Chapter 3 Testing for Serial Correlation
3.1 Example: Revisit Phillip’s Curve
Lesson: using our t-test, Durban Watson test, Breusch-Godfrey test, and Durban’s alternative statistic to test for serial correlation
We’ll use our t-test for serial correlation with strictly exogenous regressors under the assumption that our regressors are exogenous.
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1948 to 2003
delta: 1 yearWe estimated a static Phillips Curve and an augmented Phillips Curve last week. We’ll retest the models for serial correlation.
Static Model
Test for serial correlation in Static Model Noticeable serial correlation
*Static Model
reg inf unem if year < 1997
*Heteroskedastiity Test: Breusch-Godfrey Test
estat bgodfrey
*Serial Correlation Test: Simple AR(1) test
predict u, resid
reg u l.u, noconst Source | SS df MS Number of obs = 49
-------------+---------------------------------- F(1, 47) = 2.62
Model | 25.6369575 1 25.6369575 Prob > F = 0.1125
Residual | 460.61979 47 9.80042107 R-squared = 0.0527
-------------+---------------------------------- Adj R-squared = 0.0326
Total | 486.256748 48 10.1303489 Root MSE = 3.1306
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
unem | .4676257 .2891262 1.62 0.112 -.1140213 1.049273
_cons | 1.42361 1.719015 0.83 0.412 -2.034602 4.881822
------------------------------------------------------------------------------
Breusch-Godfrey LM test for autocorrelation
---------------------------------------------------------------------------
lags(p) | chi2 df Prob > chi2
-------------+-------------------------------------------------------------
1 | 18.472 1 0.0000
---------------------------------------------------------------------------
H0: no serial correlation
Source | SS df MS Number of obs = 55
-------------+---------------------------------- F(1, 54) = 29.61
Model | 161.010846 1 161.010846 Prob > F = 0.0000
Residual | 293.652171 54 5.43800317 R-squared = 0.3541
-------------+---------------------------------- Adj R-squared = 0.3422
Total | 454.663017 55 8.26660031 Root MSE = 2.332
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .5822453 .1070035 5.44 0.000 .3677161 .7967745
------------------------------------------------------------------------------First difference
Test for serial correlation in Augmented Serial correlation is removed with the first-difference in inflation
*First Difference Model
reg d.inf unem if year < 1997
predict u2, resid
*Breusch-Godfrey Test
estat bgodfrey Source | SS df MS Number of obs = 48
-------------+---------------------------------- F(1, 46) = 5.56
Model | 33.3829996 1 33.3829996 Prob > F = 0.0227
Residual | 276.305138 46 6.00663344 R-squared = 0.1078
-------------+---------------------------------- Adj R-squared = 0.0884
Total | 309.688138 47 6.58910932 Root MSE = 2.4508
------------------------------------------------------------------------------
D.inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
unem | -.5425869 .2301559 -2.36 0.023 -1.005867 -.079307
_cons | 3.030581 1.37681 2.20 0.033 .259206 5.801955
------------------------------------------------------------------------------
(1 missing value generated)
Breusch-Godfrey LM test for autocorrelation
---------------------------------------------------------------------------
lags(p) | chi2 df Prob > chi2
-------------+-------------------------------------------------------------
1 | 0.062 1 0.8039
---------------------------------------------------------------------------
H0: no serial correlationTest for Serial Correlation: Durban-Watson Test
Durbin-Watson d-statistic( 2, 48) = 1.769648\[ d_L = 1.503 \] \[ d_U = 1.585 \] \[ 1.769648 > 1.585 \] We fail to reject the null hypothesis.
Test for Serial Correlation: AR(1) Test
Source | SS df MS Number of obs = 54
-------------+---------------------------------- F(1, 53) = 0.07
Model | .267251656 1 .267251656 Prob > F = 0.7905
Residual | 198.724905 53 3.74952651 R-squared = 0.0013
-------------+---------------------------------- Adj R-squared = -0.0175
Total | 198.992157 54 3.68503994 Root MSE = 1.9364
------------------------------------------------------------------------------
u2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u2 |
L1. | -.0308131 .1154154 -0.27 0.791 -.262307 .2006808
------------------------------------------------------------------------------Test change in error terms for FD assumption on serial correlation so that the differences in errors are not correlated.
