316     5.96   0.000     4.309424    8.532426
int_av3    3.385284   .9856227     3.43   0.001     1.453499    5.317069
int_av4    3.686842   1.047583     3.52   0.000     1.633616    5.740067
int_av5    2.821071   1.026758     2.75   0.006     .8086618    4.833481
int_av6     3.16616   1.077989     2.94   0.003      1.05334    5.278979
int_av7    3.353581   1.106437     3.03   0.002     1.185004    5.522157
int_av8    2.772264   1.097081     2.53   0.012     .6220249    4.922502
int_av9    2.145313    1.12897     1.90   0.057    -.0674276    4.358054
int_av10    2.341027   1.118976     2.09   0.036     .1478745     4.53418
int_ap1           0  (omitted)
int_ap2   -2.831508   .8815614    -3.21   0.001    -4.559337   -1.103679
int_ap3   -1.190499    .733592    -1.62   0.105    -2.628312    .2473153
int_ap4   -.7628211   .8763299    -0.87   0.384    -2.480396     .954754
int_ap5   -.0457845   .8796854    -0.05   0.958    -1.769936    1.678367
int_ap6   -.9728998   .7790681    -1.25   0.212    -2.499845    .5540456
int_ap7   -.5041601   .8901142    -0.57   0.571    -2.248752    1.240432
int_ap8   -.4973147   .8063276    -0.62   0.537    -2.077688    1.083058
int_ap9    .6573118   .8566892     0.77   0.443    -1.021768    2.336392
int_ap10    .4352756   .7966329     0.55   0.585    -1.126096    1.996647
MB2_w01    2.227686   .1499215    14.86   0.000     1.933846    2.521527
fsize_w01   -1.732461   .1596916   -10.85   0.000    -2.045451   -1.419471
tang_w01   -6.244286   .8355756    -7.47   0.000    -7.881984   -4.606588
prof2_w01   -17.08799   1.537879   -11.11   0.000    -20.10218    -14.0738
LnRnD_w01     13.9455   1.288791    10.82   0.000     11.41951    16.47148
capex_w01    .0359132   2.262054     0.02   0.987     -4.39763    4.469457
abne_w01   -.0056373    .591528    -0.01   0.992    -1.165011    1.153736
indlev_w01   -1.604053   .5645413    -2.84   0.004    -2.710534   -.4975725
_cons    22.35429   1.363306    16.40   0.000     19.68226    25.02632

sigma_u   17.009718
sigma_e   19.347874
rho   .43595487   (fraction of variance due to u_i)

Regards,
Huyen

[Carlo Lazzaro]

Huyen:
please use CODE delimiters to share what you typed and what Stata gave you back (as per FAQ). Thanks.
That said, (by the way: you created both categorical variables and interactions by hand. why not exploiting the wonderful capabilities of -fvvarlist- notation?), Stata has omitted some predictors to avoid the so called dummy trap (https://en.wikipedia.org/wiki/Dummy_...le_(statistics).

[Huyen Nguyen VUW]

Hi Carlo,
Sorry for the above posts. Now I figured out that I should use the CODE delimiters (#) for the command and STATA results separately.

xtreg s_oina_v1_w01 vc pe age1-age10 int_av1-int_av10 int_ap1-int_ap10 $CONTROLSF  if nomiss==1, cluster (gvkey_n)

note: age1 omitted because of collinearity
note: int_av1 omitted because of collinearity
note: int_ap1 omitted because of collinearity

Random-effects GLS regression                   Number of obs     =     42,240
Group variable: gvkey_n                         Number of groups  =      5,188

R-sq:                                           Obs per group:
     within  = 0.0597                                         min =          1
     between = 0.3485                                         avg =        8.1
     overall = 0.3371                                         max =         52

                                                Wald chi2(37)     =    2196.75
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

