Lecture 51011

Introduction to Stata Handout 5: Time Series

Hayley Fisher

2 December 2010

Key references: Wooldridge (2009) part II, Greene (2008) chapter 21.

1 Time series

To illustrate some basic features of Stata with time series, I am using a dataset of the general fertility rateand personal tax exemptions for the US from 1913 to 1984. This is based on an example in chapter 10 ofWooldridge (2009), and ultimately on an article by Whittington, Alm and Peters from 1990. The dataset isavailable from my website.

We start off by summarizing the small dataset.

. summarize

Variable | Obs Mean Std. Dev. Min Max

-------------+--------------------------------------------------------

gfr | 72 95.63194 19.80464 65.4 126.6

pe | 72 100.4015 65.87563 0 243.83

year | 72 1948.5 20.92845 1913 1984

pill | 72 .3055556 .4638749 0 1

ww2 | 72 .0694444 .2559923 0 1

gfr is the general fertility rate, and we are looking to explain it using personal tax exemption (pe),whether World War II was occurring, and whether the contraceptive pill was available. Stata makes it easyto create variables such as lags, leads and first differences. To take advantage of these we first need to declarethe data to be time series using the tsset command:

. tsset year

time variable: year, 1913 to 1984

delta: 1 unit

Once this has been done, we can create variables using the lag (L.), lead (F.) and first difference (D.)operators:

. generate Lgfr=L.gfr

(1 missing value generated)

. generate Fgfr=F.gfr


. generate Dgfr=D.gfr


A missing value is created each time at the beginning, or end, of the dataset. We can check that these arecorrect by listing the first five values:

1

. list year gfr Lgfr Fgfr Dgfr in 1/5

+------------------------------------------+

| year gfr Lgfr Fgfr Dgfr |

|------------------------------------------|

1. | 1913 124.7 . 126.6 . |

2. | 1914 126.6 124.7 125 1.900002 |

3. | 1915 125 126.6 123.4 -1.599998 |

4. | 1916 123.4 125 121 -1.599998 |

5. | 1917 121 123.4 119.8 -2.400002 |

+------------------------------------------+

We can naively estimate the relationship between the general fertility rate and these variables usingregress.

. regress gfr pe ww2 pill, vce(robust)

Linear regression Number of obs = 72

F( 3, 68) = 51.57

Prob > F = 0.0000

R-squared = 0.4734

Root MSE = 14.685

------------------------------------------------------------------------------

| Robust

gfr | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

pe | .08254 .0269359 3.06 0.003 .0287902 .1362898

ww2 | -24.2384 3.351355 -7.23 0.000 -30.92592 -17.55087

pill | -31.59403 3.131765 -10.09 0.000 -37.84337 -25.34469

_cons | 98.68176 4.222996 23.37 0.000 90.2549 107.1086

------------------------------------------------------------------------------

As would be expected, the fertility appears to be lower during World War II and when the contraceptivepill is available. For these results to be consistent we require no serial correlation in the error terms. Wecan test for this using the Durbin-Watson statistic, implemented by typing estat dwatson after runningthe regression.

. estat dwatson

Durbin-Watson d-statistic( 4, 72) = .1768727

The statistic is far below 2 and so indicates serial correlation. Stata can also implement two LagrangeMultiplier tests for serial correlation, using estat durbinalt and estat bgodfrey.

. estat durbinalt

Durbins alternative test for autocorrelation

---------------------------------------------------------------------------

lags(p) | chi2 df Prob > chi2

-------------+-------------------------------------------------------------

1 | 255.261 1 0.0000

---------------------------------------------------------------------------

H0: no serial correlation

2

. estat bgodfrey

Breusch-Godfrey LM test for autocorrelation

---------------------------------------------------------------------------


-------------+-------------------------------------------------------------

1 | 57.031 1 0.0000

---------------------------------------------------------------------------


Again, serial correlation is detected. There are several ways of dealing with serial correlation of the errorterms. One method might be to add lags of the dependent variable. Here we can add two lags and it isnot necessary to create both lags before running the regression.

