Chapter 4 Correct for Serial Correlation
4.1 Prais-Winsten Estimation in the Event Study
Lesson: We’ll account for serial correlation with a FGLS Prais-Winsten model.
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: t, 1960m2 to 1970m12
delta: 1 monthOLS
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
------------------------------------------------------------------------------
Durbin-Watson d-statistic( 7, 131) = 1.458414Prais-Winsten
Iteration 0: rho = 0.0000
Iteration 1: rho = 0.2708
Iteration 2: rho = 0.2910
Iteration 3: rho = 0.2930
Iteration 4: rho = 0.2932
Iteration 5: rho = 0.2932
Iteration 6: rho = 0.2932
Iteration 7: rho = 0.2932
Prais-Winsten AR(1) regression -- iterated estimates
Source | SS df MS Number of obs = 131
-------------+---------------------------------- F(6, 124) = 5.24
Model | 10.3252323 6 1.72087205 Prob > F = 0.0001
Residual | 40.7593886 124 .328704747 R-squared = 0.2021
-------------+---------------------------------- Adj R-squared = 0.1635
Total | 51.0846209 130 .392958622 Root MSE = .57333
------------------------------------------------------------------------------
lchnimp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lchempi | 2.940949 .6328402 4.65 0.000 1.688381 4.193517
lgas | 1.04638 .9773356 1.07 0.286 -.8880406 2.980801
lrtwex | 1.132791 .5066578 2.24 0.027 .1299738 2.135609
befile6 | -.0164787 .3193802 -0.05 0.959 -.6486216 .6156641
affile6 | -.0331563 .3218101 -0.10 0.918 -.6701086 .603796
afdec6 | -.5768122 .3419865 -1.69 0.094 -1.253699 .1000748
_cons | -37.0777 22.7783 -1.63 0.106 -82.16235 8.006941
-------------+----------------------------------------------------------------
rho | .293217
------------------------------------------------------------------------------
Durbin-Watson statistic (original) 1.458414
Durbin-Watson statistic (transformed) 2.087181First, let’s find our critical values:
https://www3.nd.edu/~wevans1/econ30331/Durbin_Watson_tables.pdf.
For the OLS Estimator
\[ DW(6,131) = 1.458414 \]
For the Prasi-Winsten Estimator
\[ DW(6,131) = 2.087181 \]
and \[ d_L \approx 1.60 \] and \[ d_U \approx 1.81 \]
We reject the null hypothesis for the OLS estimator, but we fail to reject
the null hypothesis for the Prais-Winsten Estimator.
Our beta-hats in Prais-Winsten Estimator are similar to our OLS estimates, but our PW standard errors account for the serial correlation. Our OLS standard errors usually understate the actual sampling variation in the OLS estimators and should be treated with suspicion.
(1) (2)
OLS Prais
--------------------------------------------
lchempi 3.117*** 2.941***
(0.479) (0.633)
lgas 0.196 1.046
(0.907) (0.977)
lrtwex 0.983* 1.133*
(0.400) (0.507)
befile6 0.0596 -0.0165
(0.261) (0.319)
affile6 -0.0324 -0.0332
(0.264) (0.322)
afdec6 -0.565 -0.577
(0.286) (0.342)
_cons -17.80 -37.08
(21.05) (22.78)
--------------------------------------------
N 131 131
rho 0.293
--------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.0014.2 Inflation and Prais
Lesson: our PW estimator might not be unbiased or consistent if our assumptions fail.We’ll show another example comparing OLS and Prais-Winsten estimators. We’ll look at the static Phillips curve, which we know has serial correlation.
Set the Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1948 to 2003
delta: 1 unitCompare
(1) (2) (3) (4)
OLS PW PW_CO FD
----------------------------------------------------------------------------
unem 0.468 -0.716* -0.665*
(0.289) (0.313) (0.320)
D.unem -0.842*
(0.314)
_cons 1.424 8.296*** 7.583** -0.0782
(1.719) (2.231) (2.381) (0.348)
----------------------------------------------------------------------------
N 49 49 48 48
rho 0.781 0.774
----------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001The OLS and PW estimators might give very different estimators if our assumptions of strict exogeneity and the a \(Cov(x_(t+1)+x_(t-1),u_t)=0\) fail. If that is the case then using OLS with a first-difference might be a better method.
We can see that the difference is quite notable. With our OLS estimators, the tradeoff between inflation and employment is non-existent, but with the PW estimator our estimate of beta 1 is closer to our first-difference estimate of beta 1.
It is possible that there is no relationship with the static model, but the change in inflation and the change in unemployment are negatively related.