                            (Std. Err. adjusted for 5,188 clusters in gvkey_n)
------------------------------------------------------------------------------
             |               Robust
s_oina_v1~01 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          vc |   4.440185   .9970745     4.45   0.000     2.485955    6.394415
          pe |  -.6701737   .7138086    -0.94   0.348    -2.069213    .7288653
        age1 |          0  (omitted)
        age2 |   3.225829   .6748729     4.78   0.000     1.903102    4.548555
        age3 |   .7078983   .5949819     1.19   0.234    -.4582447    1.874041
        age4 |   .7579673   .6141664     1.23   0.217    -.4457767    1.961711
        age5 |   .1442564   .5784581     0.25   0.803    -.9895007    1.278014
        age6 |  -.3168103   .5543824    -0.57   0.568     -1.40338    .7697593
        age7 |  -.1728711   .5748216    -0.30   0.764    -1.299501    .9537585
        age8 |  -.9442892   .5692455    -1.66   0.097     -2.05999    .1714115
        age9 |  -1.310552   .5572784    -2.35   0.019    -2.402797   -.2183063
       age10 |  -.8850471   .5579147    -1.59   0.113     -1.97854    .2084456
     int_av1 |          0  (omitted)
     int_av2 |   4.174099   1.276243     3.27   0.001     1.672707     6.67549
     int_av3 |   1.312059   1.193775     1.10   0.272    -1.027696    3.651814
     int_av4 |   1.748455   1.229669     1.42   0.155    -.6616524    4.158563
     int_av5 |   1.015402   1.187707     0.85   0.393    -1.312462    3.343265
     int_av6 |   1.475805   1.217691     1.21   0.226    -.9108248    3.862435
     int_av7 |   1.776771   1.213566     1.46   0.143    -.6017747    4.155316
     int_av8 |   1.277543   1.171493     1.09   0.275    -1.018541    3.573627
     int_av9 |   .7335421   1.206518     0.61   0.543     -1.63119    3.098274
    int_av10 |   .9975508   1.170011     0.85   0.394    -1.295628     3.29073
     int_ap1 |          0  (omitted)
     int_ap2 |  -2.456276    .990555    -2.48   0.013    -4.397728   -.5148238
     int_ap3 |  -.8204365   .8314389    -0.99   0.324    -2.450027    .8091538
     int_ap4 |  -.4274948   .9719415    -0.44   0.660    -2.332465    1.477476
     int_ap5 |   .2666641   .9492772     0.28   0.779    -1.593885    2.127213
     int_ap6 |  -.6745472   .8406924    -0.80   0.422    -2.322274    .9731797
     int_ap7 |  -.2271647   .9462136    -0.24   0.810    -2.081709     1.62738
     int_ap8 |  -.2221775   .8550044    -0.26   0.795    -1.897955      1.4536
     int_ap9 |   .9197778   .9221622     1.00   0.319     -.887627    2.727182
    int_ap10 |   .6836452   .8264573     0.83   0.408    -.9361814    2.303472
     MB2_w01 |   2.216702   .1501654    14.76   0.000     1.922383    2.511021
   fsize_w01 |  -1.749969    .163783   -10.68   0.000    -2.070977    -1.42896
    tang_w01 |  -5.924007   .8266883    -7.17   0.000    -7.544286   -4.303727
   prof2_w01 |  -16.88284   1.541332   -10.95   0.000     -19.9038   -13.86188
   LnRnD_w01 |   13.69953   1.299815    10.54   0.000     11.15194    16.24712
   capex_w01 |   .2722369   2.261079     0.12   0.904    -4.159396     4.70387
    abne_w01 |  -.0600861    .590574    -0.10   0.919     -1.21759    1.097418
  indlev_w01 |  -1.078625   .5563648    -1.94   0.053    -2.169081    .0118296
       _cons |   20.59596   1.275669    16.15   0.000     18.09569    23.09622
-------------+----------------------------------------------------------------
     sigma_u |  17.012561
     sigma_e |  19.347874
         rho |  .43603708   (fraction of variance due to u_i)
------------------------------------------------------------------------------

Thanks for your suggestions.
1. About -fvvarlist notation- it seems a bit long to my command because I have 10 age dummies then would have 20 interaction terms of these age dummies and VC dummy and PE dummy. Is that still fine if I use the interaction terms that created by hand? The results table is even longer with the matrix results of each interaction term and so that I couldn't export a neat table for further use.
2. Reading your suggested link, "The removed dummy then becomes the base category against which the other categories are compared." So then I have a question about how to interpret the coefficients.
2.1.For example, the coefficient associated with age2(0/1) ( which takes the value of 1 if that is the second year since the company listed and 0 otherwise) is 3.22. So I can say the cash flow volatility of the company in the second year is 3.22 % higher than in the first year? Similarly, the coefficient associated with age3(0/1) (if significant) is to show the difference between year 3 and year 1?

2.2.The coefficient associated with the interaction term int_av2 (= age2(0/1)#vc(0/1) )

  int_av2 |   4.174099   1.276243     3.27   0.001     1.672707     6.67549

given the coefficients of VC(0/1) and age2(0/1)

      vc |   4.440185   .9970745     4.45   0.000     2.485955    6.394415

      age2 |   3.225829   .6748729     4.78   0.000     1.903102    4.548555

I can say: In year 2, the cash flow volatility of VC backed firms is 4.17% higher than non-VC backed firms. But how I can interpret related to the interaction term of age1(0/1) and vc(0/1), which now becomes the base category?

2.3. When I use the factor variable notation as you suggested

  age2#vc |
        0 1  |  -4.174099   1.276243    -3.27   0.001     -6.67549   -1.672707
        1 0  |          0  (omitted)
        1 1  |          0  (omitted)
             |

the interested thing is the coefficient associated with vc(0/1) change much, to

    vc |   18.95141   6.591246     2.88   0.004     6.032807    31.87002

Regards,
Huyen

[Carlo Lazzaro]

Huyen:
most of your interpretation problems seems to rest on your data layout, that looks -wide- instead than -long- (the latter is highy preferable for almost all the statistical procedures that you carry out.
with Stata).
At the moment, I do not have the time to skim through all your questions; hence I will reply to the first one ony:
you can say that, when adjusted for the other predictors, the cash-flow volatility due to year 2 predictor is 3.22% (by the way: is your regressand expressed in natural log terms; if that were the case, 3.22% is only an approximation of the change in the regressand explained by that predictor) higher than in year 1 (assumed that year 1 is the reference category).
Perhaps it would be more frutiful to express time as a continuous variable and search for possible non-linear regression with the dependent variable.
As an aside, I would tidy up your model, as it seems that you have too many predictors to disseminate your results in an effective way.

[Huyen Nguyen VUW]

Thank you Carlo