. regress gfr L.gfr L2.gfr pe ww2 pill

Source | SS df MS Number of obs = 70

-------------+------------------------------ F( 5, 64) = 341.66

Model | 25053.8199 5 5010.76397 Prob > F = 0.0000

Residual | 938.613043 64 14.6658288 R-squared = 0.9639

-------------+------------------------------ Adj R-squared = 0.9611

Total | 25992.4329 69 376.701926 Root MSE = 3.8296

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

gfr |

L1. | 1.076351 .1214261 8.86 0.000 .8337748 1.318928

L2. | -.175926 .1156576 -1.52 0.133 -.4069785 .0551266

pe | .0227773 .0086381 2.64 0.010 .0055207 .040034

ww2 | -2.545189 2.096259 -1.21 0.229 -6.732947 1.642569

pill | -4.824531 1.439447 -3.35 0.001 -7.700157 -1.948904

_cons | 8.143783 3.182315 2.56 0.013 1.786377 14.50119

------------------------------------------------------------------------------

. estat durbinalt

Durbins alternative test for autocorrelation

---------------------------------------------------------------------------


-------------+-------------------------------------------------------------

1 | 0.498 1 0.4805

---------------------------------------------------------------------------


. estat bgodfrey

Breusch-Godfrey LM test for autocorrelation

---------------------------------------------------------------------------


-------------+-------------------------------------------------------------

1 | 0.549 1 0.4589

---------------------------------------------------------------------------

3


Adding these lags removes the serial correlation, and reduces the point estimates of coefficients. Note thatwith a lagged dependent variable the standard Durbin-Watson statistics is not appropriate.

Instead of removing the serial correlation from the error terms, we could attempt to correct the standarderrors using newey. Here we must specify the order of serial correlation the number of lags over whichto calculate the serial correlation in the errors. I include the heteroscedasticity robust but not adjusted forserial correlation results for comparison:

. regress gfr pe ww2 pill, vce(robust)


F( 3, 68) = 51.57

Prob > F = 0.0000

R-squared = 0.4734

Root MSE = 14.685

------------------------------------------------------------------------------

| Robust


-------------+----------------------------------------------------------------

pe | .08254 .0269359 3.06 0.003 .0287902 .1362898

ww2 | -24.2384 3.351355 -7.23 0.000 -30.92592 -17.55087

pill | -31.59403 3.131765 -10.09 0.000 -37.84337 -25.34469

_cons | 98.68176 4.222996 23.37 0.000 90.2549 107.1086

------------------------------------------------------------------------------

. newey gfr pe ww2 pill, lag(2)

Regression with Newey-West standard errors Number of obs = 72

maximum lag: 2 F( 3, 68) = 25.84

Prob > F = 0.0000

------------------------------------------------------------------------------

| Newey-West


-------------+----------------------------------------------------------------

pe | .08254 .0436866 1.89 0.063 -.0046352 .1697153

ww2 | -24.2384 3.615538 -6.70 0.000 -31.45309 -17.0237

pill | -31.59403 5.016629 -6.30 0.000 -41.60456 -21.58351

_cons | 98.68176 7.003884 14.09 0.000 84.70572 112.6578

------------------------------------------------------------------------------

As the order of serial correlation in the error term increases, the standard errors also increase. Thisadjusts the standard errors for the presence of serial correlation.

Alternatively, it is possible to directly model the serial correlation in the error terms using the arimacommand. For example, if we believe that the error terms are best modelled by a first order autoregressiveprocess, and so (3) below should be estimated:

gfrt = Xt + ut (1)

ut = ut1 + t (2)

gfrt = Xt + (gfrt1 Xt1) + t (3)

4

Using arima with the option ar(1) achieves this. Extra autoregressive lags can be added, as can movingaverage components using the option ma(i) where i is the order of the moving average term.

. arima gfr pe ww2 pill, ar(1)

(setting optimization to BHHH)

Iteration 0: log likelihood = -231.52653





(switching optimization to BFGS)











(switching optimization to BHHH)



ARIMA regression

Sample: 1913 - 1984 Number of obs = 72

Wald chi2(4) = 791.75

Log likelihood = -205.9535 Prob > chi2 = 0.0000

------------------------------------------------------------------------------

| OPG

gfr | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

gfr |

pe | -.025783 .0313082 -0.82 0.410 -.087146 .0355799

ww2 | -5.033364 1.873287 -2.69 0.007 -8.704939 -1.36179

pill | -4.253375 19.34541 -0.22 0.826 -42.16968 33.66292

_cons | 98.88224 18.5853 5.32 0.000 62.45571 135.3088

-------------+----------------------------------------------------------------

ARMA |

ar |

L1. | .982849 .040655 24.18 0.000 .9031666 1.062531

-------------+----------------------------------------------------------------

/sigma | 4.129038 .3477984 11.87 0.000 3.447366 4.81071

------------------------------------------------------------------------------

Having explicitly modelled the autocorrelation we see that the personal exemption and pill variables areno longer statistically significant.