(2 missing values generated)
Source | SS df MS Number of obs = 53
-------------+---------------------------------- F(1, 52) = 2.01
Model | 13.6239993 1 13.6239993 Prob > F = 0.1623
Residual | 352.532902 52 6.77947888 R-squared = 0.0372
-------------+---------------------------------- Adj R-squared = 0.0187
Total | 366.156901 53 6.90862078 Root MSE = 2.6037
------------------------------------------------------------------------------
D.u2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u2_1 |
D1. | -.1661051 .1171734 -1.42 0.162 -.4012307 .0690204
------------------------------------------------------------------------------We fail to reject the null hypothesis that there is no serial correlation in the difference in error term.
FDL Model
Let’s add a lag in inflation to change our static model into a FDL model. We can also use the estat dwatson or dwstat
Source | SS df MS Number of obs = 48
-------------+---------------------------------- F(2, 45) = 19.72
Model | 219.518187 2 109.759094 Prob > F = 0.0000
Residual | 250.471823 45 5.56604051 R-squared = 0.4671
-------------+---------------------------------- Adj R-squared = 0.4434
Total | 469.99001 47 9.99978744 Root MSE = 2.3592
------------------------------------------------------------------------------
inf | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
inf |
L1. | .7278693 .1263167 5.76 0.000 .4734544 .9822841
|
unem | -.2443814 .26124 -0.94 0.355 -.7705457 .2817829
_cons | 2.43082 1.354276 1.79 0.079 -.2968325 5.158473
------------------------------------------------------------------------------
Durbin-Watson d-statistic( 3, 48) = 1.477467
Durbin-Watson d-statistic( 3, 48) = 1.477467First, let’s find our critical values:
https://www3.nd.edu/~wevans1/econ30331/Durbin_Watson_tables.pdf.
\[ DW(3,48)=1.48 \] and \[ d_L \approx 1.4 \] and \[ d_U \approx 1.67 \]
so our DW test is inconclusive. We’ll look at Durban-Watson and DW Tables again a bit later.
3.2 Example: Revisit Puerto Rican Employment and Minimum Wage
Lesson: We can use Durban’s alternative statistic to test for serial correlation if our regressors are not strictly exogenous.Durban’s alternative statistic
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1950 to 1987
delta: 1 unitWe’ll rerun our static model and regress the natural log of the employment-population ratio onto the natural log of the importance of minimum wage, the natural log of Puerto Rican Gross National Product, the natural log of US GNP, and a time trend.
Durbin-Watson Test
Source | SS df MS Number of obs = 38
-------------+---------------------------------- F(4, 33) = 66.23
Model | .284430187 4 .071107547 Prob > F = 0.0000
Residual | .035428331 33 .001073586 R-squared = 0.8892
-------------+---------------------------------- Adj R-squared = 0.8758
Total | .319858518 37 .008644825 Root MSE = .03277
------------------------------------------------------------------------------
lprepop | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lmincov | -.2122612 .0401524 -5.29 0.000 -.2939518 -.1305706
lprgnp | .2852386 .0804922 3.54 0.001 .121476 .4490012
lusgnp | .4860463 .2219829 2.19 0.036 .0344188 .9376739
t | -.0266633 .0046267 -5.76 0.000 -.0360765 -.0172502
_cons | -6.663432 1.257831 -5.30 0.000 -9.222508 -4.104356
------------------------------------------------------------------------------Durbin’s Alternative Statistics
Test Serial Correlation with Durbin’s alternative statistic, which is valid when we don’t have strictly exogenous regressors.
