Newey West Wald Test

[Lezgin Ay]

Hello everyone,

I am running two separate time-series regressions using Newey.

y1= a1 + b1 x1 + e
y2= a2 + b2 x1+ e

I would like to test whether b1=b2 . How can I run a Wald test to do this? I checked many other posts but the only solution offered there was to calculate

z= (b1-b2) / ((stnd err b1)^2+( stnd err b2)^2))^1/2 by hand but this is not exactly the Wald test. I appreciate your help.

[Andrew Musau]

webuse idle2, clear
tsset time
set seed 2018
gen idle2= runiformint(60, 120)
newey usr idle, lag(3)
newey usr idle2, lag(3)
rename idle idle1
reshape long idle, i(time) j(group)
replace time=_n
tsset time
gen cons=1
newey usr (c.idle c.cons)#i.group, nocons lag(3)
test _b[1.group#c.idle]= _b[2..group#c.idle]

Individual and joint regressions

. newey usr idle, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =      10.90
                                                Prob > F          =     0.0026

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        idle |  -.2281501   .0690927    -3.30   0.003    -.3696801     -.08662
       _cons |   23.13483   6.327031     3.66   0.001     10.17449    36.09516
------------------------------------------------------------------------------

. newey usr idle2, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =       0.01
                                                Prob > F          =     0.9285

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       idle2 |  -.0031678    .034996    -0.09   0.929    -.0748539    .0685182
       _cons |   4.106188   3.745698     1.10   0.282    -3.566527     11.7789
------------------------------------------------------------------------------

. newey usr (c.idle c.cons)#i.group, nocons lag(3)

Regression with Newey-West standard errors      Number of obs     =         60
maximum lag: 3                                  F(  4,        56) =      31.59
                                                Prob > F          =     0.0000

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
group#c.idle |
          1  |  -.2281501    .077743    -2.93   0.005     -.383888   -.0724121
          2  |  -.0031678   .0393758    -0.08   0.936    -.0820471    .0757115
             |
group#c.cons |
          1  |   23.13483   7.119611     3.25   0.002     8.872533    37.39712
          2  |   4.106188   3.959958     1.04   0.304     -3.82656    12.03894
------------------------------------------------------------------------------

. test _b[1.group#c.idle]= _b[2..group#c.idle]

 ( 1)  1b.group#c.idle - 2.group#c.idle = 0

       F(  1,    56) =    8.22
            Prob > F =    0.0058

[Andrew Musau]

Just to add to #2, because we are combining two time-series, you need to appropriately adjust the lag length so that the Newey-West standard errors match. In my example above, the lag length should be set at 7 to match a lag length of 3 in the individual regressions.

. newey usr idle, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =      10.90
                                                Prob > F          =     0.0026

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        idle |  -.2281501   .0690927    -3.30   0.003    -.3696801     -.08662
       _cons |   23.13483   6.327031     3.66   0.001     10.17449    36.09516
------------------------------------------------------------------------------

. 
. newey usr idle2, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =       0.01
                                                Prob > F          =     0.9285

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       idle2 |  -.0031678    .034996    -0.09   0.929    -.0748539    .0685182
       _cons |   4.106188   3.745698     1.10   0.282    -3.566527     11.7789
------------------------------------------------------------------------------


. 
. newey usr (c.idle c.cons)#i.group, nocons lag(7)

Regression with Newey-West standard errors      Number of obs     =         60
maximum lag: 7                                  F(  4,        56) =      37.09
                                                Prob > F          =     0.0000

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
group#c.idle |
          1  |  -.2281501   .0690927    -3.30   0.002    -.3665595   -.0897406
          2  |  -.0031678    .034996    -0.09   0.928    -.0732732    .0669376
             |
group#c.cons |
          1  |   23.13483   6.327031     3.66   0.001     10.46026    35.80939
          2  |   4.106188   3.745698     1.10   0.278    -3.397347    11.60972
------------------------------------------------------------------------------

. 
. test _b[1.group#c.idle]= _b[2..group#c.idle]

 ( 1)  1b.group#c.idle - 2.group#c.idle = 0

       F(  1,    56) =   10.49
            Prob > F =    0.0020

[Lezgin Ay]

Thank you very much for your help. In your example, the dependent variable is the same in both regressions and you run the regression on two different independent variables (idle and idle2). In my question, it is the opposite. I regress y1 and y2 separately on the same independent variable. I wonder if this affects your answer.

[Andrew Musau]

Not at all, but I guess seeing is believing...

clear
webuse idle2
tsset time
set seed 2018
gen usr2= runiformint(0,15)
newey usr idle, lag(3)
newey usr2 idle, lag(3)
rename usr usr1
reshape long usr, i(time) j(group)
gen cons=1
replace time=_n
tsset time
newey usr (c.idle c.cons)#group, nocons lag(7)
test 1.group#c.idle= 2.group#c.idle

Result

. newey usr idle, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =      10.90
                                                Prob > F          =     0.0026

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        idle |  -.2281501   .0690927    -3.30   0.003    -.3696801     -.08662
       _cons |   23.13483   6.327031     3.66   0.001     10.17449    36.09516
------------------------------------------------------------------------------

. newey usr2 idle, lag(3)

Regression with Newey-West standard errors      Number of obs     =         30
maximum lag: 3                                  F(  1,        28) =       0.64
                                                Prob > F          =     0.4308

------------------------------------------------------------------------------
             |             Newey-West
        usr2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        idle |  -.0579036   .0724355    -0.80   0.431    -.2062811    .0904739
       _cons |   12.23198   6.166846     1.98   0.057    -.4002351    24.86419
------------------------------------------------------------------------------

. 
. newey usr (c.idle c.cons)#group, nocons lag(7)

Regression with Newey-West standard errors      Number of obs     =         60
maximum lag: 7                                  F(  4,        56) =      64.66
                                                Prob > F          =     0.0000

------------------------------------------------------------------------------
             |             Newey-West
         usr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
group#c.idle |
          1  |  -.2281501   .0690927    -3.30   0.002    -.3665595   -.0897406
          2  |  -.0579036   .0724355    -0.80   0.427    -.2030094    .0872022
             |
group#c.cons |
          1  |   23.13483   6.327031     3.66   0.001     10.46026    35.80939
          2  |   12.23198   6.166846     1.98   0.052    -.1217007    24.58566
------------------------------------------------------------------------------

. 
. test 1.group#c.idle= 2.group#c.idle

 ( 1)  1b.group#c.idle - 2.group#c.idle = 0

       F(  1,    56) =    3.03
            Prob > F =    0.0875

[Lezgin Ay]

Thank you! I use a lag length of 12 in individual regressions. How do you decide on the lag for the combined regression? Unlike your example, my coefficients and std. errors are different for the individual and combined regressions (lag=25).

[Andrew Musau]

Without seeing your data, I cannot know what is going on. It could be several things, including improper implementation of the procedure. In addition, this method relies on the set of independent variables being the same across both models, with only the dependent variable being different. Bottom line, you should get the same coefficients irrespective of the number of lags. Double plus one lags works in my example, and I think is a good approximation. Thus if revisiting your implementation does not yield anything fruitful, you can post a sample of your data here using dataex, after verifying that results do not match based on that sample. If you cannot publicly post the data, you can send it to me via email (available at my webpage linked to my Statalist profile). If this is not feasible, I cannot help and maybe you can ask a colleague to read this post and try it out.