4.3 Differencing and Serial Correlation
4.3.1 Inflation and Deficits on Interest Rates
Lesson: First-differencing is a potential method to eliminate serial correlation.Set the Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1948 to 2003
delta: 1 unitWe’ll look at the relationship between interest rates from the 3-month T-bill rate (i3) and inflation rate (inf) and deficits as a percentage of GDP.
Static Model
Calculate a static model and test for serial correlation
Source | SS df MS Number of obs = 56
-------------+---------------------------------- F(2, 53) = 40.09
Model | 272.420338 2 136.210169 Prob > F = 0.0000
Residual | 180.054275 53 3.39725047 R-squared = 0.6021
-------------+---------------------------------- Adj R-squared = 0.5871
Total | 452.474612 55 8.22681113 Root MSE = 1.8432
------------------------------------------------------------------------------
i3 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
inf | .6058659 .0821348 7.38 0.000 .4411243 .7706074
def | .5130579 .1183841 4.33 0.000 .2756095 .7505062
_cons | 1.733266 .431967 4.01 0.000 .8668497 2.599682
------------------------------------------------------------------------------
Durbin-Watson d-statistic( 3, 56) = .7161527Critical values \(d(3,56)=.716\), at the 5% level \(dL = 1.452, dU=1.681\) \(dwstat < dL\) so we reject the null hypothesis
Source | SS df MS Number of obs = 55
-------------+---------------------------------- F(1, 54) = 32.83
Model | 64.1034163 1 64.1034163 Prob > F = 0.0000
Residual | 105.448636 54 1.95275252 R-squared = 0.3781
-------------+---------------------------------- Adj R-squared = 0.3666
Total | 169.552053 55 3.08276459 Root MSE = 1.3974
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .6228816 .1087148 5.73 0.000 .4049216 .8408415
------------------------------------------------------------------------------We find that we have notable serial correlation in the error term that is statistically significant. We’ll run a first-difference model next.
First differencing
Run a first-difference and test for serial correlation
Source | SS df MS Number of obs = 55
-------------+---------------------------------- F(2, 52) = 5.57
Model | 17.8058164 2 8.90290822 Prob > F = 0.0065
Residual | 83.1753707 52 1.59952636 R-squared = 0.1763
-------------+---------------------------------- Adj R-squared = 0.1446
Total | 100.981187 54 1.87002198 Root MSE = 1.2647
------------------------------------------------------------------------------
D.i3 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
inf |
D1. | .1494892 .0921555 1.62 0.111 -.0354343 .3344127
|
def |
D1. | -.1813151 .1476825 -1.23 0.225 -.4776618 .1150315
|
_cons | .0417738 .1713874 0.24 0.808 -.3021401 .3856877
------------------------------------------------------------------------------
Durbin-Watson d-statistic( 3, 55) = 1.796365Critical values \(d(3,56)=1.796\), at the 5% level \(dL = 1.452, dU=1.681\) \(dwstat > dU\) so we fail to reject the null hypothesis
(1 missing value generated)
Source | SS df MS Number of obs = 54
-------------+---------------------------------- F(1, 53) = 0.29
Model | .427058269 1 .427058269 Prob > F = 0.5921
Residual | 77.8807184 53 1.46944752 R-squared = 0.0055
-------------+---------------------------------- Adj R-squared = -0.0133
Total | 78.3077766 54 1.45014401 Root MSE = 1.2122
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .0717246 .133046 0.54 0.592 -.1951318 .3385811
------------------------------------------------------------------------------The serial correlation is no longer a problem with our first difference model, since first differencing can transform a unit root process \(I(1)\) into a \(I(0)\) process.
4.4 Serial Correlation Robust Standard Errors
Lesson: We can use the Newey estimator for estimates with Heteroskedastic and Autocorrelation Consistent (HAC) standard errors.
We’ll revisit our Puerto Rican examination of the impact of minimum wage importance We’ll use Heteroskedastic and Autocorrelation Consistent (HAC) standard errors, and we’ll compare OLS, OLS with HAC standard errors, and Prais-Winsten estimates We’ll allow a maximum lag order of autocorrelation up to 2 with the lag(2) option.