However, it is also hypothesised that there may be a lag in the response to the personal tax exemption.We should therefore include lagged values of the personal exemption.

. regress gfr pe L.pe L2.pe pill ww2, vce(robust)

5


F( 5, 64) = 31.21

Prob > F = 0.0000

R-squared = 0.4986

Root MSE = 14.27

------------------------------------------------------------------------------

| Robust


-------------+----------------------------------------------------------------

pe |

--. | .0726718 .0979877 0.74 0.461 -.1230812 .2684248

L1. | -.0057796 .1132301 -0.05 0.959 -.2319826 .2204235

L2. | .0338268 .0894437 0.38 0.707 -.1448575 .2125111

pill | -31.30499 3.123293 -10.02 0.000 -37.54448 -25.06549

ww2 | -22.1265 6.950919 -3.18 0.002 -36.01256 -8.24044

_cons | 95.8705 4.284189 22.38 0.000 87.31185 104.4291

------------------------------------------------------------------------------

The coefficients on the lags of pe are imprecisely estimated, but we can test their joint significance using thetest command introduced above:

. test pe L.pe L2.pe

( 1) pe = 0

( 2) L.pe = 0

( 3) L2.pe = 0

F( 3, 64) = 4.67

Prob > F = 0.0051

The three coefficients are jointly significantly different from zero at the 1% level.We may also want to include a time trend. This can be generated (plus a quadratic) if the data is sorted

by date:

. sort year

. generate t=_n

. generate t2=t^2

These are easily included in a regression model. It remains important to test for the presence of serialcorrelation in the error term.

However, we must also consider the possibility of a unit root in the series we are trying to explain. Timeseries can be simply displayed using the tsline command once the data have been declared to be time series.Graphing the fertility rate suggests that a unit root is very likely.

6

. tsline gfr

6080

100

120

140

birth

s pe

r 100

0 wo

men

15

44

1920 1940 1960 19801913 to 1984

To investigate further we can look at the autocorrelations of both the fertility rate and its first difference.These too strongly suggest a unit root which is removed by first differencing the series.

. ac gfr, lags(10)

1.

00

0.50

0.00

0.50

1.00

Auto

corre

latio

ns o

f gfr

0 2 4 6 8 10Lag

Bartletts formula for MA(q) 95% confidence bands

7

. ac Dgfr, lags(10)

0.

40

0.20

0.00

0.20

0.40

Auto

corre

latio

ns o

f Dgf

r

0 2 4 6 8 10Lag

Bartletts formula for MA(q) 95% confidence bands

A formal test for a unit root can be conducted using a Dickey-Fuller test and the dfuller command.

. dfuller gfr

Dickey-Fuller test for unit root Number of obs = 71

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -0.857 -3.551 -2.913 -2.592

------------------------------------------------------------------------------

MacKinnon approximate p-value for Z(t) = 0.8019

. dfuller Dgfr

Dickey-Fuller test for unit root Number of obs = 70

---------- Interpolated Dickey-Fuller ---------

Test 1% Critical 5% Critical 10% Critical

Statistic Value Value Value

------------------------------------------------------------------------------

Z(t) -6.161 -3.552 -2.914 -2.592

------------------------------------------------------------------------------

MacKinnon approximate p-value for Z(t) = 0.0000

Here the null hypothesis is that there is a unit root. So there is insufficient evidence to suggest that thereis no unit root in the gfr series, but the null hypothesis is rejected in the first difference of the series, assuggested by the autocorrelations.

References

Greene, William H., Econometric Analysis, 6th ed., Pearson/Prentice Hall, 2008.

8

Wooldridge, Jeffrey M., Introductory Econometrics: A Modern Approach, 4th ed., South Western /Cengage Learning, 2009.

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Documents

Lecture 51011