Source | SS df MS Number of obs = 37
-------------+---------------------------------- F(5, 32) = 1.92
Model | .007182064 5 .001436413 Prob > F = 0.1191
Residual | .023990478 32 .000749702 R-squared = 0.2304
-------------+---------------------------------- Adj R-squared = 0.1101
Total | .031172542 37 .000842501 Root MSE = .02738
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .4757945 .1653074 2.88 0.007 .1390742 .8125147
|
lmincov | .041425 .0346345 1.20 0.240 -.0291232 .1119732
lprgnp | -.0522061 .0615544 -0.85 0.403 -.1775883 .0731762
lusgnp | .0606233 .0644769 0.94 0.354 -.070712 .1919585
t | -.0005264 .0015194 -0.35 0.731 -.0036213 .0025685
------------------------------------------------------------------------------We can see that we reject the null hypothesis, since our \(\hat{rho} = 0.48\) and statistically significant.
3.3 Test AR(3) serial correlation
Lesson: We can test for serial correlation among prior periods beside AR(1).Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: t, 1960m2 to 1970m12
delta: 1 monthWe’ll look at our barium chloride model again to see if ITC complaints affect behavior of foreign exporters. We may have higher order of serial correlation since we are using monthly data.
Source | SS df MS Number of obs = 131
-------------+---------------------------------- F(6, 124) = 9.06
Model | 19.4051607 6 3.23419346 Prob > F = 0.0000
Residual | 44.2470875 124 .356831351 R-squared = 0.3049
-------------+---------------------------------- Adj R-squared = 0.2712
Total | 63.6522483 130 .489632679 Root MSE = .59735
------------------------------------------------------------------------------
lchnimp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lchempi | 3.117193 .4792021 6.50 0.000 2.168718 4.065668
lgas | .1963504 .9066172 0.22 0.829 -1.598099 1.9908
lrtwex | .9830183 .4001537 2.46 0.015 .1910022 1.775034
befile6 | .0595739 .2609699 0.23 0.820 -.4569585 .5761064
affile6 | -.0324064 .2642973 -0.12 0.903 -.5555249 .490712
afdec6 | -.565245 .2858352 -1.98 0.050 -1.130993 .0005028
_cons | -17.803 21.04537 -0.85 0.399 -59.45769 23.85169
------------------------------------------------------------------------------Test for \(H_{0}: p1=p2=p3=0\) jointly with an F-test.
Source | SS df MS Number of obs = 128
-------------+---------------------------------- F(9, 119) = 1.67
Model | 4.8836741 9 .542630456 Prob > F = 0.1024
Residual | 38.5511192 119 .323958985 R-squared = 0.1124
-------------+---------------------------------- Adj R-squared = 0.0453
Total | 43.4347933 128 .339334323 Root MSE = .56917
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .215573 .0910635 2.37 0.020 .0352581 .3958878
L2. | .1296804 .0917465 1.41 0.160 -.0519868 .3113477
L3. | .1209405 .0906816 1.33 0.185 -.0586181 .3004992
|
lchempi | -.105758 .4679356 -0.23 0.822 -1.032317 .8208012
lgas | .0125667 .1187882 0.11 0.916 -.2226459 .2477792
lrtwex | .0518992 .3452261 0.15 0.881 -.631683 .7354815
befile6 | -.0730016 .249775 -0.29 0.771 -.567581 .4215779
affile6 | -.122359 .2541453 -0.48 0.631 -.6255919 .380874
afdec6 | -.0694522 .2737453 -0.25 0.800 -.6114953 .4725909
------------------------------------------------------------------------------
( 1) L.u = 0
( 2) L2.u = 0
( 3) L3.u = 0
F( 3, 119) = 4.98
Prob > F = 0.0027Our reject the null hypothesis that all of the lags are equal to zero using an F-test.
However, when we test our last two lags, they are jointly insignificant.
( 1) L2.u = 0
( 2) L3.u = 0
F( 2, 119) = 2.42
Prob > F = 0.0930We have evidence of at least serial correlation to the order of one.