Set Time Series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: year, 1950 to 1987
delta: 1 yearPrais-Winsten
With t-stats below estimated parameters
(1) (2) (3) (4)
OLS Newey1 Newey2 PW
----------------------------------------------------------------------------
lmincov -0.212*** -0.212*** -0.212*** -0.148**
(-5.29) (-4.67) (-4.64) (-3.22)
lprgnp 0.285** 0.285** 0.285** 0.251*
(3.54) (2.85) (2.86) (2.16)
lusgnp 0.486* 0.486 0.486 0.256
(2.19) (1.79) (1.74) (1.10)
t -0.0267*** -0.0267*** -0.0267*** -0.0205**
(-5.76) (-4.86) (-4.63) (-3.50)
_cons -6.663*** -6.663*** -6.663*** -4.653**
(-5.30) (-4.52) (-4.34) (-3.38)
----------------------------------------------------------------------------
N 38 38 38 38
F 66.23 40.98 37.84 24.87
----------------------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001With standard errors below the estimated parameters
(1) (2) (3) (4)
OLS Newey1 Newey2 PW
----------------------------------------------------------------------------
lmincov -0.212*** -0.212*** -0.212*** -0.148**
(0.0402) (0.0455) (0.0457) (0.0458)
lprgnp 0.285** 0.285** 0.285** 0.251*
(0.0805) (0.100) (0.0996) (0.116)
lusgnp 0.486* 0.486 0.486 0.256
(0.222) (0.272) (0.279) (0.232)
t -0.0267*** -0.0267*** -0.0267*** -0.0205**
(0.00463) (0.00549) (0.00576) (0.00586)
_cons -6.663*** -6.663*** -6.663*** -4.653**
(1.258) (1.475) (1.536) (1.376)
----------------------------------------------------------------------------
N 38 38 38 38
F 66.23 40.98 37.84 24.87
----------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001Notice how our t-stat have fallen and our standard errors have risen for our Newey1 and Newey2 compared to OLS. Also, notice how the Prais-Winsten beta-hat on minimum coverage is closer to 0, then OLS or OLS with HAC standard errors.
4.5 Testing for Heteroskedasticity in Time Series
We’ll revisit our test of the Efficient Market Hypothesis with stock return data. We can test to see if our assumption of homoskedasticity is valid with our time series. This is a separate test than the test for serial correlation.
Set the time series
/Users/Sam/Desktop/Econ 645/Data/Wooldridge
time variable: t, 1960w2 to 1973w16
delta: 1 weekOLS
We’ll test heteroskedasticity with estat hettest
Source | SS df MS Number of obs = 689
-------------+---------------------------------- F(1, 687) = 2.40
Model | 10.6866231 1 10.6866231 Prob > F = 0.1218
Residual | 3059.73817 687 4.45376735 R-squared = 0.0035
-------------+---------------------------------- Adj R-squared = 0.0020
Total | 3070.42479 688 4.46282673 Root MSE = 2.1104
------------------------------------------------------------------------------
return | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
return |
L1. | .0588984 .0380231 1.55 0.122 -.0157569 .1335538
|
_cons | .179634 .0807419 2.22 0.026 .0211034 .3381646
------------------------------------------------------------------------------
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
Ho: Constant variance
Variables: fitted values of return
chi2(1) = 95.22
Prob > chi2 = 0.0000We’ll manually calculate the Breusch-Pagan test
(2 missing values generated)
(2 missing values generated)
Source | SS df MS Number of obs = 689
-------------+---------------------------------- F(1, 687) = 30.05
Model | 3755.56877 1 3755.56877 Prob > F = 0.0000
Residual | 85846.2961 687 124.958219 R-squared = 0.0419
-------------+---------------------------------- Adj R-squared = 0.0405
Total | 89601.8649 688 130.235269 Root MSE = 11.178
------------------------------------------------------------------------------
u2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
return |
L1. | -1.104133 .2014029 -5.48 0.000 -1.499572 -.7086934
|
_cons | 4.656501 .4276789 10.89 0.000 3.816786 5.496216
------------------------------------------------------------------------------Test for serial correlation
Source | SS df MS Number of obs = 688
-------------+---------------------------------- F(1, 687) = 0.00
Model | .00603936 1 .00603936 Prob > F = 0.9706
Residual | 3059.08227 687 4.45281262 R-squared = 0.0000
-------------+---------------------------------- Adj R-squared = -0.0015
Total | 3059.08831 688 4.44634929 Root MSE = 2.1102
------------------------------------------------------------------------------
u | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
u |
L1. | .001405 .0381496 0.04 0.971 -.0734987 .0763087
------------------------------------------------------------------------------We can see we reject the null hypothesis that the variance of the error term is constant, and heteroskedasticity is a problem.
Newey-West Estimator
We can use robust, or HAC standard errors, but since serial correlation is not a problem, then we don’t need newey estimator.
Regression with Newey-West standard errors Number of obs = 689
maximum lag: 1 F( 1, 687) = 0.74
Prob > F = 0.3895
------------------------------------------------------------------------------
| Newey-West
return | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
return |
L1. | .0588984 .0684001 0.86 0.389 -.0754 .1931968
|
_cons | .179634 .0855634 2.10 0.036 .0116369 .3476311
------------------------------------------------------------------------------Expected value does not depend on past returns, but the variance of the error term is not constant, and needs to be